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Optimal Unemployment
Insurance Requirements
Gustavo de Souza and André Victor D. Luduvice
REVISED
March 2023
WP 2022-45
https://doi.org/10.21033/wp-2022-45
*Working papers are not edited, and all opinions are the
responsibility of the author(s). The views expressed do not
necessarily reflect the views of the Federal Reserve Bank
of Chicago or the Federal Reserve System.
Optimal Unemployment Insurance Requirements*
Gustavo de Souza†
André Victor D. Luduvice‡
Federal Reserve Bank of Chicago
Federal Reserve Bank of Cleveland
This version: March 2023
Abstract
In the US, workers must satisfy two requirements to receive unemployment insurance (UI): a tenure
requirement of a minimum work spell and a monetary requirement of past minimum earnings. Using
discontinuity of UI rules at state borders, we find that the monetary requirement decreases the number of
employers and the share of part-time workers, while the tenure requirement has the opposite effect. In a
quantitative model, the monetary requirement induces workers to stay longer in unemployment because
low-paying jobs are not covered by UI. Since it mitigates moral hazard, the optimal UI design has a high
monetary requirement.
Keywords: Unemployment Insurance, UI Eligibility, Optimal UI
JEL Codes: E24, E61, J65
* First version:
October 2020. We want to thank the participants at the Brazilian Econometric Society, FRB Cleveland,
and Washington State University seminars and the SEA 2021, LuBraMacro 2022, and the Midwest Macro Dallas 2022.
The views expressed herein are solely those of the authors and do not necessarily reflect the views of the Federal
Reserve Bank of Chicago, the Federal Reserve Bank of Cleveland or the Federal Reserve System. All errors are our
own.
† E-mail: gustavo@microtomacro.net
‡ E-mail: andrevictor.luduvice@clev.frb.org.
1
Introduction
Unemployment insurance (UI) programs have attracted economists’ attention both because of their moral hazard component and because of their widespread presence in
many modern economies (Vodopivec and Raju, 2002). The design of such programs is
usually characterized by three main elements: a replacement ratio, that is, the percentage
of a past wage thatthe worker receives during the unemployment spell; a limit on how
many months the worker can collect such benefits; and some requirement related to the
worker’s labor market history that deems her eligible to enroll in the program. In this
paper, we study empirically and quantitatively how UI requirements in the US affect the
labor market and use a model to compute the optimal requirement levels.
We first conduct an empirical analysis to understand the effect of the UI requirements
on unemployment, the UI take-up, and labor market dynamics. Gathering data from
states’ UI policies since 1963, we use discontinuities in the UI requirement at state borders
to identify the causal effect of the tenure and monetary requirements on the labor market.1
We find that the introduction of a tenure requirement decreases unemployment, increases
the number of employers and the share of part-time workers, and does not affect the UI
take-up rate. On the other hand, the monetary requirement has a negative effect on the
number of employers and the share of part-time workers, with a significant and negative
effect on the UI recipient rate. These results indicate that both requirements significantly
affect the pool of unemployed workers but with opposite effects when it comes to labor
market dynamics, a finding that could have different implications for the fiscal burden of
the UI as well as its effectiveness in providing insurance for workers.
To understand these effects, we build a quantitative model to rationalize the empirical
results and find the optimal level of requirements for UI. We develop an infinite horizon model with incomplete markets, heterogeneous agents, labor market imperfections,
and a history-dependent UI program. The program in the model economy has monetary
and tenure requirements that closely mimic those of the US design. Workers choose con1 In
our empirical analysis we follow a methodology similar to the one introduced by Hagedorn et al.
(2016a).
1
sumption, savings, and the extensive margin of labor supply. They receive idiosyncratic
labor productivity shocks and unemployment shocks. Workers can collect UI benefits after becoming unemployed if they satisfy both the tenure and the monetary requirements.
The tenure requirement is a function of workers’ past labor supply, while the monetary
requirement is a function of workers’ past salaries. Therefore, UI is a function of the relevant employment history of workers. Following Hopenhayn and Nicolini (2009), the
government cannot perfectly distinguish quits from layoffs, and, with some probability,
workers quitting their job can end up receiving UI, thus capturing one of the potential
moral hazard components of UI.
To compute the model, we introduce a methodology that reduces the infinite-dimensional
state space defined by the sequences of UI recipiency, labor income, and labor supply
histories. By focusing on the relevant part of their allocation and collection history, we
can rewrite the workers’ problem in a tractable manner while still keeping it historydependent and rich enough to accommodate the details of the requirements of the UI
program and allow the computation of its optimal arrangement.
We show that the model can replicate the qualitative effect of changes in UI requirements on the labor market as found in our empirical analysis. An increase in the monetary
requirement reduces workers’ incentives to stay at low-paying jobs because these jobs are
no longer covered by UI. Therefore, to avoid jobs not protected by UI, workers decide
when to take jobs, and when they do so, they are less likely to leave them. Therefore, an
increase in the monetary requirement decreases employment transition and the share of
workers at low-paying jobs. An increase in the tenure requirement, on the other hand,
increases workers’ incentives to stay at such jobs. A worker who received an unemployment shock but still does not satisfy the tenure requirement has incentives to stay in a
low-paying job until the tenure requirement is satisfied and she is covered by UI. Therefore, an increase in the tenure requirement increases employment transition and the share
of workers at low-paying jobs.
We then conduct our quantitative normative analysis and find that the optimal unemployment insurance design has a large monetary requirement and a low tenure require2
ment. Because the monetary requirement reduces the share of workers at jobs for which
the monetary threshold is binding, it reduces the share of workers quitting their job and
reduces the financial cost of UI. The tenure requirement, on the other hand, induces workers to stay at any job, even ones with low pay and subject to the monetary requirement,
and, at a risk, to quit to receive UI. We find that an increase in the monetary requirement
from 10 percent to 23 percent of average earnings and a decrease in the tenure requirement from 24 weeks to 12 weeks would increase welfare by 0.99 percent while reducing
UI expenditure by half of its benchmark value. We also extended our analysis to include
other channels such as general equilibrium effects and changes in job transition probabilities and we find that the main intuition and effect of our optimality results are preserved.
This paper is organized as follows. In the next section, we discuss the related literature. In Section 3, we show the empirical evidence obtained on the effect of UI requirements on employment outcomes. In Section 4, we construct the setting of our quantitative
model, provide intuition about the underlying theory, and define all relevant model objects. In Section 5, we describe the calibration used to map the model to the data. Section
6 presents the results for the benchmark economy and the properties of the initial steady
state. Section 7 lays out the thought experiment and the results of counterfactual analyses. In Section 8, we conduct a normative analysis and search for the optimal UI policy
within the model environment. The last section states our conclusions.
2
Related Literature
This paper builds on the literature that assesses numerically the effects of unemployment
insurance policies on the labor market. One of the earliest and most influential references is Hansen and Imrohoroğlu (1992), who construct a quantitative incomplete markets model with moral hazard to analyze the optimal replacement ratio. In a similar environment but focusing on the search-theoretic component, Gomes et al. (2001) study the
welfare cost of business cycles. Pallage and Zimmermann (2001) extend the same model
setting to include heterogeneity of skills and study voting preferences toward UI gen-
3
erosity. Abdulkadiroğlu et al. (2002) move one step further in the canonical framework
by making unemployment insurance depend on how much time the agent has been in an
unemployed state. The authors find that the optimal UI has to decrease the number of
weeks that an agent can collect UI but it should be more extensive if agents are prevented
from saving. Young (2004) incorporates search effort in the numerical analysis and finds
that the optimal replacement rate should be zero independent of the limits on benefit
duration and that eliminating the UI system generates welfare gains, which get smaller
if one takes the transition into account. Lentz (2009) further confirms the importance of
including transitional dynamics by estimating a search model of optimal UI policy with
Danish data.
Moving to job search environments, employment requirements are known to distort
job search while employed since early work by Burdett (1979) and Mortensen (1977).
Zhang and Faig (2012) study UI eligibility in a Diamond-Mortensen-Pissarides environment and find a Ricardian equivalence type of result in which taxation of risk-neutral
households annuls the job creation effect of the employment requirement. A similar approach of quantitatively identifying the UI design in a heterogeneous agents economy but
with an endogenization of the labor market in a search and matching framework taken by
Mukoyama (2013). Mazur (2016) identifies large welfare gains from a policy that allows
quitters to receive UI benefits.
There’s a recent strand of the literature focusing specifically on UI requirements in
quantitative settings. In a standard search and matching model Andersen et al. (2018)
show how the employment requirement strengthens the re-entitlement effects, thus altering the incentives of the optimal UI scheme. Another key reference is Auray et al. (2019),
who show that endogeneizing the take-up rate of UI-eligibe workers slows the impact of
changes in the benefit on the unemployment rate and duration of unemployment. More
recently, Birinci and See (2019) analyze the UI eligibility and recipiency along the wealth
distribution and find that in a heterogeneous agents job search model the insurance benefits are larger for wealth-poor workers. In a contemporaneous paper, Chao et al. (2021)
study whether paying unemployment insurance affects the value of unemployment. The
4
authors use a regression discontinuity design to help uncover the causal effect of UI that
is confounded by the endogeneity of eligibility. They focus on the lower bound of eligibility and find that UI eligibility has a causal effect on next period earnings that ranges
from $300 to $900. They use a competitive search model to interpret their results as better
match quality and higher rents in light of the endogenous UI take-up. Their results are
complementary to ours and in line with our empirical evidence and our model showing
the relevance of UI eligibility for workers’ labor market outcomes.
We contribute to this branch of the literature by developing a quantitative model of
optimal unemployment insurance, incorporating a history-dependent UI program. In
our model, to become eligible for UI benefits, agents must satisfy the UI requirements,
which test households’ labor market history, which needs to be present in the workers’
state space. Moreover, we focus on optimizing such requirements, a question that is, to
our knowledge, still open in the literature.
