Covid-19 and Output in Japan∗
Daisuke Fujii†

Taisuke Nakata‡

University of Tokyo

University of Tokyo

First Draft: January 2021

Abstract
Using a tractable macro-SIRD model, we explore the relationship between the spread of
Covid-19 and output in Japan. We also examine the consequences of adopting alternative
criteria for ending the state of emergency currently in place in Tokyo. We will update ouranalysis weekly at https://Covid19OutputJapan.github.io/.

JEL: E17, E70, I18
Keywords: Covid-19; Emergency Declaration; Forecast Evaluation; Japan; Macro-SIRD
model; Vaccines.

∗

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
University of Tokyo for comments and suggestions. We thank Takeki Sunakawa for reviewing our Matlab
codes, designing our project’s computational architecture, and setting up the website for weekly updates.
Daisuke Fujii is supported by JSPS Grant-in-Aid for Scientific Research (KAKENHI), Project Number
17K13744. 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.u-tokyo.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

Source: Ministry of Health, Labor, and Welfare.

As shown in Figure 1, the number of new Covid-19 cases in Japan has been on an
upward trend since October 2020. The pace of the increase has accelerated in the past
few weeks, having led policymakers and the public to ask how strictly they should restrain
economic activities and how vigilant they would like to be against the risk of infection.
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 week and months.
In this paper, we develop a tractable model of Covid-19 and the economy to assist
such reassessment. We use our model to compute projections of Covid-19—conditional
on various paths of output—and construct a projected relationship between Covid-19
and output in Japan. We also use our model to examine the effectiveness of alternative
strategies for ending the state of emergency currently in place in Tokyo.

2

Model

Our model is formulated in a discrete-time with each period interpreted as a week. It
consists of two parts: the epidemiological and economic part. The epidemiological part2

is given by 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 persons, respectively. Dt denotes the number of cumulative deaths. Nt , NtIR , and NtID are the numberof 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 fromtime t to time t+1.1 γt and δt are time-varying parameters for recovery and death rates,
respectively.2
The economic part of our model is given by the following linear production function.
Yt = (1 − αt )At ({αt }t−1j=0 )(St + aI It + Rt ):= (1 − αt )Ȳt

(7)

Yt is output and depends on (i) the number of persons adjusted for the fact that some
infected persons cannot work or are less productive—given by (St + aI It + Rt )—and (ii)output per person—given by (1 − αt )At ({αt }t−1j=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 measuresagainst the risk of infection at time t. The dependence of At on the history of αt is intended
to capture 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 in detail what we intend to capture by the reference level of output as well as
how we construct it.3
1

As we will discuss later, we assume that a person becomes vaccinated with a 80 percent probability
after they receive two shots of vaccines.
2
See Atkeson (2020) and Moll (2020) for exposition of SIR models.
3
Our model abstracts from optimizing behaviors of individuals. See Eichenbaum, Rebelo, and Trabandt (2020) for an example of optimization-based macro-SIRD models, among many others.

3

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

(8)

β̃t = βt (1 − hαt )2

(9)

Nt =
where

and P OP0 := S0 + I0 . β̃t denotes the infection rate. βt denotes the “output-adjusted” or“raw” infection rate that would prevail in the absence of any decline in economic activity.
βt falls if people take actions that reduce the infection risk but do not directly affect theireconomic activities. For example, βt falls if people wear masks or wash their hands when
they return home. βt also reflects the intrinsic infection rate of Covid-19. If a coronavirus
variant with a higher infectious capacity spreads, it will appear as a higher value of βt .
(1 − hαt )2 captures the effect of a decline in economic activity on the infection rate.4
It is helpful to think of this term as a proxy of people’s mobility. While some mobility
is necessary for households to consume and businesses to produce goods and services, itleads to interactions between susceptible and infectious persons and thus helps spread thedisease. A high value of h means that the infection rate can be reduced a lot without reducing output that much. The value of 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 advantages of our modelling approach arethat the absence of tight cross-equation restrictions makes it easy for our model to fit
the past data and solve the model quickly and allows us to conduct a broad set of policy
experiments in a short amount of time. The disadvantage of our approach is that ouranalysis is subject to the Lucas critique. We judged that the advantages outweigh thedisadvantage in our work because the main goal of our project is to provide with policymakers and the public a back-of-the-envelope calculation of the effects of various policies
in a timely manner.5
4

See Alvarez, Argente, and Lippi (2020) and Farboodi, Jarosch, and Shimer (2020) for similar matching functions.
5
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. 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, Nakata, and Paustian (2018).

