Generalized Fractal Transforms with Condensation: a Macroeconomic-Epidemiological Application

We establish novel results on generalized fractal operators with condensation and apply them in the analysis of a macroeconomic-epidemiological model characterized by deep uncertainty under the assumption that it is impossible to quantify with certainty the exact number of current and future infectives. The setting is simple: the level of prevalence of a communicable disease determines the size of the healthy labor force, affecting output and consumption; health policy is publicly funded via income taxation but the availability of resources is endogenously determined since depending on disease prevalence. Since the high degree of uncertainty is reflected also in the policymakers' choice of the policy tools to limit the spread of the disease, we investigate how the peculiarities of different policymakers (a short-sighted vs far-sighted approach) affect the asymptotic invariant distribution of macroeconomic activity. Specifically, we exploit the condensation term of the fractal operator to characterize the consequence of short-sighted policies. Through numerical simulations we find that, as we would expect, far-sighted policies lead to asymptotic invariant probability distributions concentrating more mass on high levels of aggregate consumption together with small numbers of infectives, while the invariant distribution reached through short-sighted policies, besides concentrating more mass on low levels of aggregate consumption together with large numbers of infectives, exhibits an additional layer of (uniform) uncertainty generated by the condensation term.