Some of the empirical work on UI requirements and eligibility dates back to Blank and
Card (1991). The empirical strategy used in our econometric evidence can be tied in with
what is now a rich literature that analyzes changes in UI features in the context of recessions. More specifically, the literature studies the extension of UI benefits granted for up
to 73 weeks during the Great Recession. We discuss the institutional background details
in Subsection 3.1. Marinescu (2017), for example, studies the general equilibrium effects
of the extension on job applicants and vacancies. The methodology of using discontinuity in state borders is also present in a sequence of papers that use it to assess the labor
market and equilibrium effects of the recession (Hagedorn et al., 2016a,b, 2019). Recent
quantitative and theoretical approaches studying the relationship between UI and recessions are Mitman and Rabinovich (2015) and Pei and Xie (2021). Our main contribution
to this branch of the literature is applying this established methodology to the context of
UI requirements, a novel approach to the best of our knowledge.
Our paper is also connected to a long tradition of theoretical papers on optimal unemployment insurance design. Two of the earliest contributions can be found in the seminal
works by Shavell and Weiss (1979) and Wang and Williamson (1996). In an environment
5
with a repeated principal-agent problem, Hopenhayn and Nicolini (1997) characterize
the optimal contract and lay out a result that is currently well-known in the literature:
the optimal payment schedule involves a replacement ratio that decreases during the unemployment spell and a re-employment wage tax that increases with the length of such
a period. Other seminal references on the search incentives generated by UI are Chetty
(2008) and Shimer and Werning (2008).
In Hopenhayn and Nicolini (2009) the authors amplify the environment in Hopenhayn and Nicolini (1997) to account for multiple unemployment spells with asymmetric
information in order to study the optimality of the eligibility condition common in UI
programs. The paper’s main result is that, if the principal cannot distinguish quits from
layoffs, it is optimal for the principal to condition benefit payments on the agent’s employment history. A similar idea is present, though not as an endogenous outcome, with
an ad hoc formulation of the environment with the experience rating component used by
Wang and Williamson (2002) and with the effective entitlement effect erasing the moral
hazard in employment flows as in the recent work by Zhang and Pan (2019). Auray and
Fuller (2020) expand on experience rated taxes paid by firms and how endogenous UI collection costs slow the impact of changes in the UI policy. In our environment we search
for a quantitative design of a UI program that is characterized by an eligibility condition
and thus, we follow Hopenhayn and Nicolini (2009) in assuming that the government in
our model has the same informational limitation as in their paper.
3
Empirical Evidence
In this section we study empirically how UI requirements affect the labor market in the
US. In Subsection 3.1 we briefly describe the institutional background of unemployment
insurance in the different states. We describe our data in 3.2 and the empirical strategy
used in 3.3. We then outline the results of the econometric exercises in 3.4.
6
3.1
Institutional Background
Unemployment insurance in the US is regulated by the federal government, administered
by the states, and paid weekly to workers who have lost their jobs through no fault of their
own. The eligibility requirements beyond the determination of the reason for the job loss
are established by the laws of each state in reference to a base period, which is usually the
first four of the last five completed calendar quarters prior to the time the claim is filed.
The majority of the states fund the program through a tax imposed on employers.2
One of the last substantial revisions to the federal UI law was in 2009 with the American Recovery and Reinvestment Act, which largely extended the duration of benefits due
to the Great Recession with the emergency unemployment compensation (EUC) and extended benefits (EB) measures. The baseline period for most of the states was 26 weeks,
which the act extended by an additional 13 to 20 weeks. The most recent significant revision was in 2012, which extended the programs from the previous revision and added
provisions on self-employment eligibility and the possibility of short-term compensation
for employers. The EUC and EB were last extended until 2014. As mentioned previously,
these revisions have focused solely on the duration of the payment of benefits and have
received due attention in the literature.3
3.2
Data
We conduct our econometric analysis using data from the IPUMS repository of the Annual Social and Economic Supplement (ASEC) of the Current Population Survey (CPS)
between 1963 and 2016 (Flood et al., 2021). We combine the labor market statistics from
this sample with data on state unemployment insurance laws taken from the US Department of Labor (USDL). A more detailed description of the data can be found in Appendix
A.
2 The
US Department of Labor provides further details on the legislation and broader components of the
UI regulation. The website can be found via this link.
3 See our review in Section 2.
7
3.3
Empirical Strategy
We identify the causal effect of UI requirements using changes in state policy. The potential peril of this approach is the possibility of state-level shocks that correlate with such
changes in UI policy. For instance, if policymakers refrain from increasing UI requirements during recessions or if they change the tenure requirements according to the state
average job duration, a traditional difference-in-difference strategy would be biased. We
then would not be able to tease apart the effect of state-level shocks from the effect of
policy reforms.
To deal with the potential endogeneity of UI requirements, we use discontinuity of
UI requirements at state borders at the MSA level. The idea is that MSAs that span different sides of a state border are subject to the same shock but different UI policies. The
specification of our econometric model is:4
0
yi,m,s,b,t = β M Is,t {Monetary Req.} + β T Is,t {Tenure Req.} + Xi,m,s,b,t
θ + µm + γb,t + ei,m,s,b,t
(1)
where yi,m,s,b,t is a labor market outcome of agent i, in MSA m and state s, which is a
member of the border pair identified by b at time t. Is,t {Monetary Req.} is a dummy
taking a value of one if state s in year t has a monetary requirement, and Is,t {Tenure Req.}
is a dummy taking a value of one if state s has a tenure requirement for UI in year t.
0
Finally, Xi,m,s,b,t
is a set of controls, µm is an MSA fixed effect, and γb,t is a border-year
fixed effect. The set of controls used in our regression results are workers’ age, years
of education, gender, race, and marital status. Standard errors are clustered at the MSA
level.
If shocks that lead to changes in UI requirements are continuous over state borders,
it should affect the two sides of a border pair b, and hence are captured by the fixed
effect γb,t . This guarantees the identification of the two coefficients of interest, β M and β T ,
4 For a similar methodology, see Dube et al. (2010), Hanson and Rohlin (2011), Hagedorn et al. (2016a),
and Hagedorn et al. (2016b).
8
which capture the impact of the monetary and the tenure requirements.
3.4
Results
Table 1 shows that UI requirements affect unemployment, unemployment benefit recipient rates, the kinds of jobs individuals take, and number of employers.
Table 1: Effect of UI requirements on the labor market
I {Monetary Req.}
I {Tenure Req.}
N
R2
(1)
I {Unemployed}
(2)
I {UnempBene f it}
(3)
I { PartTime}
(4)
#Employers
-0.0191***
(0.000)
-0.0531***
(0.000)
-0.0256***
(0.000913)
-0.000561
(0.00231)
-0.121***
(0.00167)
0.0324***
(0.00485)
-0.0121***
(0.00336)
0.0857***
(0.00586)
136617
0.051
225853
0.902
130404
0.100
137744
0.051
Notes: This table shows the estimated parameters of model (1). Labor data are from the CPS, and UI requirement data are hand collected from reports of the US Department of Labor. The sample is from 1963 to 2016
and the number of observations varies according to variable availability. I {Unemployed} is a dummy taking
a value of one if the worker is employed, I {UnempBene f it} is a dummy taking a value of one if the worker
received UI in the current year, I { PartTime} is a dummy taking one if the worker worked at a part-time job
in the current year, and #Employers is the number of employers the worker had in the current year. Standard
errors are clustered at the MSA level. See details in Appendix A.
Column 1 of Table 1 shows that the introduction of a monetary requirement reduces
unemployment by 1.9 percent, while the introduction of a tenure requirement reduces
unemployment by 5.3 percent. Since the introduction of requirements tends to preclude
certain workers from accessing unemployment insurance, it is reasonable to expect that
removing workers from UI would then give them an incentive to work and cause a decrease in unemployment. The direction of our result holds similarly with the effect of a
decline in the unemployment and job finding rates stemming from cuts in the duration
of unemployment insurance as in Johnston and Mas (2018) and Karahan et al. (2019). As
9
shown in Column 2 of Table 1 above, we find that the monetary requirement has a significant and negative effect on UI benefit applications, whereas the tenure requirement
has a negative but small and non-significant coefficient. The introduction of a monetary
requirement reduces access to UI benefits by 2.5 percent.
The UI requirements also affect the type of jobs workers take and employment dynamics. As depicted in Column 3 of Table 1, the introduction of a monetary requirement
leads to a reduction in the share of part-time workers, while the tenure requirement increases part-time jobs. Since the monetary requirement directly establishes a minimum
UI eligibility wage, it disincentives workers to take low-paying jobs, such as part-time
jobs. The tenure requirement has a positive effect on part-time employment because even
low-paying jobs can count toward workers’ eligibility. Column 4 shows that the monetary
requirement reduces the number of employers that workers had, while the tenure requirement increases it. Therefore, a monetary requirement seems to create longer employment
spells, while the tenure requirement contributes to more transitions in employment.5
In the next section we use a model to interpret this empirical result. We show that a
monetary requirement leads workers to wait longer to accept a job and avoid low-paying
jobs. As a consequence, workers do not move across employers and do not stay in parttime jobs. The tenure requirement, on the other hand, leads workers to accept low-paying
jobs just so they become eligible for UI. As a consequence, it increases the share of parttime workers and transitions in employment.
4
The Model
This section describes the model we use to analyze the optimal degree of unemployment
insurance requirements in the US economy. The environment is an infinite horizon economy in partial equilibrium with incomplete markets and individual heterogeneity, discrete labor supply, and UI system that depends on the employment history of workers.
5 In Appendix C we show different robustness checks to our main analysis, with the inclusion of controls
for other state policy reforms, to running the regressions at the county level, and to the use of marginal
variations in a continuous measure of the requirements.
10
4.1
Preferences
The economy is populated by a continuum of households with a time-separable period
utility function. Households are risk-averse and maximize their discounted expected lifetime utility from non-durable goods consumption c and labor supply n ∈ {0, 1} with β
as the discount factor. They have access to incomplete markets and can choose to accumulate assets a ≥ b to protect themselves against idiosyncratic shocks, where b is their
borrowing limit.
4.2
Technology
There is a single good produced in this economy with technology given by a CobbDouglas production function that exhibits constant returns to scale, Y = F (Kt , Nt ) =
Ktα Nt1−α , where α ∈ (0, 1) is the output share of capital income and Yt , Kt and Nt denote,
respectively, aggregate output, physical capital, and labor. The final good can be consumed or invested in physical capital on a one-to-one basis. A competitive representative
firm uses the technology to produce Yt and rents capital at rt and labor at wt . Since the
benchmark model economy is going to be analyzed in a partial equilibrium setting, the
interest rate rt will be exogenously fixed at the steady state, with value r ∗ . From the firm’s
first-order conditions, we recover w∗ . Details are standard and can be found in Appendix
B.1.
