# Behavioral epidemiology: An economic model to evaluate
optimal policy in the midst of a pandemic111Chakrabarti:
<EMAIL_ADDRESS>University of Manchester; Krasikov:
<EMAIL_ADDRESS>Higher School of Economics Moscow; Lamba:
<EMAIL_ADDRESS>Pennsylvania State University. We are grateful to David
Argente, Shoumitro Chatterjee, Krishna Dasaratha, Elisa Giannone, Callum
Jones, Kei Hirano, Shouyong Shi, Shamim Sinnar, Jakub Steiner and Flavio
Toxvaerd for helpful comments and suggestions.

Shomak Chakrabarti Ilia Krasikov Rohit Lamba

(February 2022)

###### Abstract

This paper combines a canonical epidemiology model of disease dynamics with
government policy of lockdown and testing, and agents’ decision to social
distance in order to avoid getting infected. The model is calibrated with data
on deaths and testing outcomes in the Unites States. It is shown that an
intermediate but prolonged lockdown is socially optimal when both mortality
and GDP are taken into account. This is because the government wants the
economy to keep producing some output and the slack in reducing infection is
picked up by social distancing agents. Social distancing best responds to the
optimal government policy to keep the effective reproductive number at one and
avoid multiple waves through the pandemic. Calibration shows testing to have
been effective, but it could have been even more instrumental if it had been
aggressively pursued from the beginning of the pandemic. Not having any
lockdown or shutting down social distancing would have had extreme
consequences. Greater centralized control on social activities would have
mitigated further the spread of the pandemic.

> "Coronavirus is Germany’s greatest challenge since World War II, says Angela
> Merkel", Deutsche Welle (2020)

> "Covid-19 restrictions not affecting social distancing, says ONS: UK
> statistics agency says level of people still meeting likely to lead to
> increased hospitalisations and deaths." Financial Times (2020)

> "Vietnam abandons zero-Covid strategy after record drop in GDP: Warnings
> that lockdowns were crippling businesses heaped pressure on Communist
> government." Financial Times (2021b)

##  1 Introduction

Three instruments have been salient in the global response to the Covid-19:
government policy of lockdown and testing, and people’s decision to practice
social distancing. The objective for the collective has largely been the
minimization of direct mortality while ensuring a steady pace of the economy,
and the objective of the individual has been lowering the chance of getting
infected while ensuring some normalcy in life.

While the academic literature in epidemiology is primarily concerned with
understanding the evolution of the disease and how to control it, economists
are attempting to understand the co-evolution of mortality and economic output
using some subset of the aforementioned three instruments as choice variables
subject to some capacity constraint. In that vein, this paper builds a model
of disease dynamics where government tries to manage mortality and the economy
through lockdown and testing policies, and agents social distance balancing
the chance of getting infected with maintaining some social interactions.

While lockdown is good at reducing the spread of infection, it is often
accompanied with severe economic costs which may further jeopardize
livelihoods even after the pandemic recedes. Social distancing as a behavioral
response to the spread of the disease influences the government’s choice of
severity of the lockdown. And, testing spreads more information all around as
to who should be participating in social and economic activities.

We start from an epidemiological framework with five possible
states—susceptible, infected, hospitalized, recovered and dead. On it we build
a model of economic and social interactions where people are matched and the
infection spreads. Economic activities produce output, and can be restricted
by the government’s lockdown policy. Social activities provide utility to the
agents, and are controlled endogenously by their social distancing decisions.
Agents suffer disutility from getting infected and infecting others. Both
types of interactions– economic and social– can be mitigated by tracing and
testing and those who are found to be infected are forced to stay at home. The
pandemic can end with the arrival of a vaccine at a random time distributed
between one and two years since the inception with a mean of about a year and
a half. Finally, anticipating the behavioral response of the agents, the
government chooses the optimal testing and lockdown policies, by maximizing a
weighted sum of total output and mortality. Each agent in turn takes the
government’s lockdown and testing policy and other agents’s social distancing
decisions as given and choose their best response of social
distancing.222Section 2 details the main ingredients of the model. Section 3
describes the system of equations that quantify the propagation of the
disease. Section 4 introduce behavioral response and the solves for the
agent’s optimization problem. And, Section 5 states and solves for the
government’s optimization problem.

The main contribution of the paper is to model and analyze the three
instruments together and explore their pairwise substitutability. This
constitutes a technical challenge because we are solving for (i) a planning
problem of a forward looking government, (ii) a strategic problem between
forward looking agents, and (iii) the fixed point of the government’s and the
agents’ problems as a "stackelberg game". A key difficulty arises from the
fact that tracing-testing introduces heterogeneity in the population on who
was tested when. Our approach allows us to conclude that time since the last
test is a sufficient statistic for the agent’s choice of how much to social
distance. We then setup an optimal control problem for the agent and another
one for the government (Sections 4 and 5 respectively) and solve the two
simultaneously using the forward-backward sweep algorithm.

Two key equations summarize the government’s lockdown and agents’ social
distancing choices. The one for the agents pins down the static and dynamic
tradeoffs from social distancing. The static trade-off is described as
follows: The fraction of social distancers (or the probability of social
distancing) is directly proportional to the lump-sum cost of getting infected
and infected others normalized by the flow cost of social distancing. Further,
the dynamic trade-off adds to this proportionality the net present value of
the normalized cost from getting infected while participating in social
activities. Analogously, the government implements the lockdown while trading
off drop in current output with mortality and net present value of future
output. Both these equations also incorporate the heterogeneity of information
introduced by testing.

A secondary contribution of the paper is to calibrate the model using data on
testing and deaths in the United States due to Covid-19 (Section 6). It is by
now well understood that the standard SIR model is poorly identified with the
typical time series on the number of infected and dead (see Fernández-
Villaverde and Jones (2020) and Korolev (2021)). We take a pragmatic approach
by fixing the medical parameters using the aggregated wisdom of various
studies in medical journals and then use the data to estimate three types of
parameters; (i) prevalence or matching of the susceptible with the infected,
(ii) efficacy of tracing and testing, and (iii) cost for agents from getting
infected and infecting others. These provide a quantitative sense respectively
of how rapidly the disease was spreading in the US, how effective was the
government in tracing and testing the infected and asymptomatic, and how
agents evaluated their decisions to social distance.

For testing we feed the model with data on daily tests that have been
conducted in the US during the pandemic. By tracing we mean the efficacy to
identify those in the population who are infected but haven’t developed severe
symptoms yet— this is captured by a parameter which is calibrated using data.
Our calibrated estimate of it suggests that the US did reasonably well (on
average) over the course of the pandemic in identifying and isolating the
infected. Its main shortcoming was in the total number of tests available
early in the pandemic say in comparison to South Korea.

How should we systemically think about the role of social distancing? We
evaluate it in terms of a daily output produced by the agent, which we take
approximately to be her/his daily wage. In that sense, the daily or flow cost
of not partaking in any social activities is about 22% of the daily wage. In
addition the agent also suffers a lump-sum disutility from getting infected;
this represents the psychological cost of contracting the virus, medical
costs, and the potential (probabilistic cost) of death. The calibration
exercise pegs this to be approximately worth one and a half year of wages. A
further 50 days of wages is the lump-sum (altruistic) disutility from
potentially infecting others.

After calibrating the model, we execute the algorithm to calculate the optimal
policy (Section 7). The government shuts down about 40 percent of the economic
activity for a prolonged period of time, about 14 months. The lockdown is not
complete in that it does not hit the upper bound (of 70 percent) at any point,
but it is consistent. For the same time frame, the agents cut back around 50
percent of their social activities. So the government expects the agents to
social distance which allows it to impose a less than severe lockdown and
since the government locks down some part of the economic activity the agents
cut back on some but not all social activities.

The underlying conceptual tack on these policy choices is that they ensure the
effective $R_{t}$, i.e. the reproduction number is maintained almost constant
throughout the pandemic at one. This controls the spread of the virus while
maintaining some economic and social activities. The constancy of effective
reproduction number is also in contrast to what actually transpired in the US
between March 2020 and August 2021—multiple waves in infections and deaths. In
fact, the total number of deaths predicted by the optimal policy is less than
half of actual number for the United States as of September 2021—around
350,000 as opposed to more than 800,000 observed in the data as of writing
this draft.

The optimal policy exercise described here is conducted for a specific (widely
accepted) Pareto weight on total output and mortality or equivalently for a
specific value of life. We then also present the Pareto frontier to illustrate
the policy mix of options available with the government. For each possible
value of the weight, both optimal control problems have to be solved from
scratch yielding distinct value of final output and deaths. This Pareto
frontier is given by the solid black line in Figure 1—net present value of
total economic output is on the $y$-axis and total number of survivors is on
the $x$-axis.

The dotted-blue line in Figure 1 shows the Pareto frontier when tracing-
testing is shutdown, and lockdown and social distancing are the only
instruments for fighting the pandemic, the curve shrinks almost uniformly. In
more detailed policy experiments in the paper (in Section 8.1) we show that if
the tracing-testing was more effective and aggressively pursued since the
inception of the pandemic, the Pareto frontier shifts out away from the black
line and the gains are quite significant. Therefore, for improved tracing-
testing technology, keeping the number of survivors fixed, total output
expands, and keeping total output fixed, total mortality goes down.

Figure 1: The Pareto frontier for different specifications of the model

What if the government imposes no lockdown? The purple star in Figure 1
represents the outcome on total output and mortality. The infection runs
through the population very quickly with a single large peak during which
agents social distance almost maximally. The total number of deaths is around
750k, which is significantly higher than the optimal policy and in line with
the actual realized number as the pandemic has played out in the US. There are
of course some economic gains since no part of the economy has been closed. We
show however (Section 8.3) that even with no-lockdown the government can
mitigate the extent of the pandemic with more effective and aggressive
testing, bringing down the cumulative number to deaths to less than 400k,
while maintaining the economics gains. A provocative thought is that even with
no lockdown the total number of deaths is in ballpark of the actual realized
deaths seen in the data. The main difference is the spread of deaths over
time. This raises the question—what is the primary goal of the lockdown
policy? One interpretation of our analysis is that lockdown (as eventually
practiced in reality, not in the optimal policy of our model) simply slowed
down the pandemic so that the institutions fighting it could cope.333This is
echoed somewhat in Atkeson (2021b): ”this model forecast that if efforts to
slow transmission were applied early but were only temporary, this dramatic
first peak would be delayed but not prevented: cases and deaths would explode
again once efforts to slow transmission were relaxed.”

In the next policy experiment, we ask what happens if we (hypothetically) shut
down the channel of social distancing. One way to achieve this is to make the
flow cost from social distancing arbitrarily high and the lump sum costs from
getting infected to be low. The Pareto frontier from this exercise is given by
the dashed-red curve in Figure 1. The set of feasible output and mortality
attainable by the government shrinks considerably. In Section 8.3, we show
that the pandemic runs through the population quickly, leading again to almost
800k deaths by the end of July 2020. The government locks down for a few
months but then opens up the economy completely for the pandemic has already
run amok. That is why the output doesn’t fall much even though the number of
deaths is large. This illustrates the significance of incorporating behavioral
response in an otherwise mechanical model of disease dynamics.

In the final policy experiment, we allow the government to have greater
control over restricting social interactions (Section 8.2). This changes the
dynamics of infections and deaths dramatically. The idea being that autocratic
governments or communitarian (as opposed to individualist) societies are able
to control to social interactions of their citizenry to a much larger degree.
Unsurprisingly, the deaths decrease non-linearly as we pass on greater control
to a centralized authority in the social realm.

Related literature. In the rapidly growing literature on economics of
epidemiology various complimentary papers look at some combination of the
three forces we seek to model. Table 1 lists some of the studies done by
economists and categorizes them on the basis of policy instruments.

| lockdown | tracing-testing | social distancing
---|---|---|---
Alvarez, Argente, and Lippi (2021) | ✓ | ✓ |
Acemoglu, Chernozhukov, Werning, and Whinston (2021) |
Bergera, Herkenhoff, Huang, and Mongey (2021) |  | ✓ |
Chari, Kirpalani, and Phelan (2021) |  |
Farboodi, Jarosch, and Shimer (2021) | ✓(exogenous) |  | ✓
Toxvaerd (2019, 2020), Dasaratha (2020), Engle et al. (2021) |  |  | ✓
Table 1: A sample of recent and related economic models of epidemiology

Broadly these papers fall within the realm of augmented SIR models, where one
or more instruments from lockdown, tracing-testing and social distancing are
added to lend realism to the otherwise mechanical set of equations driving
disease dynamics. Lockdown and testing are typically added as a planning
problem for the government and social distancing through a strategic or agency
model of behavior. To the best of our knowledge, no paper so far has studied a
model that incorporates all three instruments together, which is the goal of
this paper. As the reader goes further, we hope to convince her/him that this
is both technically challenging task and a qualitatively important one, as
suggested already in Figure 1. For more detailed list of references, see
McAdams (2021) for an excellent survey on the recent progress in augmented SIR
models.

The importance of explicitly modeling behavior in SIR models has especially
been emphasized. For example, Atkeson (2021b) writes: " "behavior turns what
would be a short and extremely sharp epidemic into a long, drawn out one."
This also builds on a great body of work in epidemiology, where the importance
of behavioral responses to improve the precision of the predictive power of
the standard SIR-type model has been emphasized. For example, writing in the
Proceedings of the National Academy of Sciences, Fenichel et al. (2011) state:

> Results indicate that including adaptive human behavior significantly
> changes the predicted course of epidemics and that this inclusion has
> implications for parameter estimation and interpretation and for the
> development of social distancing policies. Acknowledging adaptive behavior
> requires a shift in thinking about epidemiological processes and
> parameters.444See also Perra et al. (2011) and Rizzo et al. (2014) for other
> approaches to modeling behavior in epidemiology.

To this discussion in particular, we add the nuance of how a uniform lockdown
policy interacts with behavioral response, exploring their substitutability.
In addition, tracing-testing introduces (i) heterogeneity in behavioral
response because social distancing now is a function of the day of the latest
test, and (ii) piecewise substitutability with both lockdown and social
distancing in a highly non-linear fashion.

There also has been a rapidly burgeoned literature in the macroeconomics of
COVID-19. Equilibrium models of the form DSGE-meets-SIR, where representative
agents make labor and consumption decisions have been proposed by Eichenbaum,
Rebelo, and Trabandt (2021a), Jones, Philippon, and Venkateswaran (2021), and
Krueger, Uhlig, and Xie (2020), amongst others, and further extended to
heterogenous agents by Kaplan, Moll, and Violante (2020) and Glover,
Heathcote, Krueger, and Rios-Rull (2021), amongst others. Closer to our
analysis Eichenbaum, Rebelo, and Trabandt (2021b) argue that lockdown type
interventions prolong the pandemic to buy time for the health infrastructure,
and there are further synergies of this force with testing and quarantining.

In non-SIR perspectives, Guerrieri, Lorenzoni, Straub, and Werning (2021)
explore demand deficiencies created by the pandemic through supply side
shortages. Also, Caballero and Simsek (2021) provide a model of asset prices
spiral when the economy is hit by severe supply shock. In contrast to these,
we look at a strategic framework to pin down the behavioral response of agents
in an augmented SIR-type model where governments incorporate these in
determining optimal policy.555There also has been recent work on understanding
the socio-economic implications of the large pandemics, and take lessons that
can inform policy decisions in dealing with the health and economic crisis,
see Òscar., Singh, and Taylor (2020), Correia, Luck, and Verner (2020) and
Barro, Ursúa, and Weng (2020). See the survey Brodeur, Gray, Islam, and
Bhuiyan (2021) for more details.

In terms of data work in the realm of the economics of epidemiology, our work
is related to Fernández-Villaverde and Jones (2020) and Korolev (2021)—the
former fits the standard SIR model to data from various parts of the world to
estimate primarily the prevalence parameter and the latter argues that a
unique identification of the SIR model is actually impossible. We proceed by
fixing medical parameters informed by medical studies and calibrate the
prevalence, tracing-testing and behavioral parameters. To the best of our
knowledge, this is the first paper that tries to tease out the parameters
driving tracing-testing and behavioral response by fitting the model to the
data on deaths and tests.

