A Quantitative Analysis of CARES Act Unemployment Insurance (Fang, Nie and Xie, RWP 20-07)
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Research working paper RWP 20-07, A Quantitative Analysis of CARES Act Unemployment Insurance, by Lei Fang, Jun Nie, and Zoe Xie, dated August 2020 and updated April 2022, which supersedes the earlier version titled Unemployment Insurance during a Pandemic. The abstract states that the authors quantitatively evaluate the impacts of the CARES Act UI policy on unemployment in 2020, finding that it raises average unemployment by 1.61 percentage points over April to December 2020 and reduces cumulative deaths by 2.09 percent. It states that the impact is larger in sectors where jobs cannot be performed remotely and that the interaction among the three CARES UI components accounts for one-third of the total policy effect.
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- Unemployment Insurance During A Pandemic
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“A Quantitative Analysis of CARES Act Unemployment Insurance (Fang, Nie and Xie, RWP 20-07).”
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A Quantitative Analysis of
CARES Act Unemployment
Insurance
Lei Fang, Jun Nie, and Zoe Xie
August 2020; updated April 2022
RWP 20-07
http://doi.org/10.18651/RWP2020-07
This paper supersedes the old version:
"Unemployment Insurance during a Pandemic"
A Quantitative Analysis of CARES Act
Unemployment Insurance∗
Lei Fang† Jun Nie‡ Zoe Xie§
April 17, 2022
Abstract
We quantitatively evaluate the impacts of the CARES Act UI policy on unemployment in
2020. More generous UI policies lead to higher unemployment, but save lives by reduc-
ing infections at the workplace. We find that the CARES UI policy raises average unem-
ployment by 1.61 percentage points over April to December 2020 and reduces cumulative
deaths by 2.09 percent, with the policy’s interaction with shutdown and COVID infection
risk playing a quantitatively important role. We also find that CARES UI’s impact on unem-
ployment is heterogeneous: it is larger in sectors where jobs cannot be performed remotely,
and it is hump-shaped over income. Decomposing the total effect into contributions by the
three CARES UI components, we find that the interaction among the components accounts
for one-third of the total policy effect.
Keywords: CARES Act; Unemployment insurance; Search and matching; Labor market
dynamics
JEL classifications: J64, J65, E24
∗
First draft: July 30, 2020. This paper was previously titled “Unemployment Insurance in a Pandemic.” We
thank seminar participants at the Fed Brown Bag, University of Nebraska Omaha, Texas A&M University, New
Hampshire University, Virtual East Asia Macroeconomics Seminar Series, Clemson University, and Indiana Uni-
versity for comments. We are especially grateful to Toni Braun, Andy Glover, Federico Mandelmann, and Juan
Rubio-Ramirez for suggestions. We thank Oske Yang for excellent assistance on the CPS data. Views expressed
are those of the authors and not necessarily those of the Federal Reserve Bank of Atlanta, Federal Reserve Bank
of Kansas City, the Federal Reserve System, the IBRD/World Bank or its executive directors.
†
Federal Reserve Bank of Atlanta (leifang@gmail.com)
‡
Federal Reserve Bank of Kansas City (jun.kc.fed@gmail.com)
§
World Bank; Federal Reserve Bank of Atlanta (xiexx196@gmail.com; lxie@worldbank.org)
1. Introduction
The COVID-19 outbreak led to widely implemented shutdown policies across the United States.
In response to the unprecedented employment and income losses, the congress passed the
CARES Act, which dramatically increased the generosity of unemployment insurance (UI).1
In addition to the usual extension in benefit duration (by 13 weeks) as in the previous reces-
sions, CARES Act also increased the weekly benefit payment by $600 and expanded UI ben-
efits to a large group of usually ineligible unemployed workers. Amid these unprecedented
changes, the U.S. unemployment rate rose rapidly from 3.5% in February 2020 to a post-war
record high in April, before it fell to a single-digit level by the end of 2020 as shutdown policies
were removed and infection risks faded. Though the spike and the following quick decline in
the unemployment rate may largely reflect changes in the shutdown policy, the increase in the
generosity of UI polices also triggered concerns that the CARES UI may have contributed to
the increased unemployment by keeping workers away from work.2
In this paper, we develop a model to quantify the effects of the CARES UI policy on the
labor market dynamics in 2020. We explore the interaction between UI policies and the shut-
down policy and infection risks, decompose the total effects of CARES UI into its three policy
components, and investigate the heterogeneous impact of CARES UI across different sectors
and income levels. The policy impact is heterogeneous because of the special nature of the
shutdown policy and the unique features of the CARES UI. First, during the pandemic, the
shutdown policy generates large job losses in sectors where jobs could not be performed re-
motely. Thus, workers in these sectors were more exposed to unemployment shocks and more
affected by the CARES UI policy. Second, the extra $600 per week generated higher UI income
than working wages for many workers and was more valuable to low-income workers. With
the eligibility expansion, low-income workers, who were ineligible for UI before the pandemic
due to minimum income requirements, were temporarily eligible for UI. The heterogeneity in
these policy aspects and the resulting heterogeneous responses of workers lead to quantita-
tively differential impacts of the CARES UI policy by sector and income.
Although we focus on the quantitative effects of the CARES UI on labor market dynam-
ics, our approach and its implications go beyond the COVID pandemic. First, our analyses of
1
CARES Act extended the UI benefit duration for 13 weeks (“Pandemic Emergency Unemployment Compen-
sation,” or PEUC), increased the weekly payment by $600 (“Federal Pandemic Unemployment Compensation,”
or FPUC), expanded the UI benefit to a large group of usually ineligible unemployed workers (“Pandemic Un-
employment Assistance,” or PUA), such as self-employed, part-time workers, and individuals who cannot work
for a wide variety of coronavirus related reasons.
2
For example, New York Times article on May 28, 2020 (https://www.nytimes.com/2020/05/28/business/
economy/coronavirus-stimulus-unemployment.html) stated that “some Republican lawmakers” were con-
cerned that “as the economy reopens, they say, the benefits could impede the recovery by providing an incentive
not to return to work.”
1
the CARES UI, including its heterogeneous effect by income and the decomposition into its
components and their interactions, are not specific to the COVID pandemic. They depend on
the structure of the UI policy and the heterogeneous responses of workers. Second, the eco-
nomic environment here is not unique to the pandemic. The shutdown policy is a large nega-
tive shock on employment which also interacts with the impact of UI policy on employment.
Other shocks, such as a natural disaster or a sectoral trade shock, can have similar interactions
with UI policy. As such, our setup is helpful in understanding the UI effects in more broader
contexts.
We embed an extended version of the epidemiological SIR model in a search-and-matching
framework. The pandemic is modeled by the risk of COVID-19 infection. Individuals with
mild or no symptoms can work and spread virus at workplaces, which in turn increases the
overall infection and deaths. Because old agents face higher probabilities of dying from the
infection than young agents, they are impacted more by higher infections. We assume that
working in a subset of industries—the contact sector—increases the worker’s infection prob-
ability, as workers in this sector have to perform their jobs at the workplace and cannot work
remotely. Infected workers face utility and income losses, and so a higher infection risk reduces
work incentives and leads to higher unemployment.
The model allows for heterogeneity in labor market and policies. In the model, unem-
ployed workers may separate from their jobs temporarily or permanently. Temporarily sepa-
rated workers can be recalled back to their old job with a probability. In addition, we allow
the exogenous job separation rates to differ by income as in the data. We model the shutdown
policy that is implemented in the U.S. as a direct destruction of jobs only in the contact sec-
tor (see also Glover et al. 2020), which leads to a drastic increase in temporary layoffs in the
contact sector. The UI policy is modeled along the three dimensions of the CARES UI policy:
eligibility, duration, and weekly benefit payment. We focus on the initial CARES UI policy
which ended in December 2020. To better capture the heterogeneous effect of the eligibility
expansion, we allow the probability that a worker receives UI pre-pandemic to depend on
their income, following a quadratic relationship to best match the data. Shutdown raises un-
employment directly, while a more generous UI policy reduces workers’ incentives to work
and in turn raises unemployment. By raising unemployment, both policies reduce workplace
infection and hence reduce the overall infection and save lives.
We calibrate the pre-pandemic steady state economy to match the sectoral distributions of
wage income, moments on labor market flows, and UI policy. Over the transition with pan-
demic, we calibrate COVID-related parameters to match reported death numbers for the young
and old at different points in time, since death numbers are arguably more accurately recorded
than infection levels, especially in the early months of the pandemic. Additionally, we calibrate
2
a drop in infection probabilities after the initial periods to reflect voluntary reduction in social
activities, such as mask wearing and social distancing. We calibrate the shutdown policy and
the share of temporary layoffs over the transition to match the dynamics of overall unemploy-
ment rate and the temporary layoff to unemployment ratio. We also calibrate the changes in
the UI policy along the three dimensions to best reflect the data and the UI policy changes.
The resulting model matches well the targeted moments and the untargeted moments both
in the steady state and over the transition. Moreover, with the calibrated wage distributions,
the $600 increase in weekly UI payment generates a distribution of UI replacement rates that
is very close to the distribution in the micro data.3
We find that the CARES UI policy raises the average unemployment rate during April to
December 2020 by 1.61 percentage points (ppt), out of a 9-ppt total increase in the average
unemployment. By raising unemployment, the policy lowers infection and reduces the total
cumulative deaths by 2.09%, or 12 thousand lives saved. Shutdown policy and infection risk
amplify the effects of CARES UI on unemployment, as they both raise unemployment and thus
increase the numbers of workers who are impacted by CARES UI. Absent these amplification
effects in a world without COVID infection risk and shutdown, the same UI policy would only
raise unemployment by 0.59 ppt.
The eligibility expansion and the $600 top-up are unprecedented UI policies and the pol-
icy changes along these two dimensions are also large. The eligibility expansion expands UI
benefits to almost all active workers and $600 per week is roughly 70% of the average personal
income in the United States (CPS 2015-2019). Accordingly, we found large impacts on un-
employment from these two policy dimensions. Specifically, we decompose the total effect of
CARES UI into the effects of its three components. The decomposition shows that out of the
1.61 ppts increase in average unemployment from April to December 2020, the $600 top-up
alone accounts for 0.33 ppt, eligibility expansion for 0.57 ppt, and duration extension for 0.17
ppt. The remaining 0.55 ppt is accounted for by the interaction among the three components.
Moreover, out of the 2.09% reduction in total cumulative deaths, the $600 top-up, eligibility
expansion, and duration extension each accounts for 0.24%, 0.93%, and 0.08%, respectively,
while the interaction effect accounts for 0.84%. The decomposition results suggest that while
most policy discussion has focused on the effect of the $600 top-up, the eligibility expansion
and the interaction effect among the three components have even larger effects.
The impacts of the CARES UI policy differ by sector and also by income. The policy effects
on unemployment and infection are larger in the contact sector, because the sector has an extra
infection risk and is directly impacted by the shutdown policy. Within each sector, the policy
3
See, for example, Ganong et al. (2020) for distribution of UI replacement rates during the pandemic based
on micro data.
3
effect on unemployment is hump-shaped over income and is the largest among workers in
the middle income range. In the equilibrium, workers with very low incomes have few jobs
to search for. So, whether more generous UI policies reduce their search, the impact on their
average job finding probabilities and unemployment rate is very small. Workers with very high
incomes respond little to the $600 UI top-up, as the money is a very small proportion of their
wage income. The eligibility expansion potentially increases the UI recipient rate the most
among the low income workers, because most of them were ineligible for UI pre-pandemic.
But in the equilibrium, the effect is not large enough to change the shape of the overall impact
of CARES UI.
Because of the heterogeneous impacts of CARES UI, shutdown, and infection on different
groups of people, the welfare effects of CARES UI also differ across the population. In general,
non-workers like the policy less than workers, because they do not receive the benefits but have
to pay the taxes used to finance the program. Among non-workers, old agents like the policy
more, or dislike it less, than young agents who are out of the labor force, because the old are
more likely to die from the infection and the policy helps reduce infection. Among workers,
those in the contact sector experience much larger welfare gains, as they are more impacted by
the shutdown-induced unemployment and the CARES UI policy provides income insurance.
Since much of the infection risk originates in the contact sector, an alternate UI policy targeting
this sector would work better. In particular, a policy that gives UI top-up only to those in
the contact-sector (while keeping other policy dimensions and total budget as the CARES UI)
would reduce total deaths by 0.7% more and generate similar unemployment effects compared
to CARES UI.
Our paper is related to several branches of literature on the effects of UI policies on the labor
market. First, our paper is closely related to the literature on the health and economic conse-
quences of the COVID-19 pandemic. Within this literature, Glover, Heathcote, Krueger, and
Ríos-Rull (2020) study the optimal shutdown and redistributional policy (such as taxes and
transfers); Kapicka and Rupert (2020) focus on the interaction between infection, wages and
unemployment; Birinci, Karahan, Mercan, and See (2021) compare the welfare implications
of UI benefit level increase and the payroll subsidies to firms (Paycheck Protection Program);
Chao (2021) looks at the welfare impacts of the CARES Act UI eligibility expansion policy
on workers of different income groups.4 Compared with these studies, our paper focuses on
quantifying the impacts of CARES UI and interactions among its components, and shows that
heterogeneous responses of workers and interactions with health and shutdown shocks are
4
See also Atkeson, Kopecky, and Zha (2020); Eichenbaum, Rebelo, and Trabandt (2020); Faria-e Castro (2021);
Aum, Lee, and Shin (2021); Gregory, Menzio, and Wiczer (2020); Mitman and Rabinovich (2021); Guerrieri,
Lorenzoni, Straub, and Werning (forthcoming); Jones, Philippon, and Venkateswaran (2020); Alon, Kim, La-
gakos, and VanVuren (2022).
4
important drivers of the overall effects.
Second, our paper is related to studies on the individual worker’s response to the $600
weekly benefit top-up, which is a component of CARES UI. Altonji et al. (2020) use an empiri-
cal approach and find that workers who experienced larger increases in UI generosity did not
experience larger declines in employment when the $600 top-up went into effect. Petrosky-
Nadeau (2020) and Boar and Mongey (2020) use partial equilibrium search models and find
that under the increased UI payments, few workers would turn down an offer to return to
work at their previous wage. In contrast to these works, we study the macro-level aggregate
effects of all three CARES UI components, and we find that the $600 top-up alone increases the
average unemployment rate over April–December 2020 by 0.33 ppt. Much of this effect comes
from the amplification effects of shutdown and infection risk, without which the $600 top-up
would only increase the average unemployment by 0.06 ppt.
Finally, although the macroeconomic shocks and the specific policies are different, our find-
ings for the total effect of CARES UI (1.61 ppts) and the individual effect of each policy compo-
nent (0.17–0.57 ppt) are in the same ballpark as the effects of UI duration extensions during the
Great Recession: Rothstein (2011) finds that extensions increased unemployment rate by 0.1-
0.5 ppt, Nakajima (2012) finds an effect of 1.4 ppts on unemployment rate, and Fujita (2010)
finds an effect of 0.8-1.8 ppts.
The rest of the paper is organized as follows. Section 2 lays out our SIR-search model. Sec-
tion 3 describes the calibration strategies in steady state and over the transition path. Section
4 presents the main results and discusses several robustness exercises. Section 5 concludes.
2. A SIR-Search Model
In this section we embed the SIR epidemiology model into a standard search-matching model.
There are two production sectors: contact sector and non-contact sector. Among other things,
the two sectors differ in the extent to which jobs can be done at home instead of at the work-
place. Contact sector has to operate at the workplace, while non-contact sector can fully op-
erate remotely. Because workers in the contact sector cannot work remotely, working in that
sector increases the probability of getting infected. Infected workers can also spread the virus
to non-workers. To evaluate the overall effects of the CARES UI policy we also model non-
workers. It is well documented that COVID affects young and old people differently, thus we
divide non-workers into young out of labor force (YOLF) and Old (65+). In addition to the
heterogeneity in infection dynamics, workers are also heterogeneous in their labor efficiency
units, the probability of separating from their jobs, and the probability of receiving UI benefits.
There is no aggregate uncertainty in the model. All off-steady state movements are driven by
5
changes in policies.
2.1. Model Environment
Population. The population size is normalized to one. There are three types of agents: young
workers, YOLF, and Old. We abstract from aging and assume workers cannot transit in and
out of the labor force or between the two sectors.5 Based on our classification of sectors, only
2% of workers switch between the two sectors in a month.6 The Old and YOLF only consume
and do not work, but they are important for the welfare evaluation of policies, because they
can be infected and policies affect the infection probabilities.
Young workers supply their labor inelastically. Each worker is born with an efficiency unit
𝑎 which does not change over time. The variation of 𝑎 across workers generates a distribu-
tion of income which helps to capture the different effects of the eligibility expansion and
the $600 UI top-up for various income levels. The distribution of the efficiency unit 𝐹𝑗 (𝑎) for
𝑗 ∈ {𝑐𝑜𝑛, 𝑛𝑐} differs by sector, where 𝑐𝑜𝑛 denotes contact sector and 𝑛𝑐 denotes non-contact
sector. A worker’s labor income is the product of her efficiency unit and the sector-specific
wage per efficiency unit 𝑤𝑗 𝑎. Agents cannot borrow or save.
Health. There are five possible health states: Susceptible, Infected Mild, Infected Severe,
Recovered, and Dead. Susceptible (type S) agents have not been infected by the virus; Infected
Mild (type M) agents are infected but with mild or no symptoms; Infected Severe (type I)
agents have more severe symptoms and are possibly hospitalized; Recovered (type R) agents
have survived the disease and acquired immunity from future infections; Dead (type D) is
the group that dies from the disease.
An infection may occur when a type S meets a type M or I. This can happen in two ways.
First, all agents can be infected at the same rate out of workplace. This “base” infection chan-
nel includes, for example, infections at home, in hospitals, and through consumption activities.
Second, contact sector workers can be infected at workplace, while the non-working popula-
tion (YOLF and Old) and non-contact sector workers, who can work at home, do not get in-
fected through this channel. To capture the reduction in infection rates following the voluntary
reduction in social activities (e.g. mask wearing, keeping social distance), we allow reduced
5
While recent works have documented a 4% fall in labor force participation in April 2020, which has since
partially recovered, evidence has also shown that much of the fall reflects women having to take up childcare
responsibilities at home, which is unrelated to changes in UI. For example, Lofton et al. (2021) find the labor
force participation of mothers fell by 5% in April 2020, recovered less than other groups in summer 2020, and
again fell to 5% below pre-pandemic level at the start of the school year.
6
Based on the CPS data, the worker transition rates between the two sectors did not change much during the
pandemic.
6
probabilities for both types of infection after the initial periods of the pandemic. Because we
focus on the labor market dynamics in 2020, we abstract from the availability of vaccine.
Once infected, the disease progresses stochastically, following age-dependent probabilities,
from M, to I, and to D. Death is only possible from I, while recovery is possible from both M
and I. R and D are both absorbing states. There is an intrinsic value to health, captured by the
utility costs of sickness and death. Let ℎ denote the health status, then the utility cost is 𝑢^ℎ ,
with 0 ≥ 𝑢^𝑀 > 𝑢^𝐼 > 𝑢^𝐷 and 𝑢^𝑆 = 𝑢^𝑅 = 0.
UI and Social Welfare Policies. The UI policy is modeled as follows. A newly separated
worker receives UI with probability 𝜆𝑗𝑎 in the first period of unemployment. The probability
𝜆𝑗𝑎 depends on the wage income of a worker (𝑤𝑗 𝑎) to capture the variation of UI recipient rate
over income in the U.S. (details in Section 3.1). An unemployed worker collecting UI loses the
UI entitlement with probability 𝜀 every period. Once she loses entitlement, she has to work to
regain eligibility. The benefit amount 𝑏𝑗,𝑎 is tied to the worker’s employment earnings and thus
differs by sector and by efficiency unit. The CARES Act UI policies are modeled as changes of
𝜆𝑗𝑎 , 𝜀, and 𝑏𝑗,𝑎 from their pre-pandemic values. The Old receive a Social Security benefit 𝑏𝑜 .
