Discussion Forum:
What descriptive statistics, if any, did the author(s) make available to the reader? If the author(s) did not make any descriptive statistics on their data available to the reader, what descriptive statistics might you have suggested that they present to their readers?
Do you think the descriptive statistics presented by the author(s) provide sufficient information about the central tendencies, dispersion, and shape of their data for you to have confidence in the appropriateness of the data for the research approach and results obtained? Why or why not?
The answers should be based on the papers,
Mack, J. G. (2021). Fraud as a Byproduct of Disaster: Relief Aid Fraud and the COVID-19 Pandemic (Doctoral dissertation, Utica College).
Fuller, David L., B. Ravikumar, and Yuzhe Zhang. 2015. “Unemployment Insurance Fraud and Optimal Monitoring.” American Economic Journal: Macroeconomics, 7 (2): 249-90.
American Economic Journal: Macroeconomics 2015, 7(2): 249–290
http://dx.doi.org/10.1257/mac.20130255
249
Unemployment Insurance Fraud and Optimal Monitoring †
By David L. Fuller, B. Ravikumar, and Yuzhe Zhang *
An important incentive problem for the design of unemployment
insurance is the fraudulent collection of unemployment benefits by
workers who are gainfully employed. We show how to efficiently
use a combination of tax/subsidy and monitoring to prevent such
fraud. The optimal policy monitors the unemployed at fixed intervals.
Employment tax is nonmonotonic: it increases between verifications
but decreases after a verification. Unemployment benefits are rela-
tively flat between verifications but decrease sharply after a verifica-
tion. Our quantitative analysis suggests that the optimal monitoring
cost is 60 percent of the cost in the current US system. (JEL D82,
H24, J64, J65)
Unemployment insurance programs insure workers against the risk of losing their jobs through no fault of their own. Such insurance, however, has many
potential incentive problems. In this paper, we study the incentive problem associ-
ated with fraudulent collection of unemployment benefits. The US Department of
Labor finds that more than 60 percent of unemployment insurance fraud overpay-
ments are attributed to concealed earnings fraud—when a worker collecting unem-
ployment benefits finds a job but continues collecting the benefits. Motivated by this
fact, we study optimal unemployment insurance in an environment where workers
can conceal earnings and collect unemployment benefits.
We study an infinitely lived worker in continuous time who has CARA prefer-
ences, is initially unemployed, and faces a stochastic arrival of employment oppor-
tunities. Employment is assumed to be an absorbing state. An employed worker can
conceal his employment status and continue to claim unemployment benefits. The
worker’s employment status can be detected using a costly monitoring technology.
In order to focus on the issue of hidden employment, we abstract from moral hazard
* Fuller: Department of Economics, University of Wisconsin Oshkosh, 800 Algoma Boulevard, Oshkosh, WI
54901 (e-mail: fullerdl@gmail.com); Ravikumar: Research Division, Federal Reserve Bank of St. Louis, PO Box
442, St. Louis, MO 63166 (e-mail: b.ravikumar@wustl.edu); Zhang: Department of Economics, Texas A&M
University, College Station, TX 77843 (e-mail: yuzhe-zhang@econmail.tamu.edu). We are grateful to Árpád
Ábrahám, Nicola Pavoni, an anonymous referee, seminar participants at the Federal Reserve Bank of St. Louis,
University of Missouri, and Toulouse School of Economics, and participants at the Workshop on Macroeconomic
Applications of Dynamic Games and Contracts, Midwest Macroeconomics Meeting, Midwest Theory Meeting,
Asia Meeting of the Econometric Society, Society for the Advancement of Economic Theory Conference, and
Tsinghua Workshop in Macroeconomics for their helpful comments. We would also like to thank George Fortier
for editorial assistance. The views expressed in this article are those of the authors and do not necessarily reflect the
views of the Federal Reserve Bank of St. Louis or the Federal Reserve System.
† Go to http://dx.doi.org/10.1257/mac.20130255 to visit the article page for additional materials and author
disclosure statement(s) or to comment in the online discussion forum.
http://dx.doi.org/10.1257/mac.20130255
mailto:fullerdl@gmail.com
mailto:b.ravikumar@wustl.edu
mailto:yuzhe-zhang@econmail.tamu.edu
http://dx.doi.org/10.1257/mac.20130255
250 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
issues by assuming that there is no search effort decision and that the wage offer
distribution is degenerate.1
In our model, there are two instruments to deter fraudulent collection of unem-
ployment benefits: tax/subsidy and monitoring. Both instruments are costly. The
first distorts consumption relative to full insurance, and the second has a direct cost.
We deliver a precommitment mechanism that optimally trades off between the two
instruments. Our mechanism allows both instruments to be fully history dependent.
As a result, the unemployed worker’s consumption (i.e., the unemployment bene-
fits) and the employed worker’s consumption vary over time.
Since employment is an absorbing state in our model, the treatment of the worker
who reports transitioning to employment is straightforward: constant consumption
forever and no monitoring. Since employment status is private information, the
worker who reports being unemployed is not fully insured and is monitored.
We consider two monitoring mechanisms: deterministic verification and stochas-
tic verification. Under deterministic verification, the worker is either verified with
probability one or not verified at all. We focus on this case for most of the paper since
it is simpler and makes the results more transparent. We show later that our results
remain the same under stochastic verification, where the worker is verified with a
probability between zero and one. That is, even though our deterministic mechanism
appears restrictive, the general mechanism of stochastic verification does not offer
any additional economic insights on unemployment insurance and monitoring.
Under deterministic verification, the optimal contract has three key features.
First, monitoring occurs at fixed intervals and is independent of history. Second,
the unemployment benefits decrease with the duration of unemployment between
monitoring dates and jump downward at every monitoring date. Third, there is a
nonmonotonic tax on employment.
The periodicity of monitoring follows from the fact that with CARA preferences
the worker’s utility flows in a new cycle are proportional to those in the previous
cycle. Hence, his incentive to commit fraud remains the same and he is monitored in
the same manner as in the previous cycle. Unemployment benefits decreasing with
duration is a familiar feature from the previous literature. Unemployment benefits
jump downward at the monitoring date because the unemployed worker’s premon-
itoring consumption is distorted upward. In our model, increasing the unemployed
worker’s premonitoring consumption benefits the truth-teller more than it benefits
the liar.2 Within a monitoring cycle, the employment tax increases with duration of
unemployment. The consumption for the worker who transitions to employment
earlier exceeds that of the worker who transitions later. However, the employment
tax decreases after the monitoring date. This is because the unemployed worker who
transitions to employment shortly after the monitoring date can conceal earnings
1 The literature on the optimal provision of unemployment insurance concentrates on moral hazard and exam-
ines incentives for optimal search effort (e.g., Baily 1978; Shavell and Weiss 1979; and Hopenhayn and Nicolini
1997). Hopenhayn and Nicolini (1997) and Wang and Williamson (2002) show that the search effort margin is
quantitatively insignificant: The unemployed worker’s optimal search effort almost equals what the current US
system implies.
2 For the same reason, in Mirrleesian taxation models with hidden ability, the labor supply of a low-ability
worker is distorted downward.
VOL. 7 nO. 2 251fuller et al.: unemployment insurance fraud
until the next monitoring date, while the worker who transitions to employment at
the monitoring date cannot.
Our optimal mechanism also deters fraud due to quits. This occurs when workers
quit their jobs, become unemployed, and start collecting unemployment benefits.
The incentives in our optimal contract ensure that the employed workers do not
engage in such behavior.3
To assess the empirical relevance of our theoretical analysis, we conduct a par-
tial equilibrium quantitative exercise similar to Hopenhayn and Nicolini (1997).
We find that the optimal monitoring cost is 60 percent of the cost incurred by the
US unemployment insurance system. Furthermore, using the same resources as
the US system, the optimal contract delivers higher utility to the average worker:
1.55 percent higher consumption at every date. This gain arises from two sources:
(i) improved consumption smoothing between employed and unemployed states,
and (ii) reduced monitoring costs (or higher average consumption). Almost all of
the gain in our optimal contract comes from (i). This is similar to the quantitative
finding in Hopenhayn and Nicolini (1997) and Wang and Williamson (2002). The
cost saving in their optimal contracts is due to improved consumption smoothing
and not due to faster transitions from unemployment to employment.
The remainder of the paper proceeds as follows. In Section I, we present the key
facts on unemployment insurance fraud. We also provide evidence that deterring
concealed earnings fraud involves a case-by-case investigation and, thus, a per case
cost, as in our model. Section II describes the model. In Section III, we establish two
properties of the optimal mechanism: scaling and periodic monitoring. In Section
IV, we use these properties to analyze the optimal unemployment insurance scheme
with exogenously given monitoring dates. Then, we characterize the optimal mon-
itoring dates in Section V. In Section VI, we show that our mechanism prevents
employed workers from quitting. In Section VII, we examine the stochastic moni-
toring case. In this section, we also describe the similarities and differences between
the insights from the deterministic mechanism and the insights from the stochastic
mechanism. We conclude in Section VIII.
I. Unemployment Insurance Fraud Data
In this section, we first briefly describe the program in place for determining
the accuracy of payments in the US unemployment insurance system. Second,
we provide details on the nature of “fraud” overpayments by category for 2007
(Appendix A provides information for more years). Third, we present data on how
these payments were detected. Finally, we discuss “off-the-books” employment.
Accuracy of Benefit payments.—Unemployment insurance benefits in the United
States are paid out by the states, with each state deciding its benefit levels and how
to finance the benefits. The US Department of Labor’s BAM (Benefit Accuracy
3 Hansen and Imrohoroglu (1992) study a model where unemployed workers can reject job offers and an exoge-
nous fraction of such workers are denied benefits. In our optimal mechanism, the unemployed worker who receives
a job offer has no incentive to refuse the offer.
252 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
Measurement) program determines the accuracy of these expenditures by choos-
ing a random sample of weekly unemployment insurance claims and determin-
ing whether there were any overpayments. The investigators also interview some
claimants if necessary. Some overpayments are simple errors in calculating benefits,
while some represent fraud overpayments.