##  2 Model

States and transitions. There is a continuum of identical agents with mass one
who interact in discrete time indexed by $t=1,2,\ldots$. The agents are
forward-looking and discount the future with a factor $\delta_{A}\in[0,1)$. At
any point, a representative agent can find herself in one of five possible
health states (or compartments):

  1. $\bf S$

_Susceptible_ , in this state the agent is non-infectious and non-immune;

  2. $\bf I$

_Infected_ , in this state the agent is infectious but is either asymptomatic
or mildly symptomatic;

  3. $\bf H$

_Hospitalized_ , in this state the agent is infectious and clearly
symptomatic, thus she is hospitalized or being treated in isolation at home;

  4. $\bf R$

_Recovered_ , in this state the agent is non-infectious and immune; it is an
absorbing state;

  5. $\bf D$

_Dead_ , in this state the agent has succumbed to the infection and is dead;
again an absorbing state.

The susceptible ($\bf S$) have not had the virus, do not have immunity and may
get infected in the future by coming in contact with an infected agent. The
agents who get infected ($\bf I$) are at first either asymptomatic or mildly
symptomatic. This assumption is especially crucial to study the Covid-19
pandemic since one of the key difficulties has been the seemingly large number
of asymptomatic carriers.666For example, Germany’s leading Covid-19 expert
informing policy, Christian Drosten, had stated in the summer of 2020: ”We now
have evidence that almost half of infection events happen before the person
passing on the infection develops symptoms – and people are infectious
starting two days prior to that,” (The Gaurdian (2020)) Without an external
intervention such as tracing/testing, agents in state $\bf I$ could
potentially be indistinguishable from those in state $\bf S$.

$\bf S$$\bf I$$\bf IT$$\bf R$$\bf H$$\bf RT$$\bf
D$infectiontestinghospitalizationdeathtestingrecovery

Figure 2: States and transitions in disease dynamics

The infected can transition to two states: they either start showing symptoms
and be hospitalized ($\bf H$) or they can recover ($\bf R$). The hospitalized
$(\bf H)$ can also transition to two states: they can either recover ($\bf R$)
or they can die ($\bf D$). Recovery and death are absorbing states. While
death is obviously an absorbing state, assuming recovery to be absorbing
implies that the agents develop immunity to the virus. Note that we are
assuming that being hospitalized is a necessary step before death. This is
done for simplicity. We could have added another state for non-hospitalized
deaths due to Covid-19. The state $\bf H$ should be broadly interpreted as
those showing clear and/or severe symptoms.

Tracing and Testing. In order to separate the infected (and asymptomatic or
mildly symptomatic) from the susceptible, we introduce tracing and testing. We
assume that in each period a fraction of individuals who are in states $\bf
S$, $\bf I$ and $\bf R$ are traced and tested. The test perfectly reveals
whether the individual is susceptible, infected or recovered. Thus, we are
assuming the availability of both the viral and antibody test. This further
splits $\bf I$ into two compartments, those _infected and asymptomatic_ and
those _infected and asymptomatic but known to be so_. These two compartments
are denoted by $\bf I$ and $\bf IT$, respectively. Similarly, $\bf R$ is split
into two compartments– $\bf R$ for the _unknown recovered_ , and $\bf RT$ for
the _known recovered_. Figure 2 summarizes the various states and transition
possibilities. Abusing notation slightly, denote by $S_{t}$, $I_{t}$,
$IT_{t}$, $R_{t}$, $RT_{t}$, $H_{t}$ and $D_{t}$ the aggregate fractions of
agents in respective states at time $t$.

Testing is modeled through the following technology. The count of available
tests is exogenous and specified by $\overline{X}_{t}$. At the end of period
$t$ a fraction $\tau_{t}$ is uniformly chosen amongst the set of “eligible”
agents, so that exactly $\overline{X}_{t}$ tests are conducted. The set of
“eligible” agents includes the asymptomatic infected $(\bf I)$ and a fraction
$\gamma\in[0,1]$ of the susceptible $(\bf S)$ and unknown recovered $(\bf R)$.
The parameter $\gamma$ pins down the efficiency of tracing. If $\gamma=0$,
tracing is perfect for only infected people are tested, and if $\gamma=1$,
then testing is blind or completely random. Finally, tracing and testing are
assumed to be history-independent, so that all agents in states $\left\\{\bf
S,I,R\right\\}$ have the same likelihood of getting traced or tested,
irrespective of the past state of tracing and testing.

This technology is of course simplified and buys us considerable tractability.
Most natural extensions, such as making anti-body tests unavailable, making
$\overline{X}_{t}$ a choice variable, and introducing history dependence in
testing, can be accommodated.

Two worlds. All agents engage in two types of activities in each period:
economic and social. In each activity, the agents are randomly matched to each
other. Matching is bilateral and each pair of agents can get matched
independently of matching outcomes of other agents. The matching probabilities
are assumed to be $\beta_{w}$ and $\beta_{s}$ for work (or economic) and
social activities, respectively. Note that $\beta_{w}+\beta_{s}$ is equivalent
to the "total prevalence" in the SIR framework, but in the context of our
model it refers to the level of interaction an agent has in each period.

For social interactions, we have in mind activities such as going to a park,
or visiting each others’ homes that may not directly produce any output.
Moreover, the agents can control this rate of matching, as will be formalized
later, by practicing social distancing. For economic activities, we have in
matches that are generated while engaging in work that directly produces
output. These can be controlled be government through lockdown.

Economic activity and lockdown. In addition to introducing tracing and
testing, the planner also implements a lockdown, modeled as the stoppage of a
fraction $1-\lambda_{t}$ of all economic activities, bounded by
$\lambda_{t}\geqslant\bar{\lambda}$.777We have in mind the necessity of
essential services such as hospitals and supply chains for food and gas, and
$\bar{\lambda}$ represents that threshold. The total output of the economy
during the pandemic, $Y_{t}$ at time $t$, is assumed to have a simple
structure:

$Y_{t}:=\lambda_{t}(S_{t}+I_{t}+R_{t}+RT_{t}).$

The functional form above states that total output is equal to the total
number of agents participating in economic activity, and that at any given
point in time only those in states $\left\\{\bf S,I,R,RT\right\\}$ are
productive. We are implicitly assuming all those in states $\left\\{\bf
IT,H\right\\}$ are quarantined and do not participate in economic activities.
A selective lockdown policy could allow only those in state $\bf RT$ to work,
but we think that is unrealistic, because society still needs essential
services to go on, and especially given what we currently see in lockdown
policies all over the world.

Social activity and behavioral response. The salient piece of the model is the
introduction of behavioral response in the agents’ social activities. We allow
the agents to voluntarily _social distance_ themselves. They have strict
preferences to participate in social activities sans the virus. To capture
this idea, we suppose that social distancing is costly, its flow costs are
denoted by $\frac{c}{2}(1-\alpha)^{2}$, where $\alpha\in[0,1]$ is a fraction
of social activities in which an agent participates. The costs are convex,
more specifically quadratic, which captures an intensive margin of the agents’
social distancing decision. Since the agents are forward-looking, they account
for the current and future costs from social distancing. In addition to
$\frac{c}{2}(1-\alpha)^{2}$, the agents also face two other (lump-sum) costs:
a threat of getting infected and altruistic concerns of not infecting others;
they suffer disutility of $\phi^{+}$ and $\phi^{-}$, respectively, from the
two scenarios.888The idea behind the three parameters $(c,\phi^{+},\phi^{-})$
is that one level we want to differentiate the opportunity cost of not
partaking in social activities from the cost of actually getting infected, and
at another level we want to decompose the lump-sum cost of getting infected
into personal costs and altruistic concerns of infecting others.

Vaccine. Finally, we also allow for the possibility of discovering a _vaccine_
which can cure the virus. For simplicity, upon its arrival all infected agents
are assumed to be cured, thus the epidemics effectively stops. Suppose the
vaccine arrives at a (random) time $T$. The total output after the end of
epidemics at $t>T$ equals to the measure of the agents who survive, that is
$1-D_{T}$.

We choose to model the arrival of vaccine through a negative binomial
distribution. This is empirically relevant for at least two reasons. First,
the distribution is parametrized by two variables: its mean $\mathbb{E}[T]$
and variance $\mathbb{V}[T]$. So, we can control both aspects of the
distribution separately to reflect the reality on vaccine consensus all
through the year 2020 before a credible claim on the existence of a vaccine
was made. Second, the negative binomial is a sort of "repeated Bernoulli"
distribution. Each period there is a probability of success or failure, and
only after an exogenously specified number of success do we deem the event
"vaccine developed" realized. Thus, there is a minimal time till which no
vaccine can be developed, which is precisely given by
$\frac{(\mathbb{E}[T])^{2}}{\mathbb{E}[T]+\mathbb{V}[T]}$. In what follows, we
will let $p_{t}$ be the probability that vaccine arrives at time $t$, that is
$T=t$, and let $q_{t}:=1-\sum\limits_{s\leqslant t}p_{t}$ be the probability
that the vaccine hasn’t arrived till time $t$.999Other leading candidate is
the Poisson or geometric arrival, which is stationary, and hence would ensure
that likelihood of the arrival of vaccine two months from the start of the
pandemic is the same for it to be developed in 14 months at the 12 month mark.
This distribution is a limiting case of the negative binomial in which the
mean and variance coincide. This has been used, for example, by Alvarez et al.
(2021) and Farboodi et al. (2021).

Tying it all together. Our main interest lies in understanding how the
tradeoff between economic well-being and mortality is shaped by the
simultaneous interaction of three forces: social distancing, tracing and
testing, and lockdown. At the outset, the planner commits to the testing and
lockdown policies which will be in effect until the development of cure. Then,
within each period the following sequence of events occurs:

  1. 1.

The agents in $\left\\{\bf S,I,R,RT\right\\}$ participate in economic and
social activities, thus creating matches and spreading the virus.

  2. 2.

The agents who were in $\\{\bf I,IT,H\\}$ may recover or transition to a
follow-up compartment.

  3. 3.

A fraction $\tau_{t}\gamma$ of $\left\\{\bf S,I,R\right\\}$ is tested
according to the technology described above, and amongst the tested, those in
states $\bf I$ and $\bf R$ transition respectively to $\bf IT$ and $\bf RT$.

  4. 4.

The vaccine maybe discovered at time $T$, which is a random variable.

Finally, the (ex ante) payoff of a (representative) agent is given by
$Y_{A}-C$, where $Y_{A}$ is the total expected output, which is computed using
the agent’s discount factor, and $C$ is the total cost. One way to think about
$Y_{A}$ is that the agent directly cares about (or is compensated for) the
amount of output produced. The total cost is driven by three parameters: $c$,
$\phi^{+}$ and $\phi^{-}$, flow cost from social distancing, and one time
disutilities from getting infected and infecting others respectively. The
payoff of the government is given by $Y_{G}+\xi\cdot(1-D)$, where $Y_{G}$ is
the same as $Y_{A}$ except we will use a (potentially) different discount
factor $\delta_{G}\in[0,1)$ for the government, $1-D$ is the expected number
of survivors, and $\xi$ is the relative Pareto weight the government puts on
them. Since the functional forms of these objects require some further
notation, we will make them precise later.

##  3 Dynamics of infection

In this section we define the set of equations that governs the dynamics of
transitions amongst $\left\\{\bf S,I,IT,R,RT,H,D\right\\}$. Importantly, we
shut down the behavioral channel for now and try to understand mechanical
aspects of the dynamics, bringing in the effect of behavioral response to the
dynamics in the next section. Two types of parameters will be introduced here:
the average time it takes for an agent to transfer from one state to another,
and conditional on the transfer away from one state, the fraction that goes to
each of the possible destination states.

Recollect that the government can control the spread of the disease by
uniformly locking down a fraction of $1-\lambda_{t}$ of productive agents,
with the maximal lockdown capped at $1-\bar{\lambda}$. Suppose that the
susceptible and infected agents always participate in the social activities,
then the stock of susceptible evolves as

$\displaystyle
S_{t+1}=S_{t}-\beta_{w}\lambda_{t}^{2}S_{t}I_{t}-\beta_{s}S_{t}I_{t},$
($\Delta S_{t}$)

where we count away the number of new infections due to matches at work and
social activities. This is of course one way of modeling economic and social
interactions within the standard epidemiology framework. As a first pass and
for tractability we consider this additively separable assumption where the
outflow of new infections is "substitutable" with coefficients $\beta_{w}$ and
$\beta_{s}$ between the two primary activities for the agents in society.

At the inception, there is an initial seed of infection $I_{0}=e_{0}$, and all
others are susceptible: $S_{0}=1-e_{0}$. Periodic matches at work and in
social activities create new infections. We assume that infected agents either
develop clear symptoms (and need to be hospitalized) or recover at a fixed
rate on average time $t_{i}$. Taking into an account the attrition from $\bf
I$ and addition to $\bf IT$ due to testing, we get that the following
equations describe the dynamics of infected cases:

$\displaystyle I_{t+1}$
$\displaystyle=(1-\tau_{t})\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}I_{t}+\beta_{w}\lambda^{2}_{t}S_{t}I_{t}+\beta_{s}S_{t}I_{t}\bigg{)},$
($\Delta I_{t}$) $\displaystyle IT_{t+1}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{i}}\bigg{)}IT_{t}+\tau_{t}\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}I_{t}+\beta_{w}\lambda_{t}^{2}S_{t}I_{t}+\beta_{s}S_{t}I_{t}\bigg{)}.$
($\Delta IT_{t}$)

Recollect that the agents can transition from being infected, ($\bf I$) or
($\bf IT$), to either developing severe symptoms and being hospitalized ($\bf
H$) or recover ($\bf R$). We assume that conditional on transitioning away
from $\bf I$ or $\bf IT$, the states $\bf H$ and $\bf R$ are reached with
probabilities of transitions given by $m_{i}$ and $1-m_{i}$, respectively.
Further, we assume that those hospitalized ($\bf H$) transition away from that
state on average time $t_{h}$. Thus, we have:

$\displaystyle
H_{t+1}=\bigg{(}1-\frac{1}{t_{h}}\bigg{)}H_{t}+\frac{m_{i}}{t_{i}}(I_{t}+IT_{t}).$
($\Delta H_{t}$)

Next, the state $\bf D$ is an absorbing state, which keeps count of the number
of fatalities amongst those agents that become hospitalized ($\bf H$), and do
not recover ($\bf R$). We assume that the hospitalized agents, recover with
the probability $1-m_{h}$ and die with the complimentary probability $m_{h}$.
Thus, we have:

$\displaystyle D_{t+1}$ $\displaystyle=D_{t}+\frac{m_{h}}{t_{h}}H_{t}.$
($\Delta D_{t}$)

The state $\bf RT$ is absorbing while the state $\bf R$ is “almost" absorbing
in the sense that the event consists agents that have recovered without ever
showing symptoms, and if tested this will be revealed and they would transfer
to the state $\bf RT$. There are three sources chipping into $\bf R$ and $\bf
RT$: infected and not tested, infected and tested, and hospitalized. These
stocks evolve as follows:

$\displaystyle R_{t+1}$
$\displaystyle=(1-\tau_{t}\gamma)\bigg{(}R_{t}+\frac{1-m_{i}}{t_{i}}I_{t}\bigg{)},$
($\Delta R_{t}$) $\displaystyle RT_{t+1}$
$\displaystyle=RT_{t}+\frac{1-m_{i}}{t_{i}}IT_{t}+\frac{1-m_{h}}{h_{t}}H_{t}+\tau_{t}\gamma\bigg{(}R_{t}+\frac{1-m_{i}}{t_{i}}I_{t}\bigg{)}.$
($\Delta RT_{t}$)

To sum up, the dynamics of infections and economic activity are jointly
determined by the system of equations ($\Delta S_{t}$) - ($\Delta RT_{t}$). In
the next section we will add behavioral responses to this otherwise mechanical
system. Some comments about the set of states we choose to model disease
dynamics are instructive. The basic framework here is based on the celebrated
SIR model which constitutes three states, namely susceptible, infected and
recovered, starting at least from Kermack and McKendrick (1927), see also
Neher et al. (2020) for a recent treatment. We enrich the setup by separating
the infected and recovered into two categories on the basis of testing, and
creating a separate state for those that show severe symptoms and thus need to
be hospitalized. These two choices allow us to (i) introduce testing in a
realistic way, (ii) match the widely reported stylized fact that asymptomatic
carries are the largest source of contagion, at least for Covid-19, and (iii)
generate at least three time series $\left(IT_{t},H_{t},D_{t}\right)$ that are
observable in the data coming out from various countries.

