COVID-19 and Output in Japan∗
Daisuke Fujii†

Taisuke Nakata‡

University of Tokyo

University of Tokyo
August 11th, 2021
Abstract

We build a tractable SIR-macro model with time-varying parameters and use it to
explore various policy questions such as when to lift the state of emergency (SOE).
An earlier departure from the SOE results in smaller output loss and more deaths inthe short-run. However, if the SOE is lifted too early, the number of new cases will
surge and another SOE may need to be issued in the future, possibly resulting in bothlarger output loss and more deaths. That is, the tradeoff between output and infectionthat exists in the short-run does not necessarily exist in the long run. Our modelbased analysis—updated weekly since January 2021, frequently reported by media, andpresented to policymakers on many occasions—has played a unique role in the policy
response to the COVID-19 crisis in Japan.

Keywords: COVID-19, Japan, SIR model, epidemiological model, vaccines, state of
emergency, variants
JEL classification: E17, E70, I18∗

We thank Yasushi Iwamoto, Munechika Katayama, Sagiri Kitao, So Kubota, Keiichiro Kobayashi, Antoine Lepetit, Haruko Noguchi, Kengo Nutahara, Fumio Ohtake, and seminar participants at the Universityof Tokyo for comments and suggestions. We also thank Takeki Sunakawa for reviewing our Matlab codes,
designing our project’s computational architecture, and setting up the website for weekly updates. We acknowledge Kenichi Nagasawa and Wataru Okamoto for their support on coding and helpful discussion. Yuta
Maeda, Kohei Machi, and Masataka Mori provided excellent research assistance. Daisuke Fujii is supported by
JSPS Grant-in-Aid for Scientific Research (KAKENHI), Project Numbers 17K13744 and 20K13475. Taisuke
Nakata is supported by JSPS Grant-in-Aid for Scientific Research (KAKENHI), Project Number 19K24682,
the Center for Advanced Research in Finance at the University of Tokyo, and the Tokyo Marine Kagami
Memorial Foundation. All errors are our own.
†
Faculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033; Email: dfujii@e.utokyo.ac.jp. Research Institute of Economy, Trade and Industry (RIETI).
‡
Faculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033; Email:taisuke.nakata@e.u-tokyo.ac.jp1

1

Introduction

Figure 1: The weekly number of new COVID-19 cases in Japan. Shaded regions indicate theperiods of SOE in Tokyo.
Source: Ministry of Health, Labor, and Welfare. As of August 8th, 2021

Since the outbreak of COVID-19 pandemic in early 2020, many countries have implemented non-pharmaceutical interventions (NPIs) such as stay-at-home orders and city-wide
lockdowns to control the spread of coronavirus. Japan is no exception. As shown in Figure 1,
Japan has experienced four epidemic waves, and is in the middle of the fifth wave as of earlyAugust in 2021. To abate the spread of infection, the Japanese government has declared thestate of emergency (SOE) four times: April 7th - May 25th in 2020 (all prefectures), January
8th - March 21st in 2021 (11 prefectures), April 25th - June 20th in 2021 (10 prefectures), andJuly 12th - August 31st (6 prefectures). Compared to strict lockdowns employed in many
countries, social and economic restrictions of SOE are weaker (Watanabe and Yabu, 2020).
Nonetheless, they had a significant impact on economy and seem to have been effective incontaining the spread of infection. These experiences sparked social debate on how much weshould suppress social and economic activities to control infection.
We build a tractable SIR-macro model with time-varying parameters and quantify the2relationship between the spread of COVID-19 and output in Japan. We then apply theframework to explore various policy questions such as when to lift the state of emergency(SOE) and how to accelerate economic activity. An earlier departure from the SOE resultsin smaller output loss and more deaths in the short-run. However, if the SOE is lifted too
early, the number of new cases will surge and another SOE may need to be issued in the
future, resulting in more deaths and larger output loss. That is, the tradeoff between output
and infection that exists in the short-run does not necessarily exist in the long run.
We also investigate the consequences of the spread of more contagious coronavirus variants
and faster vaccine rollout. A faster increase in the ratio of a more transmissive variant, beit the Alpha or Delta variant, would lead to a faster increase in the number of new cases,
necessitating an earlier invocation of the SOE and larger economic loss. A faster vaccine
rollout is associated with smaller economic loss and deaths. If we compare 0.6M/day and1.2M/day, the difference in economic loss is 1.8 trillion yen and the that of cumulative deaths
is around 400. These numbers are only for Tokyo; they are larger at the national level.
Finally, we extend our framework to a multi-group SIR model to incorporate age heterogeneity, and show simple parameter adjustment of the single-group SIR model allows it
to replicate the aggregate dynamics of the multi-group SIR model. More specifically, when
vaccines are distributed to elderlies before non-elderlies, an AR(1) adjustment on the transmission rate and a declining path of a mortality rate—that depends on the rate of vaccine
rollout—can generate aggregate dynamics that are quantitatively similar to those of a multigroup SIR model with age heterogeneity. Because of data limitation as well as the practicalneed for frequent assessments of the outlook, we argue that our single-group SIR-macro model
is a viable tool for real-time policy analyses.
The purpose of our project is to provide policy-oriented research in a timely manner.
Given unpredictable nature of the evolution of COVID-19, policymakers and the public will
likely need to frequently reassess their policies and behaviors in coming weeks and months.
To provide a timely analysis on a wide range of policy questions, we deliberately made thebaseline model as simple as possible, so as to be able to flexibly accommodate many pragmatic
extensions. In particular, our model abstracts from individuals’ optimization problem andwelfare analysis, unlike most of the SIR-macro models (e.g. Alvarez et al., 2020; Eichenbaum
et al., 2020; Farboodi et al., 2020; or Kaplan et al., 2020). As discussed in Section 2, we haveupdated our model-based projections every week since mid-January 2021 and have conducteda diverse set of policy analyses using our model. Our analyses have been presented to the
Cabinet Office, Prime Minister’s Office, Advisory Board of Ministry of Health, Labour andWelfare (henceforth, MHLW-AB), and key policymakers on many occasions. Our analyses
3have also been conveyed to the Japanese public through a number of newspaper articles andTV news.
In terms of the theoretical framework, our model resembles that of Acemoglu et al. (2020).
They also construct a multi-group SIR model without individual’s optimization problem andtrace the tradeoff curve between economic loss and deaths from infection. Their focus is
to study the optimal targeted lockdown policies which shift the tradeoff curve. Unlike theirmodel, we consider time-varying parameters, and fit past data to obtain reasonable estimatesof parameters for future projections. Also, we allow realistic vaccine rollout to different age
groups and examine its effect on the reduction in mortality.
Our analysis on the second SOE in Tokyo complements that of Kubota (2021) based on
a SIR-macro model that embeds economic agents’ optimizations. His model can decompose
the effects of government’s containment policies and individuals’ behaviors, and translate
the degree of soft lockdown to a consumption tax. His results on the SOE in Tokyo arequantitatively and qualitatively similar to our results on the same issue. Our work is different
from his work in a number of ways, including, but not limited to, the model, the estimation
method, and the range of policy experiments considered.
This paper is organized as follows. Section 2 elaborates how our analyses contributed tothe policy discussions on COVID-19 in Japan. Section 3 describes our model and data. Section 4 illustrates conditional projections of COVID-19 and the relationship between economy
and infection. Section 5 presents various applications of our framework such as the examination of an exit strategy of the SOE and different vaccine rollout scenarios. Section 6 considers
age heterogeneity by extending our model to a multi-group SIR and describes the differencesfrom a single-group SIR model. Section 7 concludes. Unlike a standard academic paper, we
describe the history of our real-time policy analyses in what follows. Retrospectively, someof the results are outdated since our specifications have been constantly evolving to reflect
fast-moving situation of COVID-19 such as the spread of mutant strains. We believe that it isworthwhile to record the entire history of our analyses including how the model specifications
have evolved. For each projection or application, we specify when it was done to clarify thetimeline.

