WINPEC Working Paper Series No.E2016
October 2020

COVID-19 Misperception and Macroeconomy

Masashige Hamano ・Munechika Katayama ・So Kubota

Waseda INstitute of Political EConomy
Waseda University
Tokyo, JapanCOVID-19 Misperception and Macroeconomy∗
Masashige Hamano†

Munechika Katayama‡

So Kubota§

Waseda University
First draft: October 23, 2020

Abstract
Uncertainty about the true state of the COVID-19 pandemic has caused substantial
difficulty in economic activities and policymaking. How does the uncertainty affectthe macroeconomy and infections? What are the policy implications? To answer
these questions, this paper presents a model that incorporates people’s misperception about the current COVID-19 spread in the market. Our baseline model showsthat underestimation about the number of infections reduces the social welfare due
to worsening the externality of economic activities on virus transmissions while overestimation improves it to some extent. In an extended model with limited medical
resources, we show that a slight breakdown of the medical system can mitigate theunderestimation of the risk of being infected. We also consider the quarantine policythat limits both infections and the fall in economic activities for various degrees ofmisperception. Finally, affecting the extent of misperception about the spread isshown to be an effective policy tool that substitutes proposed containment policies
in the literature.
Keywords: COVID-19, imperfect information, SIR-macro
JEL classification: E1, I1, H0∗

We thank all seminar and conference participants for their helpful comments. The present project
was supported by JSPS KAKENHI Grant Number 18K01520 and 18K01521. Of course all remaining
errors are our own.
†
Email: masashige.hamano@waseda.jp‡
Email: mkatayama@waseda.jp§
Email: gkubotaso@gmail.com

1

1

Introduction

Since the outbreak of COVID-19, several efforts in containing the disease has been taken.
Obviously, one of the challenges is to identify the true state of the spread of the epidemic.
Various factors, from technical reason such as insufficient PCR tests to psychologicalones such as fear, can prevent people from measuring the exact state of the spread of thevirus. Uncertainty about the true number of infections would change the tradeoff between
economic activities and human lives. Moreover, it causes a great barrier for making
effective policy decisions, on the one hand, affecting the misperception can be an effective
tool in containing the number of infections. What is the consequence of misperception
about infection for the economy? What kind of policy mitigates the economic and healthcrisis under the uncertainty about the true state of infection? In this paper, we tackle
these questions.
We consider the misperception about the number of infections in an SIR-macro model,
which incorporates the epidemiological susceptible-infected-recovered (SIR) framework
into a standard macro model. In our theoretical model, susceptible people do not know
the true number of infections and hence underestimate or overestimate the risk of being
infected. The imperfect information about the true number of infections let susceptible
people to consume and work more in case of underestimation while they consume andwork less in case of overestimation. Therefore, a tradeoff arises therefore between economic
activities and the number of infections, which depends on the extent of misperception.
In our baseline model, we show that the social welfare depends on the extent of misperception about the spread of the virus. The pandemic leads to a negative externality of
economic activities through infections. People do not internalize the transmission of virus
caused by their consumption and labor in the market. On the one hand, the underestimation about the COVID-19 infections worsens the externality because it diffuses the novel
coronavirus more with too much economic activities. On the other hand, overestimation
about the spread improves the social welfare by containing the number of infections at
the expense of economic activities. We show, however, that too much overestimation
or fear from which the loss in economic activities dominates the gain in human lives is2

detrimental to the welfare.
We extend our benchmark model into various setups. First extension is a possibility of
medical system breakdown by treating too many patients. We show that when the medical
capacity is low and susceptibles underestimate the risk of infection, it is not optimal toidentify infected people as much as possible for the medical capacity. In such a case, it isbetter to hold slightly more patients beyond the capacity. This is because the breakdownof the medical system and a higher mortality rate that follows turns out to mitigate theunderestimation of the risk of being infected. The second one is quarantine policy. We
derive the similar results as the prior researches that the quarantine possibly improves
both health and economy (see for instance, Berger et al., 2020; Eichenbaum et al., 2020b;
Brotherhood et al., 2020a). In our setup, the quarantine which is organized as a meretransfer among different groups improves substantially the social welfare by limiting the
spread of the virus and enhancing economic activities. This is the case for different extent
of misperception about the state of infection. Finally, we compute optimal containment
policy in the form of consumption tax. It is shown that affecting the misperception of thevirus spread can indeed be an effective policy tool that substitutes the tax policy.
Many researchers call attention to the estimation bias of the true infection rate forpolicy analysis (Stock, 2020; Manski and Molinari, 2020). In comparison, our interestis people’s misperception and its consequences on their economic and social behaviors.
Akesson et al. (2020) conduct an online experiment and document that people dramatically misunderstand infectiousness and deadliness of COVID-19. Simonov et al. (2020)
find that stay-at-home behavior is crucially swung by news media.
Also this paper contributes to the recent literature of macroeconomic models that
integrated with the epidemiological SIR framework. Our model stands on an influential
work by Eichenbaum et al. (2020a, henceforth ERT), which is also applied to Eichenbaum
et al. (2020b), Krueger et al. (2020), and von Carnap et al. (2020). Misperception about
the virus spread is a central issue in COVID-19 policies. A number of papers show
the effectiveness of medical tests as revealing the true state of infection and targeted
quarantine (Berger et al., 2020; Bethune and Korinek, 2020; Brotherhood et al., 2020b;