##  4 Behavioral response and the agent’s problem

We augment the epidemiological system of equations described above by giving
the agents the agency to social distance themselves. This decision is
motivated by two types of parameters: the flow cost of social distancing ($c$)
and the (one time) disutility from either getting the infection or infecting
some other person ($\phi^{+}$ and $\phi^{-}$, respectively). All three are
assumed to be fixed behavioral parameters. Further, the agent chooses a
probability of social distancing at each time. Recollect that behavioral
response is only relevant for agents in the states $\left\\{\bf
S,I,R\right\\}$ because the others are either quarantined, hospitalized, known
recovered, or dead.101010Note that $\phi^{+}$ is the disutility an agent
suffers from getting infected, which we take here to include the psychological
cost from infecting close friends and family, and the actual perceived cost of
dying or the death of these family or friends. On the other hand $\phi^{-}$
refers to the disutility the agent suffers from infecting a person he/she is
randomly matched within a workspace, buying groceries or eating out or
socializing in crowded places.

###  4.1 Modified dynamics

In Section 3 the stock values of all the seven states completely characterized
the dynamics. However, testing introduces heterogeneity in the model which is
then propagated through the agents’ behavioral response. As a consequence,
aggregate stock variables are no longer sufficient to keep track of the
disease dynamics. In fact, we need to trace the evolution of the susceptible
($\bf S$), infectious ($\bf I$) and recovered ($\bf R$) agents conditional on
the _time of last test_. That the time of last test is a sufficient statistic
for the dynamics follows from our assumption that testing perfectly reveals an
agent’s current health state. After taking a test the agent is certain if she
is still susceptible, infectious or recovered. In the last two scenarios the
agent will transit to the follow-up compartments and never be “eligible” for
testing in the future. As a result, those who are tested at time $t$ and found
to be susceptible have identical (degenerate) beliefs independently of the
exact time instance $t$.111111The assumption that testing is history-
independent, that is all agents in $\left\\{\bf S,R\right\\}$ and all agents
in $\bf I$ have the same likelihoods of getting tested, can be relaxed to
allow for conditioning on the time of last test.

Define by $S_{t}^{k}$, $I_{t}^{k}$ and $R_{t}^{k}$ the aggregate fractions of
agents who are susceptible, infected and recovered, given calendar time $t$
and time of the last test $k\leqslant t$. For example, an agent in state
$I_{50}^{35}$ was tested at the 35th day of the pandemic, found to be
susceptible, and has since acquired the virus at some point in the last 15
days, but is asymptomatic. Here, $k=1$ tracks the subpopulation of agents who
never got any evidence, and $S^{t}_{t}$ is the fraction of people who have
just been tested to be susceptible. Further, let $\alpha_{t}^{k}$ be the
_probability that the agent who received the last test at time $k$
participates in social activities at time $t$_. We note that the total number
of susceptible, asymptomatic-infected and recovered satisfy the following:

$S_{t}=\sum_{k\leqslant t}S^{k}_{t},\;I_{t}=\sum_{k\leqslant
t}I^{k}_{t},\;R_{t}=\sum_{k\leqslant t}R^{k}_{t}.$

Similarly, define the total number of susceptible, asymptomatic-infected and
recovered among those who participate in social activities as

$\widehat{S}_{t}:=\sum_{k\leqslant
t}\alpha_{t}^{k}S_{t}^{k},\;\widehat{I}_{t}:=\sum_{k\leqslant
t}\alpha_{t}^{k}I_{t}^{k},\;\widehat{R}_{t}:=\sum_{k\leqslant
t}\alpha_{t}^{k}R_{t}^{k}.$

Here, $\widehat{S}_{t}$, $\widehat{I}_{t}$ and $\widehat{R}_{t}$ are
equilibrium objects and potentially smaller than the total numbers of people
in respective compartments, viz. $S_{t}$, $I_{t}$ and $R_{t}$, respectively.

When the agents make social distancing decisions, the mechanical system
($\Delta S_{t}$) - ($\Delta RT_{t}$) has to be adjusted to incorporate the
fact that socially inactive agents cannot spread the virus. Slightly abusing
notations, we redefine certain equations taking into an account the agents’
behavioral response:

$\displaystyle S^{k}_{t+1}$
$\displaystyle=(1-\tau_{t}\gamma)\bigg{(}S^{k}_{t}-\beta_{w}\lambda^{2}_{t}S^{k}_{t}I_{t}-\beta_{s}\alpha^{k}_{t}S^{k}_{t}\widehat{I_{t}}\bigg{)}\;\text{for
}k=1,\ldots,t,$ ($\Delta S_{t}^{k}$) $\displaystyle S^{t+1}_{t+1}$
$\displaystyle=\tau_{t}\gamma\bigg{(}S_{t}-\beta_{w}\lambda^{2}_{t}S_{t}I_{t}-\beta_{s}\widehat{S_{t}}\widehat{I_{t}}\bigg{)},$
($\Delta S_{t}^{t+1}$) $\displaystyle I^{k}_{t+1}$
$\displaystyle=(1-\tau_{t})\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}I^{k}_{t}+\beta_{w}\lambda^{2}_{t}S^{k}_{t}I_{t}+\beta_{s}\alpha^{k}_{t}S^{k}_{t}\widehat{I_{t}}\bigg{)}\;\text{for
}k=1,\ldots,t,$ ($\Delta I_{t}^{k}$) $\displaystyle IT_{t+1}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{i}}\bigg{)}IT_{t}+\tau_{t}\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}I_{t}+\beta_{w}\lambda^{2}_{t}S_{t}I_{t}+\beta_{s}\widehat{S_{t}}\widehat{I_{t}}\bigg{)},$
($\Delta IT_{t}$) $\displaystyle R^{k}_{t+1}$
$\displaystyle=(1-\tau_{t}\gamma)\bigg{(}R^{k}_{t}+\frac{1-m_{i}}{t_{i}}I^{k}_{t}\bigg{)}\;\text{for
}k=1,\ldots,t.$ ($\Delta R_{t}^{k}$)

In addition, of course, $I^{t+1}_{t+1}=0$, and $R^{t+1}_{t+1}=0$. The modified
dynamics makes it clear that we additively separate the two worlds of economic
and social decisions, and allow lockdown policies to affect the former and
social distancing to influence the latter. Testing in turn, endogenously
affects both decisions by reducing those partaking in both worlds in the state
$\bf I$ and potentially increasing those in the state $\bf R$.

A quick sanity check on the system can be done by noticing that since the
total population remains constant (including counting the number of deaths),
the sum of all states at all points must equal unity, and in fact we have:
$S_{t}+I_{t}+R_{t}+IT_{t}+RT_{t}+H_{t}+D_{t}=1$.

It follows that it is without loss to ignore the equation for the state $\bf
D$, i.e., at any point in time $D_{t}$ can be uniquely determined from the
other state variables using the above identity. To save on notations in the
future, we stack together the state variables and corresponding equations as

$\displaystyle\Pi_{t}:=$ $\displaystyle\Big{(}(S_{t}^{k})_{k\leqslant
t},(I_{t}^{k})_{k\leqslant t},IT_{t},(R_{t}^{k})_{k\leqslant
t},RT_{t},H_{t}\Big{)},$ $\displaystyle\Delta\Pi_{t}:=$
$\displaystyle\Big{(}(\Delta S_{t}^{k})_{k\leqslant t+1},(\Delta
I_{t}^{k})_{k\leqslant t},\Delta IT_{t},(\Delta R_{t}^{k})_{k\leqslant
t},\Delta RT_{t},\Delta H_{t}\Big{)}.$

Equations $(\Delta\Pi_{t})$ jointly define the disease dynamics in an
equilibrium, when the tests follows the process $\overline{X}_{t}$, the
government sets the lockdown policy $\lambda_{t}$ and the agents best-respond
by choosing social distancing $(\alpha_{t}^{k})_{k\leqslant t}$ as a function
of the last time of being tested. In the parlance of dynamic optimization
theory, these state equations are termed forward because they are pinned down
by the initial conditions at the outset, i.e., $S_{1}^{1}=1-e_{0}$ and
$I_{1}^{1}=e_{0}$.

###  4.2 Agent’s optimization problem

We now turn to the agent’s problem. Since a representative agent is
infinitesimal, she cannot influence the aggregate state variables. Instead,
the agent takes the dynamics of aggregate states as well as the government’s
lockdown and testing policies as given, and maximizes her own expected payoff
by choosing the rate of social distancing $(\alpha_{t}^{k})_{k\leqslant t}$.
In this quest the agent internalizes how her current social distancing
decision will affect likelihood of getting infected in the future. It turns
out that the agent’s problem is a dynamic control problem with the set of
constraints which resembles Equations $(\Delta\Pi_{t})$ modulo the fact that
certain aggregate variables are taken to be fixed.

To formally define the agent’s optimization problem, we should distinguish
between an agent’s state vector $\pi_{t}:=\big{(}(s_{t}^{k})_{k\leqslant
t},(i_{t}^{k})_{k\leqslant t},it_{t},(r_{t}^{k})_{k\leqslant
t},rt_{t},h_{t}\big{)}$ and its aggregate counterpart $\Pi_{t}$. For example,
$s_{t}^{k}$ is the ex-ante probability that a representative agent is
susceptible at time $t$ when she last time got tested at time $k$. The
remaining variables in $\pi_{t}$ are interpreted similarly. We will make use
the following shorthand notations:
$y_{t}:=\lambda_{t}(s_{t}+i_{t}+r_{t}+rt_{t})$, and

$\displaystyle s_{t}:=\sum_{k\leqslant t}s_{t}^{k},\;i_{t}:=\sum_{k\leqslant
t}i_{t}^{k},\;r_{t}:=\sum_{k\leqslant
t}r_{t}^{k},\;\widehat{s}_{t}:=\sum_{k\leqslant
t}\alpha_{t}^{k}s_{t}^{k},\;\widehat{i}_{t}:=\sum_{k\leqslant
t}\alpha_{t}^{k}i_{t}^{k},\;\widehat{r}_{t}:=\sum_{k\leqslant
t}\alpha_{t}^{k}r_{t}^{k}.$

In addition, note that the probability of being dead $d_{t}$ at time $t$
follows from the accounting identity:
$s_{t}+i_{t}+r_{t}+it_{t}+rt_{t}+h_{t}+d_{t}=1$.

We now have all the concepts and notation in place to define the agent’s
preferences. The (ex ante) expected payoff of the representative agent is
given by $Y_{A}-C$, where

$Y_{A}:=(1-\delta_{A})\sum_{t=1}^{\infty}\delta_{A}^{t-1}\bigg{(}y_{t}q_{t}+\frac{\delta_{A}}{1-\delta_{A}}(1-d_{t+1})p_{t}\bigg{)},$

and

$\displaystyle
C:=(1-\delta_{A})\frac{c}{2}\sum_{t=1}^{\infty}\delta_{A}^{t-1}\bigg{(}(it_{t}+h_{t}+d_{t})q_{t}+\frac{\delta_{A}}{1-\delta_{A}}d_{t+1}p_{t}+\sum_{k\leqslant
t}(1-\alpha_{t}^{k})^{2}(s_{t}^{k}+i_{t}^{k}+r_{t}^{k})q_{t}\bigg{)}+$
$\displaystyle+(1-\delta_{A})\sum_{t=1}^{\infty}\delta_{A}^{t-1}\bigg{(}\phi^{+}\left(\beta_{w}\lambda_{t}^{2}s_{t}{I}_{t}+\beta_{s}\widehat{s}_{t}{\widehat{I}}_{t}\right)+\phi^{-}\left(\beta_{w}\lambda_{t}^{2}{S}_{t}i_{t}+\beta_{s}{\widehat{S}}_{t}\widehat{i}_{t}\right)\bigg{)}q_{t}.$

The functional form of $Y_{A}$ is straightforward, it simply aggregates the
total (expected) output produced by the agent using her individual discount
factor. One way to think about it is that the agent directly cares (or is
compensated for) the amount of output produced.

The total cost is driven by three parameters: $c$, $\phi^{+}$ and $\phi^{-}$.
The agents incur a flow cost $\frac{c}{2}(1-\alpha^{k}_{t})^{2}$ from social
distancing– these arise in the states $\left\\{\bf IT,H,D\right\\}$ at their
maximal level of $\frac{c}{2}$, because complete social distancing here is a
necessity, and in the states $\left\\{\bf S,I,R\right\\}$, because social
distancing here is a choice based on information of last test.

In addition, the agent suffers the one-shot disutility from getting infected
$\phi^{+}$ and another one from infecting others $\phi^{-}$. The former
represents the probabilistic cost of potentially getting very sick, or even
facing death, and infecting members of family, while latter is an altruism
parameters that captures the cost of infecting a random person at work or in
the social meetings. We want the reader (relatively speaking) to the think of
$c$ as being small, $\phi^{+}$ as being large, and $\phi^{-}$ as being
somewhat intermediate.121212There is a subtlety in how to interpret the
altruistic parameter $\phi^{-}$. Each agent is non-atomistic, so they have a
negligible impact on the aggregate. How should we then interpret the altruism
parameter? What we are doing here is plugging in the beliefs of the individual
into the preferences. So, it is an “as if” component of their utility. We want
the reader to think of it as a psychological cost from the prospect of
infecting others and since we are imposing a rational expectations equilibrium
this cost is realized ex post. It can also be regarded as a “warm glow effect”
(in the sense of Andreoni (1989)) from controlling the likelihood of infecting
others.

We now describe the laws of motion of the individual state vector. These
differ from the laws of motion of the aggregate state vector, because each
agent individually cannot influence behavior of the others on the matching
markets. We have the following system:

$\displaystyle s^{k}_{t+1}$
$\displaystyle=(1-\tau_{t}\gamma)\bigg{(}s^{k}_{t}-\beta_{w}\lambda^{2}_{t}s^{k}_{t}I_{t}-\beta_{s}\alpha^{k}_{t}s^{k}_{t}\widehat{I_{t}}\bigg{)}\;\text{for
}k=1,\ldots,t,$ ($\Delta s_{t}^{k}$) $\displaystyle s^{t+1}_{t+1}$
$\displaystyle=\tau_{t}\gamma\bigg{(}s_{t}-\beta_{w}\lambda^{2}_{t}s_{t}I_{t}-\beta_{s}\widehat{s_{t}}\widehat{I_{t}}\bigg{)},$
($\Delta s_{t}^{t+1}$) $\displaystyle i^{k}_{t+1}$
$\displaystyle=(1-\tau_{t})\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}i^{k}_{t}+\beta_{w}\lambda^{2}_{t}s^{k}_{t}I_{t}+\beta_{s}\alpha^{k}_{t}s^{k}_{t}\widehat{I_{t}}\bigg{)}\;\text{for
}k=1,\ldots,t,$ ($\Delta i_{t}^{k}$) $\displaystyle it_{t+1}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{i}}\bigg{)}it_{t}+\tau_{t}\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}i_{t}+\beta_{w}\lambda^{2}_{t}s_{t}I_{t}+\beta_{s}\widehat{s_{t}}\widehat{I_{t}}\bigg{)},$
($\Delta it_{t}$) $\displaystyle r^{k}_{t+1}$
$\displaystyle=(1-\tau_{t}\gamma)\bigg{(}r^{k}_{t}+\frac{1-m_{i}}{t_{i}}i^{k}_{t}\bigg{)}\;\;\text{for
}k=1,\ldots,t,$ ($\Delta r_{t}^{k}$) $\displaystyle rt_{t+1}$
$\displaystyle=rt_{t}+\frac{1-m_{i}}{t_{i}}it_{t}+\frac{1-m_{h}}{h_{t}}h_{t}+\tau_{t}\gamma\bigg{(}r_{t}+\frac{1-m_{i}}{t_{i}}i_{t}\bigg{)},$
($\Delta rt_{t}$) $\displaystyle h_{t+1}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{h}}\bigg{)}h_{t}+\frac{m_{i}}{t_{i}}(i_{t}+it_{t}).$
($\Delta h_{t}$)

In addition, $i_{t+1}^{t+1}=r_{t+1}^{t+1}=0$ at all dates. Stack the above
equations in a vector $\Delta\pi_{t}$, that is

$\Delta\pi_{t}:=\Big{(}(\Delta s_{t}^{k})_{k\leqslant t+1},(\Delta
i_{t}^{k})_{k\leqslant t},\Delta it_{t},(\Delta r_{t}^{k})_{k\leqslant
t},\Delta rt_{t},\Delta h_{t}\Big{)}$

The reader can verify that Equations $(\Delta\pi_{t})$ are almost identical to
$(\Delta\Pi_{t})$; the only difference is that we have $I_{t}$ and
$\widehat{I}_{t}$ instead of $i_{t}$ and $\widehat{i}_{t}$ in the first three
equations.