The goal of the program is different from the goal of unemployment insurance
fraud investigators. While the latter look to recapture overpayments, BAM investi-
gators calculate statistics of the unemployment insurance program (see BAM State
Operations Handbook ET No. 495, 4th edition). We use these statistics throughout
the paper.
Overpayments Due to Fraud.—There are several types of unemployment insur-
ance fraud. Examples include collecting unemployment benefits while being
employed, after quitting a job, or after refusing a suitable job offer. Table 1 catego-
rizes the overpayments by type of fraud.
“Concealed Earnings” refers to cases where payments are made to individu-
als who are simultaneously earning wages and collecting unemployment benefits.
“Insufficient Job Search” refers to cases where individuals did not meet the manda-
tory work search requirement (e.g., a minimum number of job applications must be
filed each week). “Refused Suitable Offer” refers to cases where individuals were
offered a job deemed suitable, but rejected it. “Quits” and “Fired,” respectively, refer
to cases where payments are made to individuals who voluntarily left their jobs or
who were fired from their jobs for a valid reason (e.g., poor performance or missing
work). “Unavailable for Work” refers to cases where payments are made to individ-
uals who cannot work (e.g., disability).
Overpayments due to concealed earnings fraud in 2007 were ten times overpay-
ments due to unemployed agents not actively searching or refusing suitable work
(see Table 1). While the data indicate that concealed earnings fraud is the dominant
source of overpayments, it does not imply that moral hazard from reduced search
effort is unimportant for the design of unemployment insurance. It might be the case
that the current unemployment insurance system provides adequate incentives to
search but does not deter concealed earnings fraud.
Table 1—Unemployment Insurance Overpayments in the United States, 2007
Category Percent of fraud overpayment
s
Concealed earnings 60.06
Insufficient job search 4.95
Refused suitable offer 0.80
Quits 13.29
Fired 4.17
Unavailable for work 7.06
Other 9.67
Total 100.00
Source: BAM program, US Department of Labor. Note that these are our calculations. Our
definitions of each type of fraud differ slightly from those used in the BAM reports available
online.
VOL. 7 nO. 2 253fuller et al.: unemployment insurance fraud
Detection Technologies.—The detection technologies used by BAM are shown in
Table 2. For example, “Verification of search contact” refers to cases when the BAM
investigator verifies the potential job contact reported by the unemployed person;
“Claimant interview” is an interview with the person collecting benefits.
Since 2003 , states have used a cross-matching technology, comparing unem-
ployment insurance records with employment records. One might think concealed
earnings fraud could be automatically detected this way; however, only 7.0 per-
cent of the fraud cases are detected by cross-matching with the state’s directory
of new hires (see Table 2). For instance, cross-matching technology would not
automatically catch a worker who is collecting unemployment benefits in one state
while employed in another state. Furthermore, the directory of new hires is updated
monthly, so even within individual states some workers who truthfully report unem-
ployment in a specific week may show up in a cross-match of employment records
and be mistakenly flagged for fraud. In most cases when a worker appears in both
unemployment insurance records and employment records, further investigation is
necessary to determine if fraud has actually occurred.
In addition, the worker could commit a more nuanced form of concealed earnings
fraud by truthfully reporting the transition to employment but underreporting the
earnings. (The worker is entitled to collect some unemployment benefits as long
as the reported earnings are sufficiently low.) In 2007, roughly 40 percent of those
committing concealed earnings fraud reported positive earnings. Less than 2 percent
of these cases were detected by cross-matching the unemployment insurance records
with wage records (updated quarterly) in each state (see Table 2). In fact, employees
working in a sector not covered by the unemployment insurance system will never
show up in the state wage records (e.g., federal employees and self-employed).
These data suggest that more than 90 percent of the overpayments due to con-
cealed earnings fraud were not detectable under the automatic procedures available
to the state authorities. Instead, detection involves a case-by-case investigation and,
thus, a per-case cost of verification.
Working “Off-the-Books”.—A worker could collect unemployment benefits
while working “off-the-books” and being paid in cash. In such cases, verifying the
true employment status might be prohibitively expensive. However, the evidence
Table 2—Detection Technologies, 2007
Percent of concealed earnings
Detection method fraud overpayments detected by method
Verification of search contact 1.20
Verification of wages and/or separation 63.99
Claimant interview 10.06
Verification of eligibility with 3rd parties 1.26
Unemployment insurance records 13.69
Job/employment service records 0.16
Verification with union 0.65
Crossmatch with state directory of new hires 7.00
Crossmatch with state wage record files 1.99
Source: Benefit Accuracy Measurement Program, US Department of Labor
254 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
suggests that concealed earnings fraud is committed by workers in “official”
employment. While the worker is committing concealed earnings fraud, his weekly
earnings are similar to the weekly earnings in the preunemployment job (which,
by design, has to be official for the worker to collect unemployment benefits). In
2007, those committing concealed earnings fraud were earning 82 percent of their
previous job’s wages, on average. One-fourth of those committing this fraud were
earning more while collecting benefits than before they became unemployed. Such
relatively high earnings while committing fraud suggest official or “on-the-books”
employment rather than “off-the-books” employment.4
II. Model
The Unemployment Insurance authority is a risk-neutral principal with a discount
rate r > 0 . She provides insurance to a risk-averse worker, whose preferences are
given by
E [ ∫ 0
∞
e −rt rv (c(t)) dt ] ,
where c(t) is consumption at time t , v(c) = − e −ρc is a CARA utility function with
risk aversion ρ , r is the discount rate, and E is the expectation operator. Note that
the flow utility is rv(c) and that the agent’s subjective discount rate is the same as
the principal’s.
A worker can be either employed with wage w > 0 or unemployed with wage
zero. The worker is unemployed at t = 0 and transitions to employment with Pois-
son rate π > 0 . We assume that employment is permanent. (For similar assump-
tions, see the unemployment insurance model of Hopenhayn and Nicolini 1997 and
the disability insurance model of Golosov and Tsyvinski 2006.)
The worker’s employment status is private information, so an employed worker
can claim to be unemployed and continue collecting the unemployment benefits. We
refer to this as fraud. The principal can verify the worker’s unemployment report
at a cost of γ units of the consumption good. Verification reveals the worker’s true
employment status.
We study precommitment mechanisms that efficiently deliver unemployment
benefits and deter fraud. In addition to the tax/subsidy instrument used by the
unemployment insurance literature, our mechanism uses the monitoring instrument
to provide incentives.5
We assume that the principal always collects the wage, so an unemployed worker
can never claim to be employed. Hence, there is no need for verification when
the worker reports a transition to employment. Furthermore, since employment is
an absorbing state, verification is unnecessary forever if the worker reports to be
4 The BAM program detects 10.0 percent of the fraud overpayments by interviewing the claimants (see Table 2).
Such interviews might reveal some cash earnings.
5 See Setty (2014) for a model of optimal unemployment insurance where the agent’s search effort is monitored.
Empirically, as noted in Table 1, fraudulent behavior in search effort is not as costly as concealment of earnings.
VOL. 7 nO. 2 255fuller et al.: unemployment insurance fraud
employed just once in the past. The incentive problem then reduces to ensuring that
an employed worker does not claim to be unemployed.
We focus on deterministic verification mechanisms. In each period the worker is
either verified with probability one or not verified at all. This mechanism is subop-
timal; it is dominated by a stochastic verification mechanism in our environment.
One may then ask why study the deterministic case? Our goal is to characterize
the optimal combination of the two instruments: tax/subsidy and monitoring. In
Section VII, we show that the key economic insights on these two instruments are
nearly identical in both the deterministic and stochastic cases. In both cases, optimal
monitoring and employment tax have the same pattern. The stochastic monitoring
case requires cumbersome notation and provides less intuition, so we start by ana-
lyzing the deterministic case.
In our deterministic mechanism, the verification in any period is based on the
history of employment status reports and past verifications outcomes. Since verifi-
cation is necessary only for agents who have been reporting unemployment in every
period in the past, a sufficient statistic for history is the duration of unemployment
reports. In other words, at t = 0 the principal commits to all future verification
periods, mapping durations of unemployment reports to {0, 1} . In a verification
period, clearly no worker would misreport. (Any penalty ϵ > 0 induces truth tell-
ing in the verification period.) Thus, the principal does not have to keep track of
the outcomes of past verifications. We represent the set of verification periods as
{ m i ; i = 1, 2, …} , where m i is the date of the i th verification.6
The timing is as follows. In the initial period, the worker is unemployed. Then
the stochastic job opportunity arrives. The worker either remains unemployed or
transitions to employment. He then chooses to report either employment or unem-
ployment to the principal. Conditional on the unemployment report, the principal
verifies the true employment status if the period is a verification period. Then, condi-
tional on the report and the outcome of the verification, the principal assigns current
and future consumptions. In subsequent periods, if the worker reported employment
in the past, he is in an absorbing state and no further reports are necessary. If the
worker reported unemployment in every period in the past, then the sequence of
events is the same as in the initial period.
If an unemployed worker transitions to employment at t , let c E (t, s) denote his
consumption at time s ≥ t . Because the principal and the worker have the same
discount rate and employment is an absorbing state, efficiency requires that the
worker’s consumption remain constant after t for all s . We therefore suppress s in
c E (t, s) and denote this constant level of consumption as c E (t) . The flow utility
from this level of consumption then is rv ( c E (t)) . We denote the discounted sum
of utilities to a worker who accepts a job offer for the first time at t as E(t) , i.e.,
E(t) = ∫ t
∞ e −r(s−t) rv ( c E (t)) ds = v ( c E (t)) . Since employment status is private
information, E(t) is also the continuation utility to a worker who accepted an offer
before t , but reports employment for the first time at t .
6 There is no loss of generality in assuming a countable collection of verification periods. Since each verification
costs γ > 0 , the principal would not want to verify infinitely many times in any finite time interval.