2

The role of our analysis in Japanese COVID-19 policies

Our model is unique in the sense that the analyses based on our model have acted as a key
input to policymakers and the public in Japan. As described in Inaba (2020), prior to the4

COVID-19 crisis, the Japanese government had invested little in research of mathematicalepidemiological modelling. As a result, a sufficient amount of model-based epidemiologicalanalyses were not available to guide key policy discussions in real-time.1
As of January 2021, when the first draft of this paper was written and our weekly update of the economic and COVID-19 outlook began, Japanese policymakers and public were
not regularly presented with any medium-term and long-term outlook of COVID-19 by epidemiologists. The lack of outlook from experts is a stark contrast with what you see in the
U.K. where a group of epidemiologists regularly provided the government and public withmedium-term and long-term outlook of infection.
Against this backdrop, our weekly projections of the economy and COVID-19 garnered
immediate attention of policymakers and the public in Japan. From early February to lateMarch, our analysis of the second SOE in Tokyo—described in Section 5—was frequently
featured by Japanese media. In early February, there was a disagreement among the government and public-health experts as to how much infection should decline before lifting theSOE, with the government seemingly inclined to lift the SOE and stimulate economic activity sooner than public-health experts would find desirable. Our analysis provided economic
justification for “being patient“ in lifting the SOE. The government ended up with extending
the second SOE twice, first from February 7th to March 7th and second, from March 7th toMarch 21st.
Our analysis also contributed to the policy debate on whether to hold the Tokyo Olympics
and, if so, how many spectators should be allowed in Olympics venues. The public debate
regarding the Olympics began to intensify in April, with the majority of public expressing
skepticism towards the feasibility of safely holding the Olympics. Yet, there was no quantitative analysis regarding how the Tokyo Olympics may affect the spread of the disease. Our
May-21st analysis based on a variation of the model presented in this paper was the firstquantitative analysis to shed light on this important policy question (Fujii and Nakata, 2021;
Fujii et al., 2021). Our analysis concluded that the marginal effects of welcoming about
100,000 foreign visitors (athletes, media, and organizers) would be very limited given thelarge number of population in Tokyo (14 million) and that if the Tokyo Olympics were tohave significant effects on infection in Tokyo, that would come from how Tokyo residents
behaves during the Tokyo Olympics. Our analysis played a crucial role in directing the at1

To fill in the gap, the Cabinet Office has funded teams of mathematical modelers from other disciplines
and has asked them to provide the future outlook of infection and quantify the effects of various policies—
from the effects of vaccine rollout to the consequences of holding the Tokyo Olympics. We joined this groupin July 2021.

5

tention of policymakers and the public to where they should focus in order to minimize therisk of Olympics-induced infection. 2
Our analysis was the first to provide Japanese policymakers and the public with theoutlook for COVID-19 infection that incorporates the effects of rising shares of Alpha andDelta variants. As discussed in Section 5.2, we began to incorporate the effects of the rising
share of the Alpha variant in generating the outlook on March 30th, 2021. Our outlook—thefirst in Japan to consider the effects of the Alpha variant—was shared with key policymakers,
including, but not limited to, Tokyo Governor.3 We began to incorporate the rising shareof the Delta variant in our outlook on May 21st, 2021, providing the Japanese public early
warning of what’s to come in the near future.4
Since April, we have received numerous requests from various parties, including the Cabinet Office, Prime Minister’s Office, and members of the Subcommittee on Novel Coronavirus
Disease Control and MHLW-AB. Table 1 summarizes the timing of our analyses and key
policy briefings for the first half of 2021.

3

Theoretical framework

This section describes our baseline model and elaborates how to back out time-varying parameters. As mentioned above, our model has developed over time to respond to fast-moving
situations of infection. Models in the following sections are slightly different each other.
Table 2 summarizes various versions of our model.

3.1

Model

Our model is formulated in a discrete time with each period interpreted as a week. It consistsof two parts: the epidemiological and economic part. The epidemiological part is given by
2

It was mid-June when several other researchers released their analyses of the Tokyo Olympics. See the39th MHLW-AB meeting materials (held on June 16th, 2021) and simulation results organized by the Cabinet
Office.
3
See the materials of Tokyo COVID-19 monitoring meeting.
4
See the slide "The risk of Delta variants". It was mid-June when other researchers began to generate theoutlook that incorporates the effect of the Delta variant. See the 39th MHLW-AB meeting materials (heldon June 16th, 2021) and COVID-19 AI & Simulation Project by Cabinet Secretariat.

6

Date

EventsJanuary 22nd

Began the weekly update of outlook

February 10th

Subcommittee on Novel Coronavirus Disease Control

March 30th

Released “How the Alpha variant affects the outlook in Tokyo”

April 8th

Tokyo COVID-19 monitoring meeting

May 8thPrime Minister’s Office

May 21st

Released “The Effects of the Tokyo Olympics on COVID-19: A Quantitative Analysis”

May 21st

Released “How the Delta variant affects the outlook in Tokyo”

May 28th

Olympics Committee Experts Round-table

June 2nd

37th MHLW Advisory Board meeting

June 16th

39th MHLW Advisory Board meeting

June 17th

Released “The Effects of the Tokyo Olympics on COVID-19: The Role of Spectators”

June 18th

Olympics Committee Experts Round-table

June 20thPrime Minister’s Office

June 30th

41st MHLW Advisory Board meeting

June 30th

Council of Ministers

Table 1: Timeline of key policy briefings and events in the first half of 2021

Date published
Adjustment of the projection path of mortality rate
Alpha variant
Delta variant
ICU patients
Adjustment of the projection path of transmission rate

Sections 3 and 4 Subsection 5.1
January 2021
January 2021
No
Yes
No
No
No
No
No
No
No
No

Subsection 5.2
March 2021
Yes
Yes
No
No
No

Subsection 5.3
June 2021
Yes
Yes
Yes
Yes
Yes

Table 2: Development of model specifications
Note: “Adjustment of the projection path of mortality rate” means whether we take into account the
decline in mortality rate due to the composition effect of the infected. Because of the prioritized vaccine
rollout for the elderly, the share of older individuals whose mortality rate is much higher is expected toshrink among the infected. “Alpha variant” and “Delta variant” means that an increase of the transmission
rate due to the spread of the variants is incorporated using a logistic function. “ICU patients” means thatthe number of severe cases is considered. “Adjustment of the projection path of transmission rate” meansthat the transmission rate is adjusted by a positive AR(1) shock process to generate comparable resultswith a multi-group SIR model.

7the following SIRD model:
St+1 − St = −Nt − Vt

(1)

It+1 − It = Nt − NtIR − NtID

(2)

Rt+1 − Rt = NtIR + Vt

(3)

Dt+1 − Dt = NtID

(4)

NtIR = γt It(5)NtID = δt It(6)

St , It , and Rt denote the number of susceptible, infectious, and recovered individuals, respectively. Dt denotes the number of cumulative deaths. The total population is denoted by
P OP0 . Since we do not consider birth and other sources of deaths, the total population ispreserved at any time t
St + It + Rt + Dt = P OP0 for any t
The flow variables Nt , NtIR , and NtID are the number of newly infected persons, newly
recovered persons, and deaths from COVID-19 between time t and time t+1, respectively. Vt
is the number of newly vaccinated persons from time t to time t+1. Time-varying parameters
γt and δt are Poisson rates for recovery and death from the infected, respectively.5
The economic part of our model is given by the following linear production function.
Yt = (1 − αt )At ({ατ }t−1
τ =0 )(St + It + Rt ):= (1 − αt )Ȳt

(7)

Yt is output and depends on (i) the population of alive individuals given by (St + It + Rt )—and (ii) output per person—given by (1 − αt )At ({ατ }t−1
τ =0 ). Output per person consists of two
components. The first component, (1 − αt ), captures the reduction in output per person due
to social-distancing or other measures aimed at reducing the risk of infection. The second
component, At , is output per person that would prevail if no person takes measures against
the risk of infection at time t. The dependence of At on the history of ατ is intended tocapture possible hysteresis effects of having restrained economic activities in the past. We
use Ȳt to denote the level of output that would prevail if no one restrained his or her economic
activities at time t and refer to it as the reference level of output. Appendix A describes indetail what we intend to capture by the reference level of output as well as how we construct
5

See Atkeson (2020) and Moll (2020) for the exposition of SIR models.

