Free response question

IEO past paper: 2020 International Economics Olympiad, open questions

As a pandemic caused by a new virus wreaks havoc in the world, no one is yet ill in town $M$ (but some people may be unknowingly infected and may be able to spread the virus). The town administration considers introducing a lockdown to prevent the spread of the virus. How strong should the lockdown be? The stronger the lockdown is, the slower is the spread of the virus, but at the same time, the stronger is the damage to the economy. There are 200 citizens in the town. Let the economic benefits of every citizen from going outside be 50. The costs of going outside are related to the probability of falling ill. This probability, in turn, depends on the number of people outside. If there are $Q$ people outside not including person $i$, the probability that the person $i$ becomes ill is equal to $Q/200$. Importantly, the costs upon falling ill are different for different people as the costs depend on age, initial health, etc. The costs for the first person are equal to 1, for the second person to 2, etc. up to 200. Thus, the expected utility of person $i$ from going outside is given by $$U_i = 50 - \frac{Q}{200} \cdot i.$$ The utility from staying home is equal to 0 (one can not get infected at home). Numbers of people can only be whole numbers. In case you forgot, $1 + 2 + 3 + \dots + n = \dfrac{n(n+1)}{2}$.