4.3
Endowments and Labor Income
Agents are born with zero assets and endowed with one unit of time. Households can
be either employed or unemployed. In either case, the household receives two types of
shocks: an unemployment or employment shock, pu or pe , respectively, and a productivity shock, z. There is no aggregate uncertainty. The component z is persistent and
follows an AR(1) process defined by zt+1 = ρzt + ε t , with ε t ∼ N (0, σε2 ). We discretize
it in a Markov chain with transition matrix πz,z0 = Pr(z j+1 = z0 |z j = z) and stationary
distribution Π(z).
11
An employed worker with productivity z in the previous period receives at the beginning of the current period an unemployment shock with probability pu and a productivity shock z0 ∈ {z1 , ..., z N } with probability Π̄(z0 |z). While an agent is working, individual
earnings depend on the competitive wage wt and the idiosyncratic persistent component.
On the other hand, an unemployed household in the previous period receives at the beginning of the current period an employment shock with probability pe and a productivity
shock z ∈ {z1 , ..., z N } with probability Π̃(z0 ).
We interpret each different shock z > 0 as the given productivity in the same job and
assume that if the agent receives an unemployment shock, then z = 0 and the agent is
laid off. Hence, in accordance with the current US unemployment insurance code, agents
quitting the labor force, i.e., z > 0 and n = 0, should not receive an unemployment
benefit, while only agents with z = 0 who meet the required eligibility criteria are able to
collect it. Without loss of generality, we can collapse all shocks into one vector and rewrite
the effective labor income process transition as Π(z0 |z), where z0 ∈ {0, z1 , ..., z N }.
4.4
Unemployment Insurance and Moral Hazard
The unemployment insurance program is designed to approximate the UI regulation in
the US. It is a function of past labor force participation, the salary of workers, and how
long a worker has already received unemployment insurance. Denote yt = wt zt nt as the
labor income at time t,and nt as the labor supply at time t, and mt is a counter of the
number of periods a worker received UI at time t. We denote bU I (ñt , ỹt , m̃t ) as the UI
benefit of an agent with labor supply history ñt = {n0 , n1 , ..., nt }, labor income history
ỹt = {y0 , y1 , ..., yt }, and UI history m̃t = {m0 , m1 , ..., mt }.6
The government pays and monitors UI benefits, bU I , which amount to a percentage of
the average past earnings characterized by a replacement ratio θ ∈ [0, 1]. It does so for a
limited number of periods {0, . . . , µb }, with µb ∈ N. It requires a minimum number of
6 For
the history variables {ñt , ỹt , m̃t }, we use here the notation with subscript t in order to facilitate
the comprehension of the definition by making explicit the history component. However, throughout the
paper, we follow the usual convention that omits the time index for individual-level variables and use it
solely for aggregate variables.
12
consecutive periods of work to be eligible for the program, µn ∈ N, as well as a minimum
threshold zmin ∈ R+ on the workers’ average earnings. Thus, the UI design is defined by
the tuple {θ, zmin , µn , µb }. We make the assumption that, in the model economy, every
worker satisfying all requirements of the UI program will automatically receive the program benefits.
Workers are also subject to two additional shocks. With probability ϕ, workers quitting the labor force, i.e., with n = 0 and z > 0, are not detected by UI authorities and
receive the UI benefit if they satisfy the UI requirements. We thus call ϕ a moral hazard
shock. Second, in order for the model to be consistent with the heterogeneity in benefit duration, there is an exogenous loss from the benefit shock: with probability η the
unemployed agent loses her UI benefit.
The government monitors the UI system and takes account of workers’ labor market
history in determining UI eligibility. In the case where a worker is caught defrauding UI,
all of her labor market history that counts toward eligibility is erased. The monitoring
agency does not keep track of job offers or workers’ decisions to take jobs after they enter
a UI spell.7 Finally, following Hopenhayn and Nicolini (2009) we impose an extra informational limitation and assume that the government cannot perfectly distinguish quits
from layoffs.8
4.5
Government
The government runs the UI system and determines its budget. The total revenue and
expenditure of the UI system are defined, respectively, by RevU I,t and ExpU I,t . On top of
that, the government issues a social security transfer Tu,t paid to all unemployed households. There is an endogenous level of aggregate expenditure Gt , which is defined residually by what is left to balance the total government’s budget. Finally, the government
7 The monitoring is conducted via a system of points that the worker accrues toward UI eligibility, which
is defined by the way we track labor market history in the state space of the problem. For now we abstract
from the notation details for the sake of better exposition of the worker’s problem but revisit them in detail
in Appendix B.2.
8 This is a condition similar to the one in Hopenhayn and Nicolini (2009) that guarantees that the eligibility requirement will arise as part of the optimal mechanism in their repeated moral hazard environment.
13
taxes labor income with an exogenously calibrated flat rate τ using the collected revenue
to fund all expenses. The government’s budget constraint is given by:
Gt + Tu,t + ExpU I,t = τ (rKt + wNt ) + RevU I,t
(2)
The universal transfer Tu in this context is important for two reasons: First, it accounts
for the fact that the government has other financial duties beyond its expenditures on unemployment insurance. Second, since the government provides insurance through other
welfare and social programs, the transfer Tu is auxiliary in interpreting the numerical
results derived in this paper as accounting for the desire for redistribution and the risk
protection provided by UI that exists on top of these other programs. This transfer works
then as a reduced-form version of the transfer programs to which households in the US
have access via the income security system.
4.6
Recursive Household Problem
t −1
Households are heterogeneous with respect to their labor income history, ỹ = w j e j n j j=0 ∈
t −1
t −1
Y t−1 , labor supply history, ñ = n j j=0 ∈ N t−1 , their UI benefit history, m̃ = m j j=0 ∈
N t−1 , their idiosyncratic productivity shock, z ∈ Z , and their asset holdings a ∈ A. The
state space of the economy is then the set S = A × Z × N t−1 × Y t−1 × Mt−1 . The individual state space is s = ( a, z, ñ, ỹ, m̃) ∈ S. Let vn (s) be the value function of an agent
dependent on the agent’s labor supply decision, n ∈ {0, 1}. Below we define all possible
value functions of the worker.
Value Function and Labor Supply Decision:
Given a certain individual state space s,
we can determine the final value function and the labor supply decision by the following
maximization:
vn ( a, z, ñ, ỹ, m̃) = max{vn=1 ( a, z, ñ, ỹ, m̃), vn=0 ( a, z, ñ, ỹ, m̃)}
14
(3)
Value Function if Working:
If an agent works, i.e., n = 1, the value function is given by
vn=1 ( a, z, ñ, ỹ, m̃) = max u(c, 1) + β pu vn=0 ( a0 , 0, ñ0 , ỹ0 , m̃0 )+
c,a0
0
0 0 0 0
0
(1 − pu ) ∑ π̃ z, z vn ( a , z , ñ , ỹ , m̃ )
z0 ∈Z
s.t.
(4)
c + a0 = (1 + (1 − τ )r ) a + (1 − τ )wz
ñ0 = {ñ, 1},
ỹ0 = {ỹ, wz},
m̃0 = {m̃, 0},
c > 0,
a0 ≥ b
where c is consumption, τ is the income tax, and wz is labor income.
Value Function if Laid Off:
If a worker is laid off i.e., z = 0, she receives unemployment
insurance bU I (ñ, ỹ, m̃) if she is eligible. Let 1(ñ, ỹ, m̃) be a dummy taking a value of one
if the worker is eligible for UI. The value function of a laid-off worker is
vn=0 ( a, 0, ñ, ỹ, m̃) = max u(c, 0) + β (1 − pe )vn=0 ( a0 , 0, ñ0 , ỹ0 , m̃0 )+
c,a0
0
0 0 0 0
0
pe ∑ π̄ z, z vn ( a , z , ñ , ỹ , m̃ )
z0 ∈Z
s.t.
(5)
c + a0 = (1 + (1 − τ )r ) a + Tu + bU I (ñ, ỹ, m̃)
ñ0 = {ñ, 0},
ỹ0 = {ỹ, 0},
m̃0 = {m̃, 1(ñ, ỹ, m̃)},
c > 0,
a0 ≥ b
where, differently from the employed worker, a laid-off worker receives welfare transfer
Tu and UI benefit bU I (ñ, ỹ, m̃), which is zero if the worker is not eligible.
15
Value Function if Quitting: If a worker quits, i.e., z > 0 and n = 0, with probability
ϕ he receives UI if eligible and with probability 1 − ϕ he does not receive UI. The value
function of a worker who quits and receives UI is
I
vU
n=0 ( a, z, ñ, ỹ, m̃ | z
I
0
0 0
0
u(c, 0) + β (1 − pe )vU
> 0) = max
n=0 ( a , z, ñ , ỹ , m̃ | z > 0)+
c,a0
0
0 0 0 0
0
pe ∑ π̄ z, z vn ( a , z , ñ , ỹ , m̃ )
z0 ∈Z
s.t.
(6)
c + a0 = (1 + (1 − τ )r ) a + Tu + bU I (ñ, ỹ, m̃)
ñ0 = {ñ, 0},
ỹ0 = {ỹ, 0},
m̃0 = {m̃, 1(ñ, ỹ, m̃)},
c > 0,
a0 ≥ b
With probability 1 − ϕ the worker is caught by the UI authority and does not receive
UI benefits. Moreover, we assume that as punishment for attempting to defraud the UI
system, the worker loses her eligibility to receive UI in the following periods. In our
notation, this is equivalent to having the worker’s labor market history erased. The value
function of a worker who quits and does not receive UI is
I
vnU
n=0 ( a, z, ñ, ỹ, m̃ | z
I 0
0 0
0
> 0) = max
u(c, 0) + β (1 − pe )vnU
n=0 ( a , z, ñ , ỹ , m̃ | z > 0)+
c,a0
0
0 0 0 0
0
pe ∑ π̄ z, z vn ( a , z , ñ , ỹ , m̃ )
z0 ∈Z
s.t.