8

it.
The epidemiological part of our model is linked to the economic part through the following
matching function for newly infected persons.
β̃t
It St
P OP0

(8)

β̃t = βt (1 − hαt )k

(9)

Nt =
where

β̃t denotes the transmission rate. βt denotes the “output-adjusted” or “raw” transmission ratethat would prevail in the absence of any decline in economic activity. βt falls if people takeactions that reduce the infection risk but do not directly affect their economic activities. For
example, βt falls if people wear masks or wash their hands when they return home. βt also
reflects the intrinsic transmission rate of COVID-19. If a coronavirus variant with a higher
infectious capacity spreads, it will appear as a higher value of βt .
The term (1−hαt )k captures the effect of a decline in economic activity on the transmission
rate. We assume quadratic matching of the susceptible and the infected and assume k = 2.
6
It is helpful to think of (1 − hαt ) as a proxy of people’s mobility. While some mobility isnecessary for households to consume and businesses to produce goods and services, it leads
to interactions between susceptible and infectious persons and thus helps spread the disease.
The elasticity of economic loss on mobility is denoted by h. A high value of h means thatthe transmission rate can be reduced a lot without reducing output that much. The valueof h captures, among others, teleworkability of office work, abilities of restaurants to raise
revenues through take-out services, or consumers’ willingness to switch from movie theaters
to online streaming services.
Our model is not micro-founded, unlike many macro-epidemiological models recently
developed in the economics profession. The advantage of our modelling approach lies in theabsence of tight cross-equation restrictions, which enables us to fit the past data well, solve
the model quickly, and conduct a broad set of policy experiments in a short period of time.
The disadvantage of our approach is that our analysis is subject to the Lucas critique. We
judge that the advantages outweigh the disadvantage in our work because the main goal of
our project is to provide with policymakers and the public future outlook and the effects ofvarious policies in a timely manner.7
6

See Alvarez et al. (2020) and Farboodi et al. (2020) for similar matching functions.
Our use of a reduced-form model is similar in spirit to the widespread use of semi-structural models
for policy analysis in central banks. For example, the FRB/US model—a semi-structural model of the U.S.
7

9

3.2

Data and identification of unobserved variables and time-varying
parameters

We use data on Nt , NtID , and Yt to recover the paths of the model variables and time-varying
parameters. Nt and NtID are the number of new positive PCR test cases and the numberof deaths due to COVID-19, respectively, from the Ministry of Health, Labour and Welfare
in Japan. Yt is based on monthly estimates of real GDP computed by the Japan Center
for Economic Research.8 We assume the following initial conditions: S0 = 125.7M , I0 = 1,
R0 = 0, and D0 = 0. Throughout the analysis, we set γt = 7/18, which is the value used in
Eichenbaum et al. (2020). This implies that the average duration for an infected individual
to recover or die is 18 days. The path of vaccinated population Vt is computed as follows.
Let E1 be the share of population with the first dose of vaccine who can move from S to R.
Define E2 analogously for the second shot of vaccine. With V1,t and V2,t be the number offirst and second shots of vaccines respectively, we have
Vt = E1 Vt,1 + (E2 − E1 ) V2,t
The time-series data of V1,t and V2,t are retrieved from the MHLW website. We assume Pfizer
vaccines are used and set E1 = 0.625 and E2 = 0.895 based on the UK’s SPI-M-O Summary
on March 31st, 2021.9
Given S0 and the paths of Vt and Nt , we can recover the path of St using equation (1).
Given I0 and the paths of NtID and γt , we can recover the path of It using equation (2).
Given R0 and the path of It , we can recover the path of Rt using equation (3). Given D0
and the path of NtID , we can recover the path of Dt using equation (4).10 Once we recover
the path of Dt , we can recover the path of the death rate (δt ) using equation (6). Once werecover the paths of It and St , we can use equation (8) to recover the path of β̃t .
We make an assumption about the path of Ȳt , which is described in detail in Appendix A.
economy—is used prominently for the analysis of forward guidance policies at the Federal Reserve, whereas
DSGE models—which suffer from the so-called forward guidance puzzle are not as frequently used as the
FRB/US model. See Chung (2015) and Chung et al. (2018) .
8
The monthly real GDP from the Japan Center of Economic Research can be accessed here: https://www.jcer.or.jp/en/economic-forecast/monthly-gdp. We assume the same GDP values for all weeks
in the same month. If a week spans over two different months, its weekly GDP is prorated using two values.
9

SPI-M-O stands for the Scientific Pandemic Influenza Group on Modelling, Operational sub-group. The
report “SPI-M-O: Summary of further modelling of easing restrictions – Roadmap Step 2” is available here.
10

See Fernandez et al. (2020) for a similar identification strategy.

10

Our assumed path grows very slowly over time and is essentially flat. Using the assumed
path of Ȳt , we can recover the path of αt using equation (7). Because of the essential flatnessof Ȳt , fluctuation of αt largely inherits that of Yt .

Figure 2: Mobility and output
Source: Japan Center for Economic Research and Google. As of August 8th, 2021

We obtain an estimate of h by regressing the Google mobility index (Mt ) on αt .11 As
shown in Figure 2, the correlation between Mt and Yt is high, and thus the correlation between
Mt and αt is also high because Ȳt is essentially flat. We run the following simple regression
Mt = h0 + h1 αt + t
where t is the residual. Using the data up April 2021, the estimated values are ĥ0 = 0.923
11
Google’s COVID-19 Community Mobility Reports provide mobility indices across different categories ofplaces at a daily frequency. We pick the following four categories as we think they compose a good measureof mobility which affects the overall transmission rate: parks, transit stations, retail and recreation, andworkplaces. For each week, we compute the average of the median values of those four series to construct
Mt . We use median values to eliminate the irregularity of holidays.

11and ĥ1 = −1.41. We then compute the estimate of h as follows
ĥ =

ĥ1
ĥ0

= −1.52

The reason why we divide the slope coefficient by the intercept is for normalization. In theabove regression specification, the intercept h0 is the mobility when there is no economic loss.
We normalize the elasticity h1 by h0 to obtain the estimate of h consistent with equation (9).
Given the path of αt and the estimated h, we can recover the path of βt using equation (9).
Figures 3 and 4 show the identified paths of time-varying parameters and model variables,
respectively.

Figure 3: History of time-varying parameters
Source: Authors’ calculation. As of August 8th, 2021

12Figure 4: History of COVID-19 and output
Source: Authors’ calculation, Japan Center for Economic Research, Ministry of Health, Labor, and Welfare.
As of August 8th, 2021
Note: In the bottom-right panel, the dashed line shows the reference level of output.

4

Conditional projections of COVID-19 (January 2021)

Analysis in this section was conducted in January, 2021.
We use our model to compute projections of COVID-19 conditional on various pathsof output. In computing these conditional projections, we make the following assumptions
regarding the evolution of time-varying parameters.
We assume that the death rate (δt ) and the output-adjusted transmission rate (βt ) will be
constant at their average values over the most recent five months. We picked the five-monthhorizon because it minimized the root mean squared error (RMSE) of the projected pathof new cases (Nt ) in the past data as of January 2021. The time horizon which minimizes
the RMSE of the past data changes over time. Since March 2021, we employ a four-monthhorizon to compute the average values of β and δ for projection.
We assume that the vaccine distribution begins in the first week of March 2021.12 The
12

Our starting-date assumption is consistent with the statement made by Prime Minister Suga during a

13

number of vaccine shots administered will increase from zero in the last week of February
2021 to 4M in the final week of May 2021. Thereafter, 4M vaccine shots will be administered
per week over the projection horizon.
We assume that a person receives the second vaccine shot 28 days after the first vaccine
shot. For simplicity, we assume that a person remains susceptible between the first andsecond shots and that 80 percent of persons who have received two vaccine shots will obtainfull immunity. With these simplifying assumptions, we obtain our baseline projection of Vt
that increases from zero in the last week of March 2021 to 1.6M (≈ 0.8 ∗ 4M/2) in the final
week of June 2021. Thereafter, Vt will remain 1.6M throughout our projection horizon.
We condition our projection on various simple paths of αt . In particular, we consider a setof paths whose initial value (the first week in the projection) is positive, declines linearly tozero in the first six months, and remains at zero thereafter. This pattern is intended to parsimoniously capture the pattern of α in the aftermath of the national emergency declarationin early April 2020.
Figure 5 shows our projection of COVID-19 conditional on three alternative paths αt .
The average output loss over the next 12 months associated with these paths are 1.5%, 2%,
and 3%—shown by the red, black, and blue lines, respectively. According to the figure,suppressing the economic activity to a level in line with that observed in the aftermath of thenational emergency declaration of April 2020 does a very good job in containing the spreadof COVID-19, as can be seen in the blue lines. The economic decline that is half of this most
severe case—shown by the red lines—would initially lead to a small transitory decline in thenumber of new infections, followed by another surge which peaks around July 2021. Under
the intermediate case—shown by the black lines—the number of new infections declines by
about half in the first three months. As the economic activity picks up suppressing again,
the number of new infections would increase again and peaks in July 2021 at a level similar
to the first two week of January 2021. Thereafter, the number of new infections declines
slowly as the vaccine distribution progresses.