256 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
An unemployed worker’s consumption at t is denoted by c U (t) and his flow utility
is rv ( c U (t)) . His continuation utility,
U(t) ≡ ∫
t
∞
e −r(x−t) e −π(x−t) rv( c U (x)) dx + ∫
t
∞
e −r(x−t) e −π(x−t) πE(x) dx,
is the sum of expected utilities before and after the transition ( e −π(x−t) in the first
integral is the conditional probability of remaining unemployed at date x and
e −π(x−t) π in the second integral is the density function of the transition time). Hence,
(1)
U(t) = ∫
t
∞
e −(r+π)(x−t) (πE(x) + ru(x)) dx
= ∫
t
s
e −(r+π)(x−t) (πE(x) + ru(x)) dx + e −(r+π)(s−t) U(s), for all t < s,
where u(x) ≡ v ( c U (x)) . We will refer to (1) as promise-keeping constraints.
The principal commits at t = 0 to verification periods { m i ; i = 1, 2, …} and
consumptions { ( c E (t), c U (t)) ; t ≥ 0} . The verification periods and consumptions
are history dependent. We denote this precommitment contract as σ .
incentive compatibility requires that a worker who transitioned to employment at
t ∈ ( m i , m i+1 ) does not have the incentive to delay the report of the transition to a
later time s ∈ (t, m i+1 ) , i.e., report unemployment and commit fraud from t to s and
then report employment from s onward:
(2) E(t) ≥ ∫
t
s
e −r(x−t) rv ( c U (x) + w) dx + e −r(s−t) E(s), ∀ s ∈ (t, m i+1 ).
Note that the worker cannot delay the report beyond the next verification period m i+1 .
We restrict contract allocations to
(3) E(t) ≥ U(t), for all t .
Restriction (3) rules out the fraud due to refusal of offers noted in Table 1 (0.8 per-
cent of total fraud overpayments). This restriction can be derived by adding a job
refusal option to our model. For ease of exposition, we have imposed the restric-
tion on the mechanism; Appendix B describes the job refusal option and derives
this restriction.
The expected cost for the principal is
c(σ) = ∫
0
∞
e −(r+π)t (π c E (t) + r c U (t)) dt + ∑
i
e −(r+π) m i γ .
There should, in fact, be an additional term in c(σ) : the discounted income obtained
by the principal, πw ____ r + π
.
However, unlike the unemployment insurance literature that
VOL. 7 nO. 2 257fuller et al.: unemployment insurance fraud
endogenizes job-finding probabilities, the discounted income in our model is a con-
stant, so it does not affect the optimal σ .
The principal’s problem is to find an incentive compatible σ that minimizes c(σ)
and delivers the initial promised utility U(0) , i.e.,
(4) min
σ
c(σ)
subject to U(0) = ∫
0
∞
e −(r+π)t (πE(t) + ru(t)) dt,
and constraints (2), (3) .
With a slight abuse of notation, denote the principal’s cost function as c(U(0)) .7
III. A Simplification of the Optimal Contract
We begin our analysis by presenting two features of the optimal contract. In
Section IIIA, we establish a “scaling” property. Then, in Section IIIB, we show that
the optimal monitoring is periodic. These properties simplify our analysis of the
optimal contract by narrowing the search of a solution to problem (4) to a smaller
space.
To help us simplify, we rewrite problem (4) in terms of continuation utilities E( · ) ,
U( · ) , and flow variable u( · ) , instead of consumptions. The objective becomes
c(σ) = ∫
0
∞
e −(r+π)t (πc(E(t)) + rc(u(t))) dt + ∑
i
e −(r+π) m i γ,
where c : (−∞, 0) → 핉 denotes the inverse of the utility function:
(5) c(v) = −log (−v)/ρ .
The incentive constraint (2) becomes
(6) E(t) ≥ ∫
t
s
e −r (x−t) e −ρw ru(x) dx + e −r (s−t) E(s), ∀ s ∈ (t, m i+1 ),
since CARA utility implies that v ( c U (x) + w) = e −ρw v ( c U (x)) = e −ρw u(x) .
7 Ravikumar and Zhang (2012) analyze the problem of tax compliance in a costly state verification model where
the verification technology is imperfect (a low-income agent might be mistakenly labeled as high income). They
solve for the principal’s cost function using the Hamilton-Jacobi-Bellman equation. In contrast, we study optimal
unemployment insurance in an environment with a perfect verification technology. We characterize the path of
unemployment benefits by formulating the optimal control problem and using the Pontryagin minimum principle.
258 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
A. Scalin
g
Our mechanism exhibits a scaling property: If the initial promise U(0) is scaled
by α > 0 , then the optimal contract is also scaled by α . More formally,
LEMMA 1: if { (U(t), E(t), u(t)) ; t ≥ 0} are optimal utilities for initial promise
U(0) , then the optimal utilities for initial promise αU(0) are
{ (αU(t), αE(t), αu(t)) ; t ≥ 0} .
Alternatively, Lemma 1 states that the consumption of the worker with initial
promise αU(0) differs from that of the worker with promise U(0) by a constant,
−log (α)/ρ , at all dates and states.
The scaling property in Lemma 1 is related to the fact that CARA utility has
no wealth effect. Although a worker with high promised utility consumes (perma-
nently) more than a worker with low promised utility, the level of promised utility
does not have an effect on the worker’s incentives to conceal earnings. In other
words, the incentive constraint (6) holds when all of the utilities are scaled by the
same factor.
Since the incentives to conceal earnings are the same for workers with different
promised utilities, the optimal sequence of monitoring dates, { m i ; i ≥ 1} , is inde-
pendent of the initial promised utility. Again, no wealth effect implies that the level
of promised utility does not change how the worker is monitored, even if it does
change the worker’s consumption.
B. periodicity
At time 0 , the principal knows the true employment status of the agent. After the
verification at m 1 , the principal again knows the true employment status. Hence, the
continuation problem at m 1 is the same as the problem at time 0 , except for the “ini-
tial” promised utility. The scaling property implies that, if U( m 1 ) = αU(0) , then
the optimal utilities from m 1 forward are scaled by α . Thus, starting with a promise
U(0) , if the principal finds it optimal to monitor the unemployed agent at m 1 , then
it must be the case that starting with the promise αU(0), the principal would again
find it optimal to monitor at m 1 . Put differently, having monitored the agent at m 1 ,
the next optimal monitoring period is 2 m 1 . We immediately conclude the following:
PROPOSITION 1: The optimal monitoring is periodic, i.e., m i = i m 1 for all i ≥ 1 .
To understand the intuition for the periodic monitoring, consider policies where
the interval between verifications is either increasing or decreasing over time. First,
it is suboptimal for the planner to verify more frequently at the beginning. Since
the worker starts out unemployed, he stays unemployed for some duration initially.
Frequent verifications early on merely incur unnecessary verification cost. Second,
one might think that it is optimal to verify more frequently later since the proba-
bility of a long duration of unemployment is small. However, this policy is also
VOL. 7 nO. 2 259fuller et al.: unemployment insurance fraud
sub
optimal.
The worker’s conditional probability of transitioning to employment is
independent of how long he has been unemployed. Moreover, because the princi-
pal knows the true employment status after each verification, the scaling property
implies that from the principal’s perspective the worker who was just verified to be
unemployed is no different from the worker at time zero. Thus, the interval between
consecutive monitoring periods is a constant.
While we have established that the optimal monitoring is periodic, finding the
optimal periodicity is difficult. To determine the optimal m 1, we must first determine
the optimal utilities in the intervals [0, m 1 ] , [ m 1 , 2 m 1 ] , etc. Toward this end, we break
the principal’s problem into two steps. First, assume that m 1 is exogenous and the
principal learns the agent’s employment status at dates m 1 , 2 m 1 , etc. Given m 1 , the
principal solves for the endogenous utility paths in [0, m 1 ] , [ m 1 , 2 m 1 ] , etc. Second,
the principal chooses m 1 optimally. We analyze the first step in the next section and
the second step in Section V.
IV. Optimal Unemployment Insurance with Exogenous Monitoring
Given the simplification in Section III, we now present the features of the optimal
unemployment insurance scheme. For a given m 1 , we first formulate the optimal
control problem in Section IVA. This allows us to analyze the time paths of the vari-
ables of interest. We then describe some features of the continuation utilities E( · )
and U( · ) in Section IVB and use these features to illustrate the employment tax in
Section IVC and unemployment benefits in Section IVD. Finally, in Section IVE
we use the Pontryagin Minimum Principle to explicitly characterize E( · ) and U( · ) .
A. Optimal control
Following Zhang (2009), we formulate the principal’s problem for interval [0, m 1 ]
as one of optimal control. Our analysis for [0, m 1 ] applies to other intervals as well.
First, we rewrite the constraints recursively. The promise-keeping constraint (1)
is equivalent to the differential equation:
U′ (t) = r (U(t) − u(t)) + π (U(t) − E(t)) .
On the right side of the differential equation, the first term is the rate of change of U
when there is no uncertainty (i.e., when there is no transition to employment), and
the second term captures the additional rate of change due to uncertainty.
The incentive constraint (6) is equivalent to the following differential inequality:
(7) r (v ( c U (t) + w) − v ( c E (t)) ) + E′(t) ≤ 0 .
That is, the short-term benefit that the agent gets from fraud, r (v ( c U (t) + w) −
v ( c E (t)) ) , is offset by lower continuation utility he receives after he delays the
employment report. Note that E( · ) could have downward jumps: When E(t) >
lim s↓t E(s) , we interpret the discontinuity as E′(t) = −∞ , and the differential
260 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
inequality (7) still holds under this interpretation. Introducing a slack variable
μ(t) ≥ 0 , we may rewrite (7) as
E′(t) = rE(t) − e −ρw ru(t) − μ(t) .
In Lemma 4 in Appendix C, we show that the above differential equation and
inequality are equivalent to (1) and (6).
Second, the scaling property implies that the cost function c( · ) satisfies
c(αU ) = c(U ) − log (α)/ρ .
Recalling the definition of c( · ) in (5), we rewrite c(U ) as
(8) c (U ) = c (|U |(−1)) = c (−1) − log (−U )/ρ ≡ ψ + c(U ),
where ψ ≡ c (−1) is the cost of private information: It is the one-time cost that
the principal is willing to pay to permanently remove private information from the
model.