4

3

Data, assumptions, 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 timevarying parameters. Nt and NtID are the number of new positive PCR test cases and thenumber of deaths due to Covid-19, respectively, from the Ministry of Health, Labour andWelfare in Japan. Yt is based on monthly estimates of real GDP computed by the Japan
Center for Economic Research.6
We assume the following initial conditions: S0 = 125.7M , I0 = 1, R0 = 0, and D0 = 0.
Throughout the analysis, we set γt = 7/5 so that the effective reproduction number overthe past four months is 1.12, the average effective reproduction number over the past
four months reported in the Toyo Keizai website, which in turn is based on mok Jung,
Akhmetzhanov, Mizumoto, and Nishiura (2020).7 Vt = 0 for all t up to today.

3.1

Finding St , It , Rt , and Dt

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).8

3.2

Finding δt , βt , β̃t , and αt

Once we recover the path of Dt , we can recover the path of the death rate (δt ) using
equation (6). Once we recover 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. 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
6

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.
7This value of γt implies the expected recovery duration of 5 days. While this duration seems too
short at a first glance, recall that we measure Nt by the number of new positive PCR test cases reported
to the Ministry of Health, Labour and Welfare and that there are typically several days between the time
when a person becomes infected and the time when that person shows symptoms and go to a doctor totake a PCR test.
8
See Fernández-Villaverde and Jones (2020) for a similar identification strategy.

5essential flatness of Ȳt , fluctuation of αt largely inherits that of Yt .9
Figure 2: Mobility and output

Source: Japan Center for Economic Research and Google.

We obtain an estimate of h by regressing the Google mobility index (Mt ) on αt .10 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. Given the path of αt andthe estimated h, we can recover the path of βt using equation (9).
The last weekly observation is the week ending in January 17th, 2021 (the seven-day
period from January 11th, 2021 to January 17th, 2021).
Figures 3 and 4 show the identified paths of time-varying parameters and model variables, respectively.

4

Conditional projections of Covid-19

We use our model to compute projections of Covid-19 conditional on various paths of
output. In computing these conditional projections, we make the following assumptions
9

The most recent monthly GDP available from the Japan Center for Economic Research is for November 2020. We impute the path of output for December 2020 and the first two weeks of January using themobility index during these periods and based on the estimated relationship between the mobility indexand GDP from August 2020 to November 2020.
10
Google’s COVID-19 Community Mobility Reports provide mobility indices across different categoriesof places at a daily frequency. We pick the following four categories as we think they compose a goodmeasure of mobility which affects the overall infection rate: parks, transit stations, retail and recreation,
and workplaces. For each week, we compute the average of the median values of those four series toconstruct Mt . We use median values to eliminate the irregularity of holidays.

6Figure 3: History of time-varying parameters

Source: Authors’ calculation.

regarding the evolution of time-varying parameters. Note that, as mentioned earlier, thelast weekly observation is the week ending in January 17th, 2021. Thus, the first week ofprojection is the week ending in January 24th, 2021.11
We assume that the death rate (δt ) and the output-adjusted infection rate (βt ) will be
constant at their average values over the most recent five months.
We assume that the vaccine distribution begins in the first week of March 2021.12 The
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 put our assumption into perspective byreviewing some international experience so far in Appendix B.
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
11

The results shown in this section and the section that follows were generated on January 20th, 2021.
Our starting-date assumption is consistent with the statement made by Prime Minister Suga duringa press conference earlier this month.
127Figure 4: History of Covid-19 and output

Source: Authors’ calculation, Japan Center for Economic Research, Ministry of Health, Labor, andWelfare.
Note: In the bottom-right panel, the dashed line shows the reference level of output.