4.1

Relationship between COVID-19 and the economy

4.1.1

Baseline Case

The solid black line in Figure 6 shows the set of pairs of cumulative deaths by the end of
the next 12 months (January 2022) and the average output loss over the next 12 months
press conference in early January, 2021.

14Figure 5: Conditional projections of COVID-19
associated with various paths of αt . As a reference, with our imputed Yt for December 2020,
the average output loss in 2020 is a touch below 4 percent. Recent projections for the growthrate of GDP in Japan by IMF, OECD, and World Bank are 2.3 percent, 2.3 percent, and
2.5 percent, respectively. The output loss in 2021 consistent with these growth projections
is about 1.6 percent.13
According to the figure, our model predicts 7,000, 17,000, and 42,000 deaths by the endof the next 12 months if the average output loss over the next 12 months is 3%, 2%, and
1.5%, respectively. As discussed later, these values are substantially higher than what ourmodel projected just two weeks ago, reflecting a sharp spike in the number of new infections
a week ago (the week ending in January 10th, 2021). Although we expect the curve to shiftdown in coming weeks, these numbers indicate just how striking the recent spike was.
13

The most recent projections by OECD, IMF, and Word Bank are published in October 2020, December
2020, and January 2020, respectively.

15

Figure 6: Projected relationship between COVID-19 and output
Source: Authors’ calculation.
Note: The vertical axis shows the number of cumulative deaths by the end of the next 12 months. The
horizontal dashed line indicates the total number of COVID-19 deaths during 2020.

Our trade-off curve is concave, reflecting the explosive dynamics inherent in the SIRD
model as well as our nonlinear specification of the matching function. One key implicationof this concavity is that we can save more lives by reducing output by one unit when the
output loss is expected to be small and the disease is out of control than when it is expected
to be large and the disease is contained pretty well. In other words, there are diminishingreturns to scale to reducing output.
Grey areas with varying darkness indicate the degree of uncertainty regarding the relationship between COVID-19 and output. The darkest and the second darkest grey areas
indicate 20- and 40-percent confidence sets, respectively. The second lightest and the lightest
grey areas indicated 60- and 80-percent confidence sets, respectively. These confidence setsare constructed as follows. We compute the standard error of the estimated h as well as
the standard errors of the average values of the raw transmission rates and deaths rates.14 .
14Here, we are essentially saying that the raw transmission rate and the death rate are randomly distributedwith population means and that we estimate their population means by the sample averages over the past
five months.

16

Assuming that they are independently and normally distributed, we draw 40,000 sets of h, β,and δ. For each draw, we compute the trade-off curve. We then look at different percentiles
at each level of output loss.
We highlight two features of uncertainty about the trade-off curve. First, uncertainty
is higher when the expected output loss is smaller. This feature is driven by the explosive
dynamics inherent in the SIRD model: a small difference in the effective reproductive number
makes a larger difference in the number of cumulative deaths when the effective reproductive
number is higher. Second, uncertainty is high at any level of output loss. Even at the rightedge of the figure in which the output loss is 3 percent and uncertainty appears relatively
low, the 80 percent confidence set ranges from 6,000 to 16,000.15
4.1.2

Sensitivity analysis

Figure 7 shows how deviations from our baseline specifications affects the relationship between
COVID-19 and the economy.
The top-left panel of Figure 7 shows how the trade-off curve depends on the assumed pathof the raw transmission rate. The panel shows that a small increase in the raw transmission
rate increases the number of cumulative deaths by tens of thousands if the output loss is
expected to be small.
The top-right panel demonstrates the importance of improving teleworkability, promoting flexible work arrangement, or encouraging households and businesses to substitute theireconomic activities in contact-intensive sectors with those in less contact-intensive sectors.
The bottom-left panel demonstrates how sensitive our projection is to the specification of
the matching function for new infections. In the baseline projection, we assume a quadratic
function for how α affects the transmission rate—recall (1 − hαt )2 in equation (9). The panel
considers two alternative functions—(1 − hαt )1.5 and (1 − hαt )2.5 .
The bottom-right panel demonstrates the benefit of distributing vaccines at a faster pace.
One key feature of the panel is that the benefit of distributing vaccines at a pace faster thanin the baseline scenario is larger when the output loss is expected to be smaller. When the
output loss is expected to be large, the disease is contained pretty well anyway so that themarginal value of a faster vaccine distribution is smaller.
These analyses demonstrate the importance of policymakers and the public to pursue
15

Note that, in constructing the confidence set, we only consider parameter uncertainty—uncertainty that
we have ways of quantifying, albeit imperfect. There is also uncertainty about the pace of vaccine distribution.
Model misspecification generates another dimension of uncertainty. Thus, our confidence set should be seen
as understating the true degree of uncertainty we currently face.

17

Figure 7: Sensitivity Analysis
Note: The vertical axis shows the number of cumulative deaths by the end of the next 12 months.

policies or individually take actions to shift the trade-off curve down. Even though short-run
lockdown policies are sometimes necessary to contain the spread of the disease in response
to a sharp increase in the number of new infections, they do adversely affect the economy atleast in the short run. Better health policies and better individual habits contain the spreadof the disease without necessarily reducing output.
4.1.3

Evolution of the relationshipThe projection of COVID-19 depends on the initial conditions as well as the projected paths of
the raw transmission rate and the death rate, which in turn depend on the recent realizationsof them because we use the average values over the most recent five months for projection. As

18

a result, the projected trade-off varies over time. To illustrate this point, Figure 8 compares
the baseline trade-off curve shown earlier (January 17th, 2021) with trade-off curves computed
one and two weeks earlier (January 10th and January 3rd, 2021) using the data available up
to that point.

Figure 8: Evolution of the tradeoff curve
Note: The vertical axis shows the number of cumulative deaths by the end of the next 12 months.

The most recent trade-off curve is a touch below the curve from one week ago and substantially above the curve from two weeks ago. Because the number of new infections was
substantially lower two weeks ago than now, the five-month average values of the raw transmission rate was substantially smaller back then. Accordingly, the number of cumulative
deaths was substantially smaller two weeks ago than now.
4.1.4

Relationship between COVID-19 and output in 2020

Figure 9 shows the relationship between the number of deaths and the average output loss in
2020, based on counterfactual simulations of COVID-19—computed conditional on various
counterfactual paths of αt . In this exercise, our counterfactual simulation starts from the third
week of January 2020 when the first COVID-19 case was reported. We consider counterfactual
paths of the economy in which the path of αt is multiplied by a constant for any time t fromthe last week of January 2020 to the last week of December 2020. In this way, we rescale the19path of economic loss preserving its shape to generate variations in average output loss. We
find that, if the average output loss in 2020 had been 3% and 5%, instead of the actual 3.9%,
the number of deaths in 2020 would have been about 38,000 and 530, respectively, insteadof the realized value of 3,598.

Figure 9: Relationship between COVID-19 and output in 2020
Note: The vertical axis shows the number of cumulative deaths by the end of 2020. The red filled circle
indicated the realized pair of death and the average output loss in 2020.

The slope of the trade-off curve at the realized pair of deaths and the output loss—shownby the red circle—can be seen as capturing the value of statistical life, albeit with manycaveats. The implied value of statistical life based on the slope of the curve in this figure
is 145 years. In a future work, we plan to apply our methodology to other countries andexplore cross-country heterogeneity in the implied value of statistical life based on realized
health and economic outcomes in 2020.16

4.2

Discussion

There are many factors absent in our analysis but that would be of utmost importance if
this type of analysis were to be used to inform the decisions of policymakers and the public.
Here, we discuss two such factors—hospital capacity and suicides.
16

See Hall et al. (2020) for discussion of the value of statistical life in the context of COVID-19.

20

We abstract from a potential increase in the death rate due to hospital congestion. Not
only do overcrowded hospitals contribute to an increase in the death rate among COVID-19
patients, they also increase the death rate from other diseases by constraining the supply of
medical resources. Taking this consideration into account would increase the overall deaths
associated with COVID-19—both direct and indirect—particularly when the output loss issmall and the number of COVID-19 cases is large.
We also abstract from a potential increase in the number of suicides associated withprolonged economic distress. According to Chen et al. (2020), the number of suicides per
capita is more responsive to the unemployment rate in Japan than in other countries. The
average unemployment rate in the first eleven months of 2020 is 2.8 percent, up from 2.4
percent in 2019. The number of suicides in the first eleven months of 2020 is 19,225—up
from 18,675 in the first 11 months of 2019—reversing the decade-long downward trend forthe first time.17 Some private-sector analysts expect the unemployment rate to edge up in
2021, which could push up the number of suicides further going forward.