(7)
c + a0 = (1 + (1 − τ )r ) a + Tu
ñ0 = {0},
ỹ0 = {0},
m̃0 = {0},
c > 0,
a0 ≥ b
Therefore, using (6) and (7) we can write the value function of a worker who quits:
16
I
vn=0 ( a, z, ñ, ỹ, m̃|z > 0) = ϕvU
n=0 ( a, z, ñ, ỹ, m̃ | z > 0)+
(8)
I
(1 − ϕ)vnU
n=0 ( a, z, ñ, ỹ, m̃ | z > 0)
Value Function if Receiving UI: If the worker has already received UI in the last period,
with probability η the worker keeps receiving UI and with probability (1 − η ), the worker
loses the benefit. The value function of a worker who is already receiving UI and has kept
receiving it is defined below
I
vU
n=0 ( a, z, ñ, ỹ, m̃ | m
I
0
0 0
0
= 1) = max
u(c, 0) + β (1 − pe )vU
n=0 ( a , z, ñ , ỹ , m̃ | m = 1)+
0
c,a
0
0 0 0 0
0
pe ∑ π̄ z, z vn ( a , z , ñ , ỹ , m̃ )
z0 ∈Z
s.t.
(9)
c + a0 = (1 + (1 − τ )r ) a + Tu + bU I (ñ, ỹ, m̃)
ñ0 = {ñ, 0},
ỹ0 = {ỹ, 0},
m̃0 = {m̃, 1(ñ, ỹ, m̃)},
c > 0,
a0 ≥ b
where we also know that the benefit, bU I (ñ, ỹ, m̃), does not depend anymore on the totality of the worker’s labor market history, rather only on the last level of earnings yt before
the worker started receiving benefits and the benefit recipient’s history, m̃. The last level
of earnings suffices for the monetary requirement constraint and determines the effective
replacement earnings during the entire unemployment spell.
The value function of a worker who is already receiving UI and has been randomly
drawn out of the UI benefit pool is given by the problem below
17
I
vnU
n=0 ( a, z, ñ, ỹ, m̃ | m
I 0
0 0
0
= 1) = max
u(c, 0) + β (1 − pe )vnU
n=0 ( a , z, ñ , ỹ , m̃ | m = 1)+
c,a0
0
0 0 0 0
0
pe ∑ π̄ z, z vn ( a , z , ñ , ỹ , m̃ )
z0 ∈Z
s.t.
(10)
c + a0 = (1 + (1 − τ )r ) a + Tu
ñ0 = {ñ, 0},
ỹ0 = {ỹ, 0},
m̃0 = {m̃, 0},
c > 0,
a0 ≥ b
Therefore, using (9) and (10) we can write the value function of a worker who is already receiving UI:
I
vn=0 ( a, z, ñ, ỹ, m̃|m = 1) =ηvU
n=0 ( a, z, ñ, ỹ, m̃ | m = 1)+
(11)
I
(1 − η )vnU
n=0 ( a, z, ñ, ỹ, m̃ | m = 1)
The solution of the dynamic programs (3) to (10) yields the decision rules for the asset
holdings a : S → R+ , consumption c : S → R++ , and labor supply n : S → {0, 1}.
In the Bellman equations written above, we have kept the notation initially introduced
for exposition purposes and to make it simpler to write down the recursive household
problem. Nonetheless, the state space contains the agents’ full labor supply, income history, and UI benefit histories, (ñ, ỹ, m̃). Given our infinite time horizon, these become
infinite dimensional objects in the steady state of the model. Hence, in the way we have
defined the problem so far a numerical solution is, by construction, intractable.
In Appendix B.2 we explain how to reduce the state space to make solving the problem feasible. The principle behind the arithmetic we describe relies on only keeping track
of the agents’ relevant labor supply history and average earnings. Furthermore, the full
definition of the partial equilibrium of the economy, together with the details on the cal18
culations of the aggregates is described in B.3.
5
Calibration
5.1
Timing, Preferences, and Technology
We define the time period of the model to be equivalent to 6 weeks. The period utility is
isoelastic in consumption c and separable with respect to the labor supply n:
u(c, n) =
c 1− γ
− χn
1−γ
(12)
where γ is the coefficient of relative risk aversion and χ controls the disutility of labor. We
calibrate the latter to match the average labor force participation (LFP) of the US economy
computed as the average between 1979 and 2014 taken from the Bureau of Labor Statistics
data for men 20 years or older. We set γ = 1, hence assuming log(c) form throughout our
numerical exercises. We endogenously calibrate β to match the average wealth to income
ratio in the US, which is taken from our own calculations using the Survey of Consumer
Finances (SCF) of 2010 and 2013.
Following Cooley (1995) we set the capital share α to 0.36, a value already standard in
the literature. We exogenously set the partial equilibrium interest rate r ∗ to the six-week
value that is equivalent to 2 percent per year. For the depreciation rate of capital δ, we
follow Gomes et al. (2001) and set it to the six-week value that is equivalent to 5 percent
per year.
5.2
Endowments and Labor Income
We prevent all workers from borrowing; hence, we set b = 0. We follow Gomes et al.
(2001) and calibrate the persistence ρ and the error variance σε2 of the AR(1) process governing the labor income shock to 0.9 and 0.052, respectively. The probability of finding
19
a job pe is calibrated endogenously to simultaneously match the average duration of UI
benefits and the measure exhausting the number of payments of UI benefits. We also calibrate endogenously the probability of losing a job pu set to match average job destruction.
Our target is based on our calculation of job losers as a share of the population from 1991
to 2014 using data from the US Department of Labor (USDL).9
5.3
Unemployment Insurance and Government
The values for the parameters governing the UI system in the benchmark economy are
calibrated exogenously. The replacement ratio θ is set to 0.4641, which is the US average
from 1989 to 2011 as provided by the USDL. The monetary requirement zmin is defined
exogenously to be 0.5573, which is the numerical value for the 4th largest shock in the grid
we use to discretize the AR(1) process in the computation. In the model, this is equivalent
to 10 percent of the average of six-weeks’ earnings. The maximum number of weeks
that workers receive the benefit µb is 24, which is the closest number consistent with the
average of 26 weeks as reported by the USDL. This is equivalent to 4 model periods,
which would otherwise amount to 30 weeks, had we considered 5 model periods. The
same number of weeks is also required for households to attain the work tenure eligibility
requirement µt .
The remaining parameters are all calibrated endogenously. The universal transfer Tu
is calibrated to match the average transfer to unemployment over average labor income
in the data. We calculate the target level for this statistic from the American Community
Survey (IPUMS-ACS). The moral hazard shock ϕ is chosen to target the share of agents
receiving UI in the US. The reference value for this moment is, once again, calculated
from the average of the USDL data from 1991 to 2014. Last, the loss of benefit shock, η,
together with pe mentioned previously, targets the average duration of UI benefits and
the measure of workers exhausting UI benefits.
9 In
Appendix D.1 and D.2, we conduct robustness checks on the effect of the changes in the calibrated
values of pe and pu on some of the benchmark aggregate statistics and the optimal policy.
20
5.4
Summary of Calibration
We summarize the information associated with the calibrated parameters in the sequence
of tables below. In Table 2, one can find the exogenously calibrated parameters and their
sources. Table 3 shows the endogenously calibrated parameters, the targeted moments
associated with each of them, the source of such moments for their data counterparts,
and the value of such statistics computed for the model economy.
Table 2: Exogenously calibrated parameters
Parameter
Value
Timing
Model’s period
t
{1, . . . ∞}
Preferences
Relative risk aversion
γ
1.0
Technology
Capital share
Interest rate
Depreciation of K
α
r
δ
0.36
0.002
0.006
{ρ, σε2 }
0.900, 0.052
θ
zmin
µb
µt
0.4641
0.5573
24
24
Labor Income
Persistence and variance of AR(1)
Government and UI
Replacement ratio
Monetary requirement
Maximum benefit periods
Eligibility requirement
Target / Source
6 weeks (Hansen and Imrohoroğlu, 1992)
Standard
Cooley (1995)
≈ 2% per year
≈ 5% per year (Gomes et al., 2001)
Gomes et al. (2001)
US Department of Labor
10% of avg six-weeks earnings
US Department of Labor
US States Data
Notes: The table shows model parameters, their numerical values, targeted moments in the model economy, and their
data sources.
21
Table 3: Endogenously calibrated parameters
Parameter
Value
Target
Data
Model
Preferences and Government
Discount factor
Labor disutility
Transfer to unemployed
β
χ
Tu
0.9974
0.3343
0.049
Wealth/Income
Labor force participation rate
Transfer to Unemp/Average Lab. Inc.
2.5
0.762
0.009
2.5
0.819
0.009
Labor Market Shocks
Probability of job offer
Probability of losing job
pe
pu
0.7977
0.0031
Weeks receiving UI & shr. exhaust. UI
Job destruction
16 & 0.371
0.028
16.1 & 0.371
0.028
Moral Hazard Shocks
Probability of UI benefit w/o being fired
Probability of losing UI exogenously
ϕ
η
0.078
0.1975
Share of agents receiving UI
Weeks receiving UI & shr. exhaust. UI
0.011
16 & 0.371
0.011
16.1 & 0.371
Notes: The table shows model parameters, their numerical values, and targeted moments in the model economy.
6
The Benchmark Economy
In Table 3 we have shown that the model is able to succesfully match the targeted data
moments. In particular, the relevant moments are exactly matched with the only exception being the labor force participation rate, which is about 6 percentage points higher
than in the data. This can be rationalized by the fact that the universal transfer Tu determined by the calibration target is relatively small and households have only UI as a source
of income beyond their own return to work. As in the design of the UI, there are explicit
incentives for households to work: the structure of the economy is one in which the forces
toward labor force participation compete directly with the preference for leisure.
For a better validation of our model, we show in Table 4 some selected non-targeted
model moments we consider relevant for our environment to be well specified for the
quantitative experiments. We can observe that the model is able to closely replicate the
mean unemployment duration and the share of workers excluded by the monetary requirement. Notice that the unemployment duration is around 22 weeks, with an average
duration that is 8 years longer than the average period in which workers receive the UI
benefit. Hence, the model is able to be precise about the dynamic behavior of the pool of
unemployed workers in and out of the labor market and of the insurance system.