With ψ + c(U( m 1 )) as the continuation cost at m 1 , we rewrite the principal’s
problem as one of optimal control with a convex objective and linear constraints:
(9) min
u(t), U(t), E(t),
0≤t≤ m 1
∫
0
m 1
e −(r+π)t (πc(E(t)) + rc(u(t))) dt
+ e −(r+π) m 1 (γ + ψ + c(U( m 1 )))
(10) subject to U′(t) = (r + π)U(t) − πE(t) − ru(t),
(11) E′(t) = rE(t) − e −ρw ru(t) − μ(t) ,
(12) E(t) ≥ U(t),
U(0) is given.
B. continuation Utilities
The continuation utilities E( · ) and U( · ) help us uncover the consumption paths
for the employed and the unemployed. We focus on the properties of E( · ) and
U( · ) in [0, m 1 ] ; those in other monitoring cycles can be obtained by scaling (see
Lemma 1).
We demonstrate five properties:
(i) E(t) > E(s) for t < s ≤ m 1 .
(ii) E(t) > U(t) for all t < m 1 .
VOL. 7 nO. 2 261fuller et al.: unemployment insurance fraud
(iii) E( m 1 ) = U( m 1 ) .
(iv) E( · ) jumps up immediately after m 1 .
(v) U( · ) declines over time.
Property (i) states that the payoff to a worker who reports the transition to employ-
ment earlier is higher than the payoff to one who reports the transition later. The
worker who transitions to employment at t but commits fraud consumes c U (t) + w
at t , whereas the worker who tells the truth consumes c E (t) . It is intuitive that
c E (t) < c U (t) + w ; otherwise deterring fraud would not be an issue. In terms of
utilities, E(t) < e −ρw u(t) . Incentive compatibility (11) requires that delaying the
report yields a lower payoff (see Figure 1). Thus, E(t) > E(s) within a monitoring
cycle.
For property (ii), recall that restriction (12) imposes E(t) must be greater than
or equal to U(t) . If the agent who transitions to employment before m 1 is offered
the same payoff as the agent who remains unemployed, then the employed agent
will claim to be unemployed and consume more than the unemployed agent. He
can continue cheating until the verification period m 1 (see Figure 2). Thus, within a
monitoring cycle, E(t) must be greater than U(t) .
To understand (iii), note that the true employment status is revealed at m 1 , so the
principal does not face an incentive problem at that instant. Hence, there is no reason
to reward the (lucky) agent who transitioned to employment at m 1 relative to the
(unlucky) agent who remains unemployed, i.e., no reason to set E( m 1 ) > U( m 1 ) .
Thus, E( m 1 ) = U( m 1 ) . (Again, recall restriction (12): E(t) ≥ U(t) for all t .)
Property (iv) states that U( m 1 ) = E( m 1 ) < E( m 1 +) , where E( m 1 +) is the utility for a worker who is unemployed at m 1, but transitions to employment immediately
Figure 1. Lower Payoff for Late Reporters ( E(t) > E(s) for t < s )
E(t )
t
E(s )
s
C
o
n
tin
u
a
tio
n
u
til
iti
e
s
Transition time
262 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
after m 1 , i.e., E( m 1 +) = lim t↓ m 1
E(t) (see Figure 3). Suppose, to the contrary, that
U( m 1 ) = E( m 1 +) . Then incentive compatibility in [ m 1 , 2 m 1 ] would be violated
because the worker employed immediately after m 1 can claim to be unemployed
and consume more than the employed worker until the next verification period, 2 m 1 .
Note that if there is no verification at date t , then an upward jump in E( · ) violates
the incentive constraint: A worker who transitions to employment prior to t would
benefit from delaying the employment report. At the moment of verification, how-
ever, the worker cannot delay the employment report since the true employment
status is revealed.
C
o
n
tin
u
a
tio
n
u
til
iti
e
s
Time
U
E
m1
C
o
n
tin
u
a
tio
n
u
til
iti
e
s
Time
U
E
m1
Figure 2. Continuation Utilities E( · ) and U( · ) in [0, m 1 ] .
Figure 3. Continuation Utility E( · ) is Nonmonotonic
VOL. 7 nO. 2 263fuller et al.: unemployment insurance fraud
To understand why U( · ) declines, suppose U( m 1 ) > U(0) . Then lowering U( m 1 )
has two benefits. First, the unemployed agent’s continuation utility path is flatter,
which implies better insurance for the unemployed. Second, lower U( m 1 ) (and
E( m 1 ) ) reduces E′( · ) , generating stronger incentives to deter fraud. In addition,
U( · ) can never jump. Because U( · ) is the promised utility to the unemployed agent,
any jump in U( · ) would violate the promise-keeping constraint.
C. Employment Tax
Here we examine the consumption allocated to the agent who reports employ-
ment earlier relative to the consumption for the agent who reports it later. Recall
that E(t) > E(s) within a monitoring cycle and the continuation utility E( · ) jumps
up after verification. Since employment is an absorbing state, any agent who reports
a transition to employment at t is allocated constant consumption c E (t) forever and
is not monitored. Thus, E(t) maps into c E (t) instant by instant and, hence, c E (t) >
c E (s) within a monitoring cycle. Furthermore, the consumption for the agent who
reports the transition to employment immediately after m 1 is higher than that for the
employed agent at m 1 (see Figure 4).
The nonmonotonicity is closely related to the way incentives are provided in
our model. Within a cycle, the principal does not monitor and relies exclusively on
consumption distortions to induce truth-telling: c E must fall sufficiently fast for the
worker not to postpone his report of employment. At m 1 , c E falls to a level such
that the agent is indifferent between transitioning to employment and remaining
unemployed. The principal can perfectly insure the agent against the unemployment
shock at m 1 because the true employment status is revealed. Immediately after
Time
C E
m1 2m1
Figure 4. Permanent Consumption for Workers
Who Transition to Employment in Different Periods
264 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
m 1 , the principal treats the worker employed right after m 1 better than the worker
employed at m 1 . This is because the worker who transitions to employment after m 1
can commit fraud until the next monitoring period, while the worker who transitions
to employment at m 1 cannot commit fraud. Hence, the principal must offer the for-
mer a higher permanent consumption to induce truth-telling.
The difference between wage w and consumption c E can be interpreted as an
employment tax. Our contract implies that within a verification cycle, the employ-
ment tax for late reporters is higher than that for the early reporters. However, unlike
the existing unemployment insurance literature, the employment tax is nonmono-
tonic: It decreases immediately following verification.
D. Unemployment Benefits
Unlike the case where c E (t) maps into E(t) at every instant, c U (t) is not pinned
down at every instant by U(t) , since the unemployed agent is not fully insured.
Instead, the path of c U ( · ) in [0, m 1 ] requires knowledge of the entire path of U( · )
in the interval. We obtain the entire trajectories of c U ( · ) and U( · ) after solving (9)
in Section IVE. However, monotonicity of U( · ) in Section IVB suggests that c U ( · )
declines with unemployment duration. As in Hopenhayn and Nicolini (1997), our
contract implies that the unemployment benefit c U eventually reaches an arbitrarily
low level with positive probability.8
Figure 5 shows that the unemployment benefits jump down at the verification
period. To understand the jump, we argue that it is optimal for the principal to set
u(t) above u( m 1 ) when m 1 − t > 0 is small. Doing this relaxes the incentive con-
straint at time t , as the following variational argument shows. The promise-keeping
constraint at m 1 − δ , for a small positive δ , is
U( m 1 − δ) = rδu( m 1 − δ) + e −rδ [(πδ)E( m 1 ) + (1 − πδ)U( m 1 )]
= rδu( m 1 − δ) + e −rδ U( m 1 ) ,
where the second equality uses the aforementioned property E( m 1 ) = U( m 1 ) . The
incentive constraint at m 1 − δ is
E( m 1 − δ) ≥ rδ e −ρw u( m 1 − δ) + e −rδ E( m 1 ) .
Suppose u( m 1 − δ) = u( m 1 ) . Then the principal can maintain the promise-keeping
constraint but relax the incentive constraint by increasing u( m 1 − δ) and decreasing
u( m 1 ) . Specifically, consider the variation
u ̃ ( m 1 − δ) = u( m 1 − δ) + e −rδ ϵ, u ̃ ( m 1 ) = u( m 1 ) − ϵ, E ̃ ( m 1 ) = E( m 1 ) − rδϵ .
8 In contrast to Hopenhayn and Nicolini (1997) and our paper, Pavoni (2007) imposes an exogenous lower
bound on promised utility and shows that the optimal benefits decrease with the duration of unemployment, but
remain constant after the promised utility reaches the lower bound. Alvarez-Parra and Sanchez (2009) show a sim-
ilar result in a model with an endogenous lower bound on promised utility.
VOL. 7 nO. 2 265fuller et al.: unemployment insurance fraud
Because the unemployed worker’s consumption after m 1 remains unchanged,
his continuation utility at m 1 is U ̃ ( m 1 ) = U( m 1 ) − rδϵ , which is equal to
E ̃ ( m 1 ) . Therefore, the promise-keeping constraint U( m 1 − δ) = rδ u ̃ ( m 1 − δ) +
e −rδ U ̃ ( m 1 ) still holds, and the incentive constraint is relaxed:
rδ e −ρw u ̃ ( m 1 − δ) + e −rδ E ̃ ( m 1 ) = rδ e −ρw u( m 1 − δ) + e −rδ E( m 1 ) − (1 − e −ρw )rδϵ
< rδ e −ρw u( m 1 − δ) + e −rδ E( m 1 ) .
Starting from u( m 1 − δ) = u( m 1 ) , the additional cost of consumption incurred by
this variation is second order, but the effect on incentive constraint is first order.
Hence, the principal always chooses u(t) above u( m 1 ) when t is close to (but
below) m 1 .
We summarize these findings in the following proposition. The proof is in
Appendix C.
PROPOSITION 2: The unemployment benefit, c U ( · ) is monotonically decreasing
with unemployment duration, with downward jumps at verification, while c E ( · ) is
nonmonotonic: it decreases between verifications with upward jumps immediately
after verification.
Unemployment insurance systems in many countries feature benefits schemes
similar to the one in Proposition 2. For example, in Spain, workers receive a replace-
ment rate of 70 percent for the first 6 months of unemployment, 60 percent for the
next 18 months, and a minimum payment thereafter.