4.3

A real-time assessment of forecasting performance

If a model were to be used as a guidepost for policy, it would be important to be aware of how
reliable the model is in explaining the past and predicting the future. Because we assume
time-varying parameters, we fit the past data perfectly by construction. To quantify how
reliable our model may be in predicting the future, we now examine a real-time forecasting
performance of our model.18
For each week from the first week of September 2020 to the first week of January 2021,
we compute the number of new infections and new COVID-19 deaths over the next week andthe next four weeks, applying the same forecasting procedure we described earlier and using
the data available up to that point. In particular, at each point in time, we re-estimate h byregressing the mobility index on α available and compute the projected paths of raw-infection
and death rates from their recent-five-month averages. We evaluate our model’s forecasts
conditional on the realized path of αt , as our paper focuses on conditional projections of
COVID-19.
Our exercise is not fully real-time for two reasons. First, the monthly GDP data producedby the Japan Center for Economic Research is revised historically every month, but we
17

Source: https://www.npa.go.jp/publications/statistics/safetylife/jisatsu.html.
For readers unfamiliar with the idea of real-time forecast evaluation, see Faust and Wright (2009) and
Clark (2011) for examples of real-time forecast evaluation exercises for macroeconomic variables.
18

21

Figure 10: Real-time forecast evaluation: New infections
abstract from that historical revision in this exercise. Second, we assume that the weekly
GDP data—which is imputed from the monthly GDP data—becomes available after six
weeks. In practice, the discrepancy between the week in which conditional projections areprepared and the week for which the most recent GDP is available depends on which weekof the month the projection is prepared. For example, if one is preparing projections in the
final week of December 2020, the most recent monthly GDP is from October 2020 and oneneeds to impute seven weeks of GDP. However, if one is preparing projections in the second
week of January, the monthly GDP for November 2020 is available and one only needs to
impute five weeks of GDP.
Top panels in Figure 10 show actual and forecasted outcomes for the number of new
infections for one- and four-week horizons, whereas bottom panels show their differences—
forecast errors. For the one-week-ahead projection, our model’s conditional forecast tracks
the overall contour reasonably well but often misses actual outcomes by a large amount. For

22

example, our model predicted about 23,000 new cases for the first week of January, 2021 in the
final week of December, which is substantially smaller than about 43,000 cases we observed.
For the four-week-ahead projection, our model systematically over-predicted the number of
new infections for most of 2020. However, our model’s four-week forecasts prepared in thefirst two weeks of December 2020 under-predicted the number of new infections.
Top panels in Figure 11 show actual and forecasted outcomes for the number of new
COVID-19 deaths for one- and four-week horizons, whereas bottom panels show their forecast
errors. For both one- and four-week horizons, our model has over-predicted the new COVID19 deaths most of the time. For the four-week horizon, the forecast errors for October andNovember 2020 are sizable.
Since we use the average of βt in the past five months to generate future projections,
we tend to over-estimate the new infections if the five-month time window includes higher
values of βt . In October 2020, we use the average of βt from May to September, and thetime window includes June and July in which a spike of βt is observed. Due to these higher
values of βt , the estimated β for projection at the time of October 2020 is high leading to theover-prediction. This applies to the estimation of δ as well. The death rate was very highfrom May to July 2020, and hence, the conditional forecasts of deaths as of October 2020
(both one-week-ahead and four-week-ahead forecasts) are higher than the actual values.
We use the simple past averages of βt and δt for projection because our main purpose is
to illustrate the relationship between economic loss and infection over one-year horizon, notprecisely forecasting the paths of new infections and deaths over one- or four-week horizons.
As shown in the bottom-left panel of Figure 3, the path of β fluctuates and does not exhibit
any predictable patterns. In this case, the past five-month average is not a bad choice forthe projection over next 12 months. If there is a factor which reasonably affects the future
paths of β or δ, we will make parameter adjustments as shown in later sections.

5

Policy analyses

5.1

Criteria for lifting the SOE in Tokyo (January 2021)

Analysis in this subsection was conducted in January 2021.
The Japanese government declared the state of emergency in Tokyo and several otherregions on January 7th, 2021, in an attempt to slow the spread of COVID-19. Since then,Governor of Tokyo has asked individuals to stay at home after 8pm, restaurants to close theirdoors by 8pm, and offices to reduce the number of workers going to the office by 70 percent,23

Figure 11: Real-time forecast evaluation: New deaths
etc. Some officials have suggested that they would like to maintain the state of emergency
until the number of new infections is down to around 500 per day in Tokyo. For the week
ending in January 17th, 2021, the average number of new daily infections was 1,504.
In this section, we use our model to examine the consequences of adopting alternative
stopping criteria for the current state of emergency in Tokyo.19 To do so, we recalibrate ourmodel using the numbers of new infections, new deaths, and the mobility index in Tokyo.
We also construct a measure of monthly output in Tokyo. To conduct our analyses,
we need high frequency (at least monthly frequency) data on economic activity for eachprefecture, but there is official statistics of the prefecture-level monthly output. Please see
Fujii and Nakata (2021) for more details on how to construct a measure of monthly prefecturelevel output in Japan by using publicly available data. Here, we lay out the basic idea
19

The results shown in this section were generated on January 20th, 2021, using the data through January
17th, 2021.

24

of how to construct such measures. Indices of Tertiary Industry Activity and Indices ofIndustrial Production, compiled by the Ministry of Economy, Trade and Industry (METI),
report monthly output indices for many sectors in service and manufacturing industries.
The Economic Census for Business Activity, published every five years, reports sectoral value
added in each prefecture. Using the sectoral value-added shares in 2016 and assuming they are
fixed over time, we compute the value-added weighted average of the monthly output indices,which is our measure of prefecture-level monthly gross domestic product (GDP). The Cabinet
Office publishes Regional Domestic Expenditure Index (RDEI) in every quarter, which is ameasure of prefecture-level monthly gross domestic expenditure (GDE). For those months inwhich these two measures, monthly GDP and GDE, are available, we take a simple average
to compute our measure of prefecture-level monthly gross domestic output (GDO). For themost recent months in which some of those data are not available yet, we nowcast monthly
GDO by using mobility and data from Teikoku Databank (TDB) as predictors.
We consider three scenarios. In the first scenario, the economic activity during the stateof emergency is such that the number of new daily cases reaches 500 in the eighth week. We
refer to this scenario as the baseline scenario. In the second and third scenarios, the numberof new daily cases reaches 500 in the fourth and twelfth week, respectively. We refer to these
two scenarios as rapid- and gradual-decline scenarios. These scenarios are implemented byadjusting the level of α during the state of the emergency.
We assume that, once the emergency period ends, α declines to 4%, its average level from
September 2020 to November 2020 according to the monthly GDP for Tokyo we constructed.
We also assume that, if the number of new daily cases increases to 2,000 after the end of
the current emergency period, there will be another emergency declaration, which would
be in place until the same stopping criterion is met. This assumption is motivated by ourobservation that the hospital capacity constraint seems to have become a pressing concern
when the number of new infections approached this threshold in the recent past. It would be
useful to examine the benefit of expanding the hospital capacity and increasing the thresholdfor the second emergency declaration, but such exercise is outside the scope of the paper.
Finally, we assume that the pace of vaccine distribution in Tokyo is such that Vt increases
from zero in the last week of March, 2021 to 160,000 in the final week of June, 2021.20
The left panel of Figure 12 shows the paths of new infections under the baseline scenario
with alternative criteria for ending the state of emergency. The solid red line shows the path20

160,000 per week in Tokyo is—in a population-adjusted term—is broadly in line with, 1.6M per week
in Japan assumed in the previous few sections because population in Tokyo is roughly 10 percent of total
population in Japan.

25

Figure 12: Baseline scenario
when the stopping criterion is 500 per day (or 3,500 per week), whereas other thin dashedlines shows the paths under alternative criteria. One important feature of this panel is that,
if the stopping criterion is sufficiently high and the current emergency period is sufficiently
short-lived, there will be a second emergency declaration down the road. Otherwise, vaccines
will arrive on time to avoid the second emergency declaration. The stopping criterion of 500
is slightly above the threshold value: it leads to another emergency declaration in late May.
The right panel of Figure 12 shows pairs of cumulative deaths and the output loss associated with various stopping criteria. Note that, when the stopping criterion leads to anotheremergency declaration, the pair of deaths and the output loss is interior to the trade-off frontier shown in the blue color. The light green curve located in the northeast of the frontier
curve consists of pairs of deaths and the output loss under stopping criteria that eventuallylead to the second emergency declaration. If the stopping criterion is 450 or below, then theeconomy avoids the second emergency declaration and the pair of deaths and the output loss
is on the trade-off frontier. The virtue of avoiding the second emergency declaration in our
exercise is reminiscent of the virtue of avoiding the second wave in standard SIR models. See(Moll 2020) who elucidates how a loose lockdown that avoids the second wave can save more
lives than a strict lockdown that eventually leads to the second wave.
Figure 13 shows the results for the rapid-decline scenario. In this scenario, the stopping
criterion of 500 again leads to another emergency declaration later on, and as a result, thedeath-output outcome is inside the frontier curve. Avoiding another emergency declaration
by setting the stopping criterion to, say 200, would improve both health and economics
26

outcomes.

