Optimal lockdown
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).
(a) (12 rp) Suppose people decide individually whether to go outside, each of them maximizes his/her expected utility. We say that people's decisions form a Nash equilibrium if no one can benefit by changing her/his decision with others' decisions fixed. How many people will go outside in the Nash equilibrium? Call this number N
We will identify a Nash equilibrium in which all people with relatively high costs of illness (high index i stay home, and others go outside.
Let N be the highest index of a person who goes outside (then, exactly N people go outside).
In the equilibrium, the expected utility from going outside for all people with indexes i = 1, \ldots, N must be nonnegative, while for people i = N+1, \ldots, 200 this utility must be nonpositive.
In particular, the expected utility of person N from going outside must be nonnegative given that N - 1 people besides her go outside, and the expected utility of person N + 1 from going outside must be nonpositive given that N people go outside.
Thus, two conditions must be satisfied: (1)\quad 50 - \frac{N - 1}{200} N \geq 0, \qquad (2)\quad 50 - \frac{(N + 1) - 1}{200} (N + 1) \leq 0.
It is easy to find that the only whole number N satisfying these conditions is N = 100.
Finally, because the expected utility from going outside of person 100 is nonnegative, so it is for all people i = 1, \ldots, 99 , because they have lower costs.
Likewise, because the expected utility from going outside of person 101 is nonpositive, so it is for all people i = 102, \ldots, 200 because they have higher costs.
Thus, we indeed found a Nash equilibrium.
(b) (12 rp) By introducing a lockdown, the town administration can mandate who can and who can not go outside. The administration knows the individual costs of falling ill of every citizen and maximizes the sum of people's expected utilities. How many people should the administration allow to go outside? Call this number K.
As staying home yields a utility of zero, the sum of everyone's utilities is just the sum of utilities of people who go outside.
Obviously, given that K people go outside, to maximize the sum, the administration should mandate that these are the people with lowest costs.
Thus, the sum of utilities (welfare) is:
W(K) = 50K - (1 + 2 + \dots + K)\frac{K - 1}{200} = 50K - \frac{K(K + 1)(K - 1)}{400}.
To maximize this function, consider a marginal increase in the welfare when K is incremented by 1:
MW(K) = W(K) - W(K - 1).
The administration should increase K as long as MW(K) is positive, and stop when it becomes negative.
We have:
MW(K) = 50 - \frac{1}{400} \left( (K + 1)K(K - 1) - K(K - 1)(K - 2) \right) =
= 50 - \frac{K(K - 1)}{400} \left( K + 1 - (K - 2) \right) = 50 - 3\cdot \frac{K(K - 1)}{400}.
Thus, MW(K) > 0 when K(K - 1) < \frac{2}{3} \cdot 10000 \approx 6666, and MW(K) < 0 otherwise.
As 80^2 = 6400 , a good guess for the K at which the sign changes is around 80.
It is straightforward to check that: 81 \cdot 82 = 6642 < 6666, \quad 82 \cdot 83 = 6806 > 6666. Thus, the optimal K is equal to 82. The administration should let 82 people go outside.
(c) (6 rp) Compare N and K. Provide intuition for why is one greater than the other. Which fundamental economic problem does this model illustrate?
As 100 > 82, N > K. This happens because (at least in this model) people do not take into account the negative externality they inflict on others when going outside (or the positive externality when they stay home). This model illustrates the classic problem of the underprovision of a public good (the public good here is the empty street). Equally correct, one may say that this model represents an instance of tragedy of commons: a common resource – the street and other common places – is, in a sense, over-exploited here when there is no regulation.
Remark. Please note that even though N>K, the difference is not that big. Thus, the administration's intervention does not have to be strong: people themselves are already rational enough so that half of the town's population stays home without any mandatory lockdown. Roughly speaking, the strength of the mandatory lockdown (N-K) should correspond only to the strength of externalities people produce but not to the total costs of the disease.