Finally, the agents’ optimization problem is given by the choice of the social
distancing $(\alpha_{t}^{k})_{k\leqslant t}$ and individual state variables
$\pi_{t}$ at each date to maximize expected output net of costs subject to the
family of state equations $(\Delta\pi_{t})$:

$\displaystyle\max\limits_{(\alpha_{t}^{k})_{k\leqslant t},\pi_{t}}\quad
Y_{A}-C+(1-\delta_{A})c\kappa P\quad\text{subject
to}\quad\text{\hyperref@@ii[eqpi A]{$(\Delta\pi_{t})$}};\;\text{given
}\big{(}S_{t},I_{t},\widehat{S}_{t},\widehat{I}_{t}\big{)}\;\text{and}\;(\lambda_{t},\;\tau_{t}).$

In the objective, $\kappa>0$ is a small number and $P$ is a punishment term to
avoid boundary solutions,131313Interior solutions for
$(\alpha_{t}^{k})_{k\leqslant t}$ make the execution of the algorithm that
finds the optimal solution to the government’s problem tractable. For context,
we actually assume $\kappa$ to be $10^{-12}$ in our numerical calculations.
that is

$P:=\sum_{t=1}^{\infty}\delta_{A}^{t-1}\sum_{k\leqslant
t}\Big{(}\ln\alpha^{k}_{t}+\ln\big{(}1-\alpha^{k}_{t}\big{)}\Big{)}q_{t}.$

The model admits as a special case the scenario where the agents are myopic:
this will constitute substituting $\delta_{A}=0$. The government, as we will
model later, will always be assumed to be forward looking. This assumption of
myopia for the agents dramatically simplifies all the calculations and some
have argued generates a model with nonetheless reasonable predictions. As
mentioned in the introduction, we do not take a call on the agent’s forward
looking capacity and allow the analyst the flexibility of varying the agents’
discount factors as she/he deems fit to the situation being studied. To
provide intuition and to point out the implications of the myopic model, we
will refer sometimes to the special case as we go along.

###  4.3 Equilibrium response and adjoint equations

Solving for the (representative) agent’s optimal social distancing rule
involves setting up the Lagrangian function. Let the Lagrange multipliers
corresponding to Equations $(\Delta\pi_{t})$ at time $t$ be $\Upsilon_{t}$,
i.e.,

$\Upsilon_{t}:=\Big{(}(\mathbbm{S}^{k}_{t})_{k\leqslant
t+1},(\mathbbm{I}_{t}^{k})_{k\leqslant
t},\mathbbm{I}\mathbbm{T}_{t},(\mathbbm{R}^{k}_{t})_{k\leqslant
t},\mathbbm{R}\mathbbm{T}_{t},\mathbbm{H}_{t}\Big{)}.$

We calculate the agent’s first order-condition with respect to the control
variable $(\alpha_{t}^{k})_{k\leqslant t}$ and then take the first-order
condition with respect to each state variable in the optimization problem to
get the agent’s adjoint equations.

The agent’s optimal social distancing decision is thus characterized as a
unique solution to the necessary first-order condition of the Lagrangian with
respect to $\alpha^{k}_{t}$, which is of course sufficient for optimality
whenever $\kappa$ is small enough:

$\displaystyle
1-\alpha^{k}_{t}+\frac{\kappa}{s_{t}^{k}+i_{t}^{k}+r_{t}^{k}}\bigg{(}\frac{1}{\alpha^{k}_{t}}-\frac{1}{1-\alpha^{k}_{t}}\bigg{)}=\frac{\phi^{+}}{c}\frac{s^{k}_{t}}{s^{k}_{t}+i^{k}_{t}+r^{k}_{t}}\beta_{s}{\widehat{I}_{t}}+\frac{\phi^{-}}{c}\frac{i^{k}_{t}}{s^{k}_{t}+i^{k}_{t}+r^{k}_{t}}\beta_{s}{\widehat{S}_{t}}+$
($FOC-\alpha_{t}^{k}$)
$\displaystyle+\frac{(1-\tau_{t}\gamma)\mathbbm{S}^{k}_{t}+\mathbbm{S}^{t+1}_{t}\tau_{t}\gamma-(1-\tau_{t})\mathbbm{I}^{k}_{t}-\tau_{t}\mathbbm{I}\mathbbm{T}_{t}}{c(1-\delta_{A})\delta_{A}^{t-1}q_{t}}\frac{s_{t}^{k}}{s_{t}^{k}+i_{t}^{k}+r_{t}^{k}}\beta_{s}{\widehat{I}_{t}}.$

One intuitive way of thinking about this condition is to first consider the
myopic case. Suppose $\delta_{A}=0$, then the second line in Equation
($FOC-\alpha_{t}^{k}$) disappears.141414This is because when $\delta_{A}=0$,
the dual variables are zero at all dates. Now, for this myopic case, take an
agent who got evidence of being susceptible at time $k$, then at date $t$ she
assigns probabilities

$f_{t}^{k}:=\frac{s_{t}^{k}}{s_{t}^{k}+i_{t}^{k}+r^{k}_{t}}\;\;\text{and}\;\;g_{t}^{k}:=\frac{i_{t}^{k}}{s_{t}^{k}+i_{t}^{k}+r^{k}_{t}}$

to the events that she is susceptible and infectious, respectively.151515If
the agent has been already discovered to be infected/recovered, then the
agent’s social distancing decision at $t$ will be mechanical. Thus, for
$\kappa$ close to zero, Equation ($FOC-\alpha_{t}^{k}$) can be re-written as
follows (whenever the solution for $\alpha_{t}^{k}$ is interior):161616In
general, as $\kappa\downarrow 0$, the solution for $1-\alpha_{t}^{k}$
converges to $\max\\{0,\min\\{1,\cdot\\}\\}$, where “$\cdot$” stays for the
right-hand side of Equation ($FOC-\alpha_{t}^{k}$). The same is true in the
more general context with $\delta_{A}>0$.

$\displaystyle\underbrace{1-\alpha_{t}^{k}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\begin{subarray}{c}\text{fraction
of }\\\ \text{social
distancers}\end{subarray}}\approx\underbrace{\frac{\phi^{+}}{c}f_{t}^{k}\beta_{s}{\widehat{I}_{t}}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\begin{subarray}{c}\text{cost
from }\\\ \text{getting
infected}\end{subarray}}+\underbrace{\frac{\phi^{-}}{c}g_{t}^{k}\beta_{s}{\widehat{S}_{t}}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\begin{subarray}{c}\text{cost
from }\\\ \text{infecting others}\end{subarray}},$

The agent experiences disutility from getting infected and infecting others
which are added up. Given the aggregate behavior, each susceptible agent gets
infected with a probability $\beta_{s}{\widehat{I}_{t}}$, and each infectious
agent infects another agent with a probability $\beta_{s}{\widehat{S}_{t}}$.
Moreover, $\frac{\phi^{+}}{c}$ and $\frac{\phi^{-}}{c}$ represent the costs of
getting infected and infecting others, respectively, normalized by the cost of
not being able to socialize for one period.

Having motivated the static component of Equation ($FOC-\alpha_{t}^{k}$),
allow now for $\delta_{A}>0$. The agents are forward looking, and thus
internalize the fact that they will transition between states in the future.
Yet, they do not internalize that their social distancing decision will affect
the aggregate dynamics.

Let $\Delta_{t}^{k}$ be the expected drop in the agent’s continuation value
due to an infection, that is
$\Delta_{t}^{k}:=(1-\tau_{t}\gamma)\mathbbm{S}^{k}_{t}+\tau_{t}\gamma\mathbbm{S}^{t+1}_{t}-(1-\tau_{t})\mathbbm{I}^{k}_{t}-\tau_{t}\mathbbm{I}\mathbbm{T}_{t}$.
Re-writing Equation ($FOC-\alpha_{t}^{k}$) for small $\kappa$ and assuming
that the social distancing decision is interior, we get:

$\displaystyle
1-\alpha_{t}^{k}\approx\underbrace{\ldots\ldots}_{\text{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
static,
$\delta_{A}=0$}}+\underbrace{\frac{\Delta_{t}^{k}}{c(1-\delta_{A})\delta_{A}^{t-1}q_{t}}f_{t}^{k}\beta_{s}{\widehat{I}_{t}}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\begin{subarray}{c}\text{dynamic
cost of}\\\ \text{an infection}\end{subarray}},$

A higher value of $\alpha_{t}^{k}$ increases the likelihood that the
susceptible agent gets infected by $\beta_{s}\widehat{I}_{t}$. In this case
the agent will transit to $it_{t}$ with the probability $\tau_{t}$ and
$i_{t}^{k}$ with the probability $1-\tau_{t}$, thus the agent’s expected
future value increases in a proportion to
$(1-\tau_{t})\mathbbm{I}_{t}^{k}+\tau_{t}\mathbbm{I}\mathbbm{T}_{t}$. At the
same time, the likelihood that in the next period the agent will still be
susceptible decreases; by the similar logic as above the agent’s expected
future value decreases in a proportion to
$(1-\tau_{t}\gamma)\mathbbm{S}_{t}^{k}+\tau_{t}\gamma\mathbbm{S}_{t}^{t+1}$.
Finally, as in the static case, the dynamic (lump-sum) cost is deflated by the
net present value of the flow cost of social distancing.

In addition to the first-order condition with respect to the control vector
$(\alpha_{t}^{k})_{k\leq t}$, we have the system of seven partial difference
equations ($\Delta\Upsilon_{t}$) which describes the dynamics of the adjoint
variables $\Upsilon_{t}$. This system is similar in spirit to Equations
$(\Delta\pi_{t})$, which define the dynamics of individual state variables
$\pi_{t}$. The main difference though is that that the system is _backward_ ,
i.e., it specifies $\Upsilon_{t}$ as a function of $\Upsilon_{t+1}$ and other
variables at time $t+1$. The “initial” conditions are given at “infinity”:

$\lim_{t\to\infty}\frac{\mathbbm{S}^{k}_{t}}{p_{t}\delta_{A}^{t-1}}=\lim_{t\to\infty}\frac{\mathbbm{I}^{k}_{t}}{p_{t}\delta_{A}^{t-1}}=\lim_{t\to\infty}\frac{\mathbbm{R}^{k}_{t}}{p_{t}\delta_{A}^{t-1}}=\lim_{t\to\infty}\frac{\mathbbm{I}\mathbbm{T}_{t}}{p_{t}\delta_{A}^{t-1}}=\lim_{t\to\infty}\frac{\mathbbm{R}\mathbbm{T}_{t}}{p_{t}\delta_{A}^{t-1}}=\lim_{t\to\infty}\frac{\mathbbm{H}_{t}}{p_{t}\delta_{A}^{t-1}}=\delta_{A}\left(1+\frac{c}{2}\right)\;\forall
k.$

As a result, the adjoint variables $\Upsilon_{t}$ have to be solved backwards.
We present Equations $(\Delta\Upsilon_{t})$ for $\Upsilon_{t}$ and their
derivations in the appendix (see Section 10.1).

Finally, equilibrium dictates that individual behavior must be consistent with
aggregate behavior at every point in time, i.e., $\pi_{t}=\Pi_{t}$,
$\widehat{s}_{t}=\widehat{S}_{t}$ and $\widehat{i}_{t}=\widehat{I}_{t}$. For
the rest of the analysis we will utilize these equilibrium conditions to
_express all the state variables in capital fonts._

##  5 The government’s problem

The government’s objective function is given by a weighted sum of total
output, which is computed using the social discount factor, and the number of
survivors: $Y_{G}+\xi\cdot(1-D)$, where

$Y_{G}:=(1-\delta_{G})\sum_{t=1}^{\infty}\delta_{G}^{t-1}\bigg{(}Y_{t}q_{t}+\frac{\delta_{G}}{1-\delta_{G}}(1-D_{t+1})p_{t}\bigg{)},\;1-D:=\sum_{t=1}^{\infty}(1-D_{t+1})p_{t},$

and the number $\xi$ is the relative Pareto weight on $1-D$.

The government has three control variables, these are the rates of lockdown
$\lambda_{t}$, testing $\tau_{t}$ and social distancing as a social planner
$(\alpha^{k}_{t})_{k\leqslant t}$. Moreover, the government also incorporates
the agents’ optimal behavior in its decision making. Thus, the Lagrangian of
the government’s optimal control problem takes as constraints the state
equations (forward system $(\Delta\pi_{t})$, now written as
$(\Delta\Pi_{t})$), the agent’s adjoint equations (backward system,
$(\Delta\Upsilon_{t})$) and the agent’s first-order conditions with respect to
the social distancing vector ($FOC-\alpha_{t}^{k}$).

As the last constraint, the government is assumed to be exogenously restricted
in its testing capacity to $\overline{X}_{t}$. We use the actual test data to
feed this time series. Making the rate of testing a choice variable,
$\tau_{t}$, then gives us the resource constraint:

$\displaystyle\overline{X}_{t}=\tau_{t}\bigg{(}\underbrace{\gamma\bigg{(}S_{t}-\beta_{w}\lambda^{2}_{t}S_{t}I_{t}-\beta_{s}\widehat{S}_{t}\widehat{I}_{t}\bigg{)}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\text{``eligible''
($S$)}}+\underbrace{\gamma\bigg{(}R_{t}+\frac{1-m_{i}}{t_{i}}I_{t}\bigg{)}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\text{``eligible''
($R$)}}+\underbrace{\bigg{(}1-\frac{1}{t_{i}}\bigg{)}I_{t}+\beta_{w}\lambda^{2}_{t}S_{t}I_{t}+\beta_{s}\widehat{S}_{t}\widehat{I}_{t}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\text{``eligible''
($I$)}}\bigg{)}.$ ($F-\tau_{t}$)

Recall that the agents are tested at the end of period $t$ after new matches,
and hence infections, are created. Then, the fraction $\gamma$ of the
susceptible $(\bf{S})$ and recovered $(\bf{R})$ added to the pool of the
infected $(\bf{I})$, and each agent in this set is tested at the same rate
$\tau_{t}$.171717A comment on modeling choice is in order here: We could have
let $\overline{X}_{t}$ be a choice variable by introducing say a convex cost
of tracing and testing, but the varied pathologies of why different countries
ended up with their own trajectory of total tests is deeply enmeshed in
political economy which requires a separate study of its own. It seems to us
more reasonable, given the complexity of the model in other dimensions, to
choose the time series of tests that actually materialized.