3

Chari et al., 2020; Charpentier et al., 2020; Hornstein, 2020; Kasy et al., 2020; Eichenbaum
et al., 2020b). These papers focus on missing information about individual-level infection
status. Different from these papers, we study a misperception about the aggregate numberof infected people. Similar underestimation about disease spread is also suggested by von
Carnap et al. (2020) as an explanation of too strict containment policy in Uganda.
The remainder of this paper is organized as follows. In the next section, we present
our benchmark SIR-Macro model with the misperception about infections. In Section 3,
we extend the model and incorporate the possibility of medical system breakdown andforced quarantine. The simulation results including the optimal containment policy are
shown in Section 4. The last section concludes the paper.

2

Baseline model

We extend the SIR-macro model presented in Eichenbaum, Rebelo, and Trabandt (2020a)
to incorporate the misperception about the true state of infection.

2.1

Infection

The spread of the novel coronavirus follows ERT’s extension of the SIR model to incorporate economic factors. Each individual transitions to four states. One is susceptible ofa mass St at period t who are not infected yet but potentially in the future. The next
one is infected of a mass It . After the disease, some people recover and join a mass Rt ,or fall into dead Dt .
Following the outbreak of an epidemic, the total number of newly infected people Tt
evolves as
Tt = πs1 (St Cts ) It Cti + πs2 (St Nts ) It Nti + πs3 St It ,(1)

where Ctj and Ntj represent total consumption and hours worked of group j = {s, i, r}.
The following SIR equations determine the dynamics of the four groups:
St+1 = St − Tt ,4

It+1 = It + Tt − (πr + πd ) It ,Rt+1 = Rt + πr It ,Dt+1 = Dt + πd It ,where πr and πd are the recovery rate and death rate, respectively.

2.2

Susceptible people

It is very often the case that susceptible people do not know the exact spread of infection.
Misperception can arise from various sources. For example, the accuracy of PCR tests,
such as false negatives or false positives, is an important factor contributing to our limited
understanding about the true number of infected people. Similarly, the limited capacity of
conducting PCR tests is another reason for underestimating the true number of infections.
Further, people may overreact or under-react to the novel coronavirus because we still
do not know the exact characteristics (transmission mechanisms, symptoms, aftereffects,
and so on) of the virus specifically when vaccine is not available yet. Therefore, manyfactors such as the incomplete medical tests, limited information, and bounded rationalityof susceptible people in assessing the exact probability of future infection can contributeto the misperception.1 We capture this fact in a simple way. The number of perceived
infections is given by I˜t ≥ 0 such that I˜t = ψIt , where the parameter ψ captures theextent of misperception given the true number of infections It . As a result, we define thenumber of misperceived infection It∗ as It∗ ≡ It − I˜t . With the above specification, when0 < ψ < 1 susceptible people underestimate the true number of infections while whenψ > 1 she overestimates it. Thus different values of ψ can capture any deviations fromthe perfect perception case (ψ =1).
The susceptible person maximizes the following perceived lifetime utility Ũts , which is1

Our specification can be considered as a special case of “imperfect information” (Morris and Shin,
1998; Angeletos and Lian, 2016; Angeletos and Huo, 2018) with which the current beliefs of the future
beliefs of others are distorted for susceptible people or “bounded rationality” (Gabaix, 2014, 2020) withwhich they believe naively the perceived number of infections and recoveries.