Unemployed workers without UI and YOLF receive social welfare benefits 𝑐. The government
balances its budget by imposing a flat proportional tax on all income to pay for the UI, welfare
and Social Security benefits. For easy exposition, we abstract from tax when describing the
worker’s value functions.
Production and Matching. A matched pair of firm and worker produces output 𝑧𝑗 𝑎 where 𝑧𝑗 is
the labor productivity in sector 𝑗 and is constant over time. Wage rate 𝑤𝑗 is sector-specific and
set exogenously.7 Workers with health status S, M, or R can work while workers with health
I cannot work.8 Firms post vacancies in the (𝑗, 𝑎) submarket with a posting cost 𝜅𝑧𝑗 𝑎 which
is proportional to the submarket productivity 𝑧𝑗 𝑎. Workers who remain unemployed (those
who are not recalled or are permanently separated) in sector 𝑗 with efficiency 𝑎 search in the
(𝑗, 𝑎) submarket. Let 𝑋𝑗𝑎 denote the aggregate search effort and 𝑉𝑗𝑎 be the aggregate number
of vacancies posted in the (𝑗, 𝑎) submarket. The number of new matches created is determined
by the matching function 𝑀𝑗 (𝑋𝑗𝑎 , 𝑉𝑗𝑎 ), where the matching parameter potentially differs by
sector. The submarket tightness is 𝜃𝑗𝑎 = 𝑉𝑗𝑎 /𝑋𝑗𝑎 . Assuming a constant returns to scale match-
ing function, worker’s per-search unit job-finding rate is 𝑓 (𝜃𝑗𝑎 ) = 𝑀𝑗 (𝑋𝑗𝑎 , 𝑉𝑗𝑎 )/𝑋𝑗𝑎 , and firm’s
job-filling rate is 𝑞(𝜃𝑗𝑎 ) = 𝑀𝑗 (𝑋𝑗𝑎 , 𝑉𝑗𝑎 )/𝑉𝑗𝑎 .
7
We assume that workers’ labor income is exogenously determined (see also Nakajima 2012; McKay and
Reis 2017), and is not an outcome of endogenous bargaining. Although past works have demonstrated that UI
generosity can affect job creation through changes in equilibrium wages (e.g. Krusell et al. 2010, Hagedorn et al.
2016), our modeling choice makes the model cleaner and more focused on the interaction between infection and
the CARES UI policy.
8
We count the type I workers as unemployed because they are eligible to collect UI benefits under CARES UI.
7
Separation and Temporary Layoff. The separation of a match depends on sector and efficiency
𝑒𝑥
unit. Let 𝛿𝑗𝑎 be the exogenous separation rate for a match in sector 𝑗 and with efficiency unit
𝑒𝑥
𝑎. without policy intervention, a match separates every period at rate 𝛿𝑗𝑎 = 𝛿𝑗𝑎 . A newly
separated worker can be on either temporary or permanent layoff. Among all the separations,
a share 𝛿˜ is temporary layoff, and the rest is permanent separation from job. This share is
the same across sector and efficiency unit. Each period, a worker on temporary layoff can
be recalled back to work with probability 𝑟 without going through the search and matching
process. If a worker on temporary layoff does not get recalled at the beginning of the period,
she can search for a new job in the period. If she does not find a job, with probability 𝜁 her
temporary layoff expires and she is permanently separated from her employer at the end of
the period.
A worker on temporary layoff who finds a new job through search will accept the job in-
stead of waiting for a recall in the next period, because the wages are the same for the two
jobs. The reason for this is that the worker’s efficiency unit stays the same over time and the
sector-specific wage is exogenous and constant. For the same reason, a recalled worker will ac-
cept the option and work for her old employer. However, a worker’s health status may change
while on temporary layoff, and we allow the recalling firm to keep track of this. We assume
that if a type I worker on temporary layoff is recalled, she becomes permanently separated
from her old employer since she cannot work and the associated job will become available to
other unemployed workers.
Shutdown Policy. During the pandemic, an unprecedented portion of unemployed workers
(up to 78% in April 2020) are on temporary layoffs, compared to an average level of no more
than 30% during the post-war period. In the model, the rise in the share of temporary layoffs
are driven by both the shutdown policy and changes in 𝛿˜. We model the shutdown policy 𝑚 ≥
0 as an increase in the contact sector’s job separation rate, and all shutdown-related separations
are assumed to be temporary layoffs. As such, the contact sector’s total job separation rate is
𝑒𝑥
𝛿𝑐𝑜𝑛,𝑎 = 𝛿𝑐𝑜𝑛,𝑎 𝑒𝑥
+ 𝑚(1 − 𝛿𝑐𝑜𝑛,𝑎 ), and separation into temporary layoff is 𝛿𝛿˜ 𝑒𝑥 + 𝑚(1 − 𝛿 𝑒𝑥 ).
𝑐𝑜𝑛,𝑎 𝑐𝑜𝑛,𝑎
The shutdown policy only applies to the contact sector and reduces workplace infection in
the sector. Shutdown may also reduce infection for consumers, for example, through reduced
activities in restaurants, hotels and retail. We only model its labor market effect to focus on
its interaction with UI policies, and capture the consumption effect through a change in the
exogenous infection probability, as a part of voluntary social distancing.9
To summarize, the contact and non-contact sector differs in the distribution of efficiency
9
While shutdown may have initiated the changes in consumer behavior, those changes have stayed on even
after shutdown is lifted. We thus capture these behavioral changes as part of the effect of voluntary social dis-
tancing after the initial periods of the pandemic.
8
unit 𝐹𝑗 (𝑎), labor productivity 𝑧𝑗 and therefore wage per efficiency unit 𝑤𝑗 , matching technol-
𝑒𝑥
ogy, separation rate 𝛿𝑗𝑎 , infection rate, and the shutdown policy.
Timing. At the beginning of a period, some employed workers lose their jobs and become
either temporarily or permanently laid-off. Workers on temporary layoff may get recalled.
Workers who remain unemployed (those who are are not recalled or are permanently sepa-
rated) search for jobs, while vacant firms post jobs. Production happens next. At the end of
the period, agents’ new health status is realized, unemployed workers with UI lose their ben-
efits with probability 𝜀, and workers on temporary layoff become permanently separated with
probability 𝜁 .10
2.2. Worker’s Problem
This subsection lays out the worker’s problem. Since the Old and YOLF do not make choices,
their value functions are simple and are included in Appendix B.2. The most important deci-
sion of workers is that unemployed workers choose how much search effort to exert. Higher
search increases job finding probability, but also comes with a utility cost. The UI policy, shut-
down, and infection risk all affect workers’ search effort and thus the labor market outcomes.
A worker’s period utility function is given by 𝑢(Income) + 𝑢^ℎ . A worker has four state
variables: sector 𝑗 , efficiency 𝑎, health status ℎ, and labor market status 𝜔 . 𝜔 is defined at
the beginning of a period and can take three values: 𝑒, 𝑏 and 𝑛, denoting employed, unem-
ployed with UI benefits, and unemployed without UI benefits, respectively. An unemployed
worker can be on temporary or permanent layoff. The value functions are denoted by 𝑊 𝑒 for
employed workers, by 𝑊 𝑏 and 𝑊 𝑛 for workers on permanent layoffs, and by 𝑊 ˜ 𝑏 and 𝑊 ˜ 𝑛 for
workers on temporary layoffs. Given the beginning-of-period labor market status, whether
the worker works in this period is determined by labor market transitions. Because a worker’s
infection probability depends on whether she works and her sector, we define the health tran-
sition probability matrix from this period’s health ℎ to next period ℎ′ : Γ1𝑗 (ℎ, ℎ′ ) and Γ0𝑗 (ℎ, ℎ′ )
for workers in sector 𝑗 who work and do not work, respectively.11
Let 𝛽 be the time discount factor. The value function for an employed worker (𝑗, 𝑎, ℎ) where
10
Appendix B.1 includes a timeline to illustrate the within-period timing.
11
Since we assume workers do not move between sectors, unemployed workers, like employed workers, also
belong to fixed sectors.
9
ℎ ∈ {𝑆, 𝑀, 𝑅} is given by:
∑︁ {︂
𝑒
𝑊 (𝑗, 𝑎, ℎ) = Γ0𝑗 (ℎ, ℎ′ ) 𝛿𝑗𝑎 𝜆𝑗𝑎 𝑢(𝑏𝑗,𝑎 ) + 𝑢 ˜ − 𝜀)𝑊
^ℎ + 𝛽 𝛿[(1 ˜ 𝑏 (𝑗, 𝑎, ℎ′ ) + 𝜀𝑊
˜ 𝑛 (𝑗, 𝑎, ℎ′ )] (1)
ℎ′ ⏟ ⏞
loses job on temporary layoff, has benefits
}︂
˜ 𝑏 ′ 𝑛 ′
+ 𝛽(1 − 𝛿)[(1 − 𝜀)𝑊 (𝑗, 𝑎, ℎ ) + 𝜀𝑊 (𝑗, 𝑎, ℎ )]
⏟ ⏞
loses job permanently separated, has benefits
∑︁
+ ^ℎ + 𝛽 𝛿˜𝑊
Γ0𝑗 (ℎ, ℎ′ ) 𝛿𝑗𝑎 (1 − 𝜆𝑗𝑎 )[𝑢(𝑐) + 𝑢 ˜ 𝑛 (𝑗, 𝑎, ℎ′ )]
˜ 𝑛 (𝑗, 𝑎, ℎ′ ) + 𝛽(1 − 𝛿)𝑊
ℎ′
⏟ ⏞
loses job (on temp or perm layoff), no benefits
∑︁
+ Γ1𝑗 (ℎ, ℎ′ ) (1 − 𝛿𝑗𝑎 )[𝑢(𝑤𝑗 𝑎) + 𝑢
^ℎ + 𝛽𝑊 𝑒 (𝑗, 𝑎, ℎ′ )] .
′
⏟ ⏞
ℎ
keeps job
We assume that if a type M worker becomes type I, she automatically becomes temporar-
ily laid-off with UI benefits.12 Hence, the probability of becoming type I affects the value of
working for a type M worker.
Workers on temporary layoff and workers on permanent separation have similar value
functions. For brevity we only show the value function for the worker on temporary layoff
with UI and include the other value functions in Appendix B.3. Let 𝑥˜ be the search effort of
a worker on temporary layoff and let 𝑣(˜𝑥) be the disutility of search. The value function for a
worker (𝑗, 𝑎, ℎ) on temporary layoff with UI, and health ℎ ∈ {𝑆, 𝑀, 𝑅} is given by:
∑︁
˜ 𝑏 (𝑗, 𝑎, ℎ) = 𝑟 Γ1𝑗 (ℎ, ℎ′ ) 𝑢(𝑤𝑗 𝑎) + 𝑢
^ℎ + 𝛽𝑊 𝑒 (𝑗, 𝑎, ℎ′ )
[︀ ]︀
𝑊 (2)
ℎ′
⏟ ⏞
recalled to job
{︂ ∑︁
Γ1𝑗 (ℎ, ℎ′ ) 𝑥 ^ℎ + 𝛽𝑊 𝑒 (𝑗, 𝑎, ℎ′ )
[︀ ]︀
+(1 − 𝑟) max −𝑣(˜
𝑥) + ˜𝑓 (𝜃𝑗𝑎 ) 𝑢(𝑤𝑗 𝑎) + 𝑢
𝑥
˜
ℎ′
⏟ ⏞
no recall, finds job through search
∑︁ (︂
′
+ 0
Γ𝑗 (ℎ, ℎ ) (1 − 𝑥 ^ℎ + 𝛽𝜁[(1 − 𝜀)𝑊 𝑏 (𝑗, 𝑎, ℎ′ ) + 𝜀𝑊 𝑛 (𝑗, 𝑎, ℎ′ )]
˜𝑓 (𝜃𝑗𝑎 )) 𝑢(𝑏𝑗,𝑎 ) + 𝑢
ℎ′ ⏟ ⏞
does not find job, temp layoff expires
)︂}︂
˜𝑏 ′
+ 𝛽(1 − 𝜁)[(1 − 𝜀)𝑊 (𝑗, 𝑎, ℎ ) + 𝜀𝑊 (𝑗, 𝑎, ℎ )] ,˜𝑛 ′
⏟ ⏞
does not find job, stays on temp layoff
From the above, setting 𝜀 = 1, we get the value function for a worker on temporary layoff
12
Because type I workers are automatically separated for health reasons, 𝑊 𝑒 (𝑗, 𝑎, 𝐼) = 𝑊 ˜ 𝑏 (𝑗, 𝑎, 𝐼). Appendix
B.3.1 gives type I worker’s values functions. In reality, some of the type I workers may be eligible for sick leave
benefits and hence do not need to be separated and matched again. However, according to the Bureau of Labor
Statistics (BLS), the average length of sick leave is only 8 days per year, substantially shorter than the 18-day du-
ration of the stage I in the model. In addition, sick leaves are also less prevalent in contact-intensive industries, for
example, in the food preparation and serving occupations (25%) and accommodation and food services (27%),
as documented by Maclean (2020). Since the effects of CARES UI on unemployment and infection mainly work
though the contact sector, the access to sick leave should have limited effect on our model results.
10
without UI (𝑊˜ 𝑛 ); setting 𝑟 = 0 and 𝜁 = 1 (no recall option), we get the value function for a
worker on permanent separation with UI (𝑊 𝑏 ); and setting 𝜖 = 1, 𝑟 = 0 and 𝜁 = 1, we get the
value function for the worker on permanent separation without UI (𝑊 𝑛 ).
The Search Channel. From equation (2) the search effort 𝑥˜𝑏 (𝑗, 𝑎, ℎ) ≥ 0 of a worker on tem-
porary layoff with UI is given by:13
𝑥𝑏 (𝑗, 𝑎, ℎ))
𝑣𝑥 (˜ ∑︁
= 𝑢(𝑤𝑗 𝑎) − 𝑢(𝑏𝑗,𝑎 ) + 𝛽 Γ1𝑗 (ℎ, ℎ′ )𝑊 𝑒 (𝑗, 𝑎, ℎ′ ) (3)
𝑓 (𝜃𝑗𝑎 )
ℎ′
∑︁ [︁ (︁ )︁]︁
−𝛽(1 − 𝜁) Γ0𝑗 (ℎ, ℎ′ ) ˜ 𝑏 ′ ˜ 𝑏 ′ ˜ 𝑛 ′
𝑊 (𝑗, 𝑎, ℎ ) − 𝜀 𝑊 (𝑗, 𝑎, ℎ ) − 𝑊 (𝑗, 𝑎, ℎ )
ℎ′
∑︁
Γ0𝑗 (ℎ, ℎ′ ) 𝑊 𝑏 (𝑗, 𝑎, ℎ′ ) − 𝜀 𝑊 𝑏 (𝑗, 𝑎, ℎ′ ) − 𝑊 𝑛 (𝑗, 𝑎, ℎ′ ) .
[︀ (︀ )︀]︀
−𝛽𝜁
ℎ′
The left-hand side is the marginal cost of search, and the right-hand side is the marginal benefit
of search, where 𝑥˜𝑏 (𝑗, 𝑎, ℎ) = 0 if RHS < 0. A higher benefit level 𝑏𝑗,𝑎 or a longer UI duration
(lower 𝜀) reduces the marginal benefit of search, assuming 𝑊 ˜𝑏>𝑊 ˜ 𝑛 and 𝑊 𝑏 > 𝑊 𝑛 which are
the case here. An eligibility expansion (larger 𝜆𝑗𝑎 ) does not directly affect individual search.
But because it increases the number of UI recipients, the expansion reduces the aggregate
search effort in submarket (𝑗, 𝑎), 𝑋𝑗𝑎 . Infection risk and shutdown policy both lower search
effort by lowering the continuation value of employment 𝑊 𝑒 : A type M worker faces the health
risk of becoming type I and thus unable to work, which reduces 𝑊 𝑒 and in turn lowers search
effort; shutdown policy increases job separation rate in the contact sector, which also reduces
𝑊 𝑒 for workers in that sector and lowers search effort.
From equation (3), setting 𝜀 = 1 gives the search effort of a worker on temporary layoff
without UI (𝑥˜𝑛 (𝑗, 𝑎, ℎ)), setting 𝜁 = 1 gives the search effort of a worker on permanent sepa-
ration with UI (𝑥𝑏 (𝑗, 𝑎, ℎ)), and setting 𝜁 = 1 and 𝜀 = 1 gives the search effort of a worker on
permanent separation without UI (𝑥𝑛 (𝑗, 𝑎, ℎ)). In particular, 𝑥𝑏 (𝑗, 𝑎, ℎ) is characterized by:
𝑣𝑥 (𝑥𝑏 (𝑗, 𝑎, ℎ)) ∑︁
= 𝑢(𝑤𝑗 𝑎) − 𝑢(𝑏𝑗,𝑎 ) + 𝛽 Γ1𝑗 (ℎ, ℎ′ )𝑊 𝑒 (𝑗, 𝑎, ℎ′ ) (4)
𝑓 (𝜃𝑗𝑎 )
ℎ′
∑︁
Γ0𝑗 (ℎ, ℎ′ ) 𝑊 𝑏 (𝑗, 𝑎, ℎ′ ) − 𝜀 𝑊 𝑏 (𝑗, 𝑎, ℎ′ ) − 𝑊 𝑛 (𝑗, 𝑎, ℎ′ ) .
[︀ (︀ )︀]︀
−𝛽𝜁
ℎ′
Because workers on temporary layoff may be recalled back to work without searching, their
value of staying unemployed is higher than their counterparts on permanent separation, i.e.
𝑊˜ 𝑏 > 𝑊 𝑏 and 𝑊˜ 𝑛 > 𝑊 𝑛 . As a result, equations (3) and (4) imply that workers on temporary
layoff search less, i.e. 𝑥˜𝑏 (𝑗, 𝑎, ℎ) < 𝑥𝑏 (𝑗, 𝑎, ℎ) and 𝑥˜𝑛 (𝑗, 𝑎, ℎ) < 𝑥𝑛 (𝑗, 𝑎, ℎ).
13
We assume the search disutility 𝑣(·) is increasing and convex. So the marginal disutility 𝑣𝑥 (·) is positive and
increasing.
11
2.3. Firm’s Problem
A producing firm in sector 𝑗 and efficiency submarket 𝑎 with a worker of health ℎ ∈ {𝑆, 𝑀, 𝑅}
will keep operating if the match is not destroyed. The value function is:
∑︁
Γ1𝑗 (ℎ, ℎ′ )𝐽(𝑗, 𝑎, ℎ′ )
[︀ ]︀
𝐽(𝑗, 𝑎, ℎ) = (1 − 𝛿𝑗𝑎 ) (𝑧𝑗 − 𝑤𝑗 )𝑎 + 𝛽 (5)
ℎ ′
{︂ ∑︁ }︂
+𝛿𝑗𝑎 𝛽 𝛿˜ 0 ′ ˜ 𝑏 ′ ˜ 𝑛 ′ ˜
Γ𝑗 (ℎ, ℎ ) [(1 − 𝜀)𝑉 (𝑗, 𝑎, ℎ ) + 𝜀𝑉 (𝑗, 𝑎, ℎ )] +(1 − 𝛿)𝑉 (𝑗, 𝑎) ,
′
⏟ ⏞
ℎ
whether or not temp layoff has UI
where if the match separates, with probability 𝛿˜ the firm has a recall option (i.e. the worker is
on temporary layoff). If a worker becomes type I at the end of a period, the match is automat-
ically dissolved.14 This implies that the firm’s value depends on the worker’s health status:
everything else equal, the firm’s value is the highest with a type R worker since she is immune
to the disease, and is the lowest with a type M worker since she may become type I in the next
period.