22
The replication of the share of unemployed workers excluded from the benefit by the
monetary requirement is key to our analysis as we have exogenously set the requirement
zmin to an arbitrarily small level defined by one of the initial points in our discretized
shock process. As we match closely the 3.8 percent share of workers who fall in this
category, we can be reassured that for the current computation, the monetary requirement
has the desired outcome within the model mechanism.10
The total expenditure by GDP is also at a level close to what is observed in the US
data. This small size of the UI program is also able to be achieved through the existence
of the universal unemployment transfer Tu , which helps households to have the correct
amount of income and insurance. Hence, with income, job displacement, and moral hazard shocks, the incompleteness of the market ends up self-selecting workers into the UI
program, thus determining the size of the insurance value given to workers through that
channel. Finally, the tenure requirement excludes fewer workers than what is observed
in the data.
Table 4: Non-targeted moments of the benchmark economy
Statistic
Data
Model
Mean Unemployment Duration
22.50
22.77
UI Expenditure/GDP
0.72%
0.55%
Share Excluded by Mon. Req.
3.8%
3.5%
Share Excluded by Tenure Req.
8.6%
2.9%
Notes: The table displays non-targeted moments computed in
the data and in the model. The mean unemployment duration
is denoted in weeks.
10 The
share of workers excluded by the monetary requirement in the data is calculated using the CPS
ASEC, the only part of this survey that has data starting in 1962. For the numbers shown in Table 4,
specifically, we take data from 2000 to 2015 and merge them with our collected database for the monetary
requirement. We then compare it to workers’ weekly earnings and compute the share that is excluded by
the threshold. More details on our calculation of the minimum weekly earnings are described in Appendix
A.
23
6.1
Connection with the Empirical Evidence
In order to understand the connection of the model with what we measured in the data in
our empirical analysis, we also calculate the change in the share of workers who receive
the unemployment insurance benefit in our model economy whenever we introduce UI
requirements. The idea is to generate an effect similar to that captured in Column 2 of Table 1 for UI take-up. For the monetary requirement, for instance, we calculate the change
in the statistic between two model economies, one in which we start it at zero and the one
with the initially calibrated level of the monetary requirement. For comparative statics
used in these counterfactual changes, we keep the partial equilibrium concept and the
initial calibration but vary only the requirement parameters while allowing an extra tax
levied on payroll to adjust the unemployment insurance budget.11
We show the results in Table 5. In our empirical analysis, we found that there is a
decrease of 2.5 percent in the UI beneficiary pool that stems from the introduction of a
monetary requirement. In our model, we preserve the same direction and magnitude of
the effect, with negative 2.71 percent. For the tenure requirement, the empirical effect
measured is null and non-significant; hence, we would expect the model to yield a small
effect. This is also true in our benchmark economy if we move the tenure requirement
from the minimum period allowed, i.e., 6 weeks, to the initially calibrated period of 24
weeks. This change generates a decrease of -0.06 percent in the share of insured between
the two economies.
Table 5: Comparison between the model and the empirical analysis
Statistic
Data
Model
Change in Share Insured with Mon. Req
-2.5%
-2.71%
Change in Share Insured with Tenure. Req
-0.0%
-0.06%
Notes: The table displays non-targeted moments computed in our empirical analysis and percent differences in the statistic between a counterfactual economy and the benchmark economy.
11 This
is identical to the thought experiment in 7.1, which we use in the comparative analyses in the next
section.
24
7
Counterfactual Analyses
In this section we outline the results of the counterfactual exercises conducted highlighting the impacts on the moral hazard component, job-taking behavior, and exhaustion of
UI benefits. First, we describe the thought experiment and the adaptations in the model
required to conduct the counterfactuals. Second, we analyze in Subsection 7.2 the effect of
all elements of the UI design on different types of employment. In Subsection 7.3, we discuss the effects of the tenure and monetary requirements on the moral hazard component
of our economy.
7.1
Thought Experiment
The idea behind the counterfactual exercises of changing the design of UI can be described
as follows: we vary the value of the policy instrument, say the monetary requirement
zmin , while keeping all other parameters of the UI constant. Naturally, the change of
regime in this ceteris paribus fashion will affect the endogenous spending and revenues
of the UI budget. In order to impose discipline on the government’s administration of
the program, we keep Tu and G fixed at their benchmark numerical level. We then add
an endogenous payroll tax τU I to finance any residual UI financing needs and close the
government’s budget. The budget constraint of the household then becomes:
c + a0 = (1 + (1 − τ )r ) a + (1 − τ − τU I )wzn + (1 − n) Tu + (1 − n)bU I (ñ, ỹ, m̃)
(13)
We also need to update the budget constraint of the government under this new
regime to add the revenue accrued from the payroll tax τU I . We can compute it as follows:
25
G + Tu +
τU I =
Z
S
b
UI
∗
∗
(ñ, ỹ) dΦ(s) − τ r K + w L +
w∗ L
Z
S
b
UI
(ñ, ỹ, m̃) dΦ(s)
(14)
We use the notion of a partial equilibrium of the model economy as described in Appendix B.3 and add τU I as an endogenous equilibrium object in our equilibrium definition. We do so by modifying the condition for the government budget constraint described in Appendix B.3 and substituting it with (14). The solution algorithm to find the
partial equilibrium now consists of iterating on the underlying fixed point defined by the
budget-clearing rate τU I . In Appendix D, we generalize the thought experiment above,
allowing for a full general equilibrium analysis, and compare the results with our main
findings.
7.2
Effects of the UI Design on Employment
In Figure 1, we show the effect on employment of varying each of the UI design parameters for two of the lowest levels of productivity, namely, z = 0.23, the lowest positive
level, and z = 0.56, the level for which we calibrated zmin in the benchmark economy.
The reason we show only this range is that from the mid-level of the idiosyncratic shock,
workers are productive enough so that they basically do not react to changes in the policy
instruments in a quantitatively significant manner. The three lowest levels of productivity
are where we identify relevant behavioral responses.
First, we observe that the share of employed workers by type changes in a way consistent with what was observed in the empirical correlations shown in Table 1. Both the
monetary and the tenure requirement in the model exhibit a sign and magnitude of their
impact that are in accordance with what we have measured in our regressions. More
specifically, the tenure requirement is positively associated with employment outcomes,
whereas the monetary requirement negatively impacts the outcomes. Moreover, the numerical order of magnitude also seems to be preserved, as the tenure requirement has a
26
smaller effect overall than the monetary requirement when measured for the same statistic.
At the benchmark level, the monetary requirement is approximately 0.56. We can
then see in the top left panel that it is exactly where both lines cross the zero level. It is
possible to observe that the negative relationship observed in the data also happens in
the model, since when the monetary requirement is made stricter or set higher, there is an
overall decrease in the employment rate, especially for the lowest level of productivity.
Though not monotone, the intuition follows that a looser monetary requirement for a
low-productivity worker essentially makes access to the UI benefit easier, which ends up
enlarging the option value of working, together with its built-in work incentives, vis-a-vis
the cost of supplying labor.
27
Figure 1: Percent variation in employment by type for different levels of UI instruments
Notes: The figure shows the share of employment by type along the ranges considered for each UI policy
element. The top panels show the variation for the monetary and tenure requirement, respectively. The
bottom panels show the variation for the replacement ratio and the benefit duration, respectively. The
solid line shows the employment share for a worker with the lowest level of productivity and the dashed
line shows the share for a worker with the third level of productivity. The latter is the level to which the
monetary requirement is calibrated in the benchmark economy.
Second, we can find an even clearer correlation when analyzing the relative changes in
employment due to the changes in the number of weeks of the tenure requirement. Once
again, as we start from 22 weeks, we can see in the top right panel that both lines for
each of the productivity levels cross zero at that number. Similarly to what we observed
in our empirical evidence, the more demanding the tenure requirement, the higher is
the share of employment by type. The intuition for this result is straightforward: with a
higher number of weeks required to be able to receive UI benefits, workers have an extra
incentive to remain attached to the labor force. Aside from the lower marginal cost of
supplying labor, the benefits are based on the average past earnings; hence, the higher
28
the productivity, even at the very bottom of the distribution, the larger the incentive to
work.
Overall, we can observe that the monetary requirement and the replacement ratio have
the strongest impact on the employment rate when making changes at the benchmark
level. The replacement ratio has different effects depending on workers’ productivity.
Among the ones depicted, we can observe that the higher the productivity, the more positive the impact of a smaller replacement ratio on the employment share of that type. With
higher productivity, a worker has more incentive to actually participate in the labor force
if the UI benefit is a smaller fraction of the worker’s earnings. Finally, the benefit duration µb has an unambiguous impact on workers: the longer the duration of the payment
stream, the smaller the incentive for households to work.
7.3
The Effects of UI Requirements on Moral Hazard
In Figure 2, we show, from top to bottom on the graph, respectively, the effect of changing
the value of the tenure and the monetary requirement on workers who are defrauding
UI benefits, the job opportunities, and the measure of workers who have exhausted the
number of periods for which they are eligible to receive the benefit.
A longer tenure requirement, i.e., a higher µt , has virtually no effect on all outcomes,
which is shown by the small percentage of impact on the shares in the figures. Despite
the small quantitative impact, we can see in the bottom graph that the measure of unemployed workers who completely exhaust their UI benefits increases as we require more
weeks of work to satisfy the tenure requirement. The intuition for this result is the following: as the requirement gets stricter, unemployed workers seizing their UI benefits realize
it is better to stay in that state until the last possible period in which the insurance is paid.
On the top graph, one can verify that a stricter monetary requirement, a higher zmin ,
overall reduces the share of workers who defraud UI over the share of workers entering
the UI program. With the lowest possible requirement, such a share achieves a level in
which about 50 percent of the workers receiving UI should not be able to receive such
29
benefits. Conversely, if we allow that level to be the highest possible level of idiosyncratic
productivity, it is possible to decrease this share to zero.
Figure 2: Moral hazard and the tenure and monetary requirement
Notes: The figure shows in the top panels the share of workers defrauding UI over entrants into UI, in
the middle panels the share of workers with a job opportunity, and in the bottom panels the measure of
workers exhausting all available periods of UI recipiency. All of those are shown along different levels for
the tenure requirement in the left column and for the monetary requirement in the right column.