Vt that increases from zero in the last week of March 2021 to 1.6M (≈ 0.8 ∗ 4M/2) inthe 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 set of paths whose initial value (the first week in the projection) is positive, declines
linearly to zero in the first six months, and remains at zero thereafter. This pattern isintended to parsimoniously capture the pattern of α in the aftermath of the national
emergency declaration in 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
the national emergency declaration of April 2020 does a very good job in containing the
spread of 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 transitory8

decline in the number 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 surpressingagain, 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.
Figure 5: Conditional projections of Covid-19

Source: Authors’ calculation.

9

5

Relationship between Covid-19 and the economy

5.1

Baseline Case
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.

The solid black line in Figure 6 shows the set of pairs of cumulative deaths by the endof the next 12 months and the average output loss over the next 12 months associatedwith various paths of αt . As a reference, with our imputed Yt for December 2020, theaverage 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 theend of 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
our model 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 the13

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

10

curve to shift down in coming weeks, these numbers indicate just how striking the recent
spike was.
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 isexpected to be large and the disease is contained pretty well. In other words, there arediminishing returns to scale to reducing output.
Grey areas with varying darkness indicate the degree of uncertainty regarding therelationship between Covid-19 and output. The darkest and the second darkest grey
areas indicate 20- and 40-percent confidence sets, respectively. The second lightest andthe lightest grey areas indicated 60- and 80-percent confidence sets, respectively. These
confidence sets are constructed as follows. We compute the standard error of the estimatedh as well as the standard errors of the average values of the raw infection rates and deaths
rates.14 . 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 right edge 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

5.2

Sensitivity analysis

Figure 7 shows how deviations from our baseline specifications affects the relationshipbetween Covid-19 and the economy.
The top-left panel of Figure 7 shows how the trade-off curve depends on the assumed
path of the raw infection rate. The panel shows that a small increase in the raw infection
14

Here, we are essentially saying that the raw infection 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.
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.

11

Figure 7: Projected relationship between Covid-19 and output:
Sensitivity Analysis

Source: Authors’ calculation.
Note: The vertical axis shows the number of cumulative deaths by the end of the next 12 months.

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 substitutetheir economic activities in contact-intensive sectors with those in less contact-intensive
sectors.
The bottom-left panel demonstrates how sensitive our projection is to the specificationof the matching function for new infections. In the baseline projection, we assume aquadratic function for how α affects the infection 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 fasterpace. One key feature of the panel is that the benefit of distributing vaccines at a pace12

faster than in 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 the marginal value of a faster vaccine distribution is smaller.
These analyses demonstrate the importance of policymakers and the public to pursue
policies or individually take actions to shift the trade-off curve down. Even though shortrun 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 affectthe economy at least in the short run. Better health policies and better individual habits
contain the spread of the disease without necessarily reducing output.

5.3

Evolution of the relationshipThe projection of Covid-19 depends on the initial conditions as well as the projected
paths of the raw infection rate and the death rate, which in turn depend on the recent
realizations of them because we use the average values over the most recent five months
for projection. As a result, the projected trade-off varies over time. To illustrate thispoint, Figure 8 compares the baseline trade-off curve shown earlier with trade-off curves
computed one and two weeks earlier using the data available up to that point.
The most recent trade-off curve is a touch below the curve from one week ago andsubstantially 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
infection rate was substantially smaller back then. Accordingly, the number of cumulative
deaths was substantially smaller two weeks ago than now.

5.4

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.16 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,
instead of the realized value of 3,598.
The slope of the trade-off curve at the realized pair of deaths and the output loss—
shown by the red circle—can be seen as capturing the value of statistical life, albeit with
16

In this exercise, we consider alternative paths of the economy in which the path of αt is multiplied
by a constant for any time t from the last week of January 2020 to the last week of December 2020.

13

Figure 8: Projected relationship between Covid-19 and output:
January 17th, 2021 versus January 3rd and 10th, 2021

Source: Authors’ calculation. The vertical axis shows the number of cumulative deaths by the end of
the next 12 months.
Note: The vertical axis shows the number of cumulative deaths by the end of the next 12 months.

many caveats. The implied value of statistical life based on the slope of the curve in thisfigure is 145 years. In a future work, we plan to apply our methodology to other countries
and explore cross-country heterogeneity in the implied value of statistical life based onrealized health and economic outcomes in 2020.17

5.5

Discussion

There are many factors absent in our analysis but that would be of utmost importanceif 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.18
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 increase17

See Hall, Jones, and Klenow (2020) for discussion of the value of statistical life in the context of
Covid-19.
18
We plan to incorporate these factors into our analysis.