The Recall Option. 𝑉 is the value of a permanent vacancy, and as in the standard search
and matching model, free entry condition applies here and 𝑉 (𝑗, 𝑎) = 0. 𝑉˜ 𝑏 and 𝑉˜ 𝑛 are the
values of a temporary vacancy where the worker on temporary layoff has or does not have UI
benefit, respectively. We assume that the firm keeps track of the worker’s labor market and
health status while she is on temporary layoff. Note that the value of a temporary vacancy is
not zero, because the firm can recall the worker back and does not post a vacancy, and so free
entry condition does not apply. If the firm does not recall and the worker finds a new job or the
recall option expires at the end of the period, then the temporary vacancy becomes permanent
(𝑉 ). The value function of a vacancy with recall where the worker on temporary layoff has UI
and health ℎ ∈ {𝑆, 𝑀, 𝑅} is given by:
∑︁
𝑉˜ 𝑏 (𝑗, 𝑎, ℎ) = 𝑟 (𝑧𝑗 − 𝑤𝑗 )𝑎 + 𝛽 Γ1𝑗 (ℎ, ℎ′ )𝐽(𝑗, 𝑎, ℎ′ )
[︀ ]︀
(6)
ℎ′
⏟ ⏞
recalls worker
∑︁
+ (1 − 𝑟)𝛽(1 − 𝑥˜𝑏 (𝑗, 𝑎, ℎ)𝑓 (𝜃𝑗𝑎 ))(1 − 𝜁) Γ0𝑗 (ℎ, ℎ′ ) (1 − 𝜀)𝑉˜ 𝑏 (𝑗, 𝑎, ℎ′ ) + 𝜀𝑉˜ 𝑛 (𝑗, 𝑎, ℎ′ )
[︀ ]︀
ℎ′
⏟ ⏞
does not recall worker, worker does not find new job and recall does not expire
𝑥𝑏 (𝑗, 𝑎, ℎ)𝑓 (𝜃𝑗𝑎 ) + (1 − 𝑥˜𝑏 (𝑗, 𝑎, ℎ)𝑓 (𝜃𝑗𝑎 ))𝜁]𝑉 (𝑗, 𝑎)
+ (1 − 𝑟)𝛽[˜
⏟ ⏞
does not recall worker, worker finds new job or recall expires
From the above equation, setting 𝜀 = 1 gives the values function of a vacancy with recall where
the worker does not have UI. We include the other firm value functions in Appendix B.4.
14
we assume that the match is permanently destroyed in this case: 𝐽(𝑗, 𝑎, ℎ′ = 𝐼) = 𝑉 (𝑗, 𝑎).
12
The Vacancy-Posting Channel. Firms with a permanent vacancy can post vacancies to hire
workers. Because of free entry condition, the value of posting a vacancy is 0:
∑︁ ∑︁
𝑑ℎ𝑗𝑎 (𝑧𝑗 − 𝑤𝑗 )𝑎 + 𝛽 Γ1𝑗 (ℎ, ℎ′ )𝐽(𝑗, 𝑎, ℎ′ ) , (7)
[︀ ]︀
𝑉 (𝑗, 𝑎) = 0 = −𝜅𝑧𝑗 𝑎 + 𝑞(𝜃𝑗𝑎 )
ℎ∈{𝑆,𝑀,𝑅} ℎ′
where 𝑑ℎ𝑗𝑎 is the probability that a firm in sector 𝑗 and submarket 𝑎 meets an unemployed
worker with health status ℎ for ℎ ∈ {𝑆, 𝑀, 𝑅}.15 We assume that a firm’s hiring policy cannot
discriminate workers by health status. Because a firm’s value is the lowest when the worker
is type M, when infection risk is high and 𝑑𝑀𝑗𝑎 is large, a firm is less willing to post vacancies.
Shutdown policy exogenously increases the job separation rate in the contact sector, which
reduces the contact sector firm’s continuation value 𝐽 and lowers vacancy posting. The CARES
Act UI policy also lowers vacancy posting but through the reduction in the aggregate search
effort.
2.4. Health and Labor Market Transitions
Within each period, recall, search, job posting, and separation happen at the beginning of the
period; the expiration of temporary layoff and the transitions in UI and health status take place
at the end of the period. We include all transition equations in Appendix B.5.
Labor Market and UI Status Transitions. Labor market transitions at the beginning of a pe-
riod are standard: Some employed workers exogenously separate from jobs; some workers
on temporary layoff are recalled; the unemployed (those on temporary layoff but not recalled
and those on permanently separated) search and some find jobs; newly unemployed receive
UI benefits with probability 𝜆𝑗𝑎 . At the end of the period, some temporary layoffs expire if the
worker did not find a job during the period, and she becomes permanently separated without
a recall option. Some of the unemployed workers with UI benefits lose their benefits and can
only regain benefit status through employment.
Health Transitions. The health transitions for the non-working groups (YOLF and Old) are
straightforward: next period’s measure of people with health ℎ is equal to today’s type ℎ less
outflows to other health types and plus inflows from other types. For example, for the group of
type M agents, the outflow consists of those who become type I or R, and the inflow consists
of the type S who are newly infected. The infection rate depends on the total measure of
infectious people in the population which includes both types M and I. Let this be Ω, then the
probability that a type S gets infected is Inf = 𝜌Ω, where 𝜌 is the per-contact infection rate in
the general population. Once a person is infected with the virus, the health transition rates are
𝑔 𝑔 𝑔
exogenous and potentially age-dependent: 𝜎𝑀 𝐼 (type M to I), 𝜎𝑀 𝑅 (type M to R), 𝜎𝐼𝑅 (type I
The probability 𝑑ℎ𝑗𝑎 is given by the search intensity-weighted fraction of the measure of type ℎ unemployed
15
workers among all unemployed workers in the (𝑗, 𝑎) submarket. Appendix B.4 gives the equation for 𝑑ℎ𝑗𝑎 .
13
𝑔
to R), 𝜎𝐼𝐷 (type I to D), where 𝑔 ∈ {𝑦, 𝑜} with 𝑦 denoting young people (workers and YOLF)
and 𝑜 denoting the Old. The assumption of age-dependency is consistent with the fact that
older agents face potentially higher risk of dying from the infection.
Infection rates for young workers depend additionally on the worker’s employment status
in the period and her sector. In particular, a worker currently employed in the contact sector
faces additional infection risk at the workplace. Let 𝜌𝑒 be the per-contact infection rate at work-
place, and Ω𝑐𝑜𝑛,𝑒 be the measure of infectious population employed in the contact sector. The
infection probability for a type S worker working in the contact sector is Inf𝑐𝑜𝑛 = 𝜌𝑒 Ω𝑐𝑜𝑛,𝑒 + 𝜌Ω.
The infection probability for all other young workers, including workers employed in the non-
contact sector, unemployed workers (both on temporary layoff and permanently separated)
are the same and is only Inf = 𝜌Ω.
Infection and progression probabilities together define the Γ transition matrices. Social
distancing is modelled as a reduction in 𝜌 and 𝜌𝑒 . The details are described in the calibration
of the health variables. Shutdown and the UI policy both reduce employment in the contact
sector and hence reduce Ω𝑐𝑜𝑛,𝑒 and new workplace infections in the contact sector. Lower
workplace infection in turn reduces future Ω and infections out of workplace.
2.5. Equilibrium
Definition 1. (Stationary Equilibrium in Health and Labor Market) Given UI policy variables
{𝑏𝑗,𝑎 , 𝜆𝑗𝑎 , 𝜀}, shutdown policy 𝑚, sector wage rates 𝑤𝑗 , and initial distribution 𝜇0 , a stationary
equilibrium is: (1) All value functions and transitions are defined as above; (2) Search levels
𝑥𝑏 (𝑗, 𝑎, ℎ), 𝑥𝑛 (𝑗, 𝑎, ℎ), 𝑥˜𝑏 (𝑗, 𝑎, ℎ), and 𝑥˜𝑛 (𝑗, 𝑎, ℎ) solve unemployed workers’ problems; (3) Mar-
ket tightness 𝜃𝑗𝑎 is consistent with firm’s free entry condition in every submarket, with 𝑓 (𝜃𝑗𝑎 )
and 𝑞(𝜃𝑗𝑎 ) determined by the matching function; (4) Stationary distribution is consistent with
workers’ and firms’ optimal decisions, equilibrium infection rates, and exogenous health and
labor market transitions; and (5) Government balances its budget.16
3. Calibration
We first calibrate an initial steady state without infection and health to the U.S. economy be-
fore the COVID-19 pandemic (averages of 2015–2019). We then calibrate the health transition
processes and the paths of UI and shutdown policies over the transitional periods.
16
Government’s budget is balanced per period in the pre-pandemic steady state. During the pandemic tran-
sition, any excess spending due to higher unemployment or discretionary policy changes are paid back with
interest in the post-pandemic steady state. In other words, the government rolls over debt during the transition
and the budget is balanced in present value.
14
Population. One period in the model is one week. We use a mortality-adjusted annual interest
rate of 4% for young agents, which gives 𝛽 = 0.961/52 . For welfare calculations, following
Glover et al. (2020) we assume a different discount rate for the Old to account for different
expected remaining life span: 𝛽𝑜 = 0.91/52 . We link young agents in the model to individuals
aged 16–64 in the Current Population Survey (CPS). This implies 81% of the population are
young; among the young, 73% are in the labor force.
Functions. We use log utility. Following Den Haan et al. (2000), we set the matching function
𝑉 𝑥1+𝜓
to 𝑀 (𝑋, 𝑉 ) = [1+(𝑉 /𝑋)𝜒 ]1/𝜒 , where 𝜒 differs by sector. The search cost function is 𝑣(𝑥) = 𝜈 1+𝜓 ,
where 𝜈 is set to 2.17 𝜓 determines how search responds to changes in UI and health. We set
𝜓 = 1.2, which implies an average micro-elasticity of unemployment duration with respect to
benefit level of 0.395 in the initial steady state. This value falls within the range of estimates
in the literature, which is from 0.3 to 0.9 (see, for example, Meyer 1990). Our model-implied
value of 0.395 is on the low end of the estimates, which means that the effect of UI on unem-
ployment and infection through search is relatively small in the model.
Classification of Sectors. Dingel and Neiman (2020) rank all 2-digit industries by the share
of workers who cannot perform their work at home. They find that 63% of all jobs in the U.S.
cannot be performed at home. We divide the 2-digit industries into contact and non-contact
sectors following their ranking, so that the contact sector consists of industries with a higher
share of workers who cannot work at home. The resulting employment share in the contact sec-
tor is 64%.18 Table A.1 in the Appendix reports the detailed industry-sector assignment. The
contact sector includes, for example, accommodation and food services, retail, transportation,
and healthcare; the non-contact sector includes utilities, federal and local government, and
finance, among others. Given the division of sectors, the distribution of efficiency units 𝐹𝑗 (𝑎)
is constructed using the sector wage distribution from the CPS and normalizing the mean to
one.19
3.1. Calibration of Initial Steady State
Exogenous Steady State Parameters. We normalize the non-contact sector productivity 𝑧𝑛𝑐
to 1. Following Hagedorn and Manovskii (2008), we set the ratio of vacancy posting cost to
submarket productivity to 0.584, which gives the value for 𝜅. We use the CPS data to construct
𝑒𝑥
the exogenous separation rate 𝛿𝑗𝑎 by income and by sector . Figure A.2 in the Appendix A.1
17
Because 𝜈 is a scale parameter, setting 𝜈 to another value and re-calibrating the model should have little effect
on the model dynamics.
18
Bick et al. (2020) find that 35.2% of workers worked from home in May 2020. Our classification implies a
value of 36%.
19
Data Appendix A.1 provides details on the construction of 𝐹𝑗 (𝑎) and shows the constructed distributions.
15
presents the results. The separation rate declines by income in both sectors and on average
separation rates in the contact sector are slightly higher than in the contact sector. Following
Birinci et al. (2021), we set the probability that temporary layoff expires each period, 𝜁 , to be
1/26. Using the Survey of Income and Program Participation (SIPP), Fujita and Moscarini
(2017) document the probabilities for workers on temporary layoff to be recalled. We take the
reported average and set the weekly probability of recall 𝑟 = 0.055.
Steady State UI Policy. The weekly UI benefit is given by the function
𝑏𝑗,𝑎 = min{𝜂 · 𝑤𝑗 𝑎, 𝑏𝑢𝑏 } + 𝑏𝑡𝑜𝑝 . (8)
𝜂 is the policy replacement rate and set to 𝜂 = 0.5 following state UI laws.20 𝑏𝑢𝑏 is the upper
bound on weekly UI payment, which is part of all states’ UI policy and is calibrated jointly
with other parameters. 𝑏𝑡𝑜𝑝 is the UI top-up as part of the CARES UI and is set to 0 in the initial
steady state. Modeling the upper bound allows the model to capture the lower replacement
rates at higher income levels.21 In normal times, UI benefits last for 26 weeks and thus we set
the UI expiration rate 𝜀 to 1/26 in the steady state.
We calibrate 𝜆𝑗𝑎 to match the UI recipient rate by income, where the recipient rate is defined
as the share of unemployed receiving UI. Using the Survey of Income and Program Participa-
tion (SIPP) Chao (2021) finds a hump-shape for the UI recipient rate by income percentile
before the pandemic. The shape is related to the UI eligibility and the willingness to file for
UI. During pre-pandemic times, the eligibility for UI requires workers to have a minimum
amount of earnings and to have a long-enough work history before unemployment. Low-
wage workers are more likely to have a short work history before unemployment because they
are more likely to be separated from their jobs as shown in Figure A.2. In addition, workers
with very high income are less likely to claim UI, because they can find jobs easily and filing
for UI incurs a time cost.
Because the UI recipient rate by income is hump-shaped, we use a quadratic function to
approximate it. First, we fit the data for the UI recipient rate by income percentile from Chao
(2021) with a quadratic function.22 Second, we assume that the probability of a newly unem-
20
In most states, weekly UI benefits for those who qualify for UI are computed using a formula. States utilize a
variety of methods, e.g. the “high-quarter method,” which is used by more than half of the states, uses a worker’s
wage in her highest-earning quarter, and apply multiple to get the weekly UI benefit. The most common multiple
is 1/26. Using this formula, weekly UI benefit = quarterly income*(1/26)≈ 0.5*weekly income. We thus use
𝜂 = 0.5. For more information on how UI benefit is computed in the U.S. please refer to Department of Labor’s
publication: https://oui.doleta.gov/unemploy/pdf/uilawcompar/2020/monetary.pdf.
21
Figure C.1 in the Appendix plots the UI benefit level in (8) for different wage income levels.
22
The quadratic function is given by:
UI Recipient Rate𝑖 = 𝛼2 log(RelInc𝑖 )2 + 𝛼1 log(RelInc𝑖 ) + 𝛼0
where “RelInc” is the ratio of the average income for the percentile 𝑖 and the economy-wide average income.
16
ployed worker receiving UI 𝜆𝑗𝑎 is also a quadratic function of income, given as follows:
^ 2 × log(RelInc𝑗𝑎 )2 + 𝜆
𝜆𝑗𝑎 = 𝜆 ^ 1 × log(RelInc𝑗𝑎 ) + 𝜆
^0, (9)
where RelInc𝑗𝑎 is a worker’s income relative to the mean income in the economy: RelInc𝑗𝑎 =
𝑤𝑗 𝑎/(average income of all workers). We calibrate the parameters in this function jointly with
other parameters in the model to target the quadratic coefficients from the empirical fit in the
first step. Because UI benefits may expire before a workers finds a job, and because workers
with and without UI search differently, the recipient rate differs from 𝜆. Figure C.2 in the
appendix plots the two together over relative wage.
Jointly Calibrated Steady State Parameters. Including the parameters in the function for 𝜆𝑗𝑎 ,
there are twelve steady state parameters left: 𝑧𝑐𝑜𝑛 , 𝑤𝑐𝑜𝑛 , 𝑤𝑛𝑐 , 𝜒𝑐𝑜𝑛 , 𝜒𝑛𝑐 , 𝛿˜, 𝜆
^2, 𝜆
^1, 𝜆
^ 0 , 𝑐, 𝑏𝑜 , and 𝑏𝑢𝑏 .
We calibrate them jointly to match the following twelve targets: (1) the contact sector’s share
of total value added; (2) economy-wide vacancy-unemployment ratio; (3) sector ratio of av-
erage income among employed workers; (4)–(5) sector unemployment rates; (6) proportion
of unemployed on temporary layoff; (7)-(9) empirical estimates of the quadratic relationship
between UI recipient rate and wage income; (10) the ratio of SNAP (Supplemental Nutrition
Assistance Program) income to average earned income; (11) the ratio of the average Social Se-
curity income to average wage income; and (12) the ratio of UI upper bound to average earned
income, averaged across states. The top panel of Table 1 reports the calibration results.23
Although these parameters are jointly calibrated, some affect certain moments more than
others. Intuitively, with 𝑧𝑛𝑐 normalized to 1, 𝑧𝑐𝑜𝑛 is used to match the sector share of value
added. The aggregate vacancy-unemployment ratio and sector income ratio of employed
workers together pin down sector wage rates 𝑤𝑐𝑜𝑛 and 𝑤𝑛𝑐 . Sectoral unemployment rates pin
down sector matching parameters 𝜒𝑐𝑜𝑛 and 𝜒𝑛𝑐 .24 The proportion of temporary laid-off work-
ers increases with the proportion of separation that is temporary 𝛿˜. The parameters 𝜆2 𝜆1
and 𝜆0 , which characterize the relationship between the probability that a newly unemployed
worker receives UI and her relative wage income in (9), directly affect the steady state UI recip-
ient rate by income. Finally, welfare income 𝑐 and the upper bound on UI 𝑏𝑢𝑏 are pinned down
using the ratio of the corresponding data moment to average earned income in the data. Be-
cause both parameters affect unemployed workers’ search choices and hence the steady state
average earned income, they need to be jointly calibrated with other parameters.
23
We use Zhang et al. (2010)’s derivative-free algorithm for least-squares minimization to perform joint cali-
bration.
24
The matching parameter 𝜒 affects the unemployment rate through affecting how efficient the matching pro-
cess is in each sector.
17
Table 1: Jointly calibrated parameters and moments
Parameter Meaning Parameter Target moment Target
value value
Economic parameters
𝑧𝑐𝑜𝑛 contact sector productivity 0.712 contact sector share of value added 0.560
𝑤𝑐𝑜𝑛 contact sector wage rate 0.690 aggregate vacancy-unemp ratio 0.926
𝑤𝑛𝑐 non-contact sector wage rate 0.979 sector income ratio of employed 0.708
𝜒𝑐𝑜𝑛 contact sector matching parame- 0.300 contact sector unemp rate 0.046
ter
𝜒𝑛𝑐 non-contact matching parameter 0.310 non-contact sector unemp rate 0.026
𝛿˜ prob. separation is temporary lay- 0.239 prop. of unemp on temp layoff 0.119
off
^2
𝜆 coef. in quadratic equation (9) -0.053 quadratic fit of UI recipient rate to rel- -0.059
ative income
^1
𝜆 same as above -0.05 same as above -0.034
^
𝜆0 same as above 0.161 same as above 0.183
𝑐 social welfare income 0.029 SNAP income / average earned in- 0.036
come
𝑏𝑜 income of the old 0.277 social security income / average 0.340
earned income
𝑏𝑢𝑏 UI benefit upper bound 0.445 UI upper bound / average earned in- 0.547
come
Health parameters
𝑦
𝜎𝐼𝐷 Young death rate from type I 0.25%*7/18 average death rate from COVID 0.6%
𝑜
𝜎𝐼𝐷 Old death rate from type I 5%*7/18 Old’s share of cum. deaths as of April 75%
4
𝜌 per-contact base infection rate 0.88 cumulative deaths as of April 4, 2020 13.6k
𝜌𝑒 per-contact infection rate at work 2.93 workplace infection/total infection 16%
1−𝛾 % fall in (𝜌, 𝜌𝑒 ) from social dis- 0.49 cumulative deaths as of June 27, 2020 120k
tancing
Note: All steady state moments are averages of 2015–2019 values. Appendix A.1 provides details on the data source and
construction of key moments.