The same inverse relationship happens with the share of workers receiving a job opportunity. With a loose monetary requirement, several workers stay in the unemployment
state due to the possibility of collecting undue benefits and thus more than 10 percent of
them are receiving opportunities for the lowest values of the requirement. Once again, by
choosing the strictest possible value of the requirement, one can reduce such a share to
zero. Finally, the monetary requirement is positively correlated with the share of households exhausting UI benefits. But the numerical range of the effect on this measure is
30
smaller than in the previously reported shares. From the last panel, it is clear that a
stricter monetary requirement makes it increasingly worthwhile for workers to use the
whole time span in which they are entitled to UI benefits, since the chance of receiving
them again is lower.
8
Optimal Policy Analysis
We conduct an optimal policy analysis by finding the UI design that maximizes welfare
in the economy described. In order to do so, we define a utilitarian social welfare function
(SWF) dependent on all relevant policy parameters as follows:
W (θ, µt , µb , zmin ) =
Z
v∗ ( a, z, n̄, m, ȳ | θ, µt , µb , zmin ) dµ∗
(15)
where {v∗ , µ∗ } are, respectively, the value function and distribution associated with a
stationary partial equilibrium.
Essentially, we hold constant the income tax τ and the expenditure components G and
Tu , as if they were fixed by the government at t = 0, and find the combination of static UI
policy parameters that optimize the social welfare function subject to its being consistent
with a stationary partial equilibrium. This set of partial equilibria to which the planner
restricts her attention will be the one defined by the household’s optimization together
with the government’s budget constraint balanced by τU I as shown in Subsection 7.1.
In light of this reasoning, the restricted social planner’s problem is thus defined as:
max
{θ,µt ,µb ,zmin }∈Γ
W (θ, µt , µb , zmin )
(16)
where Γ is the restricted set of policies for which an associated stationary partial equilibrium exists.
We report the welfare gain in terms of the consumption equivalent variation (CEV).
31
This measure defines the increment in consumption that we would need to give households in each state of the world so that they would be indifferent between their benchmark level of consumption and their level of consumption in the alternative economies.
We do so by calculating the household’s ex-ante value, hence under the veil of ignorance.
The CEV is defined for our environment as follows:
CEV (θ, µt , µb , zmin ) = 100 ∗ {exp [(1 − β) (W (θ, µt , µb , zmin ) − Wbchmk )] − 1}
(17)
where Wbchmk is the SWF associated with the benchmark partial equilibrium parameterized according to Table 2.
We show the results in Table 6 below. When welfare is optimized in each instrument
dimension separately, we can notice that the monetary requirement is the one yielding
the highest CEV, as shown in the fourth column. The intuition for this result comes from
the fact that the monetary requirement prevents households from taking low-wage jobs;
otherwise households would have a high incentive to quit their jobs in order to receive the
benefit and to defraud the UI program. This is confirmed by the fact that such an optimal
outcome exhibits the smallest number of beneficiaries of the program, as shown in the
last row of the table, hence diminishing the overall moral hazard faced by the planner.
The second highest level of welfare gain is achieved through a large reduction in the
replacement ratio from the initial calibration, as shown by the CEV in the first column of
the table. When comparing this equilibrium to the benchmark scenario, it becomes clear
that the effect of such a reduction does not have a large impact on the number of workers
receiving benefits. However, it sharply decreases the overall cost of the program, with the
lowest expenditure/GDP share of all of the other instruments at their optimal level. As
we effectively find that the budget-clearing rate τU I is negative, i.e., a transfer, the smaller
size of the UI system comes with a lower effective tax rate on payroll when taking into
account the wedge already imposed by τ.
32
Table 6: Optimal policies and statistics for each of the UI program instruments
Optimal Policies
Replacement Ratio
Benefit Duration
Monetary Requirement
Tenure Requirement
Statistics
CEV
Expenditure/GDP
Beneficiaries
Benchmark
Replacement Ratio
Benefit Duration
Monetary Requirement
Tenure Requirement
Requirements
0.46
24
10%
24
0.15
24
10%
24
0.46
6
10%
24
0.46
24
23%
24
0.46
24
10%
54
0.46
24
23%
12
0
0.55%
1.1%
0.83%
0.18%
1.07%
0.58%
0.42%
0.40%
.88%
0.31%
0.25%
0.26%
0.50%
0.93%
0.99%
0.22%
0.27%
Notes: The table displays the computed results for the model. The benefit duration and the tenure requirements are denoted in weeks. The monetary requirement is in percentage
of six-week average earnings.
We can observe that the tenure requirement yields small welfare gains when considered separately; it needs to be at a level of 50 or more weeks, beyond what we consider
in the current computation of the model. Nonetheless, when tenure is combined with
the monetary requirement, the planner is able to recover the welfare gains by setting the
latter at a level associated with the ceteris paribus optimal. At the same time, though, the
planner is able to increase the welfare gains by lowering the tenure requirement to half of
the benchmark level.
This result happens because with a high monetary requirement and low-wage workers’ defrauding behavior ruled out, a lower tenure requirement yields fewer workers who
are exhausting their UI benefits, since they anticipate they will be able to benefit from this
policy again by taking another job with less restrictive requirements for the program in
the future. Since the effective transition probabilities, which otherwise would be endogenous in a standard search and matching environment, are an important component of this
mechanism, in Appendix D.2 we allow for small perturbations in the calibrated values of
pe and pu and find that the optimal level of requirements is the same. Also, in Appendix
D.3 we show the direction of change in the results under general equilibrium.
33
9
Conclusion
In this paper, we addressed the question of what are the optimal levels of the two types
of requirements used in the UI benefits program in the US. We developed an infinite
horizon partial equilibrium model with incomplete markets and heterogeneous agents
that includes a UI system that closely mimics the rules observed in the data. The model
has a rich individual state space that includes workers’ assets, idiosyncratic shocks, and
labor supply and income histories. Furthermore, the economy has a structure of shocks
that allows a moral hazard component akin to the one studied in the theoretical literature
about optimal UI design. Our analysis focuses on the impact of changes in the UI policy
instruments on workers’ labor market outcomes
We conducted an empirical analysis to assess the UI requirements’ effects on employment outcomes and obtained stylized facts for our quantitative exercises. We used discontinuities in the UI policies to identify the requirements’ causal effect on different labor
market outcomes. The monetary requirement has a stronger effect than the tenure requirement on discouraging UI benefit applications and a negative effect on the number
of employers and part-time jobs. The tenure requirement has an opposite impact on the
latter. The intuition for these results comes from the fact that a tenure requirement does
not influence workers to stay at the same job. In contrast, the monetary provision gives
workers an incentive to keep high-paying jobs.
We calibrated the model to the US data and conducted a series of counterfactual exercises by following a thought experiment that recovered the balance in the government’s
budget constraint. We were able to recover the negative correlation between the monetary requirement and the employment outcomes and the associated positive correlation
with the tenure requirement. In the results of our exercises, we observe that a stricter
monetary requirement significantly reduces the share of workers entering the UI system
who are not technically qualified to do so. On the other hand, the tenure requirement has
a negligible numerical impact on labor market outcomes subject to moral hazard.
We have maximized a utilitarian social welfare function on a restricted Ramsey prob-
34
lem and assessed the level of CEV associated with the optimal parametric region for the
tuple that characterizes the program. In our results, we found that the highest level of
welfare is achieved by a monetary requirement when instruments are evaluated separately. A combination of the tenure and the monetary requirement can achieve a higher
welfare level than the ceteris paribus optimum.
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39
Appendix
A
Data
The time period used in our sample is from 1963 to 2016. The data on the labor market are taken
from the IPUMS source for the Annual Social and Economic Supplement (ASEC) of the Current
Population Survey (CPS) (Flood et al., 2021). All the data used regarding the unemployment
insurance law for the US states are taken from the US Department of Labor (USDL). We do not
perform any sample selection for the current analysis but rely on demographic controls in our
regressions. For the county-level data, we are able to identify respondents’ county only after
1996, as further discussed in C.2.
A.1
Construction of Historical UI Requirements
Data on UI requirements are hand-collected and standardized from the USDL historical archives.
The USDL collects data on unemployment insurance (UI) laws for all US states and publishes
it in a document called ”Comparison of State Unemployment Laws.” This document provides
detailed information on the financing schemes, eligibility criteria, employer requirements, and
replacement ratios of each state’s UI regulation. The USDL also provides the formula for determining UI eligibility and the minimum earnings required to qualify for UI. We have handcollected and standardized the data on UI requirements for all states since 1950.
Each state has its eligibility formula and criteria. In many cases, it is a function of total earnings in the 6 months before unemployment (called the base period), the earnings on the quarter
with the highest wage, and weeks of employment. But, in general, there are 4 types of eligibility
formulas: (i) multiple of high-quarter earnings; (ii) multiple of weekly benefit amount; (iii) flat
qualifying amount; or (iv) weeks of employment. On the first eligibility formula, multiple of
high-quarter earnings, workers are eligible if their base period earnings are above a multiple of
their highest quarter earnings. For instance, for a worker to be eligible for UI in Alabama in 2006,
she needs her earnings in the base period to be above 1.5 times her earnings in her highest quarter. On the second type of eligibility formula, the multiple of weekly benefit amount, a worker’s
earnings in the base period must be higher than a multiple of its weekly benefit amount. For instance, to qualify for UI in Connecticut in 2006, workers would need a base period earning above
40
40 times their weekly benefit amount. The third type of eligibility formula requires workers to
have earnings in the base period above a certain dollar amount. For instance, West Virginia required workers to have earnings above $2,200 to be eligible for UI. Finally, some states require
workers to have a minimum number of employment weeks to be eligible.
Given how complex and diverse the UI requirements are across states and over time, we need
to make assumptions to make them compatible. We estimate the tenure requirement by calculating the minimum amount of employment weeks a full-time worker continuously employed and
with constant wages would need to become eligible. If a state uses an eligibility formula based
on a multiple of high-quarter wages, the minimum amount of weeks a worker has to work is this
multiple times twelve.12 If the state has a requirement based on a minimum number of weeks or
hours of work, we can directly calculate the tenure requirement. The tenure requirement is one
week if the state is using an eligibility formula based on a flat qualifying amount or a multiple
of the weekly benefit amount.