Figure 13: Rapid-decline scenario
Figure 14 shows the results for the gradual-decline scenario. In this scenario, the pair ofdeath and the output loss associated with the stopping criterion of 500 is on the trade-off
frontier because it does not lead to the second emergency declaration.

Figure 14: Gradual-decline scenario
Figure 15 shows all three scenarios together. Several lessons emerge from the right panel.
First, the first-best strategy is the strategy of a rapid decline with a low stopping criterion27because the rapid-decline scenario is associated with the best trade-off frontier among all scenarios and a low stopping criterion allows the economy to be on the frontier curve. However,in countries like Japan in which the government lacks authorities to impose—or is reluctant
to impose—strict social-distancing measures on its citizens, a rapid decline may not be possible. The next best strategy seems to be the strategy of a gradual decline with a moderate
stopping criterion, which leads to pairs of deaths and the output loss that are in the middle
part of the frontier curve.
There are two types of strategies that appear inferior to the aforementioned two strategies.
The first is the strategy of a rapid decline with a high stopping criterion, which puts the
economy at a high risk of inducing the second emergency declaration. The second is thestrategy of a gradual decline with a low stopping criterion, which is associated with pairs ofdeaths and the output loss being in flat regions of the trade-off curve.

Figure 15: All scenarios
We end this subsection by examining the implications of alternative vaccine assumptions for the effectiveness of alternative stopping criteria. Figure 16 shows the evolution of
COVID-19 and the trade-off curves—shown in the left and right panel, respectively—underthe baseline scenario with the baseline and two alternative vaccine assumptions. In the firstalternative vaccine assumption, the value of Vt is twice as large as that under the baseline at
any t. In the second alternative vaccine assumption, the value of Vt is half of that under thebaseline at any t.
Let’s first consider the first alternative vaccine assumption with the stopping criterionof 500. In this case, once the emergency status is over, the number of new cases increases
28

Figure 16: Alternative vaccine assumptions
and peaks in May at a level comfortably below the level that triggers the new round of theemergency period. Thus, the pair of deaths and the output loss is on the frontier curve, which
is located on the southwest part of the frontier curve associated with the baseline vaccine
assumption.
The frontier curve associated with the second alternative vaccine assumption is on thenortheast part of the baseline frontier curve. Under this assumption and with the stopping
criterion of 500, the government declares the second emergency declaration in early June.
Thus, the pair of deaths and the output loss associated with the stopping criterion of 500 isinterior to the frontier curve. To be on the frontier curve, the stopping criterion has to be
300 or less.
Figure 16 underscores how important it is to distribute vaccines at a faster pace. Provided
that the government successfully avoids the second emergency declaration, the choice of the
stopping criterion is about which point to choose on a given trade-off frontier. The optimal
choice depends on various factors, including model specifications, assumptions, and one’s
philosophy about life and death–factors about which empirical evidence may not be able toprovide definite answers and two reasonable people can disagree with each other. In contrast,
policies of distributing vaccines at a faster pace are desirable regardless of what these factors
are, as better vaccine policies move the entire trade-off curve in the southwest direction inwhich both health and economic outcomes are better.21
21

The same goes to policies of reducing β and increasing h. As we saw in the sensitivity analysis in
Section 4.1.2, the economy can attain a lower number of deaths for any given output loss with lower β and29

Figure 17: Spread of a more infectious variant

5.2

Spread of coronavirus variants (March 2021)

Analysis in this subsection was conducted in March 2021.
In March, several cases of the Alpha variant have been reported. This variant exhibits
higher infection and mortality rate. One way to reflect the spread of the variants on βt is touse a logistic function. Let pt be the share of the Alpha variant in reported new cases. The
share grows as follows
pt= η0 + η1 tln
1 − ptwhere t is the duration from the start of simulation, η0 controls the variant share in theinitial period, and η1 controls the growth rate of the variant. Let 1 + κ be the relativetransmission rate of the Alpha variant to that of the then-prevalent variant (if it is 1.5 times
more infectious, κ = 0.5). Then, the path of transmission rate βt is given by the following
equationβt = (1 + κ) pt β̄
where β̄ is the past average of βt . Theoretically, η1 equals to κ, but we do not impose
this restriction. Figure 17 displays the spread of variants and associated increase of βt . We
consider several scenarios of spreading the Alpha variant and examine the risk to the economy
and death toll. The fist scenario assumes slow spread of the variant, and the logistic growthparameter is set as η1 = 0.17. The initial share of the variant is set as p0 = 0.55%. This
scenario is consistent with the spread of the Alpha variant observed in the U.S. The second
higher h.

30scenario considers fast spread of the variant by setting η1 = 0.52 and p0 = 1.1%, which isconsistent with the data in the U.K. In both scenarios, we set κ = 0.5.
Figure 18 illustrates the risk of the spread of variants. In each panel, we show variationsin the pace of economic recovery. The numbers in graph legends indicate the number ofweeks it takes for the economy to recover from the level during the SOE to a higher level,
which we set the level of autumn 2020. In the baseline case without variants (top panel), we
can avoid another SOE if we recover the economy gradually. If the Alpha variant spreads
even at a slower pace (middle panel), we cannot avoid another SOE and output loss will bemuch larger. If the speed of spread is faster (bottom panel), we may need to issue the SOE
twice, which will exacerbate the economic damage even more.

5.3

Faster vaccine rollout (June 2021)

Analysis in this subsection was conducted in June, 2021.
The third SOE was issued to Tokyo and other prefectures on April 26th, 2021 and announced to be lifted on June 20th. Compared to other major developed countries, Japan
started vaccine rollout late, but acceleration has been remarkable. The number of vaccine
shots reached a million per day for some days in June. In examining the effect of vaccine
rollout pace in this subsection, we consider its effects not only on infection, but also on thenumbers of severe cases, which we incorporated into our model in June. The laws of motion
are given by the following equations22(1 − hαt )2
It St
Nt = ωβ,t βtP OP0St+1 = St − Nt − Vt
It+1 = It + Nt − γIt − ωδ,t δt It
Ht+1 = Ht + ωδ,t δtICU Nt − γ ICU Ht − ωδ,t δt It
Dt+1 = Dt + ωδ,t δt It
where δtICU is proportional to δt , and γ ICU is picked to match the past data best. The
difference from Section 3 is the addition of a new variable Ht , the number of severe patients
22

When we introduced the severe cases into the model, we let the inflow into H depend on N . Early July,
we modified the model so that the inflow depends on It to be consistent with the literature.
Ht+1 = Ht + δtICU It − γ ICU Ht − δt It

31

(a) Baseline without the Alpha variant(b) Slow spread of the Alpha variant(c) Fast spread of the Alpha variant