5based on the number of perceived infection I˜t :
h
i
s
s
s
i
s
˜
Ut = u (ct , nt ) + β (1 − τ̃t ) Ũt+1 + τ̃t Ut+1 ,where cst and nst are consumption and hours worked for the susceptible person, Uti is the
lifetime utility in case of infection, and β represents the discount factor. The limited
perception is embedded in the perceived infection rate τ̃t , which is defined as:2τ̃t = πs1 cst I˜t Cti + πs2 nSt I˜t Nti + πs3 I˜t .

(2)With the above specification, the susceptible person underestimate (0 < ψ < 1) oroverestimates (ψ > 1) the infection rate which is based on I˜t . It is important to emphasize that the susceptible’s perceived lifetime utility differs from Uts = u (cst , nst ) +is+ τt Ut+1, which is based on the objective infection rate τt and the trueβ (1 − τt ) Ut+1
number of infections It , therefore. The equilibrium allocation with the perfect perception,in general, differs from that with misperception.
The susceptible person maximizes her lifetime utility subject to the following budget
constraint(1 + µct ) cst = wt nst + Γst ,(3)

where µct is tax rate on consumption, wt is real wage and Γst represents the lump-sumtransfer from the government.
The first-order condition with respect to cst yields:
u1 (cst , nst ) − (1 + µct ) λsbt + λsτ̃ t πs1 I˜t Cti = 0.
2

The true number of new infections is thus decomposed as






∗
∗
Tt =πs1 (St Cts ) I˜t Cti + It∗ Cti + πs2 (St Nts ) I˜t Nti + It∗ Nti + πs3 St I˜t + It∗=πs1 (St Cts ) It Cti + πs2 (St Nts ) It Nti + πs3 St It=τt StNote also by abusing the notation, Tt = (τ̃t + τt∗ ) St , T̃t ≡ τ̃t Tt , and T ≡ τt∗ Tt .

6where λsbt is the Lagrange multiplier for the budget constraint (3) and λsτ̃ t is the Lagrange
multiplier for the infection rate (2). The first-order condition with respect to nst givesu2 (cst , nst ) + wt λsbt + λsτ̃ t πs2




i
˜
It Nt = 0.

The first-order condition with respect to τ̃t ishiisβ Ut+1− Ũt+1= λsτ̃ t .
Importantly, the above first-order conditions depend on the perceived number of infections
I˜t instead of its true number It .

2.3

Infected people

The problem of an infected person is isomorphic to ERT. In the case of infection, thepatient will recover with a probability πr or stay infected if not die with a probability πd .
Her lifetime utility is given by:



i
r
Uti = u cit , nit + β (1 − πr − πd ) Ut+1
+ πr Ut+1,
where cit and nit are consumption and hours worked of the infected person, respectively,
and Utr is the lifetime utility of a recovered person. The infected person maximizes her
lifetime utility subject to the following budget constraint:(1 + µct ) cit = wt φi nit + Γit ,(4)

where φi denotes productivity in the case of infection and Γit represents the lump-sumtransfer from the government. The first-order condition with respect to cit yields:
u1 cit , nit − (1 + µct ) λibt = 0,7where λibt is the Lagrange multiplier for (4). The first-order condition with respect to nit
gives
u2 cit , nit + φi wt λibt = 0.

2.4

Recovered people

A recovered person also faces an identical problem as in ERT. Her lifetime expected utilityis:r
Utr = u (crt , nrt ) + βUt+1,

where crt and nrt are consumption and hours worked of the recovered person, respectively.
She maximizes the above utility subject to the following budget constraint:(1 + µct ) crt = wt nrt + Γrt ,(5)

where Γrt represents the lump-sum transfer from the government. The first-order condition
with respect to crt yields:
u1 (crt , nrt ) − (1 + µct ) λrbt = 0,
where λrbt is the Lagrange multiplier for (5). The first-order condition with respect to nrt
gives
u2 (crt , nrt ) + wt λrbt = 0.

2.5

Firms

As in ERT, there is a continuum of competitive representative firms of unit measure.
They produce consumption goods Ct using the following linear production technology:
Ct = ANt ,where Nt is the aggregate hours worked, and A represents the level of technology. The
firm chooses Nt by maximizing its current profits.

8

2.6

Government

The budget constraint of the government is the following:µct St Cts + It Cti + Rt Ctr = Γst St + Γit It + Γrt Rt .
Different from susceptibles that maximize Ũts , a perfectly informed government maximizes the following total welfare based on the true number of infections It :U0 = S0 U0s + I0 U0i .