Figure 5. Consumption for the Unemployed
U
n
e
m
p
lo
ym
e
n
t
b
e
n
e
fit
s
Time
c U
m1 2m1
266 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
E. pontryagin Minimum principle
We construct a solution to the optimal control problem (9) in which the incentive
constraint (11) binds (i.e., μ(t) = 0 ) for all t < m 1 . The problem faced by the
principal is to choose an initial state E(0) and a time path u( · ) to minimize the cost
in (9), given U(0) . The promise-keeping and incentive constraints (10) and (11)
then imply a time path (U( · ), E( · )) for continuation utilities. One way to think
about this problem is to think of choosing u(t) at each date, given the values of U(t)
and E(t) that have been attained by that date. The principal faces a tradeoff between
the current period cost and the cost of delivering continuation utilities. Hence, she
needs to set “prices,” Φ and λ , on increments to the continuation utilities U and E .
Because it is costly for the principal to maintain a low E as a threat, it must be the
case that λ ≤ 0 . Moreover, we have argued in Section IVB that E(t) ≥ U(t) is
slack except at m 1 , so we impose only the constraint E( m 1 ) = U( m 1 ) .
A central construct in the optimal control problem is the current value
Hamiltonian ℋ defined by
ℋ = πc(E(t)) + rc(u(t)) + Φ(t)((r + π)U(t) − πE(t) − ru(t))
+ λ(t)(rE(t) − e −ρw ru(t)) ,
which is just the sum of current period cost and the rate of increase in continuation
utilities valued at Φ(t) and λ(t) . An optimal allocation must minimize ℋ at each
date t .
The first-order condition for minimizing ℋ with respect to u is
(13) c′(u) = Φ + e −ρw λ .
The left-hand side is the marginal cost of today’s utility, while the right-hand side
is the marginal cost of starting with higher continuation utility U tomorrow, offset
by the benefit of a slacker incentive constraint (it is a benefit because λ ≤ 0 ). The
utility u must be chosen to equalize the costs at each date.
The prices Φ and λ must satisfy
(14) Φ′(t) = (r + π)Φ − ∂ℋ ___ ∂U = 0,
(15) λ′(t) = (r + π)λ − ∂ℋ ___ ∂E = π(Φ − c′(E) + λ),
at each date t if (u( · ), U( · ), E( · )) is an optimal path. Equation (14) implies that
Φ(t) is a constant. Moreover, since multiplier Φ(0) is the marginal cost of U(0) , we
have
Φ = c′(U(0)) = −(ρU(0) ) −1 > 0 .
VOL. 7 nO. 2 267fuller et al.: unemployment insurance fraud
Since the planner can choose E(0) freely,
(16) λ(0) = 0 .
At m 1 , the shadow prices Φ and λ( m 1 ) must satisfy
(17) Φ = −κ + c′(U( m 1 )),
(18) λ( m 1 ) = κ,
where e −(r+π) m 1 κ is the multiplier on the constraint E( m 1 ) = U( m 1 ) . Since the
principal’s problem is convex, these conditions (13–18) are both necessary and suf-
ficient for a minimum.
When (11) holds as equality, the states (U, E) and the costate λ satisfy differen-
tial equations:
(19) U′(t) = (r + π)U − πE − ru,
(20) E′(t) = rE − r e −ρw u,
(21) λ′(t) = π(Φ − c′(E) + λ) .
The ODE system contains three variables and would be difficult to analyze in a
general context. However, we can solve (20) and (21) regardless of (19) because
neither (20) nor (21) relies on U . Once (20) and (21) are solved, it is easy to solve
(19). Formally,
LEMMA 2: if (20) and (21) hold, then (19) holds if and only if
(22) ΦU(t) + λ(t)E(t) + ρ −1 = 0, ∀ t ∈ [0, m 1 ] .
To solve the reduced ODE system, (20) and (21), we need two boundary condi-
tions. The first is (16), λ(0) = 0 . The second cannot be a value for E(0) , as E(0)
is endogenous and unknown a priori. We obtain the second boundary condition,
E( m 1 ) = − ρ −1 (Φ + λ( m 1 )) −1 , from E( m 1 ) = U( m 1 ) and equation (22).
The following lemma shows that these two boundary conditions pin down a
unique solution curve for the system (20) and (21). Figure 6 shows the phase dia-
gram. That λ < 0 implies that the incentive constraint binds for all t < m 1 .
LEMMA 3: For any m 1 > 0 , there is a unique initial condition E(0) such that the
solution starting at (λ(0) = 0, E(0)) satisfies E( m 1 ) = − ρ −1 (Φ + λ( m 1 )) −1 .
268 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
V. Optimal Monitoring
Until this point, we have taken m 1 as exogenous. In this section, we characterize
the optimal choice of m 1 . The tradeoff in choosing m 1 is as follows. Monitoring more
frequently implies higher verification cost, but the principal can provide better insur-
ance: The consumption path for the unemployed is similar to that for the employed.
Monitoring less frequently implies lower verification cost but worse insurance.
For any m 1 > 0 , denote the minimized cost in (9) as 𝒞( m 1 ); that is,
𝒞( m 1 ) = ∫ 0
m 1
e −(r+π)t (πc(E(t)) + rc(u(t))) dt
+ e −(r+π) m 1
(γ + ψ + c(U( m 1 ))) .
Intuitively, delaying monitoring (i.e., a small increase in m 1 ) saves the principal both
the cost of monitoring and the cost of (after-monitoring) consumptions, because
the payment of γ + ψ + c(U( m 1 )) is postponed. By doing so, however, the prin-
cipal must maintain the consumptions c(E( · )) and c(u( · )) for a longer duration.
Subtracting the benefit from the cost (algebraic details in Appendix C) yields
𝒞′( m 1 ) = e −(r+π) m 1
(r ρ
−1 log (
Φ + e −ρw λ( m 1 ) _____________
Φ + λ( m 1 )
) − (r + π)(γ + ψ)) .
Figure 6. Phase Diagram for (λ, E )
λ
E
E(0)
(0, 0)
E = −ρ−1(Φ + λ)−1
VOL. 7 nO. 2 269fuller et al.: unemployment insurance fraud
Thus, the first-order condition for m 1 is
(23) r ρ −1 log (
Φ + e −ρw λ( m 1 ) ____________
Φ + λ( m 1 )
) = (r + π)(γ + ψ) .
PROPOSITION 3: The optimal m 1 is the unique solution to (23). That is, (23) is
both necessary and sufficient for the minimum of 𝒞( m 1 ) .
REMARk 1: Although our analysis relies on an undetermined parameter ψ , the
parameter can be uniquely pinned down by a fixed-point condition that the actual
cost function at time zero must equal the conjectured function ψ + c(U(0)) . Further
details are in Appendix c.
REMARk 2: Our analytical results rely on the assumption of cArA preferences.
Unlike the cArA case where the length of the monitoring cycle is independent of
history, the cycle length in the crrA case depends on the worker’s continuation
utility. However, most of the main features of the optimal contract remain valid
even if the worker has crrA preferences. We demonstrate this through a numerical
example in Fuller, ravikumar, and Zhang (2013).
VI. Quits
Another type of fraud that could arise in our model is quits. An agent in our model
could transition to employment in period t , claim to be unemployed until almost m 1 ,
and then quit to become unemployed at m 1 . The verification at m 1 would not reveal
him to be a cheater. Thus, quitting is possible in our model.
Our mechanism guarantees that the agent does not commit such a fraud. The
continuation utilities E( · ) and U( · ) are such that the agent is indifferent between
reporting the transition immediately and delaying it to the next period. By following
the path above and quitting at m 1 , he becomes truly unemployed, is subject to the
stochastic arrival rate of employment opportunity, and is worse off.
Hopenhayn and Nicolini (2009) examine a model where quits cannot be distin-
guished from layoffs and the only fraudulent behavior is quits. In their model, the
employment status is observable and nonabsorbing, and disutility from working is
greater than that from searching for employment. Employed agents might want to
opportunistically quit their job, enjoy more leisure, and collect unemployment bene-
fits. To discourage quits, the principal offers (i) higher consumption to the employed
workers who stay on the job longer and (ii) more generous benefits to unemployed
workers with longer employment spells, as quitters have shorter employment spells
on average. In our model, the utility functions for the unemployed worker and the
employed worker are the same, and employment status is private information. Since
employment is an absorbing state, quitting as considered in Hopenhayn and Nicolini
(2009) cannot arise in our model. The potential reason for quitting in our model is
to cover up the fraudulent collection of unemployment benefits before the verifica-
tion period. Our optimal mechanism provides incentives for the agent not to delay
reporting his transition to employment and not to conceal his earnings.
270 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
Overpayment due to quits is small relative to the overpayment due to concealed
earnings (see Table 1). Our mechanism deters fraud due to both concealed earnings
and quits.
VII. Stochastic Verification
Our monitoring mechanism in the previous sections was restricted to determin-
istic verification. Here we consider a more general mechanism where the principal
verifies randomly after receiving the unemployment report. As in Section II, veri-
fication reveals the worker’s true employment status. (In Appendix E, we consider
an imperfect verification technology: An unemployed worker might be erroneously
labeled as employed.) Conditional on the unemployment report at t , the principal
chooses the monitoring Poisson rate p(t) ≥ 0 . That is, over a period of length dt ,
the principal monitors with probability p(t) dt and she does not monitor with proba-
bility 1 − p(t) dt . (Since our model is in continuous time, p(t) is not the monitoring
probability.)
We assume that if a worker is monitored and caught cheating, he has to pay a
finite penalty forever. With infinite penalty, an arbitrarily small monitoring prob-
ability would deliver the full-information constant consumption. In our model, if
the principal can choose any finite penalty between 0 and ϕ > 0 , he would always
choose ϕ . Henceforth, we assume that the finite penalty is ϕ units of the consump-
tion good, forever.
Similar to (10) and (11), the promise-keeping constraint and incentive constraint
are
(24) U ′ = r(U − u) − π(E − U ) − p( U ̃ − U ),
(25) E′ ≤ rE − r e −ρw u − p( e ρϕ − 1)E,
where U ̃ is the unemployed agent’s continuation utility after monitoring. Because
the probability that monitoring does not occur in [0, t) is e − ∫ 0
t p(s)ds , the principal’s
objective is
(26) ∫
0
∞
e −(r+π)t− ∫ 0
t p(s) ds (πc(E(t)) + rc(u(t)) + p(t)(γ + c( U ̃ (t)))) dt .