To sum up, the government seeks to maximize $Y_{G}+\xi\cdot(1-D)$ by choosing
the policies $\lambda_{t}$, $\\{\alpha_{t}^{k}\\}_{k\leqslant t}$ and
$\tau_{t}$ which jointly control the dynamics of the state variables $\Pi_{t}$
and agent’s adjoint variables $\Upsilon_{t}$. The government is further
constrained by the resource constraint on testing and the fact that the
agent’s social distancing decision must be a best-response. Thus, the
government’s problem can be stated in a consolidated way as follows:

$\displaystyle\max\limits_{\lambda_{t},(\alpha_{t}^{k})_{k\leqslant
t},\tau_{t},\Pi_{t},\Upsilon_{t}}\quad Y_{G}+\xi\cdot(1-D)\quad\text{subject
to}\quad\text{\hyperref@@ii[eqpi]{$(\Delta\Pi_{t})$}},\;\text{\hyperref@@ii[equps
A]{$(\Delta\Upsilon_{t})$}},\;\text{(\ref{agent_foc})}\;\text{and}\;\text{(\ref{eq
testing capacity})}.$

It should be noted that the government is not utilitarian in the usual sense
of the word. It takes a specific stance that its main job is to ensure maximal
economic output while keeping number of fatalities in check. The agents’
utilities enter the government’s problem as a constraint since it internalizes
their social distancing decision. At a conceptual level, our analysis can thus
be regarded as theory of second-best.181818Rowthorn and Toxvaerd (2020)
explores implications of the SIS model for different objectives of the
“planner”, terming them controlled and uncontrolled decentralized equilibrium
and social optimum. In the context of that framework, we are closest to a
controlled decentralized equilibrium.

We next describe the set of optimality conditions associated with the
planner’s problem and the numerical algorithm to solve the model.

###  5.1 Optimality conditions

We use the Lagrangian approach to solve the government’s problem. We now
describe the system of constraints as well as the family of Lagrange
multipliers which we will attach to them.

  * •

Recollect $\Pi_{t}=\big{(}(S_{t}^{k})_{k\leqslant t},(I_{t}^{k})_{k\leqslant
t},IT_{t},(R_{t}^{k})_{k\leqslant t},RT_{t},H_{t}\big{)}$ satisfies Equations
$(\Delta\Pi_{t})$. Define the government’s adjoint variables corresponding to
these equations to be $\overline{\Upsilon}_{t}$, i.e.,

$\overline{\Upsilon}_{t}:=\Big{(}(\overline{\mathbbm{S}}^{k}_{t})_{k\leqslant
t+1},(\overline{\mathbbm{I}}^{k}_{t})_{k\leqslant
t},\overline{\mathbbm{IT}}_{t},(\overline{\mathbbm{R}}^{k}_{t})_{k\leqslant
t},\overline{\mathbbm{RT}}_{t},\overline{\mathbbm{H}}_{t}\Big{)}.$

Similar to the agent’s problem, we denote by $\overline{\Delta}_{t}^{k}$ the
expected drop in the government’s continuation value due to an infection, that
is

$\overline{\Delta}_{t}^{k}:=(1-\tau_{t}\gamma)\overline{\mathbbm{S}}^{k}_{t}+\tau_{t}\gamma\overline{\mathbbm{S}}^{t+1}_{t}-(1-\tau_{t})\overline{\mathbbm{I}}^{k}_{t}-\tau_{t}\overline{\mathbbm{IT}}_{t}.$

  * •

The government internalizes that each agent will best-respond. This is
incorporated in the government’s problem by taking the rate of social
distancing $(\alpha_{t}^{k})_{k\leqslant t}$ and the agent’s adjoint variables
$\Upsilon_{t}$ to be choice variables on their own. Of course, the government
is constrained by the fact that the agent’s adjoint variables $\Upsilon_{t}$
must respect the adjoint equations $(\Delta\Upsilon_{t})$, which are formally
derived in the appendix. Denote the vector of dual variables associated with
this system by $\overline{\Pi}_{t}$:

$\overline{\Pi}_{t}:=\Big{(}(\overline{S}_{t}^{k})_{k\leqslant
t},(\overline{I}_{t}^{k})_{k\leqslant
t},\overline{IT}_{t},(\overline{R}_{t}^{k})_{k\leqslant
t},\overline{RT}_{t},\overline{H}_{t}\Big{)}.$

  * •

Let $\chi_{t}$ be the vector of Lagrange multipliers associated with the
testing constraint, viz. Equation ($F-\tau_{t}$).

  * •

Finally, we define $\\{\eta_{t}^{k}\\}_{k\leqslant t}$ to be the vector of
Lagrange multipliers associated with the agent’s first-order condition with
respect to the rate of social distancing $(\alpha_{t}^{k})_{k\leqslant t}$,
viz. Equation ($FOC-\alpha_{t}^{k}$). Again, this is needed to guarantee that
the rate of social distancing constitutes the agent’s best-reply.

The next step is to solve for the Lagrange variables $\overline{\Pi}_{t}$. The
reader can verify that these variables satisfy system which resembles
Equations $(\Delta\Pi_{t})$, yet there are several noticeable differences:

$\displaystyle\overline{S}^{k}_{t+1}$
$\displaystyle=(1-\tau_{t}\gamma)\bigg{(}\overline{S}^{k}_{t}-\beta_{w}\lambda^{2}_{t}\overline{S}^{k}_{t}I_{t}-\beta_{s}\bigg{(}\alpha^{k}_{t}\overline{S}^{k}_{t}-\eta^{k}_{t}{S}^{k}_{t}\bigg{)}\widehat{I}_{t}\bigg{)}\;\text{for
}k=1,\ldots,t,$ ($\Delta\overline{S}_{t}^{k}$)
$\displaystyle\overline{S}^{t+1}_{t+1}$
$\displaystyle=\tau_{t}\gamma\bigg{(}\overline{S}_{t}-\beta_{w}\lambda^{2}_{t}\overline{S}_{t}I_{t}-\beta_{s}\bigg{(}\widehat{\overline{S}}_{t}-\sum_{k\leqslant
t}\eta_{t}^{k}S_{t}^{k}\bigg{)}\widehat{I}_{t}\bigg{)},$
($\Delta\overline{S}_{t}^{t+1}$) $\displaystyle\overline{I}^{k}_{t+1}$
$\displaystyle=(1-\tau_{t})\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\overline{I}^{k}_{t}+\beta_{w}\lambda^{2}_{t}\overline{S}^{k}_{t}I_{t}+\beta_{s}\bigg{(}\alpha^{k}_{t}\overline{S}^{k}_{t}-\eta^{k}_{t}{S}^{k}_{t}\bigg{)}\widehat{I}_{t}\bigg{)}\;\text{for
}k=1,\ldots,t,$ ($\Delta\overline{I}_{t}^{k}$)
$\displaystyle\overline{IT}_{t+1}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\overline{IT}_{t}+\tau_{t}\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\overline{I}_{t}+\beta_{w}\lambda^{2}_{t}\overline{S}_{t}I_{t}+\beta_{s}\bigg{(}\widehat{\overline{S}}_{t}-\sum_{k\leqslant
t}\eta_{t}^{k}S_{t}^{k}\bigg{)}\widehat{I}_{t}\bigg{)},$
($\Delta\overline{IT}_{t}$) $\displaystyle\overline{R}^{k}_{t+1}$
$\displaystyle=(1-\tau_{t}\gamma)\bigg{(}\overline{R}^{k}_{t}+\frac{1-m_{i}}{t_{i}}\overline{I}^{k}_{t}\bigg{)},\;\;\;\text{for
}k=1,\ldots,t,$ ($\Delta\overline{R}_{t}^{k}$)
$\displaystyle\overline{RT}_{t+1}$
$\displaystyle=\overline{RT}_{t}+\frac{1-m_{i}}{t_{i}}\overline{IT}_{t}+\frac{1-m_{h}}{h_{t}}\overline{H}_{t}+\tau_{t}\gamma\bigg{(}\overline{R}_{t}+\frac{1-m_{i}}{t_{i}}\overline{I}_{t}\bigg{)},$
($\Delta\overline{RT}_{t}$) $\displaystyle\overline{H}_{t+1}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{h}}\bigg{)}\overline{H}_{t}+\frac{m_{i}}{t_{i}}(\overline{I}_{t}+\overline{IT}_{t}),$
($\Delta\overline{H}_{t}$)

where $\overline{I}_{t+1}^{t+1}=\overline{R}_{t+1}^{t+1}=0$,
$\overline{S}_{t}:=\sum_{k\leqslant t}\overline{S}_{t}^{k}$,
$\overline{I}_{t}:=\sum_{k\leqslant t}\overline{I}_{t}^{k}$,
$\overline{R}_{t}:=\sum_{k\leqslant t}\overline{R}_{t}^{k}$ and
$\widehat{\overline{S}}_{t}:=\sum_{k\leqslant
t}\alpha_{t}^{k}\overline{S}_{t}^{k}$,
$\widehat{\overline{I}}_{t}:=\sum_{k\leqslant
t}\alpha_{t}^{k}\overline{I}_{t}^{k}$. As usual, we stack together the
aforementioned equations as

$\Delta\overline{\Pi}_{t}:=\Big{(}(\Delta\overline{S}_{t}^{k})_{k\leqslant
t},(\Delta\overline{I}_{t}^{k})_{k\leqslant
t},\Delta\overline{IT}_{t},(\Delta\overline{R}_{t}^{k})_{k\leqslant
t},\Delta\overline{RT}_{t},\Delta\overline{H}_{t}\Big{)}.$

We note that holding the path of state variables fixed, the Lagrange variables
$\overline{\Pi}_{t}$ follow the same law of motion as the state variables
$\Pi_{t}$, but adjusted proportionally to $(\eta_{t}^{k})_{k\leqslant t}$. In
particular, Equations $(\Delta\overline{\Pi}_{t})$ are forward in the sense
that $\overline{\Pi}_{t+1}$ is a function of $\overline{\Pi}_{t}$ and other
variables at time $t$. The initial condition for this system is
$\overline{\Pi}_{1}=0$, which should be contrasted with the state equations
where we have $\Pi_{1}\neq 0$, i.e., $S_{1}^{1}=1-e_{0}>0$ and
$I_{1}^{1}=e_{0}>0$.

The variables $\overline{\Pi}_{t}$ have a natural interpretation. These
variables are shadow prices that convert changes in the agent’s continuation
payoff to changes in the government’s continuation payoff. For example,
consider the first period. Recall that the agent’s social distancing decision
at this date depends on two static costs as well as the dynamic component,
which is proportional to the expected change in the agent’s value due to an
infection, i.e.,
$\Delta_{1}^{k}=(1-\tau_{1}\gamma)\mathbb{S}_{1}^{k}+\tau_{1}\gamma\mathbbm{S}_{1}^{2}-(1-\tau_{1})\mathbbm{I}_{1}^{k}-\tau_{1}\mathbbm{I}\mathbbm{T}_{1}$.
Each of these adjoint variables can be thought as a sensitivity of the agent’s
first period continuation payoff with respect to a certain state at date
$t=2$. For instance, $\mathbbm{S}_{1}^{1}$ is the sensitivity related to
$S_{2}^{1}$, and if the agent’s payoff becomes marginally more sensitive to
$S_{2}^{1}$, then the agent will social distance more. As a result, the
planner’s payoff will change in a proportion to $\eta_{1}^{1}$, that is the
multiplier on the agent’s first-order condition with respect to
$\alpha_{1}^{1}$. In fact, the total adjustment of the government’s profit due
to the marginal change in $\mathbbm{S}_{1}^{1}$ is exactly
$\overline{S}_{1}^{1}=-\eta_{1}^{1}(1-\tau_{1}\gamma)\beta_{s}S_{1}^{1}\widehat{I}_{1}$.

We now look at the problem of choosing the optimal lockdown. It turns out that
the optimal choice of $\lambda_{t}$ depends on the state variables and all
Lagrange multipliers through the following:

$\displaystyle\lambda_{t}$
$\displaystyle=\arg\max\limits_{\lambda\in[\overline{\lambda},1]}\bigg{\\{}\bigg{(}(1-\delta_{G})\delta_{G}^{t-1}Y_{t}q_{t}+(1-\delta_{A})\delta_{A}^{t-1}\overline{Y}_{t}q_{t}\bigg{)}\lambda-$
(OPT-$\lambda_{t}$) $\displaystyle-\bigg{(}\beta_{w}\sum_{k\leqslant
t}\bigg{(}\overline{\Delta}_{t}^{k}S_{t}^{k}I_{t}+\Delta_{t}^{k}\overline{S}_{t}^{k}I_{t}\bigg{)}+\beta_{w}\chi_{t}\tau_{t}(1-\gamma)S_{t}I_{t}+\beta_{w}(1-\delta_{A})\delta_{A}^{t-1}\bigg{(}\phi^{+}\overline{S}_{t}I_{t}+\phi^{-}S_{t}\overline{I}_{t}\bigg{)}q_{t}\bigg{)}\lambda^{2}\bigg{\\}}.$

To understand Equation (OPT-$\lambda_{t}$), suppose for a moment that the
agents are myopic, that is $\delta_{A}=0$. Then, the agent’s adjoint variables
$\Upsilon_{t}$, and the dual variables $\overline{\Pi}_{t}$ attached to them,
are identically equal to zero. Equation (OPT-$\lambda_{t}$) reduces to the
following:

$\displaystyle\lambda_{t}$
$\displaystyle=\arg\max\limits_{\lambda\in[\overline{\lambda},1]}\bigg{\\{}\underbrace{(1-\delta_{G})\delta_{G}^{t-1}Y_{t}q_{t}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\text{direct
output}}\lambda-\bigg{(}\underbrace{\beta_{w}\sum_{k\leqslant
t}\overline{\Delta}_{t}^{k}S_{t}^{k}I_{t}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\begin{subarray}{c}\text{direct
dynamic cost}\\\ \text{of an
infection}\end{subarray}}+\underbrace{\beta_{w}\chi_{t}\tau_{t}(1-\gamma)S_{t}I_{t}q_{t}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\begin{subarray}{c}\text{direct
dynamic cost}\\\ \text{propagated through
testing}\end{subarray}}+\underbrace{\beta_{w}(1-\delta_{A})\bigg{(}\phi^{+}\overline{S}_{t}I_{t}+\phi^{-}S_{t}\overline{I}_{t}\bigg{)}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\begin{subarray}{c}\text{indirect
static lump-sum cost}\end{subarray}}\bigg{)}\lambda^{2}\bigg{\\}}.$

The first term is the discounted marginal product at date $t$ multiplied by
the probability of having the pandemic going at this date. Consider the second
term and recall that $\overline{\Delta}_{t}^{k}$ is the net expected change of
the planner’s continuation payoff due to an additional infection at state
$k\leqslant t$. This is then multiplied by the likelihoods of transitions from
the susceptible to infected, i.e., $\beta_{w}S_{t}^{k}I_{t}$, and aggregated
across all testing states. As for the third term, note that an increase in
economic activities creates $\beta_{w}S_{t}I_{t}$ new infections in total, and
these then reduce the testing probability and decrease the planner’s
continuation payoff in a proportion to $\chi_{t}$. Finally, consider the last
term and recall that the agent suffers disutility $\phi^{+}$ from getting
infected and disutility $\phi^{-}$ from infecting others. These disutilities
are weighted by the probabilities of respective events, pre-multiplied by
their shadow prices $\overline{\Pi}_{t}$ and aggregated over the time of the
last time. The first three terms in the above equation can be thought as
direct costs, because they are unrelated to the agent’s best reply.