14

Figure 9: Relationship between Covid-19 and output in 2020

Source: Authors’ calculation.
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 overall deaths associated with Covid-19—both direct and indirect—particularly when
the output loss is small 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, Choi, and Sawada (2009), the numberof suicides per capita is more responsive to the unemployment rate in Japan than inother 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 elevenmonths of 2020 is 19,225—up from 18,675 in the first 11 months of 2019—reversing thedecade-long downward trend for the first time.19 Some private-sector analysts expect
the unemployment rate to edge up in 2021, which could push up the number of suicides
further going forward.

19

Source: https://www.npa.go.jp/publications/statistics/safetylife/jisatsu.html.

15

6

A real-time assessment of our model’s forecasting
performance

If a model were to be used as a guidepost for policy, it would be important to be awareof 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 areal-time forecasting performance of our model.20
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 and the next four weeks, applying the same forecasting procedure we describedearlier and using the data available up to that point. In particular, at each point in
time, we re-estimate h by regressing the mobility index on α available and compute theprojected 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 produced by the Japan Center for Economic Research is revised historically every month,
but we 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 are prepared and the week for which the most recent GDP is available depends
on which week of 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 one needs to impute seven weeks of GDP.21 However, if one is preparing projections in the second week of January, the monthly GDP for November 2020 isavailable 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 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 cases20

For readers unfamiliar with the idea of real-time forecast evaluation, see Faust and Wright (2009)
and ? for examples of real-time forecast evaluation exercises for macroeconomic variables.
21
See Section 3 for how we impute Yt .

16

Figure 10: Real-time forecast evaluation:
New infections

Source: Authors’ calculation.

we observed. For the four-week-ahead projection, our model systematically overpredicted
the number of new infections for most of 2020. However, our model’s four-week forecasts
prepared in the first two weeks of December 2020 underpredicted 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 overpredicted the new
Covid-19 deaths most of the time. For the four-week horizon, the forecast errors for
October and November 2020 are sizable.

17Figure 11: Real-time forecast evaluation:
New Covid-19 deaths

Source: Authors’ calculation.

7

Analysis of Covid-19 and output in Tokyo

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
their doors by 8pm, and offices to reduce the number of workers going to the office by 70
percent, 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.22 To do so, we recalibrate
our model using the numbers of new infections, new deaths, and the mobility index inTokyo. The monthly GDP in Tokyo is constructed as follows. Indices of Tertiary Industry
22

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

18

Activity and Indices of Industrial Production, compiled by the Ministry of Economy, Tradeand Industry (METI), report monthly output indices for many sectors in service andmanufacturing industries. The Economic Census for Business Activity reports sectoralvalue added in each prefecture. We merge these datasets to compute the value-added
weighted average of the monthly output indices, which is our monthly prefectural GDP
index.
We consider three scenarios. In the first scenario, the economic activity during thestate of emergency is such that the number of new daily cases reaches 500 in the eighthweek. We refer to this scenario as the baseline scenario. In the second and third scenarios,
the number of 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 by adjusting the level of α during the state of the emergency.
We assume that, once the emergency period ends, α declines to 4%, its average levelfrom September 2020 to November 2020 according to the monthly GDP for Tokyo weconstructed. 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 ismotivated by our observation that the hospital capacity constraint seems to have becomea 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 threshold for the second emergency declaration, but such exercise isoutside 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,
2021.23
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 path when the stopping criterion is 500 per day (or 3,500 per week), whereas
other thin dashed lines shows the paths under alternative criteria. One important featureof 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 theroad. 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 another23

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.

19

Figure 12: Alternative strategies in Tokyo:
Baseline scenario

Source: Authors’ calculation.
Note: The vertical axis in the right panel shows the number of cumulative deaths by the end of the next
12 months.