We estimate the monetary requirement by calculating the minimum weekly earnings that a
full-time worker continuously employed and with constant wages would need to become eligible. We use three statistics to calculate that: the minimum wages needed to qualify in the
highest earning quarter, the minimum wages needed to qualify in the base period, and the minimum weekly wage to qualify, if the state has such a requirement.13 The monetary requirement
is the maximum weekly wage required between the three different requirement types.
A.2
Statistics of UI Requirements
In this section, we show statistics of the tenure and monetary requirements over time. Table 7
shows that the average tenure requirement in the sample is 16 weeks with considerable variation
on it across time and states. The second column shows the monetary requirement in weekly
earnings in 2015 dollars. The average weekly monetary requirement is US$89 with an also large
standard deviation.
Figures 3a and 3b show the average tenure and monetary requirements over time. The tenure
requirement has been increasing since the 50’s and slowing down in more recent years. The mon12 Twelve
is the number of weeks in a quarter. In other words, if a worker needs to have earnings in the base
period equal to c times the earnings in the highest quarter, a worker who has constant weekly earnings needs to
work for c × 12 weeks to become eligible for UI.
13 These variables are already provided by the USDL.
41
etary requirement, on the other hand, has fluctuated significantly over time. The stabilization of
the increase in the two requirements after the 2000’s suggests that the UI eligibility in the US is
at an all-time high.
Table 7: Summary statistics of tenure and monetary requirements.
Mean
Std. Dev.
Median
Min
Max
Tenure Requirement (Weeks)
16.39
7.79
18.00
1.00
36.00
Monetary Requirement (US$)
89.14
48.93
82.35
0.00
531.45
Notes: This table shows statistics of the tenure and monetary requirements. The tenure
requirement is in weeks and the monetary requirement in 2015 dollars.
The average tenure and monetary requirements described in Figures 3a and 3b hide a large
degree of heterogeneity on UI requirement reforms. Figures 4 and 5 show the tenure and monetary requirements over time for different states, respectively, diving deeper into the variation of
the requirements and their reforms. These figures highlight the fact that UI requirements tend to
be sticky but exhibit several state-level reforms that introduce large fluctuations in them. Finally,
to further highlight the heterogeneity of requirements, Figure 6 plots the histogram of tenure and
monetary requirements across our whole sample.
42
Figure 3: Tenure and monetary requirements over time
(a) Tenure Requirement Over Time
(b) Monetary Requirement Over Time
Notes: The figure shows the average tenure and monetary requirements over time. The tenure requirement is in
weeks and the monetary requirement in 2015 dollars.
Figure 4: Tenure requirement over time across different states
Notes: The figure shows the tenure in weeks in different states.
43
Figure 5: Monetary requirement over time across different states
Notes: The figure shows the monetary requirement of different states in 2015 dollars.
Figure 6: Distribution of tenure and monetary requirements
(b) Distribution of monetary requirements
(a) Distribution of tenure requirements in weeks
Notes: The figure plots the histogram of tenure and monetary requirements over the whole sample. The tenure
requirement is in weeks and the monetary requirement in 2015 dollars.
44
B
B.1
Details of the Model
Firm’s Problem
The price of the consumption good is normalized to one and aggregate investment in physical
capital, It , is defined by the following law of motion:
Kt+1 = (1 − δk )Kt + It ,
(18)
where δk is the depreciation rate of physical capital.
This technology is used by a representative firm that behaves competitively, maximizing
profits at every period t by choosing labor and capital given factor prices. The profit maximization problem is:
Πt = max Ktα Nt1−α − wt Nt − (rt + δk )Kt .
Kt ,Nt
(19)
which yields the following first-order conditions:
rt = α
Kt
Nt
α −1
− δk
w t = (1 − α )
Kt
Nt
(20)
α
(21)
In the partial equilibrium setting with an exogenously given r ∗ , we recover Kt /Nt via equation (20) and, thus, use equation (21) to determine the associated steady-state wage rate w∗ .
B.2
Reduction of the State Space
We can start by reducing the size of the agents’ labor supply history ñ. Given that the UI program
needs to keep track of how many periods the agent has been working, the effective time span
needed in the state space at any period t is completely determined by the minimum tenure
requirement µt . Hence, the agent satisfies this requirement if, at any period t, n` = 1 for all
` ∈ {t − µt − 1, ..., t − 1}. Let n̄t ∈ {0, . . . , µt } denote the number of periods the worker has been
t−µ
working during her relevant labor supply history {n` }`=t−t µt −1 . Hence, we can update n̄ with
45
the following law of motion
n̄t+1 = (n̄t + 1)1{nt =1}
(22)
where the notation 1 stands for the indicator function, which also takes into account the fact that
the worker loses all of her labor history points whenever nt = 0.
Analogously, define m̄ ∈ {0, . . . , µt } as the number of periods the worker has been receiving
t−µ
benefits during her relevant UI recipiency history, {m` }`=t−b µ
b −1
. We can then compute the next
period’s benefit eligibility in terms of periods received using the following law of motion:
m̄t+1 = (m̄t + 1)1{m̄t ≤µb }
(23)
With the variables introduced above, we then have collapsed all of the relevant information from
(ñ, m̃) into (n̄, m̄).
In order to tackle the reduction in the dimensionality of the agents’ labor income history ỹ,
we construct the variable ȳ ∈ R++ . It captures the average labor income history of an agent who
has worked for t consecutive periods:
ȳt =
t
yi
, if t ≤ µt
∑
i =1 t
t−µt
∑ y µt +i
i =1
t−µ
t−µ
µt t − i
µt t
+
( µ t + 1 ) t − µ t − i +1 ( µ t + 1 ) t − µ t +1
µt
(24)
y
∑ µit , o.w.
i =1
which can be updated recursively as
ȳt+1 = (1 − τ )wzt
1
t
+ ȳt
t+1
t+1
(25)
With this summary statistic, we can implement a reduction in the state space of the labor income
history from ỹ to ȳ.
46
The set {n̄ = µt } ∩ {m̄ ≤ µb } ∩ {ȳ ≥ zmin } is thus able to fully determine whether the
agent satisfies all requirements to receive UI benefits at a given period and we can then define
an indicator function 1 over it. This allows us to write a proper algebraic characterization of the
formula for the UI benefits bU I (ñ, ỹ, m̃) as it is implemented in the quantitative solution of the
model:
bU I (n̄, m̄, ȳ) = θ ȳ 1{n̄=µt }∩{m̄≤µb }∩{ȳ≥zmin }
(26)
It is important to notice that, following US tax regulations, the benefits are subject to income
taxation. Hence, when we update the next period’s income ȳt+1 according to the law of motion
in (25), we are already including the post-tax income in the state space. Also, the law of motion
takes into account that ȳt+1 will be the last relevant earnings level before the worker enters a UI
spell, as mentioned in problem (9).
Finally, in order to computationally solve our model, it is possible to further collapse a part of
the reduced the state space into one single variable that summarizes the information in (n̄, m̄).
If we take into account that a worker only accrues counts of n̄ whenever she is not accruing
counts of m̄, and vice-versa, we can define a variable j ∈ {1, . . . , Nc }, where Nc = µt + µb , that
summarizes all possible cases of “points” accrued toward benefits.
B.3
Partial Equilibrium
Agents are heterogeneous at each point in time in the state s ∈ S. The agents’ distribution among
the states s is described by a measure of probability Φ defined on subsets of the state space S. Let
(S, B(S), Φ) be a space of probability, where B(S) is the Borel σ-algebra on S. For each ω ⊂ B(S),
Φ(ω ) denotes the fraction of agents who are in ω. There is a transition function M(s, ω ) that
governs the movement over the state space from time t to time t + 1 and that depends on the
invariant probability distribution Π(z) and on the decision rules obtained from the household’s
problem. We define such distributional shares as stationary when Φt+1 = Φt = Φ.
The definition below stands for a stationary equilibrium and we omit the arguments of the
distribution for notational convenience. Furthermore, for expositional purposes, the definition
47
is written using the notation associated with the full state space as initially defined in the description of the model.
Definition 1 (Stationary Recursive Partial Equilibrium). Given a UI program {θ, zmin , µt , µb }, a tax
τ, and exogenous prices {r ∗ , w∗ }, a partial equilibrium for this economy is an allocation of value function
v, policy functions, production plans for the firm {K, N }, residual expenditure G, and universal transfer
Tu , such that:
1. Given prices {r ∗ , w∗ }, the UI program, fiscal policy, and government transfer, v solves the workers’
problems in (4) to (3), and {c, a0 , n} are the associated policy functions;
2. The individual and aggregate behaviors are consistent:
K=
Z
C=
N=
S
a0 (s) dΦ(s)
Z
S
Z
S
c( s) dΦ(s)
z · n(s) dΦ(s)
3. The government’s budget constraint is satisfied:
Z
G + Tu + bU I (ñ, ỹ, m̃) dΦ(s) =
S
Z
Z
Z
∗
0
∗
UI
τ r
at (s) dΦ(s) + w
z · n(s) dΦ(s) + b (ñ, ỹ, m̃) dΦ(s)
S
S
S
4. Given the decision rules, Φ satisfies:
Φ(ω ) =
Z
S
M(s, ω )dΦ, ∀ω ⊂ B(S)
where M : (S, B(S)) → (S, B(S)), can be written as follows:
48
M(s, ω ) =
π
z,z0
, if a0 (s) ∈ A, ñ0 (s) ∈ N t , ỹ0 (s) ∈ Y t , m̃0 (s) ∈ Mt
0 , otherwise.
C
Robustness of the Empirical Analysis
In this section we show that the main empirical results are robust to the addition of controls, to
the variation of requirements at the intensive margin, and to the use of county borders.
C.1
Controls
If the UI requirements correlate with other policy reforms, the estimates in Table 1 would be
biased, capturing the effect of not only UI requirements but also of other policies that correlate
with them and are discontinuous at state borders. To ensure that this is not the case, Table
8 adds as controls the UI replacement rate, minimum wage, and welfare transfers. The main
results shown in Table 1 are preserved.
Table 8: Effect of UI requirements on the labor market controlling for state policies
I {Monetary Req.}
I {Tenure Req.}
Replacement Rate
Minimum Wage
Welfare Transf.