The principal chooses the utilities {U(t), E(t), u(t), U ̃ (t); t ≥ 0} and the arrival
rates of monitoring { p(t); t ≥ 0} to minimize (26) subject to (24), (25), and the
constraint E(t) ≥ U(t) , ∀ t ≥ 0 .
Since the penalty for a worker with high promised utility is the same as that for a
worker with low promised utility, we obtain a scaling property similar to the one in
Section IIIA. Thus, the incentives to conceal earnings are the same for workers with
different promised utilities. Similar to our model with deterministic verification, we
show in Proposition 4 that the optimal stochastic verification mechanism consists of
cycles. See Appendix D for the proof.
VOL. 7 nO. 2 271fuller et al.: unemployment insurance fraud
PROPOSITION 4: There exists an n > 0 such that the principal monitors the
unemployed with a constant arrival rate p > 0 if and only if t ≥ n . Before n ,
the time path (U( · ), E( · )) converges to the 45-degree line; after n , it moves
along the 45-degree line toward (−∞, −∞) until the agent is randomly drawn to
be verified. After the verification, (U, E) jumps to a new state ( U ̃ , E ̃ ) and a new
cycle starts.
The unemployed worker is in one of two states: (i) not monitored
(i.e., p(t) = 0 ) or (ii) randomly drawn to be monitored (i.e., p(t) ≡ p > 0 ).
Within each cycle, an unemployed worker is initially in the not-monitored state.
He is moved to the random monitoring state if the duration of his unemployment
report exceeds the threshold n . If he is randomly drawn to be monitored, then
he is moved to the not-monitored state after being monitored, and a new cycle
begins. While the date of monitoring is stochastic, the threshold duration is not.
That is, within each cycle, the principal guarantees that the worker will not be
monitored until the threshold duration is reached, similar to the deterministic ver-
ification case.
The intuition for why the worker is not monitored before the threshold duration
is as follows. The Unemployment Insurance agency has access to two instruments:
tax/subsidy and monitoring. Recall that at verification the true employment status
is revealed, and E is reset to a level such that its shadow price is zero, which means
that, immediately after monitoring, the employment tax can be varied at no cost.
The cost of the tax/subsidy instrument is lower than the cost of monitoring, γ > 0 ,
immediately after monitoring, and remains so until some threshold unemployment
duration is reached. Hence, it is optimal to use only the tax/subsidy instrument for
the provision of incentives before the threshold.
REMARk 3: The absence of verification until a threshold duration is unlikely to
be robust to other types of penalties. For instance, in popov (2009) there is an
exogenous lower bound on the worker’s continuation utility, and a worker who is
caught cheating is pushed to this lower bound. So the penalty for a worker with
high continuation utility is larger than that for a worker with low continuation
utility. With hidden independently and identically distributed income, he shows
that the verification probability is always positive.
The stochastic monitoring mechanism clearly dominates the deterministic
mechanism characterized in Section V. To see this, consider a stochastic moni-
toring scheme in which the arrival rate of monitoring is higher than p for workers
in the random monitoring state. Denote this higher arrival rate as p ̃ . Proposition 4
implies that p ̃ is suboptimal. By continuity, the limiting scheme as p ̃ → ∞ should
also be suboptimal. This limiting scheme is exactly the deterministic monitoring
mechanism.
We argue below that the key insights on the use of tax/subsidy and monitoring
instruments in the suboptimal deterministic mechanism are nearly identical to the
insights from the optimal stochastic mechanism. We describe in detail the similari-
ties and differences between the implications of the two mechanisms.
272 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
A. comparison of Monitoring with the Deterministic case
First, both the stochastic and deterministic mechanisms have the feature that
monitoring does not occur before a threshold unemployment duration; m 1 in the
deterministic case and n in the stochastic case. These thresholds, however, could be
different; i.e., in general m 1 ≠ n .
Second, both mechanisms feature cycles. In the deterministic case, after m 1 , a
new cycle begins; the new cycle has exactly the same length as the previous cycle.
Similarly, in the stochastic case, after monitoring occurs a new cycle begins and
verification does not occur again before the threshold n is reached. The exact date
when the monitoring occurs in the stochastic case is random. This is because, after
n , monitoring arrives according to a Poisson process and, hence, the exact length of
each cycle depends on when the worker is actually verified. As in the deterministic
case, however, the value of n is the same in each cycle.
B. comparison of Tax/Subsidy with the Deterministic case
Consumptions in the stochastic monitoring case are similar to those in the deter-
ministic case. Within each cycle, before the threshold n , the patterns of con sumption
are identical to ( c E , c U ) in Figures 4 and 5. After n , if a worker is monitored and
verified to be truly unemployed, then the unemployment benefits jump down, as in
the deterministic case.
The only difference is that in the deterministic case, continuation utilities, and
consumptions are reset when the threshold m 1 is reached. In the stochastic case, after
the threshold n and before the monitoring actually arrives, continuation utilities and
consumptions smoothly decline with the duration of unemployment. The decreas-
ing continuation utilities and the monitoring (and finite punishment) jointly provide
incentives for truth telling; the worker is indifferent between reporting a job offer
and committing fraud.
C. Quantitative Analysis
To illustrate our optimal contract, we follow Hopenhayn and Nicolini (1997)
closely and perform a quantitative exercise similar to theirs. We let the agents in
our model face a stylized version of the US unemployment insurance system. We
calibrate the model to match the observed rate of concealed earnings fraud. We then
compute the gain from switching to the optimal mechanism in our model.
To perform this exercise, we have to add some heterogeneity to our model; oth-
erwise everyone would cheat or no one would cheat, and we would not be able to
match the observed rate of concealed earnings fraud. We assume that the workers
are heterogeneous in the wages they earn and, hence, the replacement rate for unem-
ployment benefits. Concretely, we assume that the wage distribution is lognormal
with parameters μ w and σ w 2 .
The BAM data provide earnings information for an individual’s previous employ-
ment (the earnings that determine the unemployment benefits for the individual). In
the 2007 sample of BAM data, the mean weekly wage is $692 and the coefficient
VOL. 7 nO. 2 273fuller et al.: unemployment insurance fraud
of variation is 0.79 . Using these data moments, we calibrate μ w = 6.296 and
σ w 2 = 0.488 . By construction, the earnings in the BAM data are only for those who
collect unemployment benefits. Instead of using the BAM data we could use the CPS
data on earnings for the entire employed population to calibrate the wage distribution
in the model. However, individuals collecting unemployment benefits generally earn
less (while employed) than the individuals in the entire employed population.9
We calculate the unemployment benefits as a function of wages, again using the
BAM 2007 data: ln(unemployment benefits) = 1.31 + 0.65 ln(wages).
We assume that the model period is one week and that the interest rate
r = 0.001 . Since the average duration of unemployment in 2007 is 16.85 weeks, we
calibrate the job arrival rate to be π = 1/16.85 . The monitoring cost γ is calibrated as
follows. On average, the BAM investigators spend 12.6 hours per case and the average
wage of the investigators is $43 in 2012 (the only year when such data are available).
So, adjusting the average wage to 2007 dollars, we calibrate γ to be $501 . We cali-
brate the value of absolute risk aversion ρ such that the relative risk aversion for the
average wage earner is 2 . Since the average wage is $692 in our sample, ρ = 2/692 .
We then calibrate the probability of monitoring and the penalty in the US system if
caught cheating to match two targets: fraction of people committing concealed earn-
ings fraud and fraction of people caught cheating among those committing the fraud.
With CARA preferences, wage heterogeneity is not relevant for matching the two
targets, but it is relevant for computing the distribution of initial promised utility in
the baseline. In the counterfactual, we take these initial promised utilities as given,
calculate the optimal monitoring and benefits, and then compute the cost of deliver-
ing the initial promised utilities. The job arrival rate, wage distribution, and penalty
are held fixed at the same values as the baseline calibration.
The results imply that, measured in present value, the cost of optimal monitoring
is 60 percent of the cost in the current US system. In the optimal contract (averag-
ing across the initial promised utilities), n = 11.64 weeks. That is, the planner
guarantees that monitoring does not occur for roughly the first 12 weeks of the
unemployment spell and, thus, reduces the monitoring cost with an efficient use of
the monitoring technology.
To determine the magnitude of the gain from switching to the optimal mecha-
nism, suppose that the planner is restricted to use the same amount of resources
as the current US system. How much additional utility can the planner deliver to
the average worker? The answer is a utility gain equivalent to 1.55 percent more
consumption at every date than the US system provides. This gain arises from two
sources: (i) improved consumption smoothing between employed and unemployed
states and (ii) reduced monitoring costs or higher consumption on average. The US
system spends only 0.24 percent its resources on monitoring the average worker and
spends the rest on unemployment benefits (net of wages), but the same resources are
allocated differently in the optimal contract: 0.17 percent is spent on monitoring the
average worker and the rest is spent on unemployment benefits. Thus, almost all of
the gain in our model comes from improved consumption smoothing.
9 The mean weekly wage among employed workers in the March 2007 CPS is $861 , and the coefficient of
variation is 1.27 .
274 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
There are some obvious limitations to this analysis. Most notably, our exercise
is a partial equilibrium analysis, as in Hopenhayn and Nicolini (1997). To fully
quantify the welfare gains from adopting the optimal contract, we have to conduct
a general equilibrium analysis incorporating transition from employment to unem-
ployment and disciplining the model with aggregate worker flows.
VIII. Conclusion
The most prevalent incentive problem in the US unemployment insurance system
is that individuals collect unemployment benefits while being gainfully employed.
We examine a model of optimal unemployment insurance where a worker can con-
ceal his employment status and the unemployment insurance authority has a tech-
nology to verify his employment status. We find that the optimal interval between
consecutive monitoring periods is a constant, independent of history. The optimal
employment tax is nonmonotonic, increasing between verifications and decreasing
immediately after a verification. The optimal unemployment benefits decline with
unemployment duration with sharp declines after each verification. Our optimal
contract also prevents fraud due to quits.