3.2. Calibration of Infection Process and Transition Path
Health Transition Parameters. We simulate the pandemic from February 2, 2020. In the first
period, we assume that 0.02% of the population is type M and they are evenly distributed
among workers, Old and YOLF. As a robustness check, in Section 4.6 we assume alternative
values for the initial measure of type M and find similar results. Following the epidemiology
18
literature and the literature on COVID-19 models, we assume an average duration of one week
𝑔 𝑔
and 18 days spent in stage M and I, respectively, for all ages. This implies 𝜎𝑀 𝑅 + 𝜎𝑀 𝐼 = 1 and
𝑔 𝑔
𝜎𝐼𝑅 + 𝜎𝐼𝐷 = 7/18 for 𝑔 ∈ {𝑦, 𝑜}.25 In the baseline we assume that for all ages, half of type M
𝑔 𝑔
progress to type I and half to type R. This implies 𝜎𝑀 𝐼 = 0.5 and 𝜎𝑀 𝑅 = 0.5. As a robustness
check, in Section 4.6 we use a lower transition probability from type M to type I to reflect the
possible presence of many untested cases with mild or no symptom in the population. Results
are consistent with the baseline.
𝑦 𝑜
That leaves four independent parameters for the virus transmission: 𝜎𝐼𝐷 , 𝜎𝐼𝐷 , 𝜌 and 𝜌𝑒 .
Additionally, to capture the reduction in infection from voluntary reduction in social activities
(e.g. mask wearing, keeping social distance), we follow Glover et al. (2020) and assume that
after March 14, 𝜌 and 𝜌𝑒 are reduced proportionally by a fraction 1 − 𝛾 .26 We jointly calibrate
the five health parameters to match the following targets: (1) the population average uncondi-
tional death rate from the virus, which we take to be the mean value among the epidemiology
estimates surveyed by Meyerowitz-Katz and Merone (2020); (2) the cumulative deaths as of
April 4;27 (3) the cumulative deaths among people aged 65+ as a fraction of the total cumula-
tive deaths as of April 4; (4) the share of all infections that happen in the workplace, which we
take to be the median value of the workplace infection share of flu estimated in the influenza
literature;28 and (5) the cumulative deaths as of June 27. (1) and (3) help pin down the uncon-
𝑦 𝑜
ditional death rates by age group and thus 𝜎𝐼𝐷 and 𝜎𝐼𝐷 ; (2) and (4) pin down the per-contact
infection rates 𝜌 and 𝜌𝑒 ; given other policies, (5) pins down the effect of social distancing and
hence 𝛾 .
The calibration generates higher unconditional death rate for old (2.5%) than for young
(0.125%). The 𝑅0 statistic is 2.41 without social distancing and 1.23 with social distancing,
both values are within the range of estimates in the literature.29 The bottom panel of Table 1
reports the calibrated health parameters and moments.
Health Utility. In the benchmark calibration, we set 𝑢^𝑀 = 0 to reflect that type M only have
mild symptoms, set 𝑢^𝐼 to be 30% of a worker’s average utility, and derive 𝑢^𝐷 following the
25
One week is one period in the model and thus a type M worker will transit out of stage M for sure after one
period. Similarly, a duration of 18 days is 18/7 periods in the model which implies a probability of 7/18 for a type
I worker to transit out of stage I.
26
We choose the week of March 14 as the first period for social distancing because 11 states issued guidance
on recommended limitation on the size of gathering between March 12 and March 18.
27
We choose April 4 to capture all deaths due to the infection before shutdown. We use the deaths numbers
reported by the CDC.
28
Edwards et al. (2016) review the influenza literature and find that workplace infection accounts for 9–33% of
the total infection with a median of 16%. We choose the median as target. A larger number increases the effect
of shutdown and UI policy on infection, as both policies work by reducing workplace infection.
29
𝑅0 is a statistic widely used in the epidemiology literature to determine the severity of an epidemic. Appendix
A.2 provides more details on the calculation of 𝑅0 in our model.
19
value of statistical life (VSL) approach. As the death probability is small and the disease is
short-lived, the utility costs of sickness and death almost have no effect on the simulated tran-
sition path. Hence the utility costs can be computed directly from the workers’ utility values.
This results in 𝑢^𝐼 = −0.1 and 𝑢^𝐷 = −10, values that are close to Glover et al. (2020)’s flow
value of life. However, the utility costs do matter for the welfare calculations, and we explore
alternative values in the welfare analysis.
CARES UI Policy. We closely follow the provisions in the CARES Act to set the UI policy
along the transition path. All policy components take effect on March 29. The UI expiration
probability 𝜀 is set to 1/39 to capture the 13-week UI duration extension, and is set back to 1/26
at the end of 2020 when the policy is scheduled to expire. The increase in the weekly payment
of $600 is captured by 𝑏𝑡𝑜𝑝 in the UI benefit formula (8), and is set to 0.57 after normalizing by
the non-contact sector wage rate. This policy is set to expire at the end of July 2020.
The eligibility expansion is captured by an increase in the probability that newly unem-
ployed workers receive UI (𝜆𝑗𝑎 ). Since the policy expands UI coverage to almost all active
workers, we assume that all workers have the same 𝜆(𝑗, 𝑎) under the eligibility expansion. It
is calibrated to match the UI recipient rates during March–December 2020, which are com-
puted using number of UI weeks paid and imputed unemployment population as discussed
in appendix A.1. Because the recipient rates during the pandemic are computed based on the
total weeks of UI paid, which also include the weeks paid to partly employed workers and
workers out of the labor force (Department of Labor 2020 and Forsythe 2021), there could be
upward bias. In fact, the computed recipient rates are above one for some periods. Hence,
we apply a 30% shrinkage to the computed rates, as suggested by Forsythe (2021). To capture
the gradual increase in the UI recipient rate from March to May 2020 and the gradual decline
during November–December 2020, we allow 𝜆𝑗𝑎 to phase in and phase out. Figure C.3 in the
appendix shows the calibrated path of 𝜆𝑗𝑎 .
Shutdown Policy. We calibrate the maximum value of shutdown policy 𝑚𝑡 to exactly match
the level and timing of peak unemployment rate during the transition, and discipline the rise
and fall of 𝑚𝑡 around the peak using the general path of rise and fall of the unemployment
rate from April to July 2020. We use the unemployment rates reported by Bick and Blandin
(2020), which peak at 21% in mid-May. Bick and Blandin conduct their own survey and report
biweekly unemployment rates based on the survey. It has two advantages over the CPS. First,
its biweekly frequency gives us observations within a month. Second, the survey does not
suffer from the misclassification issue of the CPS. The CPS classifies all workers who are “em-
ployed but absent from work due to other reasons” as employed, even though a large number
20
of these workers should instead be classified as unemployed during the pandemic.30 This cal-
ibration yields a path of 𝑚𝑡 that sharply increases from March 21 to its peak level on March
29, 2020, and falls to 20% of the peak level in mid-May and to 0 in early July.31
Increase in Temporary Layoff. In addition to shutdown, temporary layoffs during the pan-
demic can also be caused by non-policy related reasons, such as workers taking unpaid leave,
employers choosing to furlough workers as demand is low. We allow 𝛿˜ to increase during the
pandemic to capture these additional increases in temporary layoffs. Over the transition, 𝛿˜𝑡 is
calibrated so that the temporary layoff-to-total unemployment ratio matches data computed
from the CPS. This gives an increase in 𝛿˜ from 0.227 in the steady state to 1 at the peak and
stays there until October 2021.32
Government Budget over Transition. We use a “pandemic tax” to pay for the increases in
deficit due to higher unemployment and the discretionary CARES UI policy. In the bench-
mark, this tax is levied proportionally on all income over 10 years after the economy has
reached the post-pandemic steady state. In other words, we allow the government to carry
debt during the pandemic and repay afterwards.
3.3. Model Fit
This subsection checks the fitness of the model calibration. Figure 1 compares the model-
generated UI recipient rate over relative wage income to the data before the pandemic. The
figure shows that the UI recipient rate in the model and the data match well. This implies that
a quadratic relationship between 𝜆𝑗𝑎 and relative income approximates the data well.
Next, we check moment fits over the simulated transition path in the following dimensions:
UI recipient rate, unemployment rate, temporary-unemployment ratio, job separation rate,
and the UI replacement rate. The first three are used to pin down the UI and shutdown policies,
and the non-policy related increase in temporary separation on the transition path. The rest
are untargeted in the calibration. The top left panel of Figure 2 shows that the model-simulated
path of UI recipient rate matches the data well between March and December 2020. The top
right panel shows that the rise and fall in unemployment rates generated by the model are
broadly consistent with the data.33 The bottom left panel shows that the model-simulated
30
The misclassification issue of the CPS unemployment is relatively small in normal times, but could increase
unemployment rate by 5 ppt as acknowledged in the April 2020 BLS Employment Situation report. Adding this
5 ppt to the April official unemployment gives 19.7%, close to the number reported by Bick and Blandin (2020).
31
Appendix C.1 shows the calibrated shutdown time series.
32
Appendix C.1 shows the calibrated time series for 𝛿˜.
33
As data is noisy, we mainly choose the shutdown policy 𝑚𝑡 to target the rapid rise, the peak level, and the
magnitude of the decline in the unemployment rate between April and July.
21
Figure 1: Steady state UI recipient rate: Model vs Data
25
Data
Data quadratic fit
20
Model
15
%
10
5
0
0 0.5 1 1.5 2 2.5 3 3.5
Relative income
Note: Plot shows the UI recipient rate in the data and generated by the model, defined as the proportion of
unemployed workers who are receiving UI benefits, over relative wages. Relative wages are computed relative
to economy-wide average wage. Please refer to the appendix A.1 for construction of data series.
Figure 2: Aggregate labor market statistics over the transition: Model vs Data
UI recipient rate Unemployment rate
100 25
Model
80 Data
20
60
% % 15
40
10
20
5
0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
Temporary-unemployment ratio Job-separation rate
0.8 0.08
0.6 0.06
0.4 0.04
0.2 0.02
0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
Note: Please refer to the appendix A.1 for construction of data series.
path of temporary-unemployment ratio tracks the data well. Similarly, the bottom right panel
shows that the aggregate job separation rate in the model, computed as a weighted average
of sector-level job separation rates, matches the available data well. The separation rate is
not targeted over the transition path. Hence the good fit in this dimension serves as a useful
validity check on the labor market dynamics in the model and provides confidence in using
the model to evaluate the quantitative effects of polices.
Table 2 reports statistics on the UI replacement rate, computed based on the calibrated
UI formula (8) and the sector distribution of efficiency units. The $600 UI top-up increases
the average replacement rate from 0.45 pre-CARES to 1.67 post-CARES. The post-CARES UI
22
replacement rates in the model are consistent with those in the micro data as reported by
Ganong et al. (2020). In particular, in the model the median replacement rate is 1.38, 76% of
workers have replacement rates greater than one, and 20% of workers have replacement rates
greater than two. These values are close to the data counterparts.
Table 2: Comparing changes in UI replacement rates
Implied by our calibrated UI formula
Pre-CARES vs. Post-CARES Aggregate Contact Non-contact
Pre-CARES 0.45 0.46 0.42
Post-CARES 1.67 1.85 1.34
Post-CARES: Micro data Implied by our calibrated UI formula
Data vs. Model (Ganong, Noel, and Vavra 2020) Aggregate Contact Non-contact
Median replacement rate 1.34 1.38 1.59 1.10
Share with replacement rate ≥ 1 0.67 0.76 0.84 0.60
Share with replacement rate ≥ 2 0.33 0.20 0.27 0.09
Note: Statistics calculated based on entire wage distribution using the calibrated formula for weekly UI benefit
amount: UI = min{0.5 · wage income, 0.445}.
4. Results
In this section we first discuss the effects of CARES UI and shutdown policies on health and
labor market by sector and by income. The effect of CARES UI is measured as the difference
between the economy with both shutdown and CARES UI and the economy with shutdown
alone. We then decompose the effects of CARES UI into contributions by the three policy
components. Finally, we discuss the welfare implications of the CARES UI policy.
4.1. Policy Effects on Health and Unemployment
Figure 3 shows the evolution of health types as shares of the population. Absent any pol-
icy intervention, the virus spreads rapidly, and by the end of July 2020 new infections (type
M) would have reached its peak. By lowering employment in the contact sector, both the
shutdown and the CARES UI policy reduce the peak infection and shift the infection curves
rightwards (“flatten the curve”). In particular, the combination of shutdown and CARES UI
reduces the peak infection by 0.5 percentage points (ppt), while CARES UI alone reduces the
peak by 0.1 ppt.
Without any mitigation policies, 0.2% of the population (or about 62k lives) would have
23
Figure 3: Health dynamics over transition
Susceptible (type S) Infected Mild (type M) Infected Severe (type I)
90 2
1.5
80 1.5
% % 1 %
70 1
0.5
60 0.5
0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
Recovered (type R) Dead (type D)
0.25
40 0.2
30 0.15
% %
20 0.1
10 0.05
0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
no shutdown, no CARES UI shutdown only shutdown+CARES UI
died from the virus over the entire transition path.34 Out of that, 80% are old because of their
higher death rates from the infection. The combination of shutdown and CARES UI reduces
total cumulative deaths by 7.81% (about 48K lives saved) and the CARES UI alone reduce
total cumulative deaths by 2.09% (about 12K lives saved).35 Both shutdown and the UI policy
directly reduce workplace infections in the contact sector by reducing employment there, and
indirectly reduce infections for other groups by lowering the total infected population and
thus the infection probability. Since the effects on the contact sector are direct, the percent
reduction in deaths is also largest among workers in the contact sector (-10.7%) than for other
groups (-7.4%).36
While the mitigation policies reduce infection and save lives, they come with the cost of
sharp rises in unemployment. As shown in Figure 4 Panel (A), without mitigation policies,
unemployment peaks at 10%, driven by the heightened infection risk. As discussed in Section
2, when the infection risk is high, unemployed workers reduce search effort, and firms lower
vacancy posting because of the possibility of being matched to type M workers. Additionally,
an increase in type I workers raises unemployment mechanically since a matched type I worker
will automatically separate from her job and become unemployed.
34
The economy reaches steady state when enough people have acquired immunity such that new infection
reaches zero.
35
These results are summarized in Table C.1 in the appendix.
36
The calibrated voluntary social distancing parameter 𝛾 indicates that exogenous voluntary actions, such as
mask wearing, reduce per-contact infection rates by about 50%. This translates to a three-time higher total cu-
mulative deaths in the end steady state without voluntary social distancing, compared to the economy with
voluntary social distancing. This sizable effect is consistent with Farboodi et al. (2021)’s finding that voluntary
actions substantially reduce COVID-related deaths.
24
Figure 4: Unemployment dynamics over transition: Aggregate and by sector
(A) Unemployment rate (B) Temporary-unemployment ratio
20 1
0.8
15
0.6
%
10 0.4
0.2
5
0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
(C) Sector unemployment rate
contact non-contact
25 25
20 20
% 15 % 15
10 10
5 5
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
no shutdown, no CARES UI shutdown only shutdown+CARES UI
The mitigation policies further increase unemployment and shift the peak unemployment
earlier. Shutdown effectively increases the job separation rate in the contact sector, and so
the unemployment peak increases to 19.5% with shutdown. The additional CARES UI policy
further increases the peak to 21%. Overall, shutdown and the CARES UI policy together raise
the average unemployment by 5.36 ppt, and CARES UI alone by 1.61 ppt, out of a total increase
of 9 ppt during April to December 2020. The increases in unemployment are larger in the
contact sector, as shown in Panel (C) of Figure 4, because it has an extra infection risk, is
directly impacted by the shutdown policy, and has lower wages and so is more impacted by
the $600 top-up.37
Panel (B) of Figure 4 plots the transition path for the share of workers on temporary lay-
off among all unemployment. Without policy mitigation, the temporary-unemployment ratio
initially increases above the steady state level of 11.9% as a result of a higher probability of
temporary layoff 𝛿˜ on the transition path. As the infection risk increases, vacancy posting
and hence the job finding rate fall below the steady state levels. Unemployment duration in-
creases, and more workers on temporary layoffs transit to permanent layoffs, which leads to
37
Because the non-contact sector is not directly impacted by shutdown and shutdown helps reduce the overall
infection risk, unemployment in this sector is lower with shutdown.
25
Table 3: Effects of CARES UI on unemployment and deaths in different economies
Economy scenarios Effect on Apr–Dec 2020 Effect on Total
Avg Unemployment (ppt) Cumulative Deaths (%)
Without infection and shutdown 0.59 –
With infection only 0.82 -0.52
With infection and shutdown (baseline) 1.61 -2.09
Note: The rows report the effects of CARES UI in three different economies: the economy without COVID
infection and without shutdown; the economy with COVID infection but without shutdown policy; the econ-
omy with COVID infection and with shutdown policy. The increase in temporary layoff is present in scenarios
with infection. The effects are calculated as the difference between the transitions with and without CARES
UI. The policy effect is expressed in percentage points for average unemployment rate, and in percent terms for
cumulative deaths.
a decline in the temporary-unemployment ratio. But because the transition probability from
temporary to permanent layoffs is small (𝜁 = 1/26), the decline in temporary-unemployment
ratio is gradual. Over time, as the mildly infected population go down, vacancy posting grad-
ually rises, so does the temporary-unemployment ratio. At the end of 2020, as 𝛿˜ returns to its
steady state value, the temporary-unemployment ratio also gradually falls to the steady state
level. Since shutdown-related unemployment are all temporary layoffs, the shutdown policy
raises the temporary-unemployment ratio drastically and the policy effect gradually dies out
by the summer of 2020 when the shutdown policy ends. The effect of CARES UI policies on
the temporary-unemployment ratio is small because the policies apply to both temporary and
permanent layoffs.
Amplification. The quantitative effect of CARES UI depends on infection risk and shutdown.
Infection risk and shutdown policy both increase unemployment, and higher unemployment
translates to more unemployed workers who are claiming UI. Thus, infection and shutdown
amplify the effect of the CARES UI policy on employment and health. Table 3 reports the
effects of CARES UI in economies with and without infection and shutdown. The UI effect
is measured as the difference with and without CARES UI in each of the economic scenarios.
As Table 3 shows, in a world without COVID infection risk and shutdown policy, CARES UI
only increases the average unemployment rate by 0.59 ppt during April to December 2020.
Infection risk (without shutdown) increases the effect of CARES UI on unemployment to 0.82
ppt, with a reduction in total deaths of 0.52%. With both infection risk and shutdown, the
effect of CARES UI on unemployment further increases to 1.61 ppt, and the policy reduces
deaths by 2.09%.
Vacancy Posting and Search Channels. To better understand the interaction between health,
shutdown and the UI policy, we look at individual firm’s and unemployed worker’s decisions.
26
On the firm side, Figure 5 shows the vacancy-unemployment ratio for the submarket where
workers have median efficiency in each sector. Consistent with the discussion in Section 2.3,
without any policy intervention, vacancy posting is lower at the beginning of the pandemic
when firms expect that the share of type M workers would increase. As the shutdown policy
increases the separation rate in the contact sector, it lowers the value of filling a vacancy, and
vacancy posting in the sector falls with shutdown during its policy period. CARES UI policy
indirectly reduces vacancy posting in both sectors by lowering the aggregate search effort of
unemployed workers.