In the general case with $\delta_{A}>0$, there are two additional terms:

$\displaystyle\lambda_{t}$
$\displaystyle=\arg\max\limits_{\lambda\in[\overline{\lambda},1]}\quad\underbrace{\ldots\ldots}_{\text{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
static,
$\delta_{A}=0$}}+\underbrace{(1-\delta_{A})\delta_{A}^{t-1}\overline{Y}_{t}q_{t}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\text{indirect
output}}\lambda-\underbrace{\beta_{w}\sum_{k\leqslant
t}{\Delta}_{t}^{k}\overline{S}_{t}^{k}I_{t}}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\begin{subarray}{c}\text{indirect
dynamic cost}\\\ \text{of an infection}\end{subarray}}\lambda^{2}.$

We note that the rate of economic activities at time $t$ affects the agent’s
adjoint process $\Upsilon_{t}$ in two ways. First, it changes each of
$\mathbbm{S}_{t-1}^{k}$, $\mathbbm{I}_{t-1}^{k}$, $\mathbbm{R}_{t-1}^{k}$,
$\mathbbm{R}\mathbbm{T}_{t-1}$ symmetrically and linearly, because the agent’s
next period output is linear in $\lambda_{t}$. This gives the first new term
above. Second, we have the quadratic term that captures certain indirect
dynamic cost. Note that $\Delta_{t}^{k}\beta_{w}I_{t}$ is the expected change
in the agent’s payoff multiplied by the probability of an infection at
$k\leqslant t$. These values are pre-multiplied by $\overline{S}_{t}^{k}$ in
order to convert them to the planner’s payoff, and then they are aggregated
across testing states.

The government’s first order condition with respect to
$(\alpha_{t}^{k})_{k\leqslant t}$ and $\tau_{t}$ can be found in the appendix
(see Equations (OPT-$\alpha_{t}^{k}$), (OPT-$\tau_{t}$)). In short, the first
set of conditions allows us to precisely pin down the set of Lagrange
variables $(\eta_{t}^{k})_{k\leqslant t}$ by solving a certain linear system,
whereas the first-order condition with respect to $\tau_{t}$ determines the
multiplier on the resource constraint.191919A comment on optimization over
testing is in order here. Although, the objective is linear in $\tau_{t}$, we
must have $\tau_{t}\in(0,1)$ at all dates, because the process
$\overline{X}_{t}$, which we infer from the actual testing data, is strictly
positive (and not too large). To guarantee that $\tau_{t}\in(0,1)$ is indeed
optimal we set the term in the objective in front of it to zero.

Finally, we have the system that describes the dynamics of government adjoint
variables $\overline{\Upsilon}_{t}$. We relegate this system to the appendix,
because it is rather complicated (see Equations
$(\Delta\overline{\Upsilon}_{t})$). We note that this system, as one for
$\Upsilon_{t}$, is backward with the following “initial" condition:

$\displaystyle\lim_{t\to\infty}\frac{\overline{\mathbbm{S}}^{k}_{t}}{p_{t}}-\delta_{G}^{t}=\lim_{t\to\infty}\frac{\overline{\mathbbm{I}}^{k}_{t}}{p_{t}}-\delta_{G}^{t}=\lim_{t\to\infty}\frac{\overline{\mathbbm{R}}^{k}_{t}}{p_{t}}-\delta_{G}^{t}=\lim_{t\to\infty}\frac{\overline{\mathbbm{IT}}_{t}}{p_{t}}-\delta_{G}^{t}=\lim_{t\to\infty}\frac{\overline{\mathbbm{RT}}_{t}}{p_{t}}-\delta_{G}^{t}=\lim_{t\to\infty}\frac{\overline{\mathbbm{H}}_{t}}{p_{t}}-\delta_{G}^{t}=\xi\;\forall
k.$

###  5.2 Numerical algorithm

We now briefly outline the algorithm which we use to solve for the optimal
policies. First of all, we truncate the problem by fixing a certain large
terminal time $\overline{t}$ such that $p_{\overline{t}}$, the probability
that the vaccine has not arrived by time $\overline{t}$, is sufficiently
small. The solution to the truncated problem is characterized by exactly the
same equations as above, but the terminal conditions for two backward systems
have to hold exactly at $t=\overline{t}$.

Our numerical approach is a variation of the so-called forward-backward sweep
algorithm. The forward-backward sweep is a standard iterative method to solve
an optimal control problem with control dynamics in the form of partial
difference/differential equations. The idea is to start with a policy, use it
to simulate state variables forward and then solve for adjoint variables
backward. These two sets of variables jointly produce a new policy. If it
sufficiently close to the original one, then the algorithm stops. Otherwise,
it updates the initial policy as a convex combination of the old and new ones
with the weights of $1-\varepsilon>0$ and $\varepsilon>0$, respectively.

We adapt the forward-backward sweep to two nested control problems: one for
the agent and the other is for the government. Thus, we will have two loops.
In the inner loop we solve the agent’s problem with a fixed lockdown policy,
whereas in the outer loop we update the lockdown policy and Lagrange
multipliers on the resource constraint and agent’s first-order condition with
respect to the rate of social distancing.

Let $\lambda_{t}$ for $t=1,2,...,\overline{t}$ be given. Then, the inner loop
is as follows:

  1. Step 0

Select the rate of social distancing $(\alpha^{k}_{t})_{k\leqslant t}$.

  2. Step 1

Solve for the state variables $\Pi_{t}$ and rate of testing $\tau_{t}$ using
Equations $(\Delta\Pi_{t})$ and ($F-\tau_{t}$).

  3. Step 2

Solve for the agent’s adjoint variables $\Upsilon_{t}$ using Equations
$(\Delta\Upsilon_{t})$.

  4. Step 3

Find the rate of social distancing $(\widetilde{\alpha}^{k}_{t})_{k\leqslant
t}$ using Equation ($FOC-\alpha_{t}^{k}$).

  5. Step 4

If the distance between $(\alpha^{k}_{t})_{k\leqslant t}$ and
$(\widetilde{\alpha}^{k}_{t})_{k\leqslant t}$ is small enough, then stop the
inner loop. Otherwise, update $(\alpha_{k}^{t})_{k\leqslant t}$ to
$(1-\varepsilon)\cdot(\alpha^{k}_{t})_{k\leqslant
t}+\varepsilon\cdot(\widetilde{\alpha}^{k}_{t})_{k\leqslant t}$ and go to Step
1.

The outer loop is as follows:

  1. Step 0

Select the rate of lockdown $\lambda_{t}$, the Lagrange multipliers on the
agent’s first-order condition with respect to the rate of social distancing
$(\eta_{t}^{k})_{k\leqslant t}$ and the Lagrange multipliers on the resource
constraint $\chi_{t}$.

  2. Step 1

Use the inner loop to obtain the rate of social distancing
$(\alpha^{k}_{t})_{k\leqslant t}$.

  3. Step 2

Solve for the state variables $\Pi_{t}$, shadow cost variables
$\overline{\Pi}_{t}$ and rate of testing $\tau_{t}$ using Equations
$(\Delta\Pi_{t})$, $(\Delta\overline{\Pi}_{t})$ and ($F-\tau_{t}$).

  4. Step 3

Solve for both sets of adjoint variables $\Upsilon_{t}$ and
$\overline{\Upsilon}_{t}$ using Equations $(\Delta\Upsilon_{t})$ and
$(\Delta\overline{\Upsilon_{t}})$.

  5. Step 4

Find the rate of lockdown $\widetilde{\lambda}_{t}$ using Equation
(OPT-$\lambda_{t}$), compute the Lagrange multipliers
$(\widetilde{\eta}_{t}^{k})_{k\leqslant t}$ and $\widetilde{\chi}_{t}$ using
Equations (OPT-$\alpha_{t}^{k}$) and (OPT-$\tau_{t}$), respectively.

  6. Step 5

If the distance between $\big{(}\lambda_{t},\\{\eta_{t}^{k}\\}_{k\leqslant
t},\tau_{t}\big{)}$ and
$\big{(}\widetilde{\lambda}_{t},\\{\widetilde{\eta}_{t}^{k}\\}_{k\leqslant
t},\widetilde{\tau}_{t}\big{)}$ is small enough, then stop the inner loop.
Otherwise, update $\big{(}\lambda_{t},(\eta_{t}^{k})_{k\leqslant
t},\tau_{t}\big{)}$ to
$(1-\varepsilon)\cdot\big{(}\lambda_{t},(\eta_{t}^{k})_{k\leqslant
t},\tau_{t}\big{)}+\varepsilon\cdot\big{(}\widetilde{\lambda}_{t},(\widetilde{\eta}_{t}^{k})_{k\leqslant
t},\widetilde{\tau}_{t}\big{)}$ and go to Step 1.

The inner loop takes a few seconds to terminate, whereas the outer loop needs
around ten minutes to produce the solution. Several comments are in order.
First, it is well-known that the forward-backward sweep algorithm might run
into cycles, especially, when $\varepsilon$ is large. We control for this by
fine-tuning this parameter, in fact, in our simulations with
$\varepsilon=0.01$ the algorithm converges monotonically. Second, we validate
global optimality by multi-starting the algorithm with different initial
values chosen at random. We are not aware of alternative procedures which can
be used to solve for the global optima in the case of two nested control
problems.

##  6 Calibration

Predicting disease dynamics, calculating optimal policy and evaluating policy
experiments all require reliable parametrization of the model. In principle we
could let the entire set of parameters be identified using time-series of data
on state variables. This unfortunately runs into problems for even an ideal
dataset will leave any standard SIR model unidentified (see Korolev (2021) and
Fernández-Villaverde and Jones (2020)).

We take the following pragmatic approach. We fix government policies, medical
and other miscellaneous parameters using the aggregated wisdom of research
from medicine and social science, and calibrate behavioral parameters to match
the time series of data available on the state variables. The underlying
conceptual thought here is to borrow liberally from other scientific studies,
but try our best to tease out the (i) initial prevalence—the building block of
SIR framework, (ii) social distancing costs, and (iii) efficacy of tracing-
testing by fitting the data on the progress of the pandemic thus far to the
predictions of our model.

Let the vector of parameters that will be estimated be given by
$\theta:=(\beta_{w},\beta_{s},c,\phi_{+},\phi_{-},\gamma)$. In what follows we
briefly describe our dataset, the set of fixed parameters and the estimation
process to pin down $\theta$. All additional details can be found in the
appendix.

###  6.1 Method

We combine data from two sources for our estimation exercise. Time series data
for daily deaths and positive cases are obtained from "The New York Times
Repository", while time series data for daily hospitalizations and daily tests
are sourced from "COVID Tracking Project". Our analysis covers data between
February 29, 2020 $(t=1)$, when the first death was reported in the US, till
February 14, 2021 $(t=352)$. 202020The collected data suffers from the
“weekend” effects: new observations are added in chunks with lags of several
days contributing to the high volatility. To mitigate this issue, we smoothen
the data by constructing moving averages in the window of seven days for each
variable. This smoothening of Covid-19 data is standard and has also been
employed by other studies, see for example, Fernández-Villaverde and Jones
(2020).

The primary objective of the estimation exercise is to parametrize our model
so that it rationalizes the US Covid-19 data obtained from the above mentioned
sources. This requires us to solve the agent’s optimization problem given the
actual lockdown and testing policies followed by the government. Since there
is no reliable source to estimate the actual lockdown function implemented by
the US government, we use an approximation which is roughly consistent with
the government response index published by Hale et al. (2021) (see Oxford
COVID-19 Government Response Tracker for the data). Figure 3(a) compares our
approximation of lockdown function and the Oxford GRT index.212121The Oxford
GRT reports a variety of indices to capture the policy responses undertaken by
different countries. These include policies for economic reform, stringency
measures to reduce transmission, healthcare reform and so on. To construct the
index illustrated in Figure 3(a), we use two stringency measures used by the
US government to curb economic activities, namely workplace closure and stay-
at-home requirements.

As for the testing capacity, we compute the test count from the state level
data as opposed to simply using the publicly available aggregate data, for
example, see "COVID Tracking Project". The reason is that data is collected at
the state level in the US, and the states have practiced distinct ways of
reporting negative tests. Some states report people tested at least once
(unique people), others report either testing encounters or samples collected.
To the best of our knowledge, neither "COVID Tracking Project" nor other data
providers, i.e., "The New York Times Repository", explicitly have corrected
for these differences in reporting.

Ideally, we would like to use testing encounters for our calibration exercise
but since only a few states (mainly less populated) reported this metric
imputing it might result in imprecise estimates. Another reason for confusion
is that more than one test is often conducted on a patient in a short span,
especially if she/he tests positive. So, we construct $\overline{X}_{t}$ by
imputing and aggregating the state level testing data on the number of unique
people who got tested, which can be thought as a lower bound on the total US
testing capacity, in-sample till February 14, 2021. Then, we extrapolate the
count of available tests out-of-sample by fitting a 2nd degree polynomial
function of time. Figure 3(b) plots the actual number of conducted tests in US
and its out-of-sample extrapolation.

(a) Lockdown vs Oxford GRT index

(b) Test count

Figure 3: Assumed government policies

Having set the lockdown and testing policies, we now present the set of
medical and other auxiliary parameters that are taken to be fixed to calibrate
the other parameters of interest. Table 2 summarizes these fixed parameters.
In short, we rely here on the accumulated knowledge thus far, and we discuss
each parameter in details in the appendix.

Parameter | Value | Definition | Source/Target
---|---|---|---
$t_{i}$ | 10 | Average infectious period (incl. latency) |
$t_{h}$ | 7 | Average hospitalization period | CDC Planning Scenario
$m_{i}$ | 0.0176 | Hospitalized as % of infected |  $m_{i}=IFR/m_{h}$, where $IFR=0.3\%$ (Ioannidis (2021))
$m_{h}$ | 0.1705 | Daily deaths as % of hospitalized | Estimated by OLS
$\mathbb{E}T$ | 540 | Expected day the pandemic ends |
$\mathbb{V}T$ | 180 | Variance in the day pandemic ends | New York Times (2020)
$\overline{\lambda}$ | 0.3 | Share of essential services |
$\delta_{G}$ | 0.9999 | Government’s discount factor |  Used by multiple studies incl. Alvarez et al. (2021),
$\delta_{A}$ | 0.9999 | Agent’s discount factor |  Acemoglu et al. (2021) and Farboodi et al. (2021)
$e_{0}$ | $0.007\%$ | Share of infected at the start | 23,000 infected agents given the population of 328.2 mil.
Table 2: Fixed parameters

Given the paucity and limited reliability of the Covid-19 data mentioned
above, only the flow data on deaths and positive cases are used to match the
model parameters. We employ the following loss function for this purpose:

$\omega\cdot\sum_{t=1}^{352}\Big{(}\Delta
D_{t}(\theta)-\Delta\widetilde{D}_{t}\Big{)}^{2}+(1-\omega)\cdot\sum_{t=1}^{352}\Big{(}X_{t}^{+}(\theta)-\widetilde{X}_{t}^{+}\Big{)}^{2}$

where $\theta=(\beta_{w},\beta_{s},c,\phi_{+},\phi_{-},\gamma)$ is the vector
of parameters being estimated, $\Delta D_{t}(\theta)$ is the predicted flow of
deaths at time $t$, $X_{t}^{+}(\theta)$ is the predicted flow of positive
tests, i.e., the sum of newly hospitalized and known infected, at date $t$,
$\Delta\widetilde{D}_{t}$ and $\widetilde{X}_{t}^{+}$ are their empirical
counterparts. The weights $\omega$ and $1-\omega$ ensure that both variables
get the same importance in the computation of the loss function.222222This is
required since the daily number of positive cases vastly outnumbers the number
of daily deaths in the US. For instance, the mean daily deaths in the US was
1284 compared to the mean daily positive cases of 70982.,232323It is
worthwhile to note that we fix the test capacity $\overline{X}_{t}$ at the
actual empirical level. Hence fitting $X^{+}_{t}(\theta)$ to the data ensures
that the predicted flow negative cases fit the data well too.

To avoid over-fitting the model we impose two additional restrictions. First,
we require $\beta_{w}=\beta_{s}$, which is broadly in agreement with the
American Time Use Survey that employed Americans (on average) spend about half
time at work and half time in social activities (see Eichenbaum et al.
(2021a)). Second, we fix $\phi^{-}=\frac{\phi^{+}}{10}$, i.e., we are assuming
the one-time cost of infecting another person outside of immediately friends
and family to be 10 percent of the cost that we would suffer from getting
infected ourself and potentially risking infection for our loved ones as well.
This is broadly subjective and can be adjusted in light of more evidence.