Figure 18: Risk of the spread of the Alpha variant
32

at time t, and parameter adjustment terms ωβ,t and ωδ,t .
There are two potential reasons for why the projected path of the transmission rate βdeviates from the actual path in the absence of judgmental parameter adjustment.
The first reason is that β tends to be low during the SOE periods, while it is high outside
the SOE periods. As a result, right after a SOE begins, the past four-month average of beta,
which includes periods outside the SOE, tends to overstate the future path of β. Conversely,right after a SOE ended, the past four-month average of β, which includes periods outside
the SOE, tends to understate the future path of β.
The second reason is that our single-group SIR model, without appropriate parameter
adjustment, can exaggerate the effect of the prioritized vaccine rollout to the elderly on thetransmission rate for the reasons described in Section 6. In particular, when vaccines are
distributed to elderlies first, a single-group SIR model over-predicts severe cases and deaths,
while it under-predicts new infection.
For these reasons, it is convenient to allow for the possibility of adjustment on the future
path of beta. To facilitate such adjustment, we introduce the following AR(1) adjustment
term to the future path of beta
ωβ,t+1 = 1 + λ (ωβ,t − 1)
We pick λ = 0.95 and ωβ,0 = 1.1 where t = 0 is the start of projection. This adjustment
amplifies the number of new cases, and its effect decays over time. We chose the AR(1) shockprocess because it is simple, yet flexible enough to respond to many situations. For example,
we can set λ = 1 or λ > 1 if there is any reason to believe that the positive shock does notdecay or amplify in the future.
As vaccines are distributed to the elderly, the share of older individuals whose mortality
rate is much higher compared to the young, among the infected will decline. This will befurther discussed in the next section. To reflect this composition effect, the mortality rate
δt and the risk of severe symptoms δtICU will decline as the vaccine rollout for the elderly
progresses. The adjustment term ωδ,t is tied to the share of the elderly among the infected
as shown in Subsection 6.2.
We explore the ramifications of various vaccine scenarios. The setup is as follows. Once
the SOE in Tokyo is lifted in the fourth week of June, the economy will boost to the level ofFebruary in 2020 (right before the pandemic) within 12 weeks. We assume that the thresholdof daily new cases to declare the SOE will increase to 1500 per day because of the declining
mortality rate. Only 80% of population will be vaccinated. The efficacy of vaccines is the33

Figure 19: Vaccine rollout (1M/day at the national level)same as in Section 3: E1 = 0.625 and E2 = 0.895. We incorporate the spread of Delta variant
as in the previous section. It is assumed to be 1.95 times more infectious, and will take over
80% share by the end of August. We consider various vaccine rollout paces from 0.6M/day
to 1.3M/day. The outlook of vaccine rollout of 1M/day is depicted in Figure 19.
The main results are summarized in Figure 20. The top-left panel shows the projected
path of new cases, bottom-left panel shows the projected path of seriously ill patients (ICU),
the top-right panel shows the relationship between infection and economy, and the bottomright panel shows the path of GDP. Under our assumptions, the next SOE is inevitable inalmost all cases except the 1.3M/day (9.1M/week) case. From the top-right panel, we cansee that faster pace of vaccine rollout reduces both output loss and deaths from infection. If
we compare 0.6M/day and 1.2M/day, the difference in economic loss is 1.8 trillion yen andthe that of cumulative deaths is around 400. If we can increase the rollout pace by 0.1M/day,
we can save dozens of people and hundreds of billions of yen.

6

Age heterogeneity

Analysis in this section was conducted in June, 2021.
One of the key features of COVID-19 is its differential mortality risks across age groups.
Older individuals exhibit much higher mortality than that of younger individuals. Table
3 shows mortality risk (the number of cumulative deaths/the number of cumulative PCR
positive cases) by age group. The mortality risk of younger cohorts below age 40 is nearly34

Figure 20: Various vaccine paces

35

age group
cumulative PCR positives
cumulative deaths
mortality risk

<10s
10s
20s
30s
40s
50s
60s
70s
80s<
Total
21655 49334 150642 101020 98371 89181 57428 51057 50616 669304
0
0
6
20
87
238
764
2441
6702
10258
0.0% 0.0%
0.0%
0.0%
0.1% 0.3% 1.3% 4.8% 13.2% 1.5%

Table 3: Mortality risk by age group
Source: MHLW COVID-19 trends (Kokunai Hassei Doukou) as of May 19th, 2021
zero while that of 70s is 4.8% and 80s+ is 13.2%. Also, contact patterns are different acrossage groups according to the social contact survey conducted by Munasinghe et al. (2019).
Strong age-dependent assortativity is identified; individuals in the same age group interact
more than different groups. This within-group contact rate is higher for younger people. The
contact rate between the young and the old is low implying that the benefit of vaccinating
the old does not propagate to the young much.
As discussed earlier, we take these effects of age heterogeneity into account through parameter adjustments of the single-group SIR model. In this section, we explain the rationaleof our approach.
We first compare the aggregate dynamics of a naïve single-group model without parameter
adjustment with those of a multi-group SIR model. We show that, when vaccines are distributed to elderlies before non-elderlies, a naïve single-group SIR model over-predicts severecases and deaths, while it under-predicts new infection. We then show that an appropriate
adjustment of parameters of the single-group SIR model generates aggregate dynamics thatare identical to, or very similar to, those of the multi-group SIR model. We focus on the
projections in Tokyo.

6.1

A multi-group SIR Model

Individuals are partitioned into age groups indexed by j = 1, ..., J with P j initial members.
P
The total population is given by P OP0 = j P j . We consider a SIR compartment modelfor each group so that the following relationship holds at any given time t
Stj + Itj + Rtj + Dtj = P j

36

Group
j=1
j=2
j=3
j=4

Age
0 - 1920 - 39
40 - 59
60+

Share (%)
16.7
21.3
27.6
34.4

Table 4: Four age groups
The dynamics of the model is described by the following equationsJ

Sjt X
Njt = K (1 − hj αt ) βt µj
ρjk Ikt|{z} P OP0
k=1
2

β̃t

Sjt+1 = Sjt − Njt − Vjt
Ijt+1 = Ijt + Njt − γIjt − δjt Ijt
Hjt+1 = Hjt + δjICU Ijt − γ ICU Hjt − δj Ijt

All variables are the same as in a single-group SIR model except a subscript j. The elasticity
of economic loss on mobility hj can be heterogeneous across age groups but assumed to becommon in this section for expositional purpose: hj = h for all j. A parameter µj denotes
relative susceptibility of j to reflect differential risk of infection across age groups. Contact
rates between groups j and k are captured by ρjk . By explicitly incorporating these contact
rates, we can examine the effect of group-targeted policies such as prioritized vaccine rollout
for the elderly more accurately.
We consider four age groups described in Table 4. Based on Munasinghe et al. (2019) ,parameters are set as follows23




0.02
4.59



 0.061  ρ =  0.74µ=
 0.041  0.93



0.043
0.24

0.97
2.30
1.35
0.63

1.42
1.58
2.16
0.95


0.45

0.91 

1.18 

2.10

The contact matrix ρ shows active interactions among the same age group (higher values of23

Munasinghe et al. (2019) provide a more disaggregated social contact matrix by conducting an internetbased questionnaire survey in Japan. The dataset covers all 47 prefectures and include a total of 1476
households. We aggregate their matrix to 4-by-4 using group population as weights.

37

diagonal elements). Also, we can see that the contact rate between the young and the elderly
is low. This fact implies that vaccine distribution to the elderly does not abate the spread of
infection as much as in the single-group SIR model, where all agents are mixed equally.
We use the same αt and βt as in the single-group SIR model. Other parameters are set
as follows:




0.027
0.0001
 0.054 
 0.007 
7
7
ICU


=
δ=
 0.982  × 0.01 × 7
 0.172  × 0.01 × 16 , δ




8.698

6.227

The 4-by-1 vector of mortality risks δ is computed by aggregating the numbers in Table
3. We assume that the average duration for an individual to stay in the state of I before
going to D is 16 days. The age-specific risk of aggravation δ ICU is computed based on thereport submitted to the MHLW-AB.24 The pace of vaccination is assumed to be 0.75 million
per day (5.25M/week) at the national level. Only 80% of population in all age groups will
be vaccinated. Vaccines will be distributed to the age group of 60+ first, then 2/3 will be
distributed to the group 40-59 and 1/3 will be distributed to the group 20-39. After that,age group of 0-19 will be vaccinated. At the start of simulation, the shares of each age groupin the infected are 40%, 10%, 30%, and 20% respectively.
Figure 21 shows the projected paths of S and the shares of each group in I. Since thenumber of I is very small compared to the total population, the reduction in S mainly comesfrom vaccination. From the left panel of Figure 21, we see the prioritized rollout for the group60+, then groups 40-59 and 20-39. Due to the vaccination, the share of group 60+ in I isdecreasing up to October in the right panel of Figure 21. The shares of other three groups
are increasing. Since the group 40-59 is prioritized to 20-39, its share in I starts decreasingin September. Because of the vaccination of other three groups, the share of group 60+ will
start increasing in October. Since the vaccination of group 0-19 is put off to the last, its
share is increasing toward February of 2022, but once vaccination start, it will decrease. The
shares of each group will eventually reach the steady state.

6.2

Comparison between single-group and multi-group SIR models

Our baseline model described in subsection 5.3 is a single-group SIR-macro model. In using
this model, we employ two adjustments on time-varying parameters as discussed in subsection
5.3; i) an AR-1 type shock to βt and ii) a declining path of δt , reflecting the composition24

See page 33 of Material 2-1, which was submitted to the AB meeting held on October 22nd, 2020.