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 toanother emergency declaration, the pair of deaths and the output loss is interior to thetrade-off frontier shown in the blue color. The light green curve located in the northeastof the frontier curve consists of pairs of deaths and the output loss under stopping criteriathat eventually lead to the second emergency declaration. If the stopping criterion is
450 or below, then the economy avoids the second emergency declaration and the pair ofdeaths 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 thatavoids 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,the death-output outcome is inside the frontier curve. Avoiding another emergency declaration by setting the stopping criterion to, say 200, would improve both health andeconomics outcomes.
Figure 14 shows the results for the gradual-decline scenario. In this scenario, the20

Figure 13: Alternative strategies in Tokyo:
Rapid-decline scenario

Source: Authors’ calculation.
Note: The vertical axis in the right panel shows the number of cumulative deaths by the end of the next
12 months.

pair of death 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: Alternative strategies in Tokyo:
Gradual-decline scenario

Source: Authors’ calculation.
Note: The vertical axis in the right panel shows the number of cumulative deaths by the end of the next
12 months.

Figure 15 show all three scenarios together. Several lessons emerge from the right21

panel. First, the first-best strategy is the strategy of a rapid decline with a low stopping
criterion 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 toimpose—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 ofa gradual decline with a moderate stopping criterion, which leads to pairs of deaths andthe 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
the strategy of a gradual decline with a low stopping criterion, which is associated withpairs of deaths and the output loss being in flat regions of the trade-off curve.
Figure 15: Alternative strategies in Tokyo:
All scenarios

Source: Authors’ calculation.
Note: The vertical axis in the right panel shows the number of cumulative deaths by the end of the next
12 months.

We end this section 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 thefirst alternative vaccine assumption, the value of Vt is twice as large as that under thebaseline at any t. In the second alternative vaccine assumption, the value of Vt is half of22

that under the baseline at any t.
Figure 16: Alternative strategies in Tokyo:
Alternative vaccine assumptions

Source: Authors’ calculation.
Note: The vertical axis in the right panel shows the number of cumulative deaths by the end of the next
12 months.

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
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 baselinevaccine assumption.
The frontier curve associated with the second alternative vaccine assumption is on
the northeast part of the baseline frontier curve. Under this assumption and with the
stopping criterion of 500, the government declares the second emergency declaration inearly June. Thus, the pair of deaths and the output loss associated with the stopping
criterion of 500 is interior 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, thechoice 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 to provide definite answers and two reasonable people can disagree with23

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 in which both health and economic outcomes are better.24

8Conclusion

In this paper, we used a tractable macro-SIRD model to explore the relationshipbetween Covid-19 and the economy in Japan. We also used our model to examine theconsequences of alternative stopping criteria for ending the state of emergency in Tokyo.
Because our model is simple, our analysis should be seen as that of providing a back-ofthe-envelope calculation. Yet, we believe that many qualitative insights from our analysis
are likely to survive in alternative more-elaborate models.
We plan to update our analysis every week. Data and codes used for our analysis areavailable at https://Covid19OutputJapan.github.io/. We will monitor the forecasting
performance of our model and modify our model assumptions and specifications as wedeem appropriate. We will keep track of any changes we make to our baseline assumptions
and model specifications.

24

The same goes to policies of reducing β and increasing h. As we saw in the sensitivity analysis in
Section 5.2, the economy can attain a lower number of deaths for any given output loss with lower β andhigher h.

24

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Quarterly Review 1, Federal Reserve Bank of Minneapolis.
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Chung, H. (2015): “The Effects of Forward Guidance in Three Macro Models,” Feds
notes, Board of Governors of the Federal Reserve System (U.S.).
Chung, H., T. Nakata, and M. Paustian (2018): “Optimal Monetary Policy ina DSGE Model with Attenuated Forward Guidance Effects,” Feds notes, Board ofGovernors of the Federal Reserve System (U.S.).
Eichenbaum, M. S., S. Rebelo, and M. Trabandt (2020): “The Macroeconomics
of Epidemics,” NBER Working Paper Series 26882, National Bureau of Economic Research.
Farboodi, M., G. Jarosch, and R. Shimer (2020): “Internal and External Effectsof Social Distancing in a Pandemic,” Covid Economics, Vetted and Real-Time Papers,
9, 25–61.
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Fernández-Villaverde, J., and C. I. Jones (2020): “Estimating and Simulating a
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Hall, R. E., C. I. Jones, and P. J. Klenow (2020): “Trading Off Consumption and
COVID-19 Deaths,” Quarterly Review 1, Federal Reserve Bank of Minneapolis.
mok Jung, S., A. R. Akhmetzhanov, K. Mizumoto, and H. Nishiura (2020):“Real-time Estimation of the Effective Reproduction Number of COVID-19 in Japan,”Mimeo.
Moll, B. (2020): “Lockdowns in SIR Models,” Lecture note.
25Appendix
A