Unemployment insurance in our model is a form of social insurance protecting
workers against the risk of job loss. Acemoglu and Shimer (1999, 2000); Shimer
and Werning (2008); and Alvarez-Parra and Sanchez (2009) explore another role
of unemployment insurance. They examine environments with heterogeneous jobs,
and unemployment insurance helps the worker wait for the appropriate job. Some
jobs have higher productivity than others, but such job opportunities arrive less fre-
quently. Unemployment benefits help workers wait for more productive matches
and endure longer unemployment durations. The benefits in these environments
affect the aggregate composition of jobs. An interesting direction for future research
is to extend our environment to multiple jobs and examine optimal monitoring in the
presence of the alternative role of unemployment insurance.
Finally, our model does not include any job retention effort. Incorporating the
job retention effort into our model requires employment to be stochastic. If workers
can conceal earnings, their hidden income could affect their job retention effort.
Analyzing interaction between effort and fraud is another interesting direction for
future research.
Appendix A. Data
Fraud and Overpayments.—Table A1 details the various types of fraud overpay-
ments from 2005−2009 , averaged over all US states. Concealed earnings fraud is
the dominant source of overpayments in every year.
The unemployment insurance system might incur another form of overpayment
if workers strategically delay the start date of employment. That is, workers might
accept a job offer but agree to start the job after their unemployment benefits have
expired. Gauthier-Loiselle (2011) documents that unemployment insurance expen-
ditures are higher in Canada because of such cases. In the United States, this is not
VOL. 7 nO. 2 275fuller et al.: unemployment insurance fraud
considered fraud. Thus, the BAM data include no information on such cases, so they
are not included in the fraud overpayments statistics.
Overpayments Due to insufficient Search.—In Table 1 in Section I, the over-
payments due to concealed earnings fraud were almost 12 times the overpayments
due to insufficient search fraud. Do the data understate the incidence of insuffi-
cient search? Recall that the BAM program measures only the extensive margin—
whether the individual submits the required number of applications. It is possible
that the unmeasured intensive margin—effort that turns an application into a job
offer—is large enough to make the overpayments due to insufficient search compa-
rable in magnitude to the overpayments due to concealed earnings. The following
facts, however, suggest that the unmeasured component is unlikely to be large:
• Measured overpayments due to insufficient search have been declining: In 1988
they accounted for 34 percent of the total overpayments due to all fraud, whereas
in 2007 they accounted for less than 5 percent. (The corresponding numbers for
concealed earnings fraud were 41 percent and more than 60 percent.)
• The job search requirements that make an unemployed person eligible for ben-
efits have increased over time, so the decline in the measured component is not
due to changes in eligibility criteria. Hence, for the insufficient search over-
payments to be the same in 2007 as those measured in 1988, the unmeasured
component has to be almost six times that of the measured component in 2007.
• If unmeasured efforts to translate a job application into a job offer were sub-
stantially higher in 2007, then the increase in efforts should imply a substan-
tially higher transition rate from unemployment to employment. However, the
transition rate is roughly constant: The quarterly rate was 0.31 for the period
1988–1997 and 0.33 for 1998–2007.
From a normative point of view, as noted in footnote 1, the prevailing quantitative
theory prescribes an intensive margin search effort that is less than the effort exerted
under the current unemployment insurance program in the United States. In other
words, insufficient search is not a critical incentive problem in the United States.
(Using evidence from randomized trials in four US sites, Ashenfelter, Ashmore,
and Deschenes (2005) find that insufficient job search is not a significant source of
unemployment insurance overpayments.)
Table A1—Fraud Overpayments
Percent of total fraud overpayments
Cause 2005 2006 2007 2008 2009
Concealed earnings 62.64 54.40 60.06 67.32 65.89
Insufficient job search 4.55 4.15 4.95 3.02 2.75
Refused suitable offer 1.50 1.23 0.80 0.36 0.77
Quits 12.78 16.41 13.29 12.69 9.61
Fired 4.27 4.60 4.17 4.60 7.38
Unavailable for work 4.94 6.95 7.06 5.04 5.14
Other 9.33 12.27 9.67 6.97 8.46
Total 100.00 100.00 100.00 100.00 100.00
Source: Benefit Accuracy Measurement Program, US Department of Labor
276 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
Appendix B. Microfoundations for E(t) ≥ U(t)
Suppose that the worker can privately refuse a job offer. The timing in each period is
as follows. The stochastic job opportunity arrives and the worker either receives an offer
or does not. He then chooses to report the offer (if any) to the principal. Conditional
on the report of an offer, the principal recommends that the worker either accept or
reject the offer. The worker then chooses whether to follow the principal’s recommen-
dation. (In contrast, job acceptance is implicitly imposed in our model in Section II.)
Conditional on the report, the principal assigns current and future consumptions.
In such a job-refusal model, it is optimal for the principal to always recommend
to the worker who reports an offer to accept the offer. Recommending “accept”
minimizes the cost of delivering the promised utility since the worker’s consump-
tion is constant upon job acceptance and the principal gets the perpetual wage.
Recommending “reject” means that the continuation contract involves additional
uncertainty of job offers, reports, and incentive constraints. So the consumption cost
of delivering the same promised utility is higher under “reject.” Recall that, unlike
Atkeson and Lucas (1995), we do not have disutility to working so it is optimal to
always recommend “accept.”
The incentive compatibility for an agent with a job offer is as follows. If he reports
his offer and receives a recommendation to accept, he strictly prefers “accept” to
“reject.” This is because rejecting the offer would not make him eligible for any
unemployment insurance benefits, but would make him lose his wage income. If the
agent does not report his offer, then either he rejects the offer and obtains U(t) or
he accepts the offer and commits fraud (i.e., he works and collects unemployment
benefits at the same time). For the agent to truthfully report his offer, the utility of
reporting and accepting the offer, E(t) , must be higher than both U(t) and the utility
he obtains by committing concealed earnings fraud. These incentive compatibility
constraints are exactly conditions (2) and (3) in our model in Section II.
Appendix C. Proofs
PROOF OF LEMMA 1:
Suppose that a contract σ ≡ { (U(t), E(t), u(t), c U (t), c E (t), m i ) ;
t ≥ 0, i ≥ 1}
delivers the continuation utility U . Then, a contract
σ α ≡ { (αU(t), αE(t), αu(t), c U (t) − log (α)/ρ, c E (t) − log (α)/ρ, m i ) ;
t ≥ 0, i ≥ 1}
delivers αU . The reverse is also true. Further, σ is incentive compatible if and only
if σ α is incentive compatible. Therefore, { ( U ∗ (t), E ∗ (t), u ∗ (t), c U∗ (t), c E∗ (t), m i ∗ ) ;
t ≥ 0, i ≥ 1} is the optimal contract to deliver U if and only if
{ (α U ∗ (t), α E ∗ (t), α u ∗ (t), c U∗ (t) − log (α)/ρ, c E∗ (t) − log (α)/ρ, m i ∗ ) ;
t ≥ 0, i ≥ 1}
Vol. 7 No. 2 277fuller et al.: unemployment insurance fraud
is the optimal contract to deliver αU . ∎
Lemma 4: The promise-keeping constraint (1) and the incentive constraint (6)
hold for all 0 ≤ t < s ≤ m 1 if and only if
(C1) U(s) − U(t) = ∫
t
s
((r + π)U(x) − πE(x) − ru(x)) dx,
(C2) E(s) − E(t) ≤ ∫
t
s
(rE(x) − r e −ρw u(x)) dx,
hold for all 0 ≤ t < s ≤ m 1 . Taking the limit as s goes to t yields the differential equations (10) and (11).
Proof:
We show only the equivalence between (6) and (C2), since the equivalence
between (1) and (C1) can be obtained similarly by replacing the inequalities below
with equalities.
Necessity: If (6) holds for all t < s , then
E(t) + ∫
t
s
(rE(x) − r e
−ρw u(x)) dx
≥ ∫
t
s
e −r (x−t) r e −ρw u(x) dx + e −r (s−t) E(s)
+ ∫
t
s
(r ( ∫ x
s
e −r (η−x) r e −ρw u(η) dη + e −r (s−x) E(s)) − r e
−ρw u(x)) dx
= ( e
−r (s−t) + ∫
t
s
r e −r (s−x) dx ) E(s) + ∫ t
s
( e −r (x−t) − 1) r e −ρw u(x) dx
+ ∫
t
s
r ( ∫ x
s
e −r (η−x) r e −ρw u(η) dη) dx
= E(s) + ∫
t
s
( e −r (x−t) − 1) r e −ρw u(x) dx + ∫
t
s
( ∫ t
η
r e −r (η−x) dx ) r e
−ρw u(η) dη
= E(s) + ∫
t
s
( e −r (x−t) − 1) r e −ρw u(x) dx + ∫
t
s
(1 − e −r (η−t) ) r e −ρw u(η) dη
= E(s) .
Hence, inequality (C2) is verified.
Sufficiency: Define an absolutely continuous function f ( · ) as
f (s) ≡ ∫
t
s
e −r (x−t) r e −ρw u(x) dx + e −r (s−t) ( E(t) + ∫ t
s
(rE(x) − r e −ρw u(x)) dx ) .
278 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
Because f is absolutely continuous, it is differentiable almost everywhere (a.e.), and
f ′(s) = e −r (s−t) r e −ρw u(s) − r e −r (s−t) ( E(t) + ∫ t
s
(rE(x) − r e −ρw u(x)) dx )
+ e −r (s−t) (rE(s) − r e −ρw u(s))
= r e −r (s−t) ( E(s) − E(t) − ∫ t
s
(rE(x) − r e −ρw u(x)) dx ) , a.e .
If (C2) holds, then f ′(s) ≤ 0 a.e. Then, it follows from Theorem 29.15 in Aliprantis
and Burkinshaw (1990) that
f (s) = f (t) + ∫
t
s
f ′(x) dx ≤ f (t) = E(t) .