Figure 5: Vacancy-unemployment ratio for submarket with median efficiency
contact non-contact
3 3
2 2
1 1
0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
no shutdown, no CARES UI shutdown only shutdown+CARES UI
Note: Figure shows the vacancy-unemployment ratio in the submarket where workers have median efficiency
level for each sector.
Workers’ search decisions are an important margin in our model and are jointly affected
by infection, the shutdown policy, and the UI policy.38 Figure 6 shows the individual search
level of an unemployed worker with UI and median efficiency level by health, sector, and
whether the worker is on temporary or permanent layoff. Panel (A) compares the search of
unemployed workers in the contact sector by health types. Type M workers search much less
than type S or R workers, because they face the health risk of becoming type I and thus unable
to work, which reduces the value of finding a job today. Both shutdown and CARES UI policies
reduce the search incentives of all health types in the contact sector. Panel (B) compares the
search across sectors and between permanent and temporary unemployment. As shutdown
applies only to the contact sector, it significantly reduces vacancy posting in the contact sector,
and with fewer vacancies to search for, unemployed workers in the contact sector lower search
effort. CARES UI policies reduce the search incentives of workers significantly in both sectors,
since more generous UI policies increase the relative value of unemployment. Consistent with
the discussion in Section 2.2, Panel (B) also shows that workers on temporary layoff search
38
Given the search margin, no workers in the pre-pandemic steady state would choose to quit into permanent
separation or turn down a job offer. Over the transition, despite the increased generosity of UI benefits, still no
workers would choose to quit their jobs.
27
Figure 6: Search over transition for unemployed worker with median efficiency
(A) Search of unemployed worker in contact sector by health
Susceptible (type S) Infected Mild (type M) Recovered (type R)
1 1 1
0.8 0.8 0.8
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
0 0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
(B) Search of type R unemployed worker by sector and unemployment status
Contact sector, permanently separated Non-contact sector, permanently separated
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
Contact sector, temporary layoff Non-contact sector, temporary layoff
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
no shutdown, no CARES UI shutdown only shutdown+CARES UI
Note: Figure shows the search level of an unemployed worker with median efficiency level and with different
health status, unemployment status (permanently separately or on temporary layoff), and in different sectors.
much less than permanently separated workers, because the former may be recalled back to
their old jobs.
4.2. Heterogeneous Effects of CARES UI
The impacts of the CARES UI policy differ by sector and also by income. Figure 7 plots the
effect of CARES UI on unemployment rate over income, computed as the percentage-point
difference in unemployment rate between the scenarios with and without CARES UI (both
28
with shutdown). In the figure, income is normalized by the economy-wide mean wage income
and the effect on unemployment is the average over April–December 2020. The graph first
confirms that CARES UI leads to a larger increase in the unemployment rate in the contact
sector than in the non-contact sector.
Figure 7: Impact of CARES UI on average unemployment by income and sector
3
2
ppt
1
0
0 0.5 1 1.5 2 2.5 3 3.5
Relative income
∙ contact sector non-contact sector
Note: This figure shows the impact of CARES UI on average (Apr-Dec 2020) unemployment by income and
sector. Income is normalized by the economy-wide mean wage. The impact of CARES UI is calculated as the
difference between the economy with shutdown and CARES UI and the economy with shutdown only.
More importantly, the figure shows that the impact of CARES UI on unemployment is
hump-shaped over income. The effect is small for workers with very low income, rises as
the income increases, and peaks for workers with income right below the average income.
As income rises further, the effect starts to decline. The intuition is as follows. At very low
income levels, workers have very low efficiency units. As such, there are few vacancies for
these unemployed workers to search for, leading to low job finding rates and low returns to
search. The $600 top-up is a large proportion of (or even greater than) these workers’ wage
income; the eligibility expansion potentially increases the UI recipient rate the most among
these workers, because they were mostly ineligible for UI pre-pandemic. But because of the
low return to search, CARES UI has very small impacts on individual search in the equilibrium
and thus also small impacts on the unemployment rate among these workers. As income rises,
the return to search increases, and so does the impact of the CARES UI. But at even higher
income levels, because the $600 is a much smaller proportion of workers’ wage income, search
is less responsive to the UI policy change, and so the impact of CARES UI becomes smaller.
4.3. Decomposition of CARES UI
To evaluate the contribution of the three components of CARES UI, we decompose the total
effect of CARES UI into the effect of each component and the interaction effect between the
29
Figure 8: Decomposition of CARES UI’s effects on unemployment over transition
$600 UI top-up Eligibility expansion
13-week duration extension Interaction effect
Note: Each color represents the effect of one component of the CARES UI program and the interaction effect
among the components. The effect of each component is calculated by subtracting the effect from shutdown
alone.
components. The interaction effect arises because the effect of one policy component depends
on the other two components. For example, the eligibility expansion and duration extension
both increase the number of UI recipients at a given point in time. A larger group of UI re-
cipients implies that more unemployed workers are receiving the $600 top-up, which in turn
increases the total effect of CARES UI.
Figure 8 shows the decomposition of the effects on unemployment over the transition path.
The effect of each component is measured by the difference between the economy with shut-
down alone and the economy with shutdown and the particular component of CARES UI.
The interaction effect is calculated by subtracting the effects of the three individual compo-
nents from the total effects of CARES UI. Because the $600 top-up expires at the end of July,
sooner than the other two policy components, its effect, as shown in Figure 8, is concentrated
in the early period. In comparison, the effects of eligibility expansion and duration extension
spread over a longer period. The interaction effect is also concentrated in the early period (as
shown by the pink area), implying that it mostly comes from the interaction of $600 top-up
with the other two components.
Overall, as Table 4 reports, out of the 1.61 ppt increase in average unemployment attributed
to CARES UI between April and December 2020, the $600 top-up alone accounts for 0.33 ppt,
eligibility expansion for 0.57 ppt, and duration extension for 0.17 ppt. The interaction effect
accounts for the rest of the 0.55 ppt, larger than any individual component except for the el-
igibility expansion. Accordingly, out of the 2.09% reduction in the total cumulative deaths,
the $600 top-up, eligibility expansion, and duration extension each accounts for 0.24%, 0.93%,
and 0.08% of the reduction, respectively, while the interaction effect accounts for 0.84%. Our
30
Table 4: Decomposition of CARES UI’s effects on unemployment and deaths
Components of CARES UI Effect on Apr–Dec 2020 Effect on Total
Avg Unemployment (ppt) Cumulative Deaths (%)
(1) $600 UI top-up 0.33 -0.24
(2) Eligibility expansion 0.57 -0.93
(3) 13-week duration extension 0.17 -0.08
(4) Interaction effect 0.55 -0.84
All three UI programs 1.61 -2.09
Note: The contribution of each CARES UI policy component is calculated by subtracting the effect of shutdown
alone.
results suggest that while most policy discussions have focused on the effect of the $600 top-
up, the eligibility expansion and the interaction effect among the three components also have
comparable effects.
4.4. Welfare Evaluation
As the CARES UI policy reduces infection and deaths at the cost of higher unemployment,
it is useful to look at the welfare implications to evaluate the trade-off. We compute welfare
as an agent’s discounted sum of lifetime utility, including both the transition periods and the
post-pandemic steady state. We assume a residual life of 50 years for young and 20 years for
old, with 120 weeks in transition and the rest in the end steady state. The welfare effect of the
CARES UI policy is calculated as the percent of income that a person is willing to pay every
week to move from the economy without CARES UI (with shutdown alone) to the economy
with the policy.
As Table 5 reports, the CARES UI policy is welfare improving for workers in the contact
sector, who have a 0.49% increase in lifetime welfare in the baseline case, compared to a negligi-
ble change in lifetime welfare for those in the non-contact sector.39 One reason for the sectoral
difference is the shutdown policy directly impacts workers in the contact sector, which makes
UI benefits particularly important for them. For the non-contact sector workers, the cost of the
pandemic tax offsets the benefit of CARES UI and leads to a tiny change in welfare. Among
the non-working population, the Old like the CARES UI policy more, or dislike it less, than
the Young (OLF), because the Old face a higher risk of dying from the infection and the UI
policy reduces the infection risks.
The welfare calculations depend on several assumptions. For example, if we double the
39
For the young, a 1% welfare effect, i.e. 1% weekly income for 52 weeks over 50 years, translates into half a
year income.
31
Table 5: Welfare effects (%) of CARES UI under different assumptions
Workers (16–64) Non-workers
Assumptions Contact Non-contact YOLF (16–64) Old (> 65)
Baseline* 0.49 0.00 -0.12 -0.08
Double the cost of death 0.50 0.01 -0.11 0.09
Old does not pay pandemic tax 0.47 -0.02 -0.14 0.13
Young does not pay pandemic tax 0.61 0.13 0.01 -1.75
Deficit paid up over 5 years 0.48 -0.01 -0.13 -0.12
Note: We use a residual lifetime of 50 years for young and 20 years for old, including 120 weeks on transition and
the rest in the end steady state. Numbers are percent (weekly) income equivalent welfare change relative to the
case with shutdown but without CARES UI. A negative number indicates the CARES UI policy reduces welfare
relative to the case with shutdown only and no CARES UI.
^𝐷 = −10. Increases in government deficit due to higher unemployment and
*In the Baseline case, cost of death 𝑢
the CARES UI policy are financed by a proportional pandemic tax on all income in the post-pandemic steady state,
over 10 years.
utility cost of death, everyone likes the CARES UI policy more than in the baseline, especially
the Old; if only young agents pay the pandemic tax used to finance the policy, then the Old
also like the policy more; and using a higher pandemic tax to pay off the deficit in 5 instead of
10 years slightly reduces the welfare gains.
4.5. Targeted Policy
CARES UI policies were designed to help workers during the pandemic since many businesses
were forced to shut down. However, business shutdown was concentrated in the contact sec-
tor. Hence a more targeted policy would be to raise the generosity of UI benefits only for
workers in the contact sector. To explore the effect of such a targeted policy, we simulate the
economy with a counterfactual policy that gives UI top-up to only workers in the contact sec-
tor, while keeping the total expenditure of the policy the same as the CARES UI policy. All
unemployed workers still have the same duration extension and eligibility expansion as in the
CARES UI package.
The targeted policy leads to an increase in the top-up amount from $600 to $665 for all
unemployed workers in the contact sector. Table 6 reports the policy effects. Compared to
the CARES UI policy, the targeted policy increases the average unemployment in the contact
sector by an additional 0.09 ppt between April and December 2020 because of the elevated UI
top-up. It also lowers unemployment in the non-contact sector by 0.23 ppt because without
UI top-up, workers in this sector search more. Overall, the policy marginally lowers aggregate
unemployment compared to the CARES UI policies. More importantly, because the policy
32
Table 6: Effects of CARES UI and targeted policy
Effect on Apr–Dec 2020 unemployment rate (ppt) Effect on Total
Policy Aggregate Contact Non-contact Cumulative Deaths (%)
CARES UI 1.61 2.30 0.41 -2.09
Targeted policy 1.59 2.40 0.18 -2.16
Difference -0.02 0.09 -0.23 -0.07
Note: The rows report the effects of two UI policy combinations: the CARES UI policy and the targeted policy.
The targeted policy gives UI top-up of $665 to unemployed workers in the contact sector and no top-up for those
in the non-contact sector. Otherwise, the two policy combinations are the same. The effects are calculated as the
difference between the transition with the UI policy and the transition without the UI policy but with shutdown
in both cases. The policy effect is expressed in percentage points for average unemployment rate, and in percent
terms for cumulative deaths.
raises unemployment in the contact sector, it further lowers infection through the work chan-
nel, which is only present in the contact sector. As a result, the policy has a larger effect on
infection and deaths, lowering total cumulative deaths by an additional 0.07%, compared to
the CARES UI policy.
4.6. Robustness Exercises
This subsection discusses a few robustness checks. The detailed results for each exercise are
included in Appendix C.3.
Alternative health calibration: Larger shares of type M agents. In the baseline calibration, we
use 𝜎𝑀 𝐼 = 0.5 by assuming half of type M agents recover without becoming severely sick, and
half progress to type I. Without comprehensive testing, it is hard to know the actual number
of type M agents and hence their recover rate. Antibody tests conducted by the CDC have
found potentially more cases with mild or no symptom among the untested population. As
an alternative, we use 𝜎𝑀 𝐼 = 0.2 for both young and old agents, which implies a higher share of
type M agents among all the infected. We then re-calibrate the health parameters targeting the
same moments as before.40 The implied initial 𝑅0 is 1.99 and with social distancing 𝑅0 falls
to 1.19. Overall, CARES UI increases the average unemployment rate over April–Dec 2020
by 1.78 ppt and reduces total cumulative deaths by 2.94%, compared to 1.61 ppt and 2.09%
40
We use the same unconditional death rates as we use in baseline calibration for calibration targets. Because
now the transition rate from type M to type I (𝜎𝑀 𝐼 ) is lower than in the baseline, the resulting conditional death
𝑦 𝑜
rates (𝜎𝐼𝐷 and 𝜎𝐼𝐷 ) are higher. An alternative way is to use the same conditional death rates as calibrated in
𝑦 𝑜
the baseline (i.e. the same 𝜎𝐼𝐷 and 𝜎𝐼𝐷 as in the baseline). This means lower unconditional death rates than
in the baseline, which would require larger per-contact infection rates 𝜌 and 𝜌𝑒 to match death numbers. Larger
infection rates would then make the effects of mitigation policies on infection stronger, but the effects of policies
on death would be similar as shown here since the unconditional death rates are lower. Appendix C.3.1 gives the
calibrated parameters.
33
in the baseline calibration. The larger effects on unemployment and deaths are because, by
assumption, this alternative calibration has proportionally more type M agents who can work
and spread the virus at the workplace, which leads to more infections. As infection amplifies
the effect of the UI policy, more infection leads to larger effects of CARES UI on unemployment
and deaths.
Alternative health calibration: Different initial size of infected population. In the baseline
calibration, we assume 0.02% of population (about 600 thousand people) are infected with
mild or no symptoms (type M) at the start of the model simulation in February 2020. Because
there is no epidemiological evidence for the exact number of infections early on, as robust
checks, we use alternative numbers for the initial size of type M population: 0.01% or 0.03%.
In each case, we re-calibrate the health parameters to target the same set of moments as in the
baseline. In particular, when we assume a smaller initial infection share (0.01% of population
infected), we need larger initial per-contact infection rates 𝜌 and 𝜌𝑒 to generate the cumulative
deaths numbers as of April 4, 2020, and smaller social distancing parameter 𝛾 to generate
the cumulative deaths as of June 27, 2020. Overall, as Table C.2 in the appendix shows, the
policy effects on the average unemployment rate are very similar across different cases. The
effect on deaths is larger when the assumed initial infected population is smaller. This is very
intuitive: As infection grows exponentially (one infected person can infect many at a time),
policy interventions generate larger impacts on total infection and deaths when there are fewer
cases early on.
Workplace infection in the non-contact sector. In the baseline case, we have assumed that
workers in the non-contact sector do not get infection from work. The underlying assumption
is that these workers have access to working-from-home options, and so even without the shut-
down policy, they avoid workplace infection by working from home. An alternative assump-
tion is that these workers can only work from home during shutdown, and when shutdown
ends, they have to work on-site. As such, after shutdown ends, there is also work-related in-
fection in the non-contact sector. We assume that workplace infection in the non-contact sector
has the same per-contact infection rate 𝜌𝑒 as in the contact sector. We re-calibrate the infection
rates 𝜌, 𝜌𝑒 and the social distancing parameter 𝛾 to match the same set of targets as before. The
overall health and unemployment dynamics are very similar to the baseline. Because work-
ers in the non-contact sector also face the additional infection risk at workplace, this higher
infection risk increases unemployment in the non-contact sector without any policy interven-
tion, which peaks at a higher level than in the baseline. The CARES UI policy increases the
average unemployment over April–Dec 2020 by 1.64 ppt, similar to the result in the baseline,
and reduce total cumulative deaths by 1.54%, smaller than in the baseline. The smaller effect
34
on deaths is because the calibrated infection risk at workplace 𝜌𝑒 is lower than in the baseline.
Since the effect of the UI policy works through workplace infection, a smaller 𝜌𝑒 leads to a
smaller effect of CARES UI on infection and deaths.
5. Conclusion
This paper embeds SIR-type infection dynamics into a labor market search-matching model
to study the effects of the CARES UI policy on health and unemployment, in the presence of
COVID-like infection risk and shutdown policy. Workers in the contact sector face higher in-
fection risk as they have to perform their work at the workplace. In the model, policies and
infection risk interact with each other. A higher risk of infection at workplace reduces workers’
incentives to work and raises unemployment. Shutdown and UI policies increase unemploy-
ment and thus reduce workplace infection and save lives. As shutdown and infection risk
both increase unemployment, they increase the number of UI recipients and thus amplify the
effects of the UI policy.
Quantitatively, our calibrated model suggests that the CARES UI policy raises unemploy-
ment by an average of 1.61 percentage points out of a total increase of 9 percentage points over
April to December 2020, but also reduces cumulative deaths by 2.09%. Absent from the am-
plification effects in a world without COVID infection risk and shutdown, the same UI policy
would only raise unemployment by 0.59 ppt. Out of the 1.61 ppts increase in average unem-
ployment, the $600 top-up alone accounts for 0.33 ppt, eligibility expansion for 0.57 ppt, dura-
tion extension for 0.17 ppt, and the interaction among the three components for 0.55 ppt. The
policy effects are larger in the contact sector and are hump-shaped by income. Overall, CARES
UI improves welfare of workers by providing income insurance and reducing infection, and
is more beneficial to the Old than the Young because of its health effect. Our findings such
as the heterogeneous responses of workers and the relative importance of CARES UI compo-
nents depend on model elements, including the interaction of UI with other shocks and the
structure of UI policy, and are therefore not specific to the COVID pandemic.
Our model abstracts from one potentially important margin. The generous CARES UI pol-
icy, especially the $600 top-up could generate a sizeable aggregate demand effect, whereby
unemployed workers receiving UI benefits may increase spending drastically which in turn
boosts firm’s labor demand. This channel would reduce the net disincentive effect of the UI
policy. But its size is likely limited, because of reduced consumption activities in response to
COVID and shutdown during the period we focus on. As evidence, from February to April
2020, the Personal Consumption Expenditure (PCE) declined by 19%, and it was still 5% lower
in July compared to February; and accordingly, personal savings rate went up from 8.3% to
35
33.7% from February to April and was 17.8% in July. We leave this for future research.
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payments,” San Francisco Fed Working Paper 2020-28.
Rothstein, J. (2011): “Unemployment Insurance and Job Search in the Great Recession,” NBER Working
Paper No. 17534.
Zhang, H., A. R. Conn, and K. Scheinberg (2010): “A Derivative-Free Algorithm for Least-Squares
Minimization,” SIAM Journal on Optimization, 20, 3555–3576.
38
Appendix for
“Unemployment Insurance during a Pandemic”
A. Data Appendix
A.1. Construction of data moments
• Classification of Industries: Based on Dingel and Neiman (2020), it is easy to assign 17 of the
industries: The lowest 11 with a teleworkable share ≤ 31% goes to contact and the highest 6
with a teleworkable share ≥ 51% goes to non-contact. The rest three are in the middle which
have similar teleworkable shares (37% − 41%). They are utility, government, and real estate.
Presumably, industries that have more jobs requiring in-person interactions with coworkers and
customers are impacted more from the pandemic and shutdown policies, and thus experience
larger employment losses. According to employment data, there are large job losses in real estate
(9.7% of total industry employment) and small losses in government (4.4%) and utility (0.5%)
between Feb. and April of 2020. Hence we assign real estate to contact and utility and government
to non-contact. This leads to a 64% employment share in contact sector which is close to 63% of the
share of jobs that can not be performed at home as reported by Dingel and Neiman (2020). Table
A.1 gives the industry assignment in the contact and non-contact sectors, their teleworkable index
and employment change between Feb and April 2020. The reported employment changes further
confirm the conjecture that industries with smaller shares of workers who can work at home
experience larger employment losses. The correlation coefficient between the remote workable
employment share and the loss in employment is 46%.