###  6.2 Estimation

Recollect that we are trying to fit the following two time series: the number
of daily deaths and the number of daily positive cases. The inner loop of the
numerical algorithm described in Section 5.2 is able to generate the fixed
point. The robustness of the fixed point is verified by multi-starting the
algorithm. Figures 4(a) and 4(b) compares the model fit and data for daily
deaths and daily positive cases, and Table 3 reports the calibrated
parameters.

(a) Deaths

(b) Positive confirmed

Figure 4: Model fit vs data Parameter | Value | Definition
---|---|---
$\beta_{w}$ | $0.2046$ | Prevalence in economic activities
$\beta_{s}$ | $0.2046$ | Prevalence in social activities
$\gamma$ | $0.0976$ | Tracing-testing efficacy
$c$ | $0.4403$ | Daily opportunity cost of infection
$\phi_{+}$ | $523.932$ | Disutility of getting infected
$\phi_{-}$ | $52.393$ | Disutility of infected someone else
Table 3: Calibrated parameters.

Overall, the model with just 4 free parameters fits the data reasonably well.
It turns out that our estimates match almost perfectly the cumulative numbers
of deaths and positive cases by February, 14. However, they do not pick up
magnitudes of some waves smoothing them over time. This can be partially
attributed to factors not present in the model such as overload of the
healthcare system, information percolation about the pandemic, seasonality in
contact rates, i.e., academic year and holidays, etc.242424Note that we could
substantially “improve the fit” visibly by freeing up some more parameters
from Table 2. However, this would surely result in an overfitting of the
model.

Our estimate of the total prevalence $\beta_{w}+\beta_{s}$ turns out to be
$0.4092$ that implies that the initial rate of growth of infection is slightly
more than $0.3$. In other words, the number of infected agents doubles in $3$
days at the beginning of our analysis. This is in line with previous studies.
For example, Farboodi et al. (2021) calibrated the total prevalence parameter
to $0.43$, and Eichenbaum et al. (2021a) estimated the transmission rate in
social activities to be around $0.16$, which is close to our estimate of
$0.5\times 0.4092=0.2046$.

The efficacy of trace-testing $\gamma$ is estimated to be $0.0976$. Recall
that a fraction $1-\gamma$ of the susceptible ($\mathbf{S}$) and unknown
recovered ($\mathbf{R}$) is excluded from testing based on tracing, so here
$\gamma=0$ is perfect tracing technology and $\gamma=1$ denotes completely
random testing. The estimate of efficacy of trace-testing is novel to the
literature, and it essentially tells us that conditional on the number of
tests, the tracing and testing infrastructure in the United States has been
reasonably effective in targeting the infected population. However, as we will
argue later, the delay in scaling up testing had significant consequences.

For the behavioral parameters, we estimate three types of costs which each
agent incurs. The daily opportunity cost of missing work and social activities
due to being quarantined, measured by $c$, is estimated to be $0.4403$. We
find $\phi^{+}$ to be estimated to be around $523.9$. This represents the
psychological cost from knowing that one has been infected plus the cost of
facing a potential death from the virus or risking the health of a close
family member. Then, $\phi^{-}$ is set to $53.4$, which is the (ex ante) cost
of getting someone else infected and contributing indirectly to all their
direct costs. As mentioned in the introduction, one way to interpret these
numbers is in terms of daily wage as the unit of measurement.

##  7 The optimal policy

In this section we report the results from executing the numerical algorithm
described in Section 5.2 to calculate the optimal lockdown and testing
policies for parameters specified in the previous section and the value of
life $\xi$ set to $20$. This specific value of $\xi$ is in line with Hall,
Jones, and Klenow (2020) and has been chosen in other papers that have
followed. We use this number for the simulation exercise and reporting of some
of the results, but then we also map out the Pareto frontier to exposit the
full set of policy options available to the government.

Figure 5(a) plots the optimal lockdown and average fraction of social
distancers at each date over the course of 20 months. The first thing to
notice is that government internalizes that individuals will voluntarily
social distance at moderate levels in order to avoid getting infected. Towards
the beginning of the pandemic in March 2020, agents are willing to give up
around 55% of their social activities (dashed red line). Interestingly this is
the highest level of social distancing that agents engage in throughout the
pandemic. Over time the extent of social distancing declines very gradually
till August 2021 to a 50%. As the possibility of the pandemic ending
increases, the optimal social distancing levels decline rather steeply to 0
between the period of August 2021 and October 2021.252525The small spike in
social distancing before the sharp decline is due to the uncertainty in
vaccine arrival or modeling end of the pandemic. Since the government starts
to ease the lockdown, if the vaccine is very likely to arrive but hasn’t quite
arrived yet, agents may social distance more to control the infection rate.

Given this moderate but persistent level of social distancing, the government
imposes an early lockdown of around 40% which then remains relatively stable
(solid black line). It rationally internalizes that the pandemic is going to
spread, and under the expectation of vaccine arriving in Summer/Fall 2021 it
chooses an intermediate but persistent lockdown to avoid a large outbreak of
infections. The rate of lockdown declines gradually to around 35% till July
2021, followed by a rapid decline as the pandemic is expected to end. However,
the government never imposes anywhere close to the maximal lockdown levels in
the economy throughout the period of the pandemic.

(a) Lockdown and social distancing

(b) Effective reproductive number

Figure 5: Optimal lockdown and social distancing

(a) Expected flow output

(b) Expected cumulative deaths

Figure 6: Flow output and cumulative deaths

The rationale behind the optimal lockdown can be seen by examining the famed
reproductive number $R_{t}$. It is an approximation of the expected number of
secondary cases produced by a single infected agent at a specific date
$t$:262626A variant of this expression without testing is reported in studies
like Farboodi et al. (2021) and Hall et al. (2020).

$R_{t}:=\underbrace{\frac{1}{1-(1-\tau_{t})(1-\nicefrac{{1}}{{t_{i}}})}}_{\text{``avg.
duration'' of state
$\mathbf{I}$}}\cdot\underbrace{(1-\tau_{t})\bigg{(}\beta_{w}\lambda^{2}_{t}S_{t}+\beta_{s}\widehat{S}_{t}\frac{\widehat{I}_{t}}{I_{t}}\bigg{)}}_{\text{\\#
of new infections}}.$

Recall that an infected agent can transmit the disease through two channels:
economic activities, in which case she/he infects
$\beta_{w}\lambda^{2}_{t}S_{t}$, and social activities, in which case she/he
infects $\beta_{s}\widehat{S}_{t}\frac{\widehat{I}_{t}}{I_{t}}$ agents. Taking
into an account that $\tau_{t}$ of these new secondary cases will be test
positive immediately at the end of date $t$, the total number of secondary
cases produced is given by
$(1-\tau_{t})\left(\beta_{w}\lambda^{2}_{t}S_{t}+\beta_{s}\widehat{S}_{t}\frac{\widehat{I}_{t}}{I_{t}}\right)$.
Then, we note that this infected agent stays in $\mathbf{I}$ if and only if
she/he has not moved to $\mathbf{IT}$ (tested positive), $\mathbf{R}$
(recovered) or $\mathbf{H}$ (hospitalized). Thus, if testing rate is close to
be being a constant, she/he is expected to spend approximately
$\frac{1}{1-(1-\tau_{t})(1-\nicefrac{{1}}{{t_{i}}})}$ days in the compartment
of unknown infected agents. We call $R_{t}$ the effective reproductive number.

Figure 5(b) plots the effective reproductive number implied by the optimal
policy and agents’ social distancing decision. The government chooses the
lockdown function internalizing the agent’s best response in a way that
$R_{t}$ is approximately $1$ for almost 17 months. This in turn means that
modulo the variation in total available tests the number of active cases is
targeted to be approximately constant over this time frame, which can be seen
by examining Equation ($\Delta I_{t}$). Importance of reducing the effective
reproductive number to one has been extensively addressed in public discourse,
and we believe that our results provide arguments in favor of keeping this as
the target of policy.

The two outcomes that define the objective function of the government are
depicted in Figure 6. Recollect that the total number of survivors is weighted
by the $\xi=20$ for these calculations. Figure 6(a) shows the flow of output
as the pandemic evolves. Total output falls almost by 45% percent initially
according to the optimal policy and then gradually claws its way back to its
full potential as the government eases lockdown. This graph of course reflects
the lockdown policy in Figure 5(a) because total output in our model is simply
equal to the labor supply.

Figure 6(b) plots the cumulative number of deaths predicted by the optimal
policy (solid black line) and the actual realized number (dashed red line).
The two series depart fairly quickly around the two-three month mark, and
while the slope of the actual number gets a bump as winter 2020 approaches,
the optimal policy ensures the slope of the number of deaths remains more or
less constant rising up to about three hundred thousand deaths, less than half
of the actual number of deaths seen in the data, which has already passed
seven hundred and fifty thousand as of writing this draft of the paper.

Figure 7: Mapping the Pareto frontier on output and lives saved

Finally, going beyond the specific value of $\xi$, in Figure 7 the entire
Pareto frontier is presented. This graph is produced by varying $\xi$ on a
grid of thirty points ($\xi=0$ to 30), and then solving the government’s
problem for each possible value. Thus, the curve traces all efficient
combinations of total output and mortality by changing the value of the Pareto
weight $\xi$. That the curve is downward sloping is obvious. Interestingly, it
first has an approximately constant slope followed by a concave transition.
Mathematically speaking, the optimization problem here is mapping out the
concave closure of the actual Pareto frontier. So, the initial straight part
of the frontier is representative of a weakly convex frontier—hence increasing
returns in terms of lives saved for every unit drop in output. Eventually the
concave part diminishes these returns quite substantially.

For low values of $\xi$, especially $1\leq\xi\leq 10$, the trade-off is stark:
the first one percent permanent drop in output increases the expected number
of survivors by 281 thousand. For larger values of $\xi$, for example
$13\leq\xi\leq 30$, a one percent drop in output increases the expected number
of survivors by 158 thousand. Thus, even a moderate emphasis on mortality in
the objective function produces significant gains in lives saved with a
concomitant linear (or even lesser) decline in output. However, as the Pareto
weight is increased the trade-off becomes less rewarding—the number of lives
saved per unit drop in output decreases marked by the concave rate of
transition.

##  8 Policy experiments

In this section we explore policy experiments by changing two parameters—
efficiency of tracing and testing and the fraction of prevalence the
government can directly control—and also removing lockdown and social
distancing piecemeal from the optimization problem.

###  8.1 Improving tracing and testing

There has been substantial variation in the way different countries have
approached large scale testing to fight the pandemic. Some seem to have
delayed in making testing widely available and others in making contact
tracing particularly effective. Here we do two experiments on the baseline.

First, we allow for early testing. As can be seen in Figure 3(b), testing
starts late in the pandemic, almost at the two and half month mark. We expand
the total set of test capacity at each date to be its highest attained value
in the data, which is one million tests a day. We call this the high testing
counterfactual. Second, we allow the tracing and testing to be much more
efficient. We do that by reducing $\gamma$ by half—from 0.09 to 0.045. This
ensures a larger proportion of infected are now being tested and quarantined.
Figure 8 plots the key variables for both counterfactuals and also for the
combined scenario of high and efficient testing.

The pairwise substitutability of better testing with lockdown and social
distancing is immediately clear from Figures 8(a) and 8(b) respectively. When
testing is scaled earlier and is more efficient the optimal extent of lockdown
and social distancing diminish significantly. On average, lockdown drops by
almost 40 percent, and social distancing by about 20 percent. On the social
distancing dimension, if an agent is not selected for testing, she/he is
relatively more confident of not being infected and thus is willing to engage
more in social activities. For lockdown too, since the government is more
certain that the pool of agents who are not quarantined or recovered are not
infected, it needs to lockdown less.

(a) Lockdown

(b) Social Distancing

(c) Flow of infections

(d) Cumulative deaths

Figure 8: Counterfactuals for early testing, and effective tracing and testing
Figure 9: The Pareto frontier for high and improved testing

Improved tracing-testing leads to less total infections; however, lower
lockdown and social distancing lead to more infections. So the aggregate
effect of improved tracing-testing on mortality is ambiguous. In fact in
Figures 8(c) and 8(d), we can see that for fixed level of testing, introducing
greater efficiency keeps the number of deaths to almost the same level, in
fact slightly increases it. Recall these numbers are being determined
endogenously through the solutions to the government and the agents’ problems
for a fixed Pareto weight. It turns out that the government is trading off
some mortality for a much larger increase in output.

Figure 9 plots the Pareto frontier for four different specifications of
tracing-testing. Introducing either high number of tests or higher efficiency
of testing pushes out the Pareto frontier. And, introducing both considerably
expands it. A simple way to think about this expansion is that for a fixed
level of mortality, the government now achieves a much higher output and
similarly for a fixed level of output it can now achieve considerably more
survivors (or less deaths). These findings are analogous to Acemoglu et al.
(2021)—they feature testing levels tagged to age, we have a uniform testing
policy; however, agents in our model choose their level of social
interactions, in theirs they are mechanical.

###  8.2 Greater control of the government

There has been intense discussion through the pandemic that autocracies have
been able to control the diseases better than democracies (see Narita and Sudo
(2021)). While the moral and institutional intricacies of that questions are
outside the scope of this paper, we can shed some light on it through a simple
policy experiment. How do the basic results change by increasing the control
of the government over social interactions?

(a) Lockdown

(b) Social Distancing

(c) Flow of infections

(d) Cumulative deaths

Figure 10: Counterfactuals for different levels of $\eta$, where
$\beta_{w}=\eta\beta$ and $\beta_{s}=(1-\eta)\beta$

We do this through the following reduced form experiment. Recollect that
$\beta_{w}=\eta\beta$ and $\beta_{s}=(1-\eta)\beta$, where $\beta$ can be
considered as the aggregate prevalence parameter which sets the contagiousness
of this disease, and $\eta$ is the fraction of infections that arise at work
and $1-\eta$ the fraction that arise in social interactions, respectively. We
started out by setting $\eta=0.5$, which is motivated by the American Time Use
Survey (see Eichenbaum, Rebelo, and Trabandt (2021a)). We now present the
results for $\eta=0.4,0.6$ and 0.9. The idea is to vary the matching intensity
in the social interactions part of Equation ($\Delta S_{t}$) and its
subsequent avatars. So, we interpret $\eta=0.4$ to mean that the fraction of
matches within the direct control of the government decreases. And, the way we
interpret $\eta=0.6,0.9$ is that government has been able to enforce masking
and other other such policies (which decreases matches) more effectively. The
agents must follow these policies to a varying degree, as $\eta$ increases.

As $\eta$ increases the government essentially tries to kill the pandemic
through a more extensive lockdown for a larger fraction of infections matches
are under its control. Why does the government do this? Remember that saving
lives has a direct value in the government’s objective function and every life
saved produces more output in the future. Even though the flow of output
decreases, its net present value is higher because of lives saved through the
stricter lockdown. And since the government has greater control on matches the
stricter lockdown has a greater bang for buck. The agents on the other hand
social distance less because given the strict lockdown in place, the
probability of them contracting the infection in social interactions is
lower—they try to take advantage of whatever social interactions are
permitted. Overall, greater governmental control reduces infections and total
deaths. See Figure 10 for the results.

###  8.3 No lockdown and social distancing

Several governments have faced the dilemma of whether to lockdown economic
activities and by how much. Sweden is perhaps the most widely discussed case,
it went in for little or almost no lockdown over the early part of the
pandemic (Financial Times (2021a)). In this section we first evaluate the
policy experiment where lockdown is exogenously fixed at zero. We analyze this
scenario for both the benchmark constellation of parameters and for the case
where testing-tracing is more extensive and efficient.