2.7

(6)

Equilibrium

The model is completed by the following two market-clearing conditions: The goods
market clears as
St Cts + It Cti + Rt Ctr = Ct .
The labor market clears as
St Nts + It Nti φi + Rt Ntr φr = Nt .

3

Extensions

In this section, we consider two extensions. First one is the impact of possible breakdownof medical system depending on the number of treatments. Second, we examine theimpact of forced quarantine.

3.1

The medical preparedness model

As in ERT, we introduce a possibility of the deteriorating medical system due to a substantial number of infected people. Here, we assume a time-varying mortality rate πdt (I˜t , I t )that depends on the maximum number of infections cared by medical capacity I t as well9

as the number of infected people It .3 We assume that∂πdt /∂ I˜t < 0

if 0 < I˜t < I t ,∂πdt /∂ I˜t > 0

if I˜t > I t .

In the former case, a higher mortality rate is associated with insufficient medicalcare brought by underestimation of medical capacity. The latter one implies a possible
breakdown of the medical system and increasing mortality rate brought by overestimationof medical capacity. Specifically, it is assumed that

πdt = πd + I˜t − I t

2

,

(7)

where I t ≡ ψIt and ψ captures the capacity of medical system.
 2
In equation (7) which is transformed as πdt = πd + ψ − ψ It , it is useful to define2κ ≡ ψ − ψ for a better comparison with the models in ERT. When ψ = 1, the perception is perfect as in the medical preparedness model in ERT. When ψ = ψ, the mortality
rate becomes a constant, πd , at the lowest level as in the benchmark model.

3.2

Forced quarantine

With a possible quarantine, some people are separated from economic activity. A quarantined person is not allowed to work but receives income compensation, Γĩ . She is facing
the same expression of lifetime utility with the following exogenous path of consumption
and labor supply:
cit = Γĩ ,nit = 0,eewhere cit and nit are consumption and hours worked for the quarantined people. The aboveeepolicy would reduce the social interaction with susceptible people hence serving to reduce
the number of new infections while mitigating the fall in consumption.
3

I t is also interpreted as the maximum number of emergency medical facilities or equipment such as
ICU beds or artificial respirators.

10

4

Numerical simulations

In this section, we calibrate the model and provide a numerical simulation of the benchmark model, the medical preparedness model, and the model with quarantine argued
previously. Further, we discuss the optimal containment policy, which takes a form of
consumption tax in our baseline setting.

4.1

Calibration

Following ERT, we assume that a group-j has the following utility function:θu(cjt , njt ) = ln(cjt ) − (njt )2 ,2where θ captures the disutility from working. For our choice of parameter values, weclosely follow ERT in order to maintain compatibility with their results. As in ERT,
one period in our model corresponds to one week. For the parameter values that control
new infections, we set πs1 = 7.8 × 10−8 , πs2 = 0.000124, and πs3 = 0.390186, so that
economic decisions either through consumption or hours of work account for 1/3 of theinitial infection rate (1/6 each), and the herd immunity is eventually obtained when 60%of the total population is infected. In addition, as in ERT, we assume and φi = 0.8, which
implies lower productivity in the case of infection. The discount factor is β = 0.961/52 .
We set the level of technology A = 39.8352 and the weight on disutility from working
θ = 0.0013 so that the model’s pre-pandemic steady-state matches observed 28 hours ofwork and $58, 000/52 weekly income in the United States.
In the simulation, the pandemic is started by a 0.1% jump of infection at the initialepidemic period. Then, the economy is simulated over 250 weeks until convergence to the
new stead state with the herd immunity.

11Infected, I

4
2

80

60

40
50

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150

Weeks
Deaths, D

0.2

0.1

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Recovered Hours

0
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% Dev. from Init. SS% Dev. from Init. SS

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Weeks
Recovered Consumption

Weeks
Identified Hours

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Susceptibles Hours

0

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% Dev. from Init. SS

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Aggregate Hours, H

0

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Infected Consumption% Dev. from Init. SS

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Susceptibles Consumption
0

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% Dev. from Init. SS

100

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Aggregate Consumption, C% Dev. from Init. SS

0.3

40

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0

% Dev. from Init. SS

0

Recovered, R

60% of Init. Pop.

60

% of Init. Pop.

Susceptibles, S

100% of Init. Pop.

% of Init. Pop.