• Value-added share is computed using industry value-added data from BEA.
• CPS data and the efficiency unit distribution: We use data from the Monthly Current Population
Survey to construct population shares, sectoral employment shares, sectoral unemployment rates,
sectoral average income ratio and the efficiency unit distribution. The classification of industries
follows Table A.1. We drop observations with missing information on either the labor-market
status or the industry information. We also drop the observations with weekly earnings below
$50. Consistent with the definition of young workers (i.e., the workers who haven’t reached the
retirement age), we restrict the ages to be above 15 and below 65. We calculate weekly labor
income using the hourly pay rate and weekly hours whenever they are available, and we use
the reported aggregate weekly earning otherwise. We use data from 2015–2019 to calibrate our
benchmark economy prior to the pandemic. To make labor income comparable across years,
we deflate nominal income by CPI. We use the income distribution in the CPS data to construct
the efficiency distribution 𝐹𝑗 (𝑎) for each sector (𝑗 = 𝑐𝑜𝑛, 𝑛𝑐). Specifically, we first obtain the
distribution of weekly labor income in each sector normalized by the average labor income in
that sector. In practice, we use a density distribution of 20 grid points to approximate this relative
1
income distribution where the log values of these 20 grid points are evenly distributed. That is, let
the minimum and maximum levels of the relative income level of the whole sample is 𝑎 and 𝑏. The
20 bins used to calculate the density are [𝑎𝑖 , 𝑏𝑖 ] (𝑖 = 1, 2, ..., 20), where 𝑎𝑖 = 𝑒𝑥𝑝(𝐿𝑛(𝑎)+(𝑖−1)*𝑑),
𝑏𝑖 = 𝑒𝑥𝑝(𝐿𝑛(𝑎) + 𝑖 * 𝑑), and 𝑑 = (𝐿𝑛(𝑏) − 𝐿𝑛(𝑎))/20; the corresponding 20 grid points are given
by 𝑔𝑖 = 𝑒𝑥𝑝((𝐿𝑛(𝑎𝑖 ) + 𝐿𝑛(𝑏𝑖 ))/2). These distributions are show in Figure A.1. We have conducted
robustness checks and confirmed that increasing the number of grids won’t qualitatively change
our results.
• UI recipient rate and relative income: Over the transition, UI recipient rate is computed as the
ratio of total weeks of UI paid in all programs (state and federal) to the number of unemployed
workers. Weekly UI payment data come from Department of Labor’s Employment and Training
Administration (DOLETA). The number of unemployed workers is computed using Bick and
Blandin (2020)’s survey-based unemployment rate and the level of civilian labor force.
• Steady state vacancy-unemployment ratio is computed using vacancy numbers from JOLTS. The
number of unemployed workers is computed as above.
• Separation rates differ by income and by sector in the model. We measure them using the same
CPS data in 2015-2019 used to construct the efficiency unit. Specifically, for each sector, the
monthly separation rate in a given relative wage income bucket is defined as the share of em-
ployed workers in this income bucket who will become non-employed next month. These monthly
separation rates are shown in Figure A.2. They have been converted to weekly frequency when
used in the model.
• Retirement income/Average earned income: As reported by the Social Security Administration,
the monthly benefit for retired workers is $1342 in 2016. This amount to a ratio of (1342*12)/(850*
52) = 36% relative to the average labor income, where $850 is the average income during 2015–
2019 (deflated) from CPS. The survivor benefit of deceased workers is in general smaller than
the payment to workers. Hence the actual ratio is likely to be slightly lower than 36%. We use a
target of 34%.
• SNAP/Average earned income: We use SNAP benefit amount to target the social welfare income
of the unemployed without UI benefits and YOLF, 𝑐. The Center on Budget and Policy Priorities
reports that the average monthly benefit level in 2019 for a one-person household is $131. This
amounts to 131 * 12/(850 * 52) = 3.56% of average labor income during 2015–2019.
• UI upper bound/Average earned income: all states have a dollar amount upper bound for the
UI weekly benefit amount. We normalize it using each state’s average weekly wage income, and
then take simple average across states to get an aggregate measure for this upper bound 𝑏𝑢𝑏 .
2
Figure A.1:
Density Distribution
Distribution of Efficiency
of Efficiency Unit (a) Unit
0.16 0.16
0.14 0.14
0.12 0.12
0.10 0.10
0.08 0.08
0.06 0.06
0.04 0.04
0.02 0.02
0.00 0.05
0.07
0.08 0.00
0.10
0.13
0.16
0.20
0.25
0.31
0.39
0.48
0.60
0.75
0.94
1.17
1.45
1.81
2.26
2.81
3.50
Sector Specific Relative Wage Income
Contact Sector Non-Contact Sector
Note: The sector specific relative wage income is defined as the ratio of a worker’s wage income to the average
wage income in the sector that this worker belongs to. The average wage income in the contact sector is about
0.71 of the average wage income in the non-contact sector.
3
Figure A.2: Separation Rates by Income
Percent
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
0 0
0.05 0.07 0.08 0.10 0.13 0.16 0.20 0.25 0.31 0.39 0.48 0.60 0.75 0.94 1.17 1.45 1.81 2.26 2.81 3.50
Sector Specific Relative Wage Income
Contact Sector Non-Contact Sector
Note: The sector specific relative wage income is defined as the ratio of a worker’s wage income to the average
wage income in the sector that this worker belongs to. The average wage income in the contact sector is about
0.71 of the average wage income in the non-contact sector.
A.2. Calculation of 𝑅0 and workplace infection share
𝑅0 measures the total number of infections generated by one infected person assuming everyone else in
the economy is susceptible and there is no policy mitigation. The higher is 𝑅0 , the faster is the spread of
the virus. Thus 𝑅0 contains information on the infection rate. In our model, 𝑅0 differs by age because
the health transition rates differ. 𝑅0 also differs for employed contact sector workers since they face an
additional infection risk. In the context of our model, 𝑅0 can be computed as follows. For workers in
the non-contact sector:
𝑦
𝜌 𝜎𝑀 𝜌
𝑦 𝑅0𝑛𝑐 =
𝑦 + 𝑦
𝐼
𝑦 𝑦 𝑦 (A.1)
𝜎𝑀 𝐼 + 𝜎 𝑀𝑅 𝜎 𝑀𝐼 + 𝜎 𝜎
𝑀 𝑅 𝐼𝐷 + 𝜎𝐼𝑅
Because workers in the non-contact sector have the same transition rates as the non-working young
(YOLF), and they both spread the disease with rate 𝜌, 𝑅0 for YOLF is the same as 𝑅0𝑛𝑐 , 𝑅0𝑦 = 𝑅0𝑛𝑐 . The
Old has different disease progression rates conditional on infection, so 𝑅0 for old has the same form:
𝑜
𝜌 𝜎𝑀 𝜌
𝑅0𝑜 = 𝑜 𝑜 + 𝑜
𝐼
𝑜 𝑜 𝑜 (A.2)
𝜎𝑀 𝐼 + 𝜎𝑀 𝑅 𝜎𝑀 𝐼 + 𝜎𝑀 𝑅 𝜎𝐼𝐷 + 𝜎𝐼𝑅
Contact sector workers have higher infection rates:
𝜌 + 𝜌𝑒 𝐸𝑐𝑜𝑛 𝜎𝑜 𝜌
𝑅0𝑐𝑜𝑛 = 𝑜 𝑜 + 𝑜 𝑀𝐼 𝑜 𝑜 + 𝜎𝑜 (A.3)
𝜎𝑀 𝐼 + 𝜎𝑀 𝑅 𝜎𝑀 𝐼 + 𝜎𝑀 𝑅 𝜎𝐼𝐷 𝐼𝑅
where 𝐸𝑐𝑜𝑛 is the contact sector employed workers as a share of total population. Aggregate 𝑅0 is the
weighted average of the above values using the shares of population for YOLF, Old, contact and non-
contact sector workers.
The workplace infection as a share of total infection is determined by the relative size of 𝜌 and 𝜌𝑒 ,
4
and is calculated as the ratio of workplace infection in the contact sector to the aggregate 𝑅0 :
(︂ )︂
𝑤𝑜𝑟𝑘𝑝𝑙𝑎𝑐𝑒 1 𝜌𝑒 𝐸𝑐𝑜𝑛
= 𝐸𝑐𝑜𝑛 𝑦 𝑦 (A.4)
𝑡𝑜𝑡𝑎𝑙 𝑅0 𝜎𝑀 𝐼 + 𝜎𝑀 𝑅
Table A.1: Classification of Industries
Dingel and Neiman (2020) Employment Change
Industry teleworkable𝑒𝑚𝑝 Feb–April, 2020
Contact sector
Accommodation and Food Services 0.035 -0.473
Agriculture, Forestry, Fishing and Hunting 0.076 –
Retail Trade 0.143 -0.137
Construction 0.186 -0.132
Transportation and Warehousing 0.186 -0.104
Manufacturing 0.225 -0.106
Health Care and Social Assistance 0.253 -0.104
Mining, Quarrying, and Oil and Gas Extraction 0.254 -0.080
Arts, Entertainment, and Recreation 0.297 -0.545
Administrative and Support
and Waste Management and Remediation Services 0.311 -0.173
Other Services (except Public Administration) 0.312 -0.220
Real Estate and Rental and Leasing 0.418 -0.097
Non-contact sector
Utilities 0.370 -0.005
Federal, State, and Local Government 0.415 -0.044
Wholesale Trade 0.518 -0.062
Information 0.717 -0.089
Finance and Insurance 0.762 -0.005
Management of Companies and Enterprises 0.792 -0.033
Professional, Scientific, and Technical Services 0.803 -0.056
Educational Services 0.826 -0.129
Contact -0.193
Non-contact -0.053
Total Non-farm -0.140
Note: Federal, State, and Local Government excludes state and local schools and hospitals and the U.S. Postal
Service (OES Designation).
5
B. Model Appendix
This appendix contains additional details for the model laid out in Section 2.
B.1. Timing illustration
˜ 𝑏, 𝑊
We define the value functions by labor market status (𝑊 𝑒 , 𝑊 𝑏 , 𝑊 𝑛 , 𝑊 ˜ 𝑛 ) at the beginning of a pe-
riod. The infection probabilities in the health transition matrices (Γ0𝑗 , Γ1𝑗 ) are defined based on the
measures of total infected population and the infected workers who are working. Figure B.1 illustrates
the sequence of events, given government policies. Since the Old and Young OLF are not part of the
labor force, only the health transition at the end of the period concerns them.
Figure B.1: Timeline within period
value functions health transition
˜ 𝑏, 𝑊
(𝑊 𝑒 , 𝑊 𝑏 , 𝑊 𝑛 , 𝑊 ˜ 𝑛) (Γ0𝑗 , Γ1𝑗 )
Labor market Production and consumption UI, Temp status & Health
𝑡 recall, search, job posting workers produce & consume UI expires with prob 𝜀 𝑡+1
job separation (𝛿𝑗 , 𝛿˜𝑗 ) health utility (𝑢
^ℎ ) temp layoff expires with prob 𝜁
receives benefits (𝜆𝑗𝑎 ) health ℎ → ℎ′
health-related separation
B.2. Value functions of non-workers
Young out of labor force (YOLF) with health ℎ consume base income 𝑐, do not make any choices:
∑︁
𝑊 𝑦 (ℎ) = 𝑢(𝑐) + 𝑢^ℎ + 𝛽 Γ𝑦 (ℎ, ℎ′ )𝑊 𝑦 (ℎ′ ). (B.1)
ℎ′
Old people with health ℎ consume retirement income 𝑏𝑜 , do not make any choices
∑︁
𝑊 𝑜 (ℎ) = 𝑢(𝑏𝑜 ) + 𝑢
^ℎ + 𝛽𝑜 Γ𝑜 (ℎ, ℎ′ )𝑊 𝑜 (ℎ′ ). (B.2)
ℎ′
where Γ𝑦 (ℎ, ℎ′ ) and Γ𝑜 (ℎ, ℎ′ ) are the health transition matrices for the young and old non-workers,
respectively.
B.3. Additional value functions of workers
Here we define the other value functions of workers in addition to the value function of worker (1) and
the value function of worker on temporary layoff with UI benefits (2). Similar to (2), the value function
6
of a worker (𝑗, 𝑎, ℎ) on temporary layoff but without UI, where health ℎ ∈ {S, 𝑀, 𝑅}is given by:
∑︁
˜ 𝑛 (𝑗, 𝑎, ℎ) = 𝑟 Γ1𝑗 (ℎ, ℎ′ ) 𝑢(𝑤𝑗 𝑎) + 𝑢
^ℎ + 𝛽𝑊 𝑒 (𝑗, 𝑎, ℎ′ )
[︀ ]︀
𝑊 (B.3)
ℎ′
⏟ ⏞
recalled to job
{︂ ∑︁
Γ1𝑗 (ℎ, ℎ′ ) 𝑥𝑓 (𝜃𝑗𝑎 ) 𝑢(𝑤𝑗 𝑎) + 𝑢
^ℎ + 𝛽𝑊 𝑒 (𝑗, 𝑎, ℎ′ )
[︀ ]︀
+(1 − 𝑟) max −𝑣(𝑥) +
𝑥
ℎ′
⏟ ⏞
no recall, finds job through search
∑︁ }︂
0 ′ 𝑛 ′ ˜ 𝑛 ′
[︀ ]︀
+ Γ𝑗 (ℎ, ℎ ) (1 − 𝑥𝑓 (𝜃𝑗𝑎 )) 𝑢(𝑐) + 𝑢
^ℎ + 𝛽𝜁𝑊 (𝑗, 𝑎, ℎ ) + 𝛽(1 − 𝜁)𝑊 (𝑗, 𝑎, ℎ )] .
ℎ′
⏟ ⏞
does not find job, temp layoff expires with prob 𝜁
Let 𝑥 be the search effort of a permanently separated worker and 𝑣(𝑥) be the disutility of search.
The value function for a permanently separated worker (𝑗, 𝑎, ℎ) with UI, where ℎ ∈ {𝑆, 𝑀, 𝑅} is given
by:
∑︁
𝑊 𝑏 (𝑗, 𝑎, ℎ) = max −𝑣(𝑥) + Γ1𝑗 (ℎ, ℎ′ ) 𝑥𝑓 (𝜃𝑗𝑎 ) 𝑢(𝑤𝑗 𝑎) + 𝑢
^ℎ + 𝛽𝑊 𝑒 (𝑗, 𝑎, ℎ′ )
[︀ ]︀
(B.4)
𝑥
ℎ′
⏟ ⏞
finds job
∑︁
+ Γ0𝑗 (ℎ, ℎ′ ) (1 − 𝑥𝑓 (𝜃𝑗𝑎 ))[𝑢(𝑏𝑗,𝑎 ) + 𝑢
^ℎ + 𝛽(1 − 𝜀)𝑊 𝑏 (𝑗, 𝑎, ℎ′ ) + 𝛽𝜀𝑊 𝑛 (𝑗, 𝑎, ℎ′ )],
ℎ′
⏟ ⏞
no job
and the value function for a permanently separated worker without UI is given by:
∑︁
𝑊 𝑛 (𝑗, 𝑎, ℎ) = max −𝑣(𝑥) + Γ1𝑗 (ℎ, ℎ′ ) 𝑥𝑓 (𝜃𝑗𝑎 ) 𝑢(𝑤𝑗 𝑎) + 𝑢
^ℎ + 𝛽𝑊 𝑒 (𝑗, 𝑎, ℎ′ )
[︀ ]︀
(B.5)
𝑥
ℎ′
⏟ ⏞
finds job
∑︁
+ Γ1𝑗 (ℎ, ℎ′ ) (1 − 𝑥𝑓 (𝜃𝑗𝑎 ))[𝑢(𝑐) + 𝑢^ℎ + 𝛽𝑊 𝑛 (𝑗, 𝑎, ℎ′ )] .
′
⏟ ⏞
ℎ
no job
B.3.1 Value functions of Infected Severe (type I) workers
In the model we assume that workers of health type I cannot work and do not search if unemployed.
Specifically, we assume that if a type M worker on temporary layoff becomes type I, she keeps her
temporary layoff status; but if a type M worker becomes type I while she is employed, she becomes
permanently separated with benefits: 𝑊 𝑒 (𝑗, 𝑎, ℎ′ = 𝐼) = 𝑊 𝑏 (𝑗, 𝑎, ℎ′ = 𝐼). Similarly, if a type I worker
on temporary layoff is recalled back to work, she becomes permanently separated but keeps her UI
status. The value functions of type I workers on temporary layoff are:
7
With UI
∑︁
˜ 𝑏 (𝑗, 𝑎, 𝐼) = 𝑟 Γ0𝑗 (ℎ = 𝐼, ℎ′ ) 𝑢(𝑏𝑗,𝑎 ) + 𝑢
^ℎ + 𝛽(1 − 𝜀)𝑊 𝑏 (𝑗, 𝑎, ℎ′ ) + 𝛽𝜀𝑊 𝑛 (𝑗, 𝑎, ℎ′ )
[︀ ]︀
𝑊 (B.6)
ℎ′
⏟ ⏞
recalled to job but unable to work
∑︁ (︂
′
+(1 − 𝑟) 0
^ℎ + 𝛽𝜁[(1 − 𝜀)𝑊 𝑏 (𝑗, 𝑎, ℎ′ ) + 𝜀𝑊 𝑛 (𝑗, 𝑎, ℎ′ )]
Γ𝑗 (ℎ = 𝐼, ℎ ) 𝑢(𝑏𝑗,𝑎 ) + 𝑢
ℎ′
⏟ ⏞
not recalled, temp layoff expires
)︂
˜𝑏 ′
+ 𝛽(1 − 𝜁)[(1 − 𝜀)𝑊 (𝑗, 𝑎, ℎ ) + 𝜀𝑊 (𝑗, 𝑎, ℎ )] , ˜𝑛 ′
⏟ ⏞
not recalled, stays on temp layoff
Without UI
∑︁
˜ 𝑛 (𝑗, 𝑎, 𝐼) = 𝑟 Γ0𝑗 (ℎ = 𝐼, ℎ′ ) ^ℎ + 𝛽𝑊 𝑛 (𝑗, 𝑎, ℎ′ )
[︀ ]︀
𝑊 𝑢(𝑐) + 𝑢 (B.7)
ℎ′
⏟ ⏞
recalled to job but unable to work
∑︁
Γ0𝑗 (ℎ = 𝐼, ℎ′ ) ˜ 𝑛 (𝑗, 𝑎, ℎ′ )] .