Constructing the reference level of output

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
their economic 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 theCovid-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 alpha would be positive even at the steady state when people are
not taking any social-distancing measures.
The reference level of output is different from the potential output because it reflects
the cyclical position of the economy right before the Covid-19 crisis occurred. Supposethat the 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 to think that the 5% decline in output, and an associated decline in themobility, reduced the infection 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,
and March 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
for these 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, consistent with the most recent OECD estimate.27 For 2021 and 2022, we letpotential GDP grow at a rate of 0.24 percent and 0.21 percent, respectively, taking on
board the December-2020 projection of OECD.
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).
27
OECD: Economic Outlook No. 108 (December 2020).

26

Let us provide some background information on why we used two sources, the Bankof Japan and OECD, in constructing the path of potential GDP. The estimates of theoutput gap from these two institutions for the past three years are very similar. In
particular, their estimates 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’squarterly estimate because OECD computes annual estimates only. As the fourth quarter
output declined nontrivially in the fourth quarter of 2019 due to the consumption tax hikeat the end of September 2019, using the fourth-quarter estimate of the output gap fromthe 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 OECDin 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 thepotential GDP path calculated in the first step. We take the level of the output gap inJanuary 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 essentiallyflat throughout our projection horizon.

B

Distribution of vaccines

In constructing the baseline path of Vt , we assume that the 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.
To put our assumed path of vaccine shots into perspective, let us discuss the evolutionof the numbers of vaccine shots administered in G7 countries, except Japan (Canada,
France, Germany, Italy, the United Kingdom, and the United States)—shown in Figure 17.
Each panel in the figure shows the daily number of vaccine shots administered per day
per 1,000 people over the past few weeks in each country.
If we take the population-weighted average of this statistics across the six countries
on January 13, 2021—the most recent observation at the time of writing—and multiply
it by 7 to translate that number into a weekly value, we get 1.5M. It appears from thevariable’s contour that it is likely to continue to increase in the future in all six countries
considered. If it increases at the same pace as in the first few weeks for the next five27

Figure 17: The daily number of vaccine shots administered per 1,000 people
in G7 countries

Source: Our World in Data.

or six weeks, the pace will be about 4M per week. As discussed above, under a set ofsimplifying assumptions regarding the effectiveness of each one of the two vaccine shots
individuals are expected to receive, 4M vaccines shots per week means the number of
newly vaccinated persons is 1.6M per week (0.8 ∗ 4M/2).
Our baseline path of Vt is admittedly highly judgmental. It is not clear to what
extent the experiences of these countries will be informative for predicting the evolutionof vaccine distribution in Japan. Our conversation with medical experts suggests that it
is logistically possible to administer vaccine shots at a substantially more rapid pace than
4M per week. However, administrative hiccups can slow down the distribution process,and the cultural aversion of the Japanese people to new and unfamiliar vaccines may
mean a substantially slower pace of vaccine distribution than in other countries.
We will closely monitor the experience of other countries as well as how the Japanese
governments—national and local—and medical communities prepare for the upcoming
vaccine distribution in Japan. We will modify our baseline assumption on vaccine distribution as we deem appropriate.

28

CMonthly GDP in Tokyo

Figure 18 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 thanin Japan as a whole, likely reflecting the high population density of Tokyo which makes
the mobility reduction imperative for the task of reducing mobility to contain the spreadof Covid-19. The magnitude of the output decline in Tokyo was about the same as thatin Japan, and the economic recovery in the fall of 2020 in Tokyo is not as strong as thatin Japan.
Figure 18: Monthly GDP in Tokyo

Source: Authors’ calculation.

29