(a) Social distancing

(b) Flow of infections

(c) Cumulative deaths

Figure 11: Counterfactuals for no lockdown

Figures 11(a) and 11(b) show that pandemic speeds through the population with
one large peak of infections at the inception in response to which the agents
social distance to a peak level of shutting down 90% of all social activities
(red lines in both graphs). Correspondingly, the number of deaths rises
rapidly to about 750 thousands by July 2020, and then stays constant as the
population reaches herd immunity (see red line in Figure 11(c)). This makes
one ponder because the actual number of deaths in the US as of writing this
draft is also in the range of 750 thousands. A provocative thought then is
that with its current capacity in testing, the US could have ended up with the
same numbers of total deaths even without a lockdown—the difference would be
in the distribution of fatalities over time.272727There are obvious caveats
here: Agents need to have understood the implications of the pandemic at the
outset and were forward looking in their social distancing decisions.
Moreover, hospitals all over the country could have handled the extreme peak
in infections shown in Figure 11(b). Subject to these caveats, this point
about the equivalence of deaths is made to highlight that one of the goals of
optimal policy is in-fact to spread out the pandemic to allow the government
and society to manage it better.

The solid purple line in Figure 11 plots social distancing, infection and
deaths for the no lockdown case when it is accompanied with more extensive and
effective tracing-testing. It is clear that if testing-tracing capacity is
ramped up then even with no lockdown the peak of infections is much smaller,
and the total number of deaths is reduced significantly, the number being
lower than what we see actually realized in the data.

(a) Lockdowm

(b) Flow of infections

(c) Cumulative deaths

Figure 12: Counterfactuals for no social distancing

In the next policy experiment we ask the question: what if social distancing
is shutdown in the model? This scenario is not meant to be taken literally, it
is simply a theoretical exercise to understand what the analyst or the modeler
may miss if agency or behavioral response is not directly modeled in an SIR
framework. Figure 12 plots the results for this exercise. A pattern similar to
the no lockdown case emerges, which highlights the complementarity of the two
forces. However, things are more severe when social distancing is shut down in
the sense that government locks down maximally while the infections peak
(dotted blue in Figures 12(a) and 12(b) respectively). The total number of
deaths rises quickly to more than 750 thousands (dotted blue line in Figure
12(c)). Again increasing efficacy of testing-tracing reduces the peak of
infections and total number of deaths (dashed red lines in Figures 12(b) and
12(c)). The reduction in the peak is much more extreme in the no lockdown case
in comparison to the no social distancing case. This is because better testing
technology is used to draw out the pandemic longer through social distancing;
whereas is in the no social distancing case, because of the concern for total
output, the pandemic is made to run through faster. This contrast can be seen
by comparing the shape of the cumulative deaths curve (dashed red line) in
Figures 11(b) and 12(c)).

##  9 Final remarks

In this paper we set out to incorporate two policy instruments, namely
lockdown and tracing-testing, into a standard epidemiology framework augmented
by behavioral response. The theoretical analysis tackles the challenge of
capturing the pairwise substitutability of each of the three forces and the
heterogeneity introduced by testing. The model is calibrated to the data on
the time series of deaths and tests conducted in the US (from Feb 2020 to Feb
2021) to calculate the three key parameters—prevalence of the virus,
efficiency of tracing-testing, and costs from social distancing. The testing
capacity is fixed by the available data on tests conducted in the US during
the same time frame. Under the optimal policy, the government locks down at a
moderate but persistent rate and agents social distance too at a moderate but
persistent rate, each picking up the slack of the other. These two together
contribute in keeping the effective reproductive number at one throughout the
pandemic. Pareto frontier maps the feasible set of economic output and total
fatalities. Three different policy experiments are conducted to evaluate how
the predictions of the model would change under some commonly debated policy
prescriptions.

In this final section, we now briefly discuss some limitations of our
analysis, and extensions of the model that could be useful in addressing these
concerns.

Modeling behavioral response. We model behavioral response by introducing
costs from social distancing that constitute a flow component as well as a
lumps-sum component from getting infected and infecting others. This seems
like a natural way to incorporate the incentives for social distancing, as has
been done, for example by Toxvaerd (2020) and Farboodi et al. (2021). However,
we depart from existing work by assuming that those who have recovered without
showing severe symptoms cannot know so unless they are tested. There are
potentially interesting micro-foundations to these decisions of social
distancing worth thinking about carefully. What role do authorities and peer
effects play? This requires a better modeling of information
percolation.282828See Bandyopadhyay, Chatterjee, Das, and Roy (2021) for a
model of optimal government policy where agents learn about the pandemic from
the intensity of the lockdown. Also, are individualistic societies less or
more likely to internalize social distancing into their daily realities than
those that are community driven?

Modeling testing. We introduced a very simple tracing and testing technology
wherein a fraction of susceptible, and infectious and recovered without
symptoms can be tested every period. The efficacy of testing is controlled by
a single parameter. Testing technology can be enriched by first separating
rapid tests that only discover infection from antibody tests that also
discover whether the person was infected in the past. Restricting testing to
only the former, which is more realistic, would change the effectiveness of
testing, but by how much is an open question, at least in the context of our
model. Another avenue of future research is to focus tracing, testing and
quarantining more towards the elderly, who are significantly more
vulnerable.292929In response to the question, should all countries be testing
uniformly, Germany’s leading Covid expert informing policy, Christian Drosten,
had stated early in the pandemic: ”I’m not sure. Even in Germany, with our
huge testing capacity, and most of it directed to people reporting symptoms,
we have not had a positivity rate above 8%. So I think targeted testing might
be best, for people who are really vulnerable… This is not fully in place even
in Germany, though we’re moving towards it. The other target should be
patients in the first week of symptoms, especially elderly patients who tend
to come to hospital too late at the moment… And we need some kind of sentinel
surveillance system, to sample the population regularly and follow the
development of the reproduction number” (The Gaurdian (2020)).

Modeling heterogeneity in the population. It is now clear Covid-19 is much
more lethal to the elderly and people with certain specific co-morbidities.
One way to model this richness is to create separate compartments for
different sections of the population, define system of equations for the
evolution of disease dynamics in each compartment with matching functions
within and across them. Work along these lines has been done by Baqaee et al.
(2020) and Acemoglu et al. (2021) amongst others. Marrying features of
behavioral response to these frameworks or introducing age dependent
compartments in our model remains an open question.

Modeling economic activity and lockdown. We modeled lockdown as as a time
dependent policy that shuts down a fraction of economic activity. Total output
in turn is modeled simply as the total labor supply. These are of course
abstractions that do capture the gist of the problem, but miss out some
important ingredients. First, lockdown can be selective across various regions
of a country, and in varying intensities; see Fajgelbaum et al. (2020) and
Giannone, Paixao, and Pang (2020) on how this matters for relaxing lockdown
policies over time. Second, drop in output itself can kill lives, especially
in developing countries where daily wage workers are abundant and social
safety nets are weak, see for example Barnett-Howell and Mobarak (2020) for a
discussion. This would, amongst other things, change the way we choose a
Pareto-weight for optimal policy. Third, a richer model of production could be
interesting to examine. How capital and supply chains are impacted by the
pandemic and governmental response to it are salient too.

Modeling the end of the pandemic. The end of the pandemic is modeled as a
negative binomial distribution. The parameters are chosen so that the pandemic
could not have ended 360 days after its inception and it almost surely ends
720 days after with an average duration of 540 days. In reality it is
difficult to define what the "end of the pandemic" means. We gave it the
interpretation of the arrival of a vaccine. To that end, a more careful
modeling of the roll out of the vaccine and related factors such as vaccine
hesitancy, and mutations to the virus also seems important.303030In a
different approach Giannitsarou, Kissler, and Toxvaerd (2021) model the
disease dynamics as SIRS which includes the possibility of reinfection, and
the pandemic there can last for a century where dynamics of infections are
characterized by dampened oscillations.

On the Pareto frontier and value of life. Putting a Pareto-weight on the
tradeoff between economic output (or consumption) and rates of mortality is a
deep philosophical question perennially relevant to how we structure society
and institutions, see Schelling (1968) for a classic reference. More recently
Hall, Jones, and Klenow (2020) have taken up the question in the context of
the recent Covid-19 crises. For our analysis, we were reluctant to pick a
particular number and thus eventually provided an entire frontier. A
potentially interesting question here is to measure how much societies value
life through a revealed preference argument with the pandemic as a natural
experiment.

On pandemic fatigues, second waves and new variants. A salient feature of the
pandemic across countries has been the rise of infection in waves. While our
model is flexible enough to match the data in the US which went through at
least two waves of the pandemic, it does not explicitly model the underlying
behavioral reasons for it. Atkeson (2021a) takes first steps in this direction
by introducing “a one-time reduction in the semi-elasticity of the
transmission rate to the daily death rate late in the year," as pandemic
fatigue. Much more work needs to be done in understanding behavioral
underpinnings of the pandemic fatigue and how optimal policy should take it
into account. Moreover, the successive waves of the pandemic have been
associated with the rise of new "variants". At the time of the writing of this
draft the Delta variant of Covid has created much havoc and the Omicron
variant is on the rise. Incorporating the ebb and flow of variants precisely
because of the global nature of the spread of the virus is another challenge
for modeling.

On mental health consequences. We have abstracted here from the mental heath
consequences of lockdown and to some extent of social distancing as well.
There is widespread evidence of a spurt in mental health troubles during the
course of Covid-19, see for example COVID-19 Mental Disorders Collaborators
(2021) and Mayo Cinic (2021). To the best of our knowledge, these haven’t been
incorporated in the economic models of the pandemic yet. It presents an
important challenge for policymakers for blanket lockdowns policies may have
other perverse consequences not immediately observable, which need to be
incorporated in trade-offs of policymaking.

In conclusion, the most important caveat is that while we hope this paper
helps economists and maybe perhaps even epidemiologists and public health
researchers think carefully about the ways in which to incorporate behavioral
response in models of disease dynamics and evaluate optimal policy, great care
should be taken to interpret any of these insights to inform actual policy. As
social scientists, we are in the long-haul of understanding socio-politico-
economic contexts of this deep health crisis.

##  10 Appendix

###  10.1 Derivation of the agent’s optimal behavior

The agent’s optimization problem is described in details in Section 4.2:
Taking the aggregate variables $(S_{t},I_{t},\widehat{S}_{t},\widehat{I}_{t})$
and $(\lambda_{t},\;\tau_{t})$ as given, the agent solves

$\max\limits_{(\alpha_{t}^{k})_{k\leqslant t},\pi_{t}}\quad
Y_{A}-C+(1-\delta_{A})c\kappa P\quad\text{subject
to}\quad\text{\hyperref@@ii[eqpi A]{$(\Delta\pi_{t})$}}.$

We now set up the Lagrangian for the agent’s problem denoting the undiscounted
dual variables associated with Equations $(\Delta\pi_{t})$ by
$\Upsilon_{t}=\big{(}\mathbbm{S}^{k}_{t})_{k\leqslant
t+1},(\mathbbm{I}^{k}_{t})_{k\leqslant
t},\mathbbm{I}\mathbbm{T}_{t},(\mathbbm{R}^{k}_{t})_{k\leqslant
t},\mathbbm{R}\mathbbm{T}_{t},\mathbbm{H}_{t}\big{)}$. The Lagrangian can be
unpacked as follows:

$\displaystyle\underbrace{(1-\delta_{A})\sum_{t=1}^{\infty}\delta_{A}^{t-1}\bigg{(}y_{t}q_{t}+\frac{\delta_{A}}{1-\delta_{A}}(1-d_{t+1})p_{t}\bigg{)}}_{Y_{A}}-\mathop{\vtop{\m@th\halign{#\cr$\hfil\displaystyle{(1-\delta_{A})\frac{c}{2}\sum_{t=1}^{\infty}\delta_{A}^{t-1}\bigg{(}(it_{t}+h_{t}+d_{t})q_{t}+\frac{\delta_{A}}{1-\delta_{A}}d_{t+1}p_{t}+\sum_{k\leqslant
t}(1-\alpha_{t}^{k})^{2}(s_{t}^{k}+i_{t}^{k}+r_{t}^{k})q_{t}\bigg{)}}\hfil$\crcr\kern
3.0pt\p@\nointerlineskip\cr$\m@th@\hbox{$\braceld$}\bracelu\leaders\vrule\@height\z@\@depth\z@\hfill\kern
0.0pt\p@\vrule\@width\p@\kern 0.0pt\p@\vrule\@width\p@\kern
0.0pt\p@\vrule\@width\p@$\crcr\kern 3.0pt\p@\cr}}}\limits-$
$\displaystyle-\mathop{\vtop{\m@th\halign{#\cr$\hfil\displaystyle{(1-\delta_{A})\sum_{t=1}^{\infty}\delta_{A}^{t-1}\bigg{(}\phi^{+}(\beta_{w}\lambda_{t}^{2}s_{t}{I}_{t}+\beta_{s}\widehat{s}_{t}{\widehat{I}}_{t})+\phi^{-}(\beta_{w}\lambda_{t}^{2}{S}_{t}i_{t}+\beta_{s}{\widehat{S}}_{t}\widehat{i}_{t})\bigg{)}q_{t}}\hfil$\crcr\kern
3.0pt\p@\nointerlineskip\cr$\m@th@\hbox{$\braceld$}\vrule\@width\p@\kern
0.0pt\p@\vrule\@width\p@\kern 0.0pt\p@\vrule\@width\p@\kern
0.0pt\p@\bracerd\braceld\leaders\vrule\@height\z@\@depth\z@\hfill\braceru$\crcr\kern
3.0pt\p@\cr}}}\limits_{C}+(1-\delta_{A})c\kappa\underbrace{\sum_{t=1}^{\infty}\delta_{A}^{t-1}\sum_{k\leqslant
t}\big{(}\ln\alpha^{k}_{t}+\ln\big{(}1-\alpha^{k}_{t}\big{)}\big{)}q_{t}}_{P}+$
$\displaystyle+\sum_{t=1}^{\infty}\sum_{k\leqslant
t}\mathbbm{S}^{k}_{t}\bigg{(}-\underbrace{s^{k}_{t+1}+(1-\tau_{t}\gamma)\bigg{(}s^{k}_{t}-\beta_{w}\lambda^{2}_{t}s^{k}_{t}I_{t}-\beta_{s}\alpha^{k}_{t}s^{k}_{t}\widehat{I_{t}}\bigg{)}}_{\Delta
s_{t}^{k}}\bigg{)}+\sum_{t=1}^{\infty}\mathbbm{S}^{t+1}_{t}\bigg{(}-\underbrace{s^{t+1}_{t+1}+\tau_{t}\gamma\bigg{(}s_{t}-\beta_{w}\lambda^{2}_{t}s_{t}I_{t}-\beta_{s}\widehat{s_{t}}\widehat{I_{t}}\bigg{)}}_{\Delta
s_{t}^{t+1}}\bigg{)}+$ $\displaystyle+\sum_{t=1}^{\infty}\sum_{k\leqslant
t}\mathbbm{I}^{k}_{t}\bigg{(}-\underbrace{i^{k}_{t+1}+(1-\tau_{t})\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}i^{k}_{t}+\beta_{w}\lambda^{2}_{t}s^{k}_{t}I_{t}+\beta_{s}\alpha^{k}_{t}s^{k}_{t}\widehat{I_{t}}\bigg{)}}_{\Delta
i_{t}^{k}}\bigg{)}+\sum_{t=1}^{\infty}\mathbbm{I}\mathbbm{T}_{t}\bigg{(}-\underbrace{it_{t+1}+\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}it_{t}+\tau_{t}\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}i_{t}+\beta_{w}\lambda^{2}_{t}s_{t}I_{t}+\beta_{s}\widehat{s_{t}}\widehat{I_{t}}\bigg{)}\bigg{)}}_{\Delta
it_{t}}\bigg{)}+$ $\displaystyle+\sum_{t=1}^{\infty}\sum_{k\leqslant
t}\mathbbm{R}^{k}_{t}\bigg{(}-\underbrace{r^{k}_{t+1}+\bigg{(}(1-\tau_{t}\gamma)\bigg{(}r^{k}_{t}+\frac{1-m_{i}}{t_{i}}i^{k}_{t}\bigg{)}\bigg{)}}_{\Delta