38

(a) Paths of S(b) Ratios of each group in I

Figure 21: S and I in by age groupeffect of the prioritized vaccine rollout. To understand the parameter adjustment we make,consider aggregating a multi-group SIR model as follows.

Nt = β̃t St It ωβ,t , where ωβ,t =

J X
J
X

µj ρjkj=1 k=1

Ikt Sjt
It St
J

1 X δjt Ijt
It+1 = It + Nt − γIt − δt It ωδ,t , where ωδ,t =δt j=1 It
Ht+1 = Ht + δtICU It ωδ,t − γ ICU Ht − It δt ωδ,tFor each aggregate variable, the functional form of the multi-group model is the same asthat of the single-group model. The difference between them is that there are time-varying
wedges (ωβ,t and ωδ,t ) that depend on the time-varying ratios of I and S in the multi-groupmodel. Thus, if we compute these time-varying wedge parameters by first simulating fromthe multi-group model and feed them into the single-group model, the resulting aggregate
dynamics of the “adjusted” single-group model are identical to those of the original multigroup model. Even without such perfect adjustment, a simpler computation of time-varying
wedges allows the single-group model to generate aggregate dynamics that are quantitatively
similar to those of the multi-group model. In what follows, we substantiate these two claims.

39

6.2.1

A naive single-group SIR model

Here, we compare the prediction of the multi-group model and a naïve single-group SIR
model. In order to compare the two models in a consistent manner, we need to pick anappropriate value of K. The constant should be
K=P P
j

1
k pj µj ρjk qk

where pj is a population share of group j and qk = PIk0Ij0 is the initial share of group k inj
I. With this value of K, the projection in the first period will be the same between the twomodels. Other parameters and settings are assumed to be the same as well except the two
adjustments mentioned above.
Blue and black lines in Figure 22 show the dynamics of the multi-group and naïve singlegroup models, respectively. According to the figure, this naïve single-group model overpredicts the path of deaths and severe cases and under-predicts the path of new infection
compared to the multi-group model when vaccines rollout are more concentrated on elderlies.
6.2.2

Adjusted single-group SIR models

While a naïve single-group SIR model cannot replicate the aggregate dynamics of the multigroup SIR model, parameter-adjusted single-group SIR model can.
Blue lines in Figures 22 show that the aggregate dynamics from the multi-group model
coincides with those from the perfectly-adjusted SG SIR model where the simulation of themulti-group model was used to compute the sequence of the time-varying wedge. Figures 23shows the aggregate dynamics from the multi-group model, the naïve single-group model,
and a single-group SIR model where we use simple procedures to capture the effects of theuneven distribution of vaccine rollout on the time-varying wedge. According to the figure,the single-group SIR model with a simple parameter adjustment delivers aggregate dynamics
that are quantitatively similar to those of the multi-group model.
Although the single-group SIR model with simple parameter adjustments can generate
similar aggregate dynamics of the multi-group SIR model, it is essential for researchers to use
the multi-group model if the question of interest entails disaggregate dynamics themselves,or group-targeted policies as in Acemoglu et al. (2020). For instance, if estimates of theage-specific elasticity of economic loss on mobility (hj ) are available, one can investigate theeffects of group-targeted lockdown policies or “vaccine passports”. Multi-group SIR models
can provide more convincing analyses on these targeted policies than single-group SIR models.
40

Figure 22: Multi-group SIR model, a naïve single-group SIR model, an adjusted single-group
SIR model (perfect adjustment)
In our weekly analysis, we use a single-group SIR model with simple parameter adjustment
due to data limitation. In Japan, data on age-distribution of newly infected, severe cases,
and deaths are not made publicly available in a timely manner. A single-group model withadjustments can buy both accuracy of projections and practicality for timely policy analyses.

41

Figure 23: Multi-group SIR model, a naïve single-group SIR model, an adjusted single-group
SIR model (simple adjustment)42

7Conclusion

We build a tractable SIR-macro model with time-varying parameters and quantify the relationship between the spread of COVID-19 and output in Japan. We then used the framework
to investigates various policy questions such as when to lift the state of emergency (SOE).
As discussed in Section 2, our model has contributed to the policy debate on how to control
infection while minimizing the economic loss in Japan. We hope that this type of model cancontribute to the policy debate in other countries as well. We hope that future research willlead to richer models that can better assist policymakers combat the next pandemic while
minimizing social and economic costs of NPIs.

43

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45Appendix
A

Constructing the reference level of output

This section was written in January 2021.
Our reference level of output at time t is the level of output at time t that would prevail
if people did not impose any economic restraints at time t (that is, if people conducted theireconomic activities as if COVID-19 suddenly and magically disappeared at time t), taking
as given the fact that the economy has suffered from the COVID-19 up to time t.
The reference level of output is different from the level of output consistent with precrisis trend (that is, the level of output that would have prevailed if there had been no
COVID-19 crisis at all) because the reference level of output reflects the possibility that the
COVID-19 crisis will leave permanent effects on the level of potential output. We would likeour reference level of output to reflect the permanent effect of COVID-19 crisis on output
because, otherwise, our α would be positive even at the steady state when people are nottaking any social-distancing measures.
The reference level of output is different from the potential output because it reflects thecyclical position of the economy right before the COVID-19 crisis occurred. Suppose thatthe output gap was 5 percent right in the first quarter of 2020 and that output declines by
5 percent in the second quarter of 2020 due to social distancing measures. We would like tothink that the 5% decline in output, and an associated decline in the mobility, reduced thetransmission rate through αt in this example.
We construct the reference level of output in two steps.
(i) In the first step, we construct the path of potential output. We apply the Bank ofJapan’s estimate of the output gap in the fourth quarter of 2019 (1.2 percent) to GDP in
December 2019 to construct potential GDP in December 2019.25 For January, February, andMarch of 2020, we apply the expected growth rate of potential GDP computed before the
COVID-19 crisis by OECD (0.63 percent, annualized) to construct the potential output forthese three months.26
For the remaining 9 months of 2020, we let potential GDP grow at a rate of 0.21 percent(annualized) so that the average growth rate of potential GDP in 2020 is 0.35 percent,25

The BOJ’s measure of the output gap can be accessed in: https://www.boj.or.jp/en/research/
research_data/gap/index.htm/.
26
OECD: Economic Outlook No. 106 (November 2019).

46

consistent with the most recent OECD estimate.27 For 2021 and 2022, we let potential GDP
grow at a rate of 0.24 percent and 0.21 percent, respectively, taking on board the December2020 projection of OECD.
Let us provide some background information on why we used two sources, the Bank ofJapan and OECD, in constructing the path of potential GDP. The estimates of the output
gap from these two institutions for the past three years are very similar. In particular, theirestimates of the output gap for 2019 are both 1.6 percent. We apply the fourth-quarter estimate of the Bank of Japan to our monthly GDP for December 2020 to compute the starting
point of our monthly potential output. We rely on the BOJ’s quarterly estimate because
OECD computes annual estimates only. As the fourth quarter output declined nontriviallyin the fourth quarter of 2019 due to the consumption tax hike at the end of September 2019,using the fourth-quarter estimate of the output gap from the BOJ is more sensible than using
the average estimate of the output gap for 2019 from OECD in constructing potential GDP
in December 2019. We primarily rely on OECD in constructing the path of potential GDP
from 2020 to 2022 because the Bank of Japan does not provide the growth rate of potential
output.
(ii) In the second step, we add the expected cyclical component of output to the potential
GDP path calculated in the first step. We take the level of the output gap in January 2020
as given and assume that the output gap will stay at that level throughout our simulation
horizon.
All in all, our reference level of output grows very slowly over time and is essentially flatthroughout our projection horizon.

B

Monthly GDP in Tokyo

This section was written in January 2021.
Figure 24 shows the Google mobility index and our monthly estimate of GDP in Tokyo.
The figure looks qualitatively similar to the one for Japan as a whole shown in Figure 2.
Both mobility and output declined sharply in April 2020 and recovered gradually thereafter.
The magnitude of the decline in the mobility was much larger in Tokyo than in Japan asa whole, likely reflecting the high population density of Tokyo which makes the mobility
reduction imperative for the task of reducing mobility to contain the spread of COVID-19.
The magnitude of the output decline in Tokyo was about the same as that in Japan, and the27

OECD: Economic Outlook No. 108 (December 2020).

47

economic recovery in the fall of 2020 in Tokyo is not as strong as that in Japan.
Figure 24: Monthly GDP in Tokyo

Source: Authors’ calculation.

48