8

0
= 0.5
=1
= 1.5
=2
= 2.5
=3
= 3.5
=4
SIR

-5
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0

50

100

Weeks

150

0

50

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Figure 1: Responses of Groups with Different Values of ψ
12

100

150

4.2

Results of the benchmark model

Given the calibrated parameters, we conduct a comparative statics according to the misperception parameter ψ in the baseline model. Figure 1 shows the dynamics of theeconomy after the pandemic shock. The dynamic paths with ψ = 1 (dark blue lines)in
Figure 1 show the case of original ERT model without misperception, and “SIR” in the
Figure 1 provides the case of the classic SIR model. Specifically, the classic SIR model
is obtained by setting πs1 = πs2 = 0 in (1) in the original setup of the ERT model andthus considered as a special case of ERT having no economic interaction in determining
the number of new infection. As can be seen for ψ = 1, a susceptible person reduces
substantially consumption and hours worked by assessing correctly the probability of new
infection. As a result, we see a tradeoff between the number of infections, and economic
activities such as aggregate consumption and hours worked. The number of infections
and deaths are lower at the sacrifice of economic activities in case of ψ = 1 compared tothose in “SIR” with which we cannot expect the mitigation of new infection from lower
economic activities.
Compared to the case of ψ = 1, the misperception dramatically changes the allocationof the economy. When 0 < ψ < 1, susceptible people underestimate the probabilityof infection. They do not react to the pandemic so much and keep consumption andhours relatively stable. Under ψ = 0.5, as red lines indicate, reductions in consumption
and hours work are substantially less compared to those in ERT (ψ = 1). As a result,the number of infections and deaths are more pronounced with underestimation. The
tradeoff is controlled by the extent of misperception: as ψ increases, susceptible people
reduce consumption and hours worked more and make virus transmission less.
Figure 1 also compares responses with different degrees of overestimation (ψ > 1). As
the degree of overestimation increases as is the case with ψ = 1.3 and ψ = 1.6 in the
figure, susceptible person reacts more than it is necessary under the correct perception
case (ψ = 1). As a result, the number of infected people decreases in exchange for more
reductions in economic activities of susceptible people.
Our model shows clearly the tradeoff between economy and health, the extent of which
13Social Welfare (U0 )8282.88282.78282.68282.58282.48282.38282.2
= 0.5

=1

= 1.5

=2

= 2.5

=3

= 3.5

=4

Figure 2: Social Welfare with Different Values of ψ
depends on the degree of misperception. Which one dominates in terms of social welfare?
Figure 2 documents the sum of expected life time utility of susceptible and infected
people at the beginning of the epidemic given misperception defined in (6). We computethe welfare under the different value of ψ, and the case of ERT (ψ = 1) for the purpose ofcomparison. As the infection is less underestimated, the social welfare increases. A better
informed susceptible person consumes and works less, and thus contribute to mitigate therise in the number of new infections. In other words, the welfare loss by infection and death
overcomes the economic damage. Interestingly, welfare keeps increasing beyond ψ > 1
and peaks around ψ = 3.4 Overreaction or excessive fear against infections would improvesocial welfare up to around ψ = 3. Excessive overreaction (ψ > 3) eventually deteriorate
the social welfare, however. Without any realistic cost arising from overestimation, tolarger extents, overestimation of infected person is welfare improving since it contributes
to reduce infection and mortality despite its recessionary impact.5

4.3

Results of the medical preparedness model

As is the case in ERT, the allocation of the model changes dramatically by considering thepossibility of deteriorating the medical system as in (7). As the benchmark calibration of
4

To be precise, the social welfare reaches its maximum at ψ = 2.88.
An example of such cost would be exhausting medical resources due to the overestimation of infection,that results in a breakdown of medical system and hence induces a higher mortality as we capture in
the medical preparedness model. Furthermore, we don’t explore in the paper, however, increasing deathwith lower economic activities induced by the overestimation would be also a realistic possibility.
5