^ℎ + 𝛽𝜁𝑊 𝑛 (𝑗, 𝑎, ℎ′ ) + 𝛽(1 − 𝜁)𝑊
[︀ ]︀
+(1 − 𝑟) 𝑢(𝑐) + 𝑢
ℎ′
⏟ ⏞
not recalled, temp layoff expires with prob 𝜁
The value functions of type I workers who are permanently separated are:
With UI
∑︁
𝑊 𝑏 (𝑗, 𝑎, 𝐼) = 𝑢(𝑏𝑗,𝑎 ) + 𝑢 Γ0𝑗 (ℎ, ℎ′ ) (1 − 𝜀)𝑊 𝑏 (𝑗, 𝑎, ℎ′ ) + 𝜀𝑊 𝑛 (𝑗, 𝑎, ℎ′ ) ,
[︀ ]︀
^ℎ + 𝛽 (B.8)
ℎ′
Without UI
∑︁
𝑊 𝑛 (𝑗, 𝑎, 𝐼) = 𝑢(𝑐) + 𝑢
^ℎ + 𝛽 Γ0𝑗 (ℎ, ℎ′ )𝑊 𝑛 (𝑗, 𝑎, ℎ′ ). (B.9)
ℎ′
B.4. Additional value functions of firms
In addition to equations (5)-(7), the value function of a vacancy with recall where the worker (𝑗, 𝑎, ℎ),
ℎ ∈ {𝑆, 𝑀, 𝑅} does not have UI is given by:
∑︁
𝑉˜ 𝑛 (𝑗, 𝑎, ℎ) = 𝑟 (𝑧𝑗 − 𝑤𝑗 )𝑎 + 𝛽 Γ1𝑗 (ℎ, ℎ′ )𝐽(𝑗, 𝑎, ℎ′ )
[︀ ]︀
(B.10)
ℎ′
⏟ ⏞
recalls worker
∑︁
+ ˜𝑛 (𝑗, 𝑎, ℎ)𝑓 (𝜃𝑗𝑎 ))(1 − 𝜁)
(1 − 𝑟)𝛽(1 − 𝑥 Γ0𝑗 (ℎ, ℎ′ )𝑉˜ 𝑛 (𝑗, 𝑎, ℎ′ )
ℎ′
⏟ ⏞
does not recall worker, worker does not find new job and recall does not expire
+ 𝑥𝑛 (𝑗, 𝑎, ℎ)𝑓 (𝜃𝑗𝑎 ) + (1 − 𝑥
(1 − 𝑟)𝛽[˜ ˜𝑛 (𝑗, 𝑎, ℎ)𝑓 (𝜃𝑗𝑎 ))𝜁]𝑉 (𝑗, 𝑎)
⏟ ⏞
does not recall worker, worker finds new job or no new find but recall expires
𝑛
where 𝑥
˜ is the search effort of worker (𝑗, 𝑎, ℎ) on temporary layoff without UI.
The probability 𝑑ℎ𝑗𝑎 in the vacancy posting condition (7) is the probability that a firm in sector 𝑗 and
submarket 𝑎 meets an unemployed worker with health status ℎ ∈ {𝑆, 𝑀, 𝑅}. It is given by the search
intensity-weighted fraction of the measure of type ℎ unemployed workers among all unemployed work-
8
ers in the (𝑗, 𝑎) submarket:
𝜇𝑗𝑎ℎ𝑏 𝑥𝑏 (𝑗, 𝑎, ℎ) + 𝜇𝑗𝑎ℎ𝑛 𝑥𝑛 (𝑗, 𝑎, ℎ) + (1 − 𝑟)[˜ ˜𝑏 (𝑗, 𝑎, ℎ) + 𝜇
𝜇𝑗𝑎ℎ𝑏 𝑥 ˜𝑛 (𝑗, 𝑎, ℎ)]
˜𝑗𝑎ℎ𝑛 𝑥
𝑑ℎ𝑗𝑎 = ∑︀ (︁ )︁
𝜇 𝑥 𝑏 (𝑗, 𝑎, ^
ℎ) + 𝜇 𝑥 𝑛 (𝑗, 𝑎, ^
ℎ) + (1 − 𝑟)[˜𝜇 𝑥
˜ 𝑏 (𝑗, 𝑎, ^
ℎ) + 𝜇
˜ 𝑥˜ 𝑛 (𝑗, 𝑎, ^
ℎ)]
^ℎ∈{𝑆,𝑀,𝑅} ^ 𝑗𝑎ℎ𝑏 ^ 𝑗𝑎ℎ𝑛 ^ 𝑗𝑎ℎ𝑏 ^ 𝑗𝑎ℎ𝑛
where 𝜇𝑗𝑎ℎ𝑏 and 𝜇𝑗𝑎ℎ𝑛 (𝜇
˜𝑗𝑎ℎ𝑏 and 𝜇
˜𝑗𝑎ℎ𝑛 ) are the measures of workers on permanent (temporary) layoff
with and without UI benefits, respectively, in sector 𝑗 , with efficiency 𝑎, health ℎ, as defined in the next
subsection. 𝑥𝑏 and 𝑥𝑛 (𝑥
˜𝑏 and 𝑥
˜𝑛 ) are the search effort of workers on permanent (temporary) layoff.
B.4.1 Value functions of firms with type I worker
Since type I workers do not work, there is no operating firm with type I worker. But a vacancy with recall
keeps track of the health status of the worker on temporary layoff. If the worker becomes type I, she
keeps the temporary layoff until she is recalled, at which point she becomes permanently separated,
and the vacancy becomes one without recall, i.e. 𝑉 . So the value functions of a vacancy with recall
associated with a type I worker are: If worker has UI
∑︁
𝑉˜ 𝑏 (𝑗, 𝑎, 𝐼) = 𝑟𝛽 Γ1𝑗 (ℎ = 𝐼, ℎ′ )𝑉 (𝑗, 𝑎) + (1 − 𝑟)𝛽𝜁𝑉 (𝑗, 𝑎) (B.11)
⏟ ⏞
ℎ′
⏟ ⏞ does not recall, but recall option expires
recalls worker unable to work
∑︁
Γ0𝑗 (ℎ = 𝐼, ℎ′ ) (1 − 𝜀)𝑉˜ 𝑏 (𝑗, 𝑎, ℎ′ ) + 𝜀𝑉˜ 𝑛 (𝑗, 𝑎, ℎ′ ) ,
[︀ ]︀
+ (1 − 𝑟)𝛽(1 − 𝜁)
ℎ′
⏟ ⏞
does not recall worker, recall does not expire
If the worker does not have UI
∑︁
𝑉˜ 𝑛 (𝑗, 𝑎, 𝐼) = 𝑟𝛽 Γ1𝑗 (ℎ = 𝐼, ℎ′ )𝑉 (𝑗, 𝑎) + (1 − 𝑟)𝛽𝜁𝑉 (𝑗, 𝑎) (B.12)
⏟ ⏞
ℎ′
⏟ ⏞ does not recall worker, but recall option expires
recalls worker unable to work
∑︁
+ (1 − 𝑟)𝛽(1 − 𝜁) Γ0𝑗 (ℎ = 𝐼, ℎ′ )𝑉˜ 𝑛 (𝑗, 𝑎, ℎ′ )
ℎ′
⏟ ⏞
does not recall worker, recall does not expire
9
B.5. Transition equations
The distribution transitions within each period take place in two stages, as described in Section 2.4. In
Stage 1 (beginning of period), recall, search, job posting, and separation take place; in Stage 2 (end of
the period) the expiration of temporary layoff and the transitions in UI and health status take place.
To distinguish the two stages, we define 𝜇 as the beginning-of-period distribution of population: 𝜇𝑗𝑎ℎ𝜔
and 𝜇
˜𝑗𝑎ℎ𝜔 are the measures of workers in sector 𝑗 , with efficiency 𝑎, health ℎ, and labor market status
𝜔 for permanent and temporary layoffs respectively; 𝜇𝑜ℎ and 𝜇𝑦ℎ are the measures of Old and Young
˜ 𝑏 and
OLF with health ℎ, respectively. At the end of Stage 1 and before Stage 2, let 𝐸𝑗𝑎ℎ , 𝑈 𝑏 , 𝑈 𝑛 , 𝑈 𝑗𝑎ℎ 𝑗𝑎ℎ 𝑗𝑎ℎ
˜ 𝑛 denote the measures for the group of workers (𝑗, 𝑎, ℎ) who are employed, permanently separated
𝑈𝑗𝑎ℎ
with and without benefits, and on temporary layoff with and without benefits, respectively.
We characterize the Stage 1 transitions first.
B.5.1 Stage 1 transitions
Given distribution at the beginning of the period 𝜇, transitions in the labor market, including recall,
search, matching, and separation, are given by:
+ 𝜇𝑗𝑎ℎ𝑏 𝑓 (𝜃𝑗𝑎 )𝑥𝑏 (𝑗, 𝑎, ℎ) + 𝜇𝑗𝑎ℎ𝑛 𝑓 (𝜃𝑗𝑎 )𝑥𝑛 (𝑗, 𝑎, ℎ)
[︀ ]︀
𝐸𝑗𝑎ℎ = 𝜇𝑗𝑎ℎ𝑒 (1 − 𝛿𝑗𝑎 )
⏟ ⏞ ⏟ ⏞
employed not separated permanently separated found a job
+ 𝑟[˜
𝜇𝑗𝑎ℎ𝑏 + 𝜇
˜𝑗𝑎ℎ𝑛 ] + (1 − 𝑟)[˜ 𝑥𝑏 (𝑗, 𝑎, ℎ) + 𝜇
𝜇𝑗𝑎ℎ𝑏 𝑓 (𝜃𝑗𝑎 )˜ 𝑥𝑛 (𝑗, 𝑎, ℎ)]
˜𝑗𝑎ℎ𝑛 𝑓 (𝜃𝑗𝑎 )˜
⏟ ⏞ ⏟ ⏞
temp laid-off recalled temp laid-off not recalled, found a job
𝑏
𝑈𝑗𝑎ℎ = 𝑏
𝜇𝑗𝑎ℎ𝑏 (1 − 𝑓 (𝜃𝑗𝑎 )𝑥 (𝑗, 𝑎, ℎ)) + ˜
𝜇𝑗𝑎ℎ𝑒 𝛿𝑗𝑎 (1 − 𝛿)𝜆𝑗𝑎
⏟ ⏞ ⏟ ⏞
eligible unemployed not found a job newly perm unemployed receive benefits
𝑛
𝑈𝑗𝑎ℎ = 𝑛
𝜇𝑗𝑎ℎ𝑛 (1 − 𝑓 (𝜃𝑗𝑎 )𝑥 (𝑗, 𝑎, ℎ)) + ˜ − 𝜆𝑗𝑎 )
𝜇𝑗𝑎ℎ𝑒 𝛿𝑗𝑎 (1 − 𝛿)(1
⏟ ⏞ ⏟ ⏞
ineligible unemployed not found a job newly perm unemployed not receive benefits
˜𝑗𝑎ℎ
𝑈 𝑏
= 𝑥𝑏 (𝑗, 𝑎, ℎ))
˜𝑗𝑎ℎ𝑏 (1 − 𝑟)(1 − 𝑓 (𝜃𝑗𝑎 )˜
𝜇 + ˜ 𝑗𝑎
𝜇𝑗𝑎ℎ𝑒 𝛿𝑗𝑎 𝛿𝜆
⏟ ⏞ ⏟ ⏞
eligible temp laid-off not recalled, not found a job newly temp laid-off receive benefits
˜𝑗𝑎ℎ
𝑈 𝑛
= 𝑛
˜𝑗𝑎ℎ𝑛 (1 − 𝑟)(1 − 𝑓 (𝜃𝑗𝑎 )𝑥 (𝑗, 𝑎, ℎ)) +
𝜇 𝜇𝑗𝑎ℎ𝑒 𝛿𝑗𝑎 ˜
𝛿(1 − 𝜆𝑗𝑎 ) .
⏟ ⏞ ⏟ ⏞
ineligible temp laid-off not found a job newly unemployed not receive benefits
10
Type I workers do not work or search, and if a type I worker on temporary layoff is recalled, she becomes
permanently separated with benefits:
𝐸𝑗𝑎𝐼 = 0
𝑏
𝑈𝑗𝑎𝐼 = 𝜇𝑗𝑎𝐼𝑏 + 𝑟𝜇
˜𝑗𝑎𝐼𝑏
⏟ ⏞
temp layoff recalled but unable to work
𝑛
𝑈𝑗𝑎𝐼 = 𝜇𝑗𝑎𝐼𝑛 + 𝑟𝜇
˜𝑗𝑎𝐼𝑛
⏟ ⏞
temp layoff recalled but unable to work
˜𝑗𝑎𝐼
𝑈 𝑏
˜𝑗𝑎𝐼𝑏 (1 − 𝑟)
= 𝜇
˜𝑗𝑎𝐼
𝑈 𝑛
˜𝑗𝑎𝐼𝑛 (1 − 𝑟).
= 𝜇
B.5.2 Stage 2 transitions
Given distribution at the beginning of the period 𝜇, and distribution of workers at the end of Stage
˜ 𝑏 and 𝑈
1 𝐸𝑗𝑎ℎ , 𝑈 𝑏 , 𝑈 𝑛 , 𝑈 ˜ 𝑛 , Stage 2 transitions consist of transitions in health, UI status, and the
𝑗𝑎ℎ 𝑗𝑎ℎ 𝑗𝑎ℎ 𝑗𝑎ℎ
expiration of temporary layoff, and they give next period’s distribution 𝜇′ . We use Inf𝑐𝑜𝑛 = 𝜌𝑒 Ω𝑐𝑜𝑛,𝑒 +𝜌Ω
to denote the total probability of infection for workers employed in the contact sector, and Inf = Inf𝑛𝑐 =
𝜌Ω for the infection probability for all other groups, including workers employed in the non-contact
sector, unemployed workers, the Old and the YOLF. Once a person is infected with the virus, the health
𝑔 𝑔
transition rates are exogenous and potentially age-dependent (𝑔 ∈ {𝑦, 𝑜}): 𝜎𝑀 𝐼 (type M to I), 𝜎𝑀 𝑅
𝑔 𝑔
(type M to R), 𝜎𝐼𝑅 (type I to R), 𝜎𝐼𝐷 (type I to D).
Next period’s distribution of type S (Susceptible) agents:
YOLF or Old (𝑔 ∈ {𝑦, 𝑜}): 𝜇′𝑔𝑆 = 𝜇𝑔𝑆 − 𝜇𝑔𝑆 Inf
Employed: 𝜇′𝑗𝑎𝑆𝑒 = 𝐸𝑗𝑎𝑆 − 𝐸𝑗𝑎𝑆 Inf𝑗
Perm unemp, UI eligible: 𝜇′𝑗𝑎𝑆𝑏 = 𝑏
(1 − 𝜀)𝑈𝑗𝑎𝑆 + ˜𝑏
𝜁(1 − 𝜀)𝑈 𝑗𝑎𝑆
⏟ ⏞ ⏟ ⏞
perm unemp, UI not expired temp layoff expired, UI not expired
[︂ ]︂
𝑏
− (1 − 𝜀)𝑈𝑗𝑎𝑆 + 𝜁(1 − 𝜀)𝑈˜𝑗𝑎𝑆 Inf
𝑏
𝜇′𝑗𝑎𝑆𝑛 = ˜𝑗𝑎𝑆 + 𝜀𝑈
˜𝑗𝑎𝑆
(︀ 𝑛 𝑏
)︀ (︀ 𝑛 𝑏
)︀
Perm unemp, UI ineligible: 𝑈𝑗𝑎𝑆 + 𝜀𝑈𝑗𝑎𝑆 + 𝜁 𝑈
⏟ ⏞ ⏟ ⏞
perm unemp, no UI or UI expired temp layoff expired, no UI or UI expired
[︂ ]︂
𝑛 𝑏 ˜𝑗𝑎𝑆 + 𝜀𝑈
˜𝑗𝑎𝑆
(︀ )︀ (︀ 𝑛 𝑏
)︀
− 𝑈𝑗𝑎𝑆 + 𝜀𝑈𝑗𝑎𝑆 +𝜁 𝑈 Inf
Temp layoff, UI eligible: ˜′𝑗𝑎𝑆𝑏 =
𝜇 ˜𝑏
(1 − 𝜁)(1 − 𝜀)𝑈 ˜ 𝑏 Inf
−(1 − 𝜁)(1 − 𝜀)𝑈
𝑗𝑎𝑆 𝑗𝑎𝑆
⏟ ⏞
temp layoff not expired, UI not expired
˜′𝑗𝑎𝑆𝑛 = ˜𝑗𝑎𝑆 + 𝜀𝑈
˜𝑗𝑎𝑆 ˜𝑗𝑎𝑆 + 𝜀𝑈
˜𝑗𝑎𝑆
(︀ 𝑛 𝑏
)︀ (︀ 𝑛 𝑏
)︀
Temp layoff, UI ineligible: 𝜇 (1 − 𝜁) 𝑈 −(1 − 𝜁) 𝑈 Inf
⏟ ⏞
temp layoff not expired, no UI or UI expired
11
Next period’s distribution of type M (Infected Mild) agents:
𝑔 𝑔 )︀
YOLF or Old (𝑔 ∈ {𝑦, 𝑜}): 𝜇′𝑔𝑀 = 𝜇𝑔𝑀 − 𝜇𝑔𝑀 (𝜎𝑀 𝐼 + 𝜎𝑀 𝑅 + 𝜇𝑔𝑆 Inf
(︀ 𝑦 𝑦 )︀
Employed: 𝜇′𝑗𝑎𝑀 𝑒 = 𝐸𝑗𝑎𝑀 − 𝐸𝑗𝑎𝑀 𝜎𝑀 𝐼 + 𝜎𝑀 𝑅 + 𝐸𝑗𝑎𝑆 Inf𝑗
Perm unemp, UI eligible: 𝜇′𝑗𝑎𝑀 𝑏 = 𝑏
(1 − 𝜀)𝑈𝑗𝑎𝑀 + 𝜁(1 − 𝜀)𝑈˜𝑗𝑎𝑀
𝑏
⏟ ⏞ ⏟ ⏞
perm unemp, UI not expired temp layoff expired, UI not expired
[︂ ]︂ [︂ ]︂
(︀ 𝑦 𝑦 )︀
𝑏 ˜ 𝑏 𝑏 ˜ 𝑏
− (1 − 𝜀)𝑈𝑗𝑎𝑀 + 𝜁(1 − 𝜀)𝑈𝑗𝑎𝑀 𝜎𝑀 𝐼 + 𝜎𝑀 𝑅 + (1 − 𝜀)𝑈𝑗𝑎𝑆 + 𝜁(1 − 𝜀)𝑈𝑗𝑎𝑆 Inf
𝜇′𝑗𝑎𝑀 𝑛 = 𝑏 ˜𝑗𝑎𝑀
˜𝑗𝑎𝑀 + 𝜀𝑈
(︀ 𝑛 )︀ (︀ 𝑛 𝑏
)︀
Perm unemp, UI ineligible: 𝑈𝑗𝑎𝑀 + 𝜀𝑈𝑗𝑎𝑀 + 𝜁 𝑈
⏟ ⏞ ⏟ ⏞
perm unemp, no UI or UI expired temp layoff expired, no UI or UI expired
[︂ ]︂
)︀ (︀ 𝑦 𝑦 )︀
˜ ˜ 𝑏
(︀𝑛 𝑏
)︀ (︀ 𝑛
− 𝑈𝑗𝑎𝑀 + 𝜀𝑈𝑗𝑎𝑀 + 𝜁 𝑈𝑗𝑎𝑀 + 𝜀𝑈𝑗𝑎𝑀 𝜎 𝑀 𝐼 + 𝜎𝑀 𝑅
[︂ ]︂
˜𝑗𝑎𝑆
˜𝑗𝑎𝑆 + 𝜀𝑈 𝑏
(︀ 𝑛 𝑏
)︀ (︀ 𝑛 )︀
+ 𝑈𝑗𝑎𝑆 + 𝜀𝑈𝑗𝑎𝑆 +𝜁 𝑈 Inf
Temp layoff, UI eligible: ˜′𝑗𝑎𝑀 𝑏 =
𝜇 ˜𝑗𝑎𝑀
(1 − 𝜁)(1 − 𝜀)𝑈 𝑏
⏟ ⏞
temp layoff not expired, UI not expired
(︀ 𝑦 𝑦 )︀
˜𝑗𝑎𝑀
−(1 − 𝜁)(1 − 𝜀)𝑈 𝑏 ˜𝑏
𝜎𝑀 𝐼 + 𝜎𝑀 𝑅 + (1 − 𝜁)(1 − 𝜀)𝑈𝑗𝑎𝑆 Inf
˜′𝑗𝑎𝑀 𝑛 = ˜𝑗𝑎𝑀 + 𝜀𝑈˜𝑗𝑎𝑀
(︀ 𝑛 𝑏
)︀
Temp layoff, UI ineligible: 𝜇 (1 − 𝜁) 𝑈
⏟ ⏞
temp layoff not expired, no UI or UI expired
)︀(︀ 𝑦 𝑦 )︀
˜𝑗𝑎𝑀
𝑛 ˜𝑗𝑎𝑀
𝑏 ˜ ˜𝑏
(︀ (︀ 𝑛 )︀
− 𝑈 + 𝜀𝑈 𝜎𝑀 𝐼 + 𝜎𝑀 𝑅 + 𝑈𝑗𝑎𝑆 + 𝜀𝑈𝑗𝑎𝑆 Inf
Because type I (Infected Severe) workers do not work, there are no employed workers in this health
group, i.e. 𝜇′𝑗𝑎𝐼𝑒 = 0. If a type M worker becomes type I while employed, she becomes permanently
12
separated with UI. Next period’s distribution of type I agents:
(︀ 𝑔 𝑔 )︀ 𝑔
YOLF or Old (𝑔 ∈ {𝑦, 𝑜}): 𝜇′𝑔𝐼 = 𝜇𝑔𝐼 − 𝜇𝑔𝐼 𝜎𝐼𝑅 + 𝜎𝐼𝐷 + 𝜇𝑔𝑀 𝜎𝑀 𝐼
Perm unemp, UI eligible: 𝜇′𝑗𝑎𝐼𝑏 = 𝑏
(1 − 𝜀)𝑈𝑗𝑎𝐼 + ˜𝑗𝑎𝐼
𝜁(1 − 𝜀)𝑈 𝑏
⏟ ⏞ ⏟ ⏞
perm unemp, UI not expired temp layoff expired, UI not expired
[︂ ]︂
(︀ 𝑦 𝑦 )︀
𝑏 ˜ 𝑏
− (1 − 𝜀)𝑈𝑗𝑎𝐼 + 𝜁(1 − 𝜀)𝑈𝑗𝑎𝐼 𝜎𝐼𝑅 + 𝜎𝐼𝐷
[︂ ]︂
𝑏 ˜𝑗𝑎𝑀
𝑏 𝑦
+ (1 − 𝜀)𝑈𝑗𝑎𝑀 + (1 − 𝜀)𝑈 + 𝐸𝑗𝑎𝑀 𝜎𝑀 𝐼
⏟ ⏞
employed type M turns type I
𝜇′𝑗𝑎𝐼𝑛 = ˜𝑗𝑎𝐼 + 𝜀𝑈
˜𝑗𝑎𝐼
(︀ 𝑛 𝑏
)︀ (︀ 𝑛 𝑏
)︀
Perm unemp, UI ineligible: 𝑈𝑗𝑎𝐼 + 𝜀𝑈𝑗𝑎𝐼 + 𝜁 𝑈
⏟ ⏞ ⏟ ⏞
perm unemp, no UI or UI expired temp layoff expired, no UI or UI expired
[︂ ]︂
)︀ (︀ 𝑦 𝑦 )︀
𝑛 𝑏 ˜ 𝑛 ˜ 𝑏
(︀ )︀ (︀
− 𝑈𝑗𝑎𝐼 + 𝜀𝑈𝑗𝑎𝐼 + 𝜁 𝑈𝑗𝑎𝐼 + 𝜀𝑈𝑗𝑎𝐼 𝜎𝐼𝑅 + 𝜎𝐼𝐷
[︂ ]︂
)︀ 𝑦
˜ ˜
(︀ 𝑛 𝑏
)︀ (︀ 𝑛 𝑏
+ 𝑈𝑗𝑎𝑀 + 𝜀𝑈𝑗𝑎𝑀 + 𝜁 𝑈𝑗𝑎𝑀 + 𝜀𝑈𝑗𝑎𝑀 𝜎𝑀 𝐼
Temp layoff, UI eligible: ˜′𝑗𝑎𝐼𝑏 =
𝜇 ˜𝑗𝑎𝐼
(1 − 𝜁)(1 − 𝜀)𝑈 𝑏
⏟ ⏞
temp layoff not expired, UI not expired
(︀ 𝑦 𝑦 )︀ 𝑦
˜𝑗𝑎𝐼
−(1 − 𝜁)(1 − 𝜀)𝑈 𝑏
𝜎𝐼𝑅 + 𝜎𝐼𝐷 ˜𝑗𝑎𝑀
+ (1 − 𝜁)(1 − 𝜀)𝑈 𝑏
𝜎𝑀 𝐼
′ ˜ ˜
(︀ 𝑛 𝑏
)︀
Temp layoff, UI ineligible: 𝜇
˜𝑗𝑎𝐼𝑛 = (1 − 𝜁) 𝑈𝑗𝑎𝐼 + 𝜀𝑈𝑗𝑎𝐼
⏟ ⏞
temp layoff not expired, no UI or UI expired
)︀(︀ 𝑦 𝑦 )︀ )︀ 𝑦
˜𝑗𝑎𝐼 + 𝜀𝑈
˜𝑗𝑎𝐼 ˜𝑗𝑎𝑀 + 𝜀𝑈
˜𝑗𝑎𝑀
(︀ 𝑛 𝑏
(︀ 𝑛 𝑏
−(1 − 𝜁) 𝑈 𝜎𝐼𝑅 + 𝜎𝐼𝐷 + (1 − 𝜁) 𝑈 𝜎𝑀 𝐼
