1. Find the optimal activity of person i as a function of a where f (ai) = ai if the society is so large that i's impact on the average activity is negligible and provide an intuition for the impact of a on the optimal social activity of i. Now assume that the population consists of two groups of equal size, the vulnerable and the non-vulnerable with v and n, respectively.2. Calculate the equilibrium activity av, an and indirect utility for members of both groups. (Note that a = (av + an)/2)3. Suppose a utilitarian social planner who observes each person's resilience can dictate social activity levels. Determine the socially optimal levels asv and asn she would choose.4. Calculate the resulting utilities for the socially optimal choices and compare them to the indirect utilities without intervention. Discuss.

Suppose individuals react to the threat of getting infected by COVID by reducing their social activity. Denote person i's level of social activity as ca 6 [0.1]. Further denote the average activity level in the society as EL. A person's resilience to the virus is given by 92-. The disutility an infection causes for person 2' is 1/62. Finally, the infectiousness of the current variant is given by p. A person chooses her activity level in order to maximize her enjoyment from social activity net of expected health consequences of an infection. The probability p1- of contracting the virus depends on activity level, average activity level in the society and on the infectiousness of the variant. Hence, the objective function of person i is _ PAW: a; P) EU(ai,a) := ag) 6- For simplicity use the following functional forms: pi (0,135,) : pagan 1. Find the optimal activity of person 2' as a function of a if the society is so large that i's impact on the average activity is negligible and provide an intuition for the impact of EL on the optimal social activity of 2'. Now assume that the population consists of two groups of equal size, the vulnerable and the non-vulnerable with (9,) and 9\