14

the time-varying mortality rate, we assume that ψ = 0.0513 and consider various valuesof ψ. Figure 3 provides the results. Since our benchmark model is identical to the basic
SIR-macro model by assuming perfect perception, “ψ = 1” is the case which corresponds
to “the medical preparedness model” in ERT.6 When the medical system breaks down
and the mortality rate starts to increase significantly, the susceptible person has a more
incentive to reduce her consumption and work less. In case of “ψ = 1”, the mortality
rate peaks out at 1 % in approximately 30 weeks. Given such a surge in the death rate,
consumption and hours worked decrease substantially as an optimal decision of susceptible
people so that the number of infections is contained. Remember that in our setting whenψ > ψ, the mortality rate is further increasing reflecting the deterioration of the medical
system. Thus, the containment in infection and the fall in consumption and hours worked
are more pronounced with overestimation of infection, that is, the case with ψ = 1.3 inthe figure. When 0 < ψ < 1, however, a rise in the mortality rate is milder and thus
weaker reductions in consumption and hours worked are realized (ψ = 0.0513, ψ = 0.09,
ψ = 0.4 and ψ = 0.7 in the figure). Note that when ψ = ψ = 0.0513, the mortality rateis constant with πd at the lowest level and the allocation of the model is the same as thebasic ERT without medical preparedness.
Finally, Figure 4 provides a welfare comparison for different values of ψ. The highest
welfare is achieved when ψ = 0.09. Under misperception of susceptibles, it is thus optimal
to let them perceived slightly a higher number of infected person than the number of
infections treated at its medical limit as Iˆt > I t , that is, ψ > ψ = 0.0513. Simply
setting Iˆt = I t (ψ = ψ) with which the lowest level of death rate is achieved does notprevent susceptibles from underestimating the probability of infection. Therefore, there
is an incentive to let them perceived the slightly higher number of infections beyond the
medical capacity to mitigate the welfare loss due to the underestimation. However, it
is important to notice that a too much number of infections reduces welfare because of
the breakdown of medical system and increasing mortality. Our result thus suggests thatimportance of expanding the medical capacity when the number of infections is being
6

Given ψ = 0.0513 and ψ = 1, we have ψ − ψ

2= 0.9 which corresponds to κ as calibrated in ERT.

15

Infected, I

6

Susceptibles Consumption% Dev. from Init. SS

4
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Susceptibles Hours

Identified Hours% Dev. from Init. SS

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% Dev. from Init. SS% of Init. Pop.

Identified Consumption

-5
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= 0.0513= 0.09
= 0.4
= 0.7
=1
= 1.3

-20
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Figure 3: Medical Preparedness with ψ̄ = 0.0513
improved. Finally, note that with an introduction of a realistic cost related to a higher
number of infections as we see in the medical preparedness model, the tradeoff between
economy and health changes and the optimal level of ψ is different with or without sucha cost.

4.4

Results of the forced quarantine

In this simulation, we conduct comparative statics about quarantine’s efficiency and misperception. In our simple setting, the number of perceived infections is also the numberof quarantined people. As more people are quarantined, it reveals the perception about
the true number of infections to some extent. Or as more people is perceived as infected,
the number of people put in quarantine tends to increase. Therefore, the quarantine effi16

Social Welfare (U0 )

8280

8275

8270

= 0.0513= 0.09

= 0.4

= 0.7

=1

= 1.3

Figure 4: Social Welfare of Medical Preparedness Modelciency is assumed to be systematically associated with a measure of perception about theinfection.
Figure 5 shows the result of this policy. Once one is quarantined, she is forced not towork and her minimum level of consumption is financed by income compensation. As aresult, there is less contact with the infected person at consumption and working place,which reduces the possibility of getting infected for susceptibles. Without the quarantine
policy, consumption and hours worked for susceptibles fall substantially as the value ofψ increases as Figure 1 indicates. With the quarantine policy, however, as the share of
infected person in quarantine increases, the risk of being infected is substantially reduced
for susceptibles and they no longer need to reduce their economic activities as the value ofψ increases. As a result, consumption and hours worked fall less compared to the model
without quarantine for a given value of ψ. Thus, the quarantine policy turns out to be very
effective since it works to mitigate the fall in economic activities simultaneously realizing
a lower number of infections. Moreover, with the quarantine policy, the peak of infection
is squeezed and becomes flatter, the extent of which is more pronounced for a higher levelof quarantine efficiency.This is because people are put into quarantine sequentially eachtime they are perceived as infected.
Finally, we compare the welfare with forced quarantine and without for a given valueof ψ. Figure 6 indicates two important results. First, the economy with forced quarantine
dominates the economy without for any level of the quarantine efficiency. Note that our17quarantine policy is a pure transfer among agents. Income compensation of infected isfinanced through a uniform lump-sum tax. Second, as is the case in the baseline model,
welfare is increasing monotonically with a higher level of test efficiency.
Welfare increasing is the direct consequence of having no cost in implementing a forced
quarantine policy. Forced quarantine policy is, however, so powerful that welfare continues
to improve with a higher level of its efficiency even under the possibility of medical system
breakdown. Figure 10 and Figure 9 in the Appendix show the dynamic paths of themodel with forced quarantine and the possibility of medical breakdown and comparisonof welfare between the medical preparedness model argued previously and the model withforced quarantine with the possibility of medical breakdown. The dynamic paths of theextended model are very similar to those obtained for the model with forced quarantine
only. Further, welfare is systematically improving beyond the capacity of medical limit.
The result would indicate that forced quarantine policy is so powerful to such an extent
that it overcomes the breaking down of the medical system.