13
Next period’s distribution of type R (Recovered) agents:
𝑔 𝑔
YOLF or Old (𝑔 ∈ {𝑦, 𝑜}): 𝜇′𝑔𝑅 = 𝜇𝑔𝑅 + 𝜇𝑔𝐴 𝜎𝑀 𝑅 + 𝜇𝑔𝐼 𝜎𝐼𝑅
𝑦
Employed: 𝜇′𝑗𝑎𝑅𝑒 = 𝐸𝑗𝑎𝑅 + 𝐸𝑗𝑎𝐴 𝜎𝑀 𝑅
Perm unemp, UI eligible: 𝜇′𝑗𝑎𝑅𝑏 = 𝑏
(1 − 𝜀)𝑈𝑗𝑎𝑅 + ˜𝑗𝑎𝑅
𝜁(1 − 𝜀)𝑈 𝑏
⏟ ⏞ ⏟ ⏞
perm unemp, UI not expired temp layoff expired, UI not expired
[︂ ]︂ [︂ ]︂
𝑏 ˜ 𝑏 𝑦 𝑏 ˜ 𝑏 𝑦
+ (1 − 𝜀)𝑈𝑗𝑎𝑀 + (1 − 𝜀)𝑈𝑗𝑎𝑀 𝜎𝑀 𝑅 + (1 − 𝜀)𝑈𝑗𝑎𝐼 + (1 − 𝜀)𝑈𝑗𝑎𝐼 𝜎𝐼𝑅
𝜇′𝑗𝑎𝑅𝑛 = 𝑏 ˜𝑗𝑎𝑅 + 𝜀𝑈˜𝑗𝑎𝐼
(︀ 𝑛 )︀ (︀ 𝑛 𝑏
)︀
Perm unemp, UI ineligible: 𝑈𝑗𝑎𝑅 + 𝜀𝑈𝑗𝑎𝑅 + 𝜁 𝑈
⏟ ⏞ ⏟ ⏞
perm unemp, no UI or UI expired temp layoff expired, no UI or UI expired
[︂ ]︂ [︂
)︀ 𝑦
˜ ˜ 𝑏
𝑛
(︀ 𝑏
)︀ (︀ 𝑛 𝑏
(︀ 𝑛 )︀
+ 𝑈𝑗𝑎𝑀 + 𝜀𝑈𝑗𝑎𝑀 + 𝜁 𝑈𝑗𝑎𝑀 + 𝜀𝑈𝑗𝑎𝐼𝑀 𝜎𝑀 𝑅 + 𝑈𝑗𝑎𝐼 + 𝜀𝑈𝑗𝑎𝐼
]︂
)︀ 𝑦
˜𝑗𝑎𝐼 𝜎
˜𝑗𝑎𝐼 + 𝜀𝑈 𝑏
(︀ 𝑛
+𝜁 𝑈 𝐼𝑅
Temp layoff, UI eligible: ˜′𝑗𝑎𝑅𝑏 =
𝜇 ˜𝑗𝑎𝑅
(1 − 𝜁)(1 − 𝜀)𝑈 𝑏
⏟ ⏞
temp layoff not expired, UI not expired
˜𝑗𝑎𝑀
𝑏 𝑦 ˜𝑏 𝑦
+(1 − 𝜁)(1 − 𝜀)𝑈 𝜎𝑀 𝑅 + (1 − 𝜁)(1 − 𝜀)𝑈𝑗𝑎𝐼 𝜎𝐼𝑅
˜′𝑗𝑎𝑅𝑛 = ˜𝑗𝑎𝑅 + 𝜀𝑈
˜𝑗𝑎𝑅
(︀ 𝑛 𝑏
)︀
Temp layoff, UI ineligible: 𝜇 (1 − 𝜁) 𝑈
⏟ ⏞
temp layoff not expired, no UI or UI expired
)︀ 𝑦 )︀ 𝑦
˜𝑗𝑎𝑀 + 𝜀𝑈
˜𝑗𝑎𝑀 ˜𝑗𝑎𝑅 + 𝜀𝑈
˜𝑗𝑎𝑅
(︀ 𝑛 𝑏
(︀ 𝑛 𝑏
+(1 − 𝜁) 𝑈 𝜎𝑀 𝑅 + (1 − 𝜁) 𝑈 𝜎𝐼𝑅
Finally, next period’s measure of Dead agents:
𝑔
YOLF or Old (𝑔 ∈ {𝑦, 𝑜}): 𝜇′𝑔𝐷 = 𝜇𝑔𝐷 + 𝜇𝑔𝐼 𝜎𝐼𝐷 (B.13)
∑︁ ∑︁ (︀ )︀ 𝑦
Workers of sector 𝑗 : 𝜇′𝑗𝐷 = 𝜇𝑗𝐷 + 𝜇𝑗𝑎𝐼𝜔 + 𝜇
˜𝑗𝑎𝐼𝜔 𝜎𝐼𝐷 (B.14)
𝑎 𝜔{𝑏,𝑛}
Both Recovered and Dead are absorbing states.
14
C. Results Appendix
C.1. Additional figures for Section 3
Figure C.1 shows the weekly UI benefit level for different wage income levels, with and without the
$600 benefit top-up:
𝑏𝑗,𝑎 = min{𝜂 · 𝑤𝑗 𝑎, 𝑏𝑢𝑏 } + 𝑏𝑡𝑜𝑝 .
The flat portion represents the calibrated upper bound on benefits level 𝑏𝑢𝑏 . The highlighted part of the
curve presents income levels where UI income with the $600 top-up is higher than wage income.
Figure C.1: Weekly UI benefit level for different wage levels (in dollars)
4000
pre-CARES Act
45-degree
3000 median income ($687)
UI income
post-CARES Act
2000
1000
0
500 1000 1500 2000 2500 3000 3500
weekly wage income
Figure C.2 shows the calibrated probability of receiving UI for newly unemployed workers (𝜆𝑗𝑎 )
and the model-generated UI recipient rate over relative wages. Because UI benefits may expire before
a worker finds a job, and because workers with and without UI search differently, the recipient rate
differs from 𝜆.
Figure C.2: Probability of receiving UI and UI recipient rate over wages
25
parameter
UI recipient rate
20
15
%
10
5
0
0 0.5 1 1.5 2 2.5 3 3.5
Relative income
Figure C.3 shows the calibrated path for the shutdown policy 𝑚𝑡 , the probability that a separation
is temporary, and the CARES UI policy. Note that the last panel shows the eligibility expansion, cap-
tured by increases in the probability that newly unemployed workers receive UI benefits. Because the
parameter 𝜆𝑗𝑎 depends on income, it takes a range in the pre- and post-pandemic steady states with-
15
out the CARES UI policy. We use the shaded areas to show this range. With the eligibility expansion
introduced by CARES UI during the pandemic, the parameter increases to 0.98 for everyone.
Figure C.3: Calibrated parameter path
Shutdown policy, m
0.1 1
0.08 0.8
0.06 0.6
0.04 0.4
0.02 0.2
0 0
03/21/20 07/31/20 12/31/20 03/21/20 07/31/20 12/31/20
FPUC ($600), btop PEUC (13wk extension), 1/ PUA (elig expansion), ja
1 60 1
0.8 0.8
40
0.6 0.6
0.4 0.4
20
0.2 0.2
0 0 0
03/21/20 07/31/20 12/31/20 03/21/20 07/31/20 12/31/20 03/21/20 07/31/20 12/31/20
16
C.2. Additional results for Section 4.1
Table C.1 shows the unemployment and total cumulative deaths under each policy scenario and the
computed total effect of CARES UI and shutdown and the computed effect of CARES UI only. All
scenarios reported here are with infection risk.
Table C.1: Unemployment and deaths in difference policy scenarios
Policy scenarios Apr–Dec 2020 Avg Total Cumulative
Unemployment (%) Deaths (%)
a. No shutdown, No CARES UI 7.79 0.21
b. shutdown only 11.54 0.19
c. CARES UI only 8.61 0.21
d. shutdown + CARES UI 13.16 0.19
Effect on Avg Effect on Total
Unemployment (ppt) Cumulative Deaths (%)
Effect of CARES UI & Shutdown (d-a) 5.36 -7.81
Effect of CARES UI (d-b) 1.61 -2.09
Note: The first panel reports the average unemployment and total cumulative deaths under each policy sce-
nario. The second panel shows the computed effects of CARES UI and shutdown, and of CARES UI only. All
policy scenarios are with infection risk.
17
C.3. Robustness check results for Section 4.6
C.3.1 Alternative health calibration: Larger shares of type M agents
Figure C.4 shows the health and unemployment dynamics under the assumption of larger shares of
type M agents. We use 𝜎𝑀 𝐼 = 0.2 here (𝜎𝑀 𝐼 = 0.5 in the baseline calibration) and re-calibrate the
𝑦 𝑜
health parameters. The re-calibration gives 𝜎𝐼𝐷 = 0.625% * 7/18, 𝜎𝐼𝐷 = 12.5% * 7/18, 𝜌 = 1.1, 𝜌𝑒 = 2.43
and 𝛾 = 0.6.
18
Figure C.4: Health and unemployment with larger shares of Infected Mild agents
(A) Health distribution
Susceptible (type S) Infected Mild (type M) Infected Severe (type I)
80 0.8
1.5
60 0.6
% % 1 %
40 0.4
0.5
20 0.2
0 0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
Recovered (type R) Dead (type D)
40 0.2
30 0.15
% %
20 0.1
10 0.05
0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
(B) Unemployment rate (C) Temporary-unemployment ratio
20 1
0.8
15
0.6
%
10 0.4
0.2
5
0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
(D) Sector unemployment rate
contact non-contact
30 30
25 25
20 20
% %
15 15
10 10
5 5
04/19/20 07/28/20 11/05/20 02/13/21 04/19/20 07/28/20 11/05/20 02/13/21
no shutdown, no CARES UI shutdown only shutdown+CARES UI
19
C.3.2 Alternative health calibration: Different initial size of infected population
Table C.2 compares the effects of CARES UI with different assumptions about the size of the initial
infected population. In the baseline we assume 0.02% of population are type M at the start of simulation.
Alternatively, we use 0.01% and 0.03% and re-calibrate the health parameters. With 0.01%, the re-
𝑦 𝑜
calibration gives 𝜎𝐼𝐷 = 0.25% * 7/18, 𝜎𝐼𝐷 = 5% * 7/18, 𝜌 = 1.05, 𝜌𝑒 = 3.49, 𝛾 = 0.41. With 0.03%, the
𝑦 𝑜
re-calibration gives 𝜎𝐼𝐷 = 0.25% * 7/18, 𝜎𝐼𝐷 = 5% * 7/18, 𝜌 = 0.79, 𝜌𝑒 = 2.63, 𝛾 = 0.58. Figures C.5
and C.6 show the health and unemployment dynamics under the alternative assumptions, which are
similar to the baseline.
Table C.2: Effects of CARES UI under different assumptions about initial infection
Assumptions about Initial share of type M Effect on Apr–Dec 2020 Effect on Total
Avg Unemployment (ppt) Cumulative Deaths (%)
Baseline (0.02% of population are type M) 1.61 -2.09
Smaller initial infection (0.01% of population) 1.64 -2.40
Larger initial infection (0.03% of population) 1.60 -1.95
Note: Effect of CARES UI (with shutdown) is calculated relative to shutdown only without CARES UI. The pol-
icy effect is expressed in percent terms for cumulative deaths, and in percentage points for unemployment rate.
The alternative scenarios are re-calibrated to match the same set of targets (especially deaths) as in the baseline
calibration.
20
Figure C.5: Health and unemployment with smaller share of initial infection
(0.01% of population type M)
(A) Health distribution
Susceptible (type S) Infected Mild (type M) Infected Severe (type I)
1.5
80
1.5
1
60
% % % 1
40
0.5
0.5
20
0 0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
Recovered (type R) Dead (type D)
40 0.2
30 0.15
% 20
%
0.1
10 0.05
0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
(B) Unemployment rate (B) Temporary-unemployment ratio
20 1
0.8
15
0.6
%
10 0.4
0.2
5
0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
(C) Sector unemployment rate
contact non-contact
25 25
20 20
% 15 % 15
10 10
5 5
04/19/20 07/28/20 11/05/20 02/13/21 04/19/20 07/28/20 11/05/20 02/13/21
no shutdown, no CARES UI shutdown only shutdown+CARES UI
21
Figure C.6: Health and unemployment with larger share of initial infection
(0.03% of population type M)
(A) Health distribution
Susceptible (type S) Infected Mild (type M) Infected Severe (type I)
2 2.5
80
2
1.5
60 1.5
% % 1 %
40 1
0.5
20 0.5
0 0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
Recovered (type R) Dead (type D)
50
0.25
40
0.2
30 0.15
% %
20 0.1
10 0.05
0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
(B) Unemployment rate (C) Temporary-unemployment ratio
20 1
0.8
15
0.6
%
10 0.4
0.2
5
0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
(C) Sector unemployment rate
contact non-contact
25 25
20 20
% 15 % 15
10 10
5 5
04/19/20 07/28/20 11/05/20 02/13/21 04/19/20 07/28/20 11/05/20 02/13/21
no shutdown, no CARES UI shutdown only shutdown+CARES UI
22
C.3.3 Workplace infection in the non-contact sector
Figure C.7 shows the health and unemployment dynamics when workers in the non-contact sector also
get infected at workplace. We assume the per-contact infection rate in the non-contact sector is also
𝜌𝑒 , the same as the contact sector, and we re-calibrate the health parameters. The re-calibration gives
𝑦 𝑜
𝜎𝐼𝐷 = 0.25% * 7/18, 𝜎𝐼𝐷 = 5% * 7/18, 𝜌 = 0.92, 𝜌𝑒 = 2.32 and 𝛾 = 0.5.
23
Figure C.7: Health and unemployment with workplace infection also in non-contact sector
(A) Health distribution
Susceptible (type S) Infected Mild (type M) Infected Severe (type I)
2
1.5
80
1.5
60 1
% % % 1
40
0.5
20 0.5
0 0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
Recovered (type R) Dead (type D)
0.25
40
0.2
30
0.15
% %
20
0.1
10 0.05
0 0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
(B) Unemployment rate (C) Temporary-unemployment ratio
1
0.8
15
0.6
%
10 0.4
0.2
5
0
04/01/20 07/01/20 10/01/20 01/01/21 04/01/21 04/01/20 07/01/20 10/01/20 01/01/21 04/01/21
(D) Sector unemployment rate
contact non-contact
25 25
20 20
% 15 % 15
10 10
5 5
04/19/20 07/28/20 11/05/20 02/13/21 04/19/20 07/28/20 11/05/20 02/13/21
no shutdown, no CARES UI shutdown only shutdown+CARES UI
24
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