N
R2
(1)
I {Unemployed}
(2)
I {Unemployed}
(3)
I {UnempBene f it}
(4)
I {UnempBene f it}
(5)
I { PartTime}
(6)
I { PartTime}
(7)
#Employers
(8)
#Employers
-0.0331***
(0.00522)
-0.0223***
(0.00365)
-0.0326***
(0.00536)
-0.0244***
(0.00625)
-0.0123***
(0.000305)
0.0321***
(0.000729)
-0.0124***
(0.000442)
0.0322***
(0.00232)
-0.0744***
(0.00198)
0.0799***
(0.00230)
-0.0760***
(0.00207)
0.100***
(0.00966)
0.00803***
(0.00195)
0.116***
(0.00333)
0.00582*
(0.00336)
0.140***
(0.0125)
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
101062
0.047
71853
0.045
167712
0.013
113393
0.014
92774
0.087
67006
0.090
105484
0.048
75484
0.050
Notes: This table shows the estimated parameters of model 1. Labor data are from the CPS and UI requirement data are hand collected from USDL reports. The sample is from 1963 to 2016,
varying according to variable availability. I {Unemployed} is a dummy taking a value of one if the worker is employed, I {UnempBene f it} is a dummy taking a value of one if the worker
received UI in the current year, I { PartTime} is a dummy taking the value of one if the worker worked in a part-time job in the current year, and #Employers is the number of employers the
worker had in the current year. Columns 1,3,5, and 7 add to the baseline controls the average state UI replacement rate. Columns 2,4,6, and 8 add to the baseline controls the average UI
replacement rate, minimum wage, and state-level welfare transfers. Standard errors are clustered at the MSA level.
C.2
County Discontinuity and Marginal Variation in Requirements
A worker’s county is available in the CPS only after 1996. But after 1996, there were no introductions of monetary requirements and only a few states introduced tenure requirements. There49
fore, we cannot identify the effect of the introduction of UI requirements on the labor market
using county-level border discontinuity.
To study if our results are robust at the county-level border discontinuity, we exploit the
marginal variation in the tenure and monetary requirements after 1996. Due to the smaller variation in requirements in this period, we expect standard errors to be larger. Table 9 displays
the estimated effect of UI requirements using county- or MSA-level state border discontinuities.
Columns 1 and 2 show that the tenure requirement has a negative effect on unemployment but
there is no significant effect from the monetary requirement. Columns 4 and 5 show that the
tenure requirement increases the share of part-time jobs, while the monetary requirement reduces it. Columns 7 and 8 show that the monetary requirement reduces employment transition,
while the tenure requirement increases it. Therefore, the effect of UI requirements are still robust
to the use of the marginal variation in requirements and county-level discontinuity.
Table 9: Effect of UI requirements on the labor market using county and MSA discontinuities
log ( MonetaryReq.)
log ( TenureReq.)
(1)
I {Unemployed}
(2)
I {Unemployed}
(3)
I {UnempBene f it}
(4)
I {UnempBene f it}
(5)
I { PartTime}
(6)
I { PartTime}
(7)
#Employers
(8)
#Employers
0.00395
(0.719)
-0.00618
(0.509)
-0.00306
(0.650)
-0.00814***
(0.000)
0.000729***
(0.009)
0.0000949
(0.659)
-0.00119
(0.685)
0.0101***
(0.000)
-0.0323*
(0.058)
0.0263*
(0.062)
0.0116
(0.233)
0.0233***
(0.000)
-0.0292**
(0.012)
0.00697
(0.562)
-0.0424***
(0.005)
0.0402***
(0.000)
Region
County
MSA
County
MSA
County
MSA
County
MSA
N
R2
108310
0.048
77624
0.044
193712
0.004
129986
0.013
124584
0.073
71042
0.090
111422
0.038
80389
0.040
Notes: This table shows the estimated parameters of model 1. Columns 1, 3, 5, and 7 compare counties that share borders, while the other columns compare the two sides of an MSA. Labor
data are from the CPS and UI requirement data are hand collected from USDL reports. The sample is from 1996 to 2016, varying according to variable availability. I {Unemployed} is a dummy
taking the value of one if the worker is employed, I {UnempBene f it} is a dummy taking the value of one if the worker received UI the current year, I { PartTime} is a dummy taking the value
of one if the worker worked at a part-time job in the current year, and #Employers is the number of employers the worker had in the current year. Standard errors are clustered at the MSA level
for columns 2, 4, 6, and 8, and at the county level for the other columns.
D
D.1
Robustness of the Model
Change in Employment Probabilities - Partial Equilibrium
In this robustness exercise we show the change in key model statistics when conducting a 10
percent reduction and increase in the parameters of the probability of a job offer and the probability of losing a job, pe and pu , respectively. In Table 10 we conduct these changes in our
benchmark economy in partial equilibrium to measure the movement in the aggregates relative
to the ones of the calibrated benchmark. We can observe that some of our calibration targets as
well as the share of unemployed workers are more sensitive to pe than to pu . This result is not
50
surprising since most of the different possible statuses of the worker shown in the value functions are affected by exiting unemployment, hence directly by pe . This highlights the key role
of the unemployment friction, especially when moving out of the unemployed state, created by
our augmented productivity process that includes the state in which z = 0.
Table 10: Comparison of model statistics for different employment probabilities.
Benchmark 0.9 ∗ pe
Variable
Unemployment
Duration Unemp.
Share Exhausting UI
Job Destruction
100
100
100
100
135.1
94.5
88.8
156.9
1.1 ∗ pe
0.9 ∗ pu
1.1 ∗ pu
85.0
101.7
116.8
59.6
99.2
100.0
99.3
99.2
94.1
99.6
99.8
99.9
Notes: The table shows model-generated statistics. The column “Benchmark” shows the
results of the benchmark model; the column “0.9 ∗ pe ” shows the results for the model with
90 percent of the calibrated value for the probability pe . All other columns are analogous.
D.2
Change in Employment Probabilities - Optimal Requirements
In the next robustness check for the model, we repeat the 10 percent changes in pe and pu , exactly
as done in the previous exercise but for the calculation of the optimal unemployment insurance
requirements with simultaneous choice of both the tenure and the monetary requirement as
shown in the last column of Table 6. As can be seen in Table 11 below, the variation in the
unemployment and employment probabilities does not affect the optimal choice of requirements
as they remain at their original level with the benchmark calibration, with zmin = 23% and
µt = 12.
Table 11: Comparison of optimal requirements for different employment probabilities.
Variable
Monetary Requirement
Tenure Requirement
CEV
Expenditure/GDP
Beneficiaries
Optimal Benchmark 0.9 ∗ pe
23%
12
0.99%
0.22%
0.27%
23%
12
1.46%
0.26%
0.34%
1.1 ∗ pe
0.9 ∗ pu
1.1 ∗ pu
23%
12
0.42%
0.30%
0.47%
23%
12
0.97%
0.20%
0.25%
23%
12
1.01%
0.24%
0.30%
Notes: The table shows model-generated statistics. The column “Benchmark” shows the results of the
benchmark model; the column “0.9 ∗ pe ” shows the results for the model with 90 percent of the calibrated
value for the probability pe . All other columns are analogous.
51
D.3
General Equilibrium Effects
In order to understand the potential general equilibrium effects of our welfare analysis, we generalize the partial equilibrium defined in Subsection B.3 and the thought experiment in Subsection 7.1. Our generalization works as follows: (i) we keep the same calibration as in the partial
equilibrium analysis shown in Tables 2 and 3; (ii) we start with the initial value of τU I = 0 as in
the benchmark economy; (iii) as in Subsection 7.1, G and Tu are fixed at the benchmark level; and
(iv) prices r and w vary in each counterfactual, in which we find a fixed point in the capital-labor
ratio, K/N.
We then compute an equilibrium for two variations, one of an increase and another one of
a decrease, of each of the UI instruments in the model. In this way, we can understand the
direction of a change in welfare in general equilibrium when we vary the instruments and we
can see how it compares with the optimal analysis with respect to the benchmark shown in Table
6. The results are in Table 12:
Table 12: Comparison of different UI requirements in general equilibrium
Variable
w
Benchmark θ = 35% θ = 55% µb = 18 µb = 30 zmin = 6% zmin = 15%
µt = 18
µt = 30
4.960
4.959
4.961
4.960
4.960
4.960
4.963
4.960
4.960
τU I
0
- 1.46%
-1.23%
-1.40%
-1.29%
-1.33%
-1.40%
-1.31%
-1.36%
CEV
0
-0.007%
0.004%
0.004%
0.012%
0%
0.125%
-0.009%
0.027%
0.55 %
0.32%
0.52%
0.37%
0.48%
0.43%
0.37%
0.45%
0.42%
1.1%
1.15%
1.16%
0.99%
1.27%
1.15%
0.82%
1.21%
1.10%
Expenditure/GDP
Beneficiaries
Notes: The table shows model-generated statistics. The column “Benchmark” shows the results of the benchmark model; the column “θ = 35%”
shows the results for the model with a replacement ratio of 35%. All other columns show analogous changes in the UI design instruments. The
benefit duration and the tenure requirements are denoted in weeks. The monetary requirement is in percentage of six-week’s average earnings.
The first clear result we can observe across all changes in UI instruments is that the general
equilibrium effect shown by the change in the wage rate w is of a small magnitude. The margin
that changes the most is the one captured by the UI budget as shown by the changes in the
tax rate τU I . Nonetheless, similarly to our partial equilibrium analysis, the rate is negative in
all of our counterfactual equilibria, reflecting the surplus of UI revenues and the need for extra
insurance in the form of a payroll subsidy.
Since these characteristics were already present in our analysis with the partial equilibrium
52
concept, we are able to conclude that the extension to a general equilibrium setting does not
bring much action at the aggregate level. Moreover, we can observe that the effect of the change
in the instruments is straightforward in terms of the UI budget, as more expansionary policy
options, such as a higher replacement rate θ or a longer collection duration, induce the system to
require a less negative τU I , i.e., more financing needs. Analogously, stricter requirements induce
the system to a cheaper rate or a more generous subsidy.
Finally, we can observe that the relative improvement in terms of welfare is preserved in a
similar way as shown in our optimality analysis in Table 6, leaving us more reassured of the main
approach we proposed. The movement toward stricter requirements shows an improvement not
only in welfare but also in the highest levels attained in the exercises considered. The result that
the increase in the monetary requirement has the highest gradient is preserved.
53
@FedFRASER