4.5

Optimal containment policy

Figure 7 shows the optimal time-varying consumption tax policy called the optimal containment policy by ERT. We consider the optimal path of µc,t so as to maximize thesocial welfare in (6) given different values of ψ in the benchmark model. As the numberof perceived infections increases with a higher value of ψ, the required optimal consumption tax µc,t is lower along dynamics. We can see that misperception and the optimal
containment policy are substituting each other. Figure 8 compares responses of different
groups under the optimal containment policies for different values of ψ. As is the case forthe benchmark model, the fall in economic activities is more pronounced with a higher
value of ψ in exchange of a lower number of infections and deaths. The results suggestthat policy makers can resort to some policies that affect the public’s perception about
the true number of infections.

18Infected, I

2
1

90
80
70
60
50

150

% Dev. from Init. SS

0.1
0.05
0
50

100

100

-1

-2

-3

150

0

Weeks
Susceptibles Consumption-18% Dev. from Init. SS

2
0

-2
-4

50

100

-20
-21

-2
-4

50

100

50

100

Weeks

150

150

Weeks
Aggregate Hours, H

-2

-3
0

50

100

150

Weeks
Not-Identified Consumption

-18
-19
-20
-21

150

0

Weeks
Identified Hours

-99.5
-100
-100.5

50

100

150

Weeks
Not-Identified Hours

2

-101
0

100

-22
0

-99

% Dev. from Init. SS

0

50

-1

150

-19

150

10

0

% Dev. from Init. SS

100

Weeks
Susceptibles Hours

2% Dev. from Init. SS

50

20

0

Weeks
Identified Consumption

-22
0

30

150

0

0.15

0

50

Weeks
Aggregate Consumption, C

0.2

40

0
0

% Dev. from Init. SS

100

Weeks
Deaths, D

0.25% of Init. Pop.

50

% Dev. from Init. SS

0

Recovered, R

50% of Init. Pop.

30

% Dev. from Init. SS

Susceptibles, S

100% of Init. Pop.

% of Init. Pop.

4

0
= 0.4
= 0.7
=1
= 1.3
= 1.6

-2

-4
0

50

100

150

0

Weeks

Figure 5: Responses with Forced Quarantine
19

50

100

Weeks

150

Social Welfare (U0 )8300
With Forced Quarantine
No Froced Quarantine

8295
8290
8285

= 0.4

= 0.7

=1

= 1.3

= 1.6

Figure 6: Social Welfare with Forced Quarantine

5Conclusion

The spread of the novel coronavirus has caused great uncertainty about possible consequences to our lives and economies around the world. Using an SIR-macro model, westudy the role of imperfect information regarding the nature of the COVID-19 pandemic.
For various reasons, people will overestimate or underestimate the true number of infections and such misperception can affect how agents in the economy react to the pandemic.
We find that overestimating the true number of infections increases the social welfare
by mitigating the externality of economic activities on virus transmission. In particular,
it tends to contain the spread of COVID-19 at the cost of economic outcomes. , When weare facing limited medical capacity, we find that welfare loss due to the underestimation
can be alleviated by having a slight breakdown of the medical system. We also findthat quarantine policies appear to be a powerful policy option regardless of the degree ofmisperception, and that optimal containment policies can mitigate misperception.

20

0.8
= 0.4
= 0.7
=1
= 1.3
= 1.6

c,t

0.6

0.4

0.2

0
0

50

100

150

Figure 7: Optimal Containment Policies
Infected, I

6

3
2
1

50

% of Init. Pop.

4

80
70
60
50

0
100

150

50

100

Weeks

Deaths, D

Aggregate Consumption, C% Dev. from Init. SS

0.2
0.15
0.1
0.05
0
50

100

Weeks

150

0

50

100

150

Weeks

0

0

-2

-2

-4
-6
-8
-10
-12

0

20

150

Weeks

0.25

30

0
0

% Dev. from Init. SS

0.3

50

40

10

40
0

Recovered, R

60

90% of Init. Pop.

% of Init. Pop.

