In order to better understand externalities in efforts to slow the march of COVID-19 such as social distancing and wearing facemasks, I wrote down the following control theory model. It simplifies in many ways from reality. For example, the model

• treats the course of the disease and the transmission to others during the course of the disease as if it all happened in an instant.

• treats everything as happening so quickly that discounting is unimportant

• acts as if the number of people who will become immune before the introduction of a vaccine remains a very small fraction of the population (that can be ignored)

• pretends that a vaccine will be deployed instantly at a known moment in the future.

• takes the perspective of the social planner, except in some of my commentary. The social planner is assumed to be able to require certain behaviors. This is not entirely unreasonable since behaviors such as social distancing and mask-wearing—or failures to do so—are fairly visible.

Still, I think this model gets the logic of the externalities right.

Among logically equivalent representations of the control theory model, I tried to choose the most convenient representation. I think in particular that making the state variable the natural logarithm of the number of people infected at any moment makes the math simpler. Here are the key definitions:

The optimization problem itself is this:

The Hamiltonian, first order condition, Euler equation and transversality condition are:

The Euler equation can be integrated with the transversality condition as terminal condition to get this integral for lambda, the marginal burden of the log level of infections:

What are the two dimensions of externalities?

1. Efforts to either not get the disease in the first place or to not transmit it to others have an externality in keeping the disease from spreading. This externality is shown in the first-order condition by lambda[f’(x)+g’(x)]. This is an externality on the assumption that you yourself become immune once you have the disease, see any efforts you make as a drop in the bucket, and are egocentric, so the future spread of the disease is not a concern to you once you have the disease.

2. Even if the vaccine were coming tomorrow, efforts to keep from transmitting the disease to others benefit those others. This externality is shown by the term e^k g’(x).

In the first-order condition, only e^k f’(x) is a direct benefit to oneself.

When is it most important to make great efforts at social distancing, facemask wearing, etc? The first-order condition gives an answer. First, it is most important to make efforts when the prevalence of the disease is high—that is, when e^k is big. Second, holding e^k fixed, efforts to reduce the spread of the disease are more important the further away the vaccine is; lambda is bigger the earlier it is in the course of the disease. Third, this is not a concave problem, so there may be multiple local minima. In particular, there may be two policies that both satisfy the necessary conditions, one of which lets the disease explode and another that keeps it under control. One of these is likely to be better than the other. Steering a course toward the lower minimum is important.

Don’t miss these other posts on the coronavirus pandemic:

• Avoiding Economic Carnage from the Coronavirus: There are Better Policies than Sending Everyone $1000

• How Does This Pandemic End?

• Logarithms and Cost-Benefit Analysis Applied to the Coronavirus Pandemic

• Why We are Likely to Need Strong Aggregate Demand Stimulus after Tight Social Distancing Restrictions are Over

• Seconding Paul Romer's Proposal of Universal, Frequent Testing as a Way Out