5

% of Init. Pop.

Susceptibles, S

100

Aggregate Hours, H

-4
-6
= 0.4
= 0.7
=1
= 1.3
= 1.6

-8
-10
-12

0

50

100

Weeks

150

0

50

Figure 8: Dynamic Responses with Optimal Containment Policies

21

100

Weeks

150

References
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——— (2020b): “An Economic Model of the Covid-19 Epidemic: The Importance ofTesting and Age-Specific Policies,” CESifo Working Paper.
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On the economics of indiscriminate versus targeted isolation policies during pandemics,”
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22

Hornstein, A. (2020): “Social Distancing, Quarantine, Contact Tracing, and Testing:
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Appendix
Solving the model
r H−1
s
i
To compute the equilibrium dynamics, given {µct }H−1
t=0 , we will find the sequence {nt , nt , nt }t=0that satisfies the following system of equations.

(1 + µct )cit = φi Anit + Γit(1 + µct )crt = φr Anrt + Γrt
Γt (St + It + Rt ) = µct St cst + It cit + Rt crt
nst = Aλsbt + λτ̃ t πs2 ψIt nit + λτ̃ t πs5 ψIt cit
To evaluate the above system, we need to compute the following.
θnrt = (1 + µrwt )φr Aλrbt(crt )−1 = (1 + µct )λrbt
23θ r 2(n )2 tr
Utr = u(crt , nrt ) + βUt+1(1 + µct )crt = Anrt + Γrt
urt = ln crt −θnit = φi Aλibt(cit )−1 = (1 + µct )λibt
θ i 2nuit = ln cit −2 tTt = πs1 (St Cts ) It Cti + πs2 (St Nts ) It Nti + πs3 St It2
πdt = πd + (ψ − ψ̄)It
St+1 = St − Tt
It+1 = It + Tt − {πr + πdt } It
Rt+1 = Rt + πr It
Dt+1 = Dt + πdt It
Tt
τt =
St
τ̃t = πs1 cst I˜t Cti + πs2 nst I˜t Nti + πs3 I˜t(1 + µct )cst = Anst + Γstθ
ust = ln cst − (nst )2
2


i
r
Uti = u(cit , nit ) + β (1 − πr − πd ) Ut+1
+ πr Ut+1
h
i
s
s
s
i
s
˜
Ut = u(ct , nt ) + β (1 − τ̃t )Ũt+1 + τ̃t Ut+1hiisλτ̃ t = β Ut+1
− Ũt+1
 (cst )−1 = (1 + µct )λsbt − λτ̃ t πs1 I˜t cit − λτ̃ t πs4 I˜t nit

24Results with medical preparedness and forced quarantine
Infected, I

2

80

60

40
100

150

50

100

% Dev. from Init. SS

0

-1
-2
-3

100

150

Weeks
Susceptibles Consumption-18% Dev. from Init. SS

2
0

-2
-4

50

100

100

-19
-20
-21

150

Weeks
Susceptibles Hours

0
-2
-4

50

100

50

100

Weeks

150

50

100

150

Weeks
Not-Identified Consumption

-18
-19
-20
-21

150

0

Weeks
Identified Hours

-99.5
-100
-100.5

50

100

150

Weeks
Not-Identified Hours

2

-101
0

-3

-22
0

-99

% Dev. from Init. SS

2

-2

0

Weeks
Identified Consumption% Dev. from Init. SS

50

150

-1

150

-22
0

100

-4
0

% Dev. from Init. SS

50

50

Weeks
Aggregate Hours, H

0

-4
0

% Dev. from Init. SS

0

0

0.1

20

150

Weeks
Aggregate Consumption, C

0.2

40

0
0

Weeks
Deaths, D

0.3% of Init. Pop.

50

% Dev. from Init. SS

0

Recovered, R

60% of Init. Pop.

40

% Dev. from Init. SS

Susceptibles, S

100% of Init. Pop.

% of Init. Pop.

60
= 0.0513= 0.09
= 0.4
= 0.7
=1
= 1.3

-2

-4
0

50

100

Weeks

150

0

50

100

150

Weeks

Figure 9: Responses with Medical Preparedness and Forced Quarantine

25Social Welfare (U0 )

8295
8290

With Forced Quarantine
No Froced Quarantine

8285
8280
8275
8270
= 0.0513= 0.09

= 0.4

= 0.7

=1

= 1.3

Figure 10: Social Welfare with Medical Preparedness and Forced Quarantine

26