Therefore,
∫
t
s
e −r (x−t) r e −ρw u(x) dx + e −r (s−t) E(s) ≤ f (s) ≤ E(t),
which verifies inequality (6). ∎
PROOF OF LEMMA 2:
If (19), (20), and (21) all hold, we can substitute them into ( ΦU + λE )
′ and
obtain
(ΦU + λE )
′
= ΦU ′ + λ′E + λE ′
= Φ ((r + π)U − πE − ru) + π (Φ − c′(E ) + λ) E + λ(rE − r e −ρw u)
= (r + π) (ΦU + λE ) − πc′(E)E − r (Φ + e −ρw λ)u .
Because −c′(E )E = ρ −1 and − (ρu) −1 = c′(u) = Φ + e −ρw λ , we have
(C3) (ΦU + λE )
′ = (r + π) (ΦU + λE + ρ −1 ) .
Because ΦU(0) + λ(0)E(0) + ρ −1 = 0 , it follows from (C3) that ΦU(t) +
λ(t)E(t) + ρ −1 = 0 for all t ∈ [0, m 1 ] .
On the other hand, if (20) and (21) hold and
ΦU(t) + λ(t)E(t) + ρ −1 = 0, ∀ t ∈ [0, m 1 ],
VOL. 7 nO. 2 279fuller et al.: unemployment insurance fraud
then (ΦU + λE )
′ = 0 for all t ∈ [0, m 1 ] . Then (19) can be derived by reversing
the above steps. ∎
PROOF OF LEMMA 3:
First, it is convenient to transform the state variable E , which may approach −∞ ,
into a bounded one. To do so, we replace E with
g ≡ c′(E ) = −(ρE ) −1 .
Now, the ODE system consists of (21) and
(C4) g′ = E′ ____
ρ E 2
=
r g 2
________ Φ e ρw + λ − rg,
with boundary condition g( m 1 ) = Φ + λ( m 1 ) (Figure C1 shows the phase dia-
gram). Let m(g(0)) be the time to hit the straight line g = Φ + λ starting with
(λ(0) = 0, g(0)) .
Second, we show that lim g(0)↓Φ
m(g(0)) = 0 . If λ = 0 and g = Φ , then
(g − λ)′(t) = (
r g 2
________ Φ e ρw + λ − rg + π(g − λ − Φ)) | (λ, g)=(0, Φ)
= r Φ
2 ____ Φ e ρw − r Φ < 0 .
λ
(0, 0)
g(0)
g
Φ
Φe ρw
line g = Φe ρw + λ
line g = Φ + λ
Figure C1. Phase Diagram for (λ, g )
280 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
Continuity of the ODE system (21), (C4) implies that (g − λ)′(t) < 0 in a small neighborhood of (0, Φ) . If λ(0) = 0 and g(0) approaches Φ from above, then g(0) − λ(0) − Φ approaches zero. Since the solution curve starting with (0, g(0)) will remain in the small neighborhood of (0, Φ) for a while, it will decrease and hit the line g = Φ + λ quickly if g(0) − λ(0) − Φ is sufficiently small.
Third, we show that m(g(0)) is strictly increasing in g(0) . Consider two paths
that start with initial conditions (0, g 1 (0)) and (0, g 2 (0)) , where Φ < g 1 (0) <
g 2 (0) . We will show that g 1 (t) − λ 1 (t) < g 2 (t) − λ 2 (t) for all t . By contradiction,
suppose ( g 1 − λ 1 )(t) = ( g 2 − λ 2 )(t) for the first time at t = t ∗ . Because the two
paths cannot cross, we cannot have that g 1 ( t ∗ ) ≤ g 2 ( t ∗ ) . Then g 1 ( t ∗ ) > g 2 ( t ∗ ) and
λ 1 ( t ∗ ) > λ 2 ( t ∗ ) . Hence,
( g 1 − λ 1 )′( t ∗ ) = −
r g 1 ________ Φ e ρw + λ 1
(Φ e ρw + λ 1 − g 1 ) − π(Φ + λ 1 − g 1 )
< − r g 2 ________ Φ e ρw + λ 2
(Φ e ρw + λ 2 − g 2 ) − π(Φ + λ 2 − g 2 )
= ( g 2 − λ 2 )′( t ∗ ) ,
where the inequality follows from
g 1 _______ Φ e ρw + λ 1
> g 2 _______ Φ e ρw + λ 2
. That ( g 1 − λ 1 )′( t ∗ ) <
( g 2 − λ 2 )′( t ∗ ) contradicts the facts that ( g 1 − λ 1 )( t ∗ ) = ( g 2 − λ 2 )( t ∗ ) and
( g 1 − λ 1 )(t) < ( g 2 − λ 2 )(t) for all t < t ∗ . Thus, g 1 (t) − λ 1 (t) < g 2 (t) − λ 2 (t)
for all t , and the path ( λ 1 (t), g 1 (t)) reaches g = Φ + λ sooner.
Finally, we show there exists a unique g(0) to satisfy m(g(0)) = m 1 for any
m 1 > 0 . The second step in this proof shows that lim g(0)↓Φ
m(g(0)) = 0 . Part (ii)
in Lemma 5 shows that m(g(0)) can be arbitrarily large with high values of g(0) .
Hence, the existence of a unique solution to m(g(0)) = m 1 follows from the inter-
mediate value theorem and the monotonicity of m(g(0)) in g(0) . ∎
PROOF OF PROPOSITION 2:
First, we show that E , c U , U , and U __
E
all fall on [0, m 1 ] . It follows from g′(t) < 0
that E ′(t) = ρ E 2 (t)g′(t) < 0 . Equation (13) implies that u′(t) = e −ρw λ′(t) ______
c″(u)
< 0 ,
or ( c U )
′ (t) < 0 . Equation (22) implies that U′(t) = − Φ −1 (λ(t)E(t) )
′ < 0 .
Equation (22) also implies that U __
E
= Φ −1 (g − λ) . Hence, part (i) in Lemma 5
implies that (
U __
E
)
′ (t) < 0 . Second, to see the downward jump in c U ( · ) at m 1 , we show that
lim
t↑ m 1
c′(u(t)) > lim
t↓ m 1
c′(u(t)) .
The left side is Φ + e −ρw λ( m 1 ) according to (13). To obtain the right side, we apply
(13) to the interval [ m 1 , 2 m 1 ) and obtain
c′(u(t)) = c ′(U( m 1 )) + e −ρw λ ̃ (t), t ≥ m 1 ,
VOL. 7 nO. 2 281fuller et al.: unemployment insurance fraud
where λ ̃ denotes the multiplier λ for the problem on the interval [ m 1 , 2 m 1 ) . Because
λ ̃ ( m 1 ) = 0 , we have lim t↓ m 1
c′(u(t)) = c′(u( m 1 )) = c ′(U( m 1 )) + 0 = Φ +
λ( m 1 ) . Therefore,
lim
t↑ m 1
c′(u(t)) = Φ + e −ρw λ( m 1 ) > Φ + λ( m 1 ) = lim
t↓ m 1
c′(u(t)) . ∎
PROOF OF PROPOSITION 3:
First, because (i) Φ + e
−ρw λ _______ Φ + λ decreases in λ and (ii) λ( m 1 ) decreases in g(0) and
m 1 , there is a unique value for g(0) (as well as m 1 ) for a given ψ .
Second, to show that (23) is sufficient, we prove that
𝒞′( m 1 ) {
< 0, m 1 < m 1 ∗ ;
> 0, m 1 > m 1 ∗ .
This is because
Φ + e −ρw λ( m 1 ) _________
Φ + λ( m 1 )
strictly increases in m 1 :
Φ + e −ρw λ( m 1 ) _________
Φ + λ( m 1 )
decreases in
λ( m 1 ) and the proof of Lemma 3 shows that λ( m 1 ) decreases in g(0) and m 1 . ∎
Details in the computation of 𝒞′( m 1 ).—
Rewrite 𝒞′( m 1 ) as
∫
0
m 1
e −(r+π)t (πc( E m 1 ) + rc( u m 1 ) + Φ ((r + π) U m 1 − π E m 1 − r u m 1 − ( U m 1 ) ′ )
+ λ m 1 (r E m 1 − r e −ρw u m 1 − ( E m 1 ) ′ ) ) dt
+ e −(r+π) m 1 (γ + ψ + c( U m 1 ( m 1 ))) + e −(r+π) m 1 λ m 1 ( m 1 )( E m 1 ( m 1 ) − U m 1 ( m 1 )),
where we put a superscript m 1 on U( · ) , E( · ) , u( · ) , and λ( · ) because these opti-
mal paths rely on m 1 . We use the envelope theorem to simplify the computation of
𝒞′( m 1 ). Since U m 1 (t) , E m 1 (t) , u m 1 (t) are already optimally chosen at each t , we may
view them as fixed when we vary m 1 . Further, U m 1 ( m 1 ) and E m 1 ( m 1 ) can be viewed
as varying only with the terminal date in the parenthesis.10 Viewed in this light, a
small increment of m 1 is just an extrapolation of all time paths over a longer duration
of unemployment, while the paths themselves are fixed. That is, we view all super-
scripts as being fixed and omit them when we calculate derivatives. Because E( m 1 ) −
U( m 1 ) = 0 , we have
𝒞′( m 1 ) = e −(r+π) m 1 (πc(E( m 1 )) + rc(u( m 1 )) − (r + π)(γ + ψ + c(U( m 1 )))
+ c′(U( m 1 ))U ′( m 1 ) + λ( m 1 )(E ′( m 1 ) − U ′( m 1 ))) .
10 This is because U m ̃ 1 ( m 1 ) and E m ̃ 1 ( m 1 ) can be viewed as being fixed when we vary m ̃ 1 .
282 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
It follows from c′(U( m 1 )) = Φ + λ( m 1 ) , λ′( m 1 ) = 0 and Lemma 2 that
c ′ (U( m 1 ))U ′( m 1 ) + λ( m 1 ) ( E ′( m 1 ) − U ′( m 1 ))
= ΦU ′( m 1 ) + λ( m 1 )E ′( m 1 ) = (ΦU( m 1 ) + λ( m 1 )E( m 1 )) ′ = 0 .
Therefore,
𝒞′( m 1 ) = e −(r+π) m 1 (πc(E( m 1 )) + rc(u( m 1 )) − (r + π)(γ + ψ + c(U( m 1 ))))
= e −(r+π) m 1 (r ρ
−1 log (
Φ + e −ρw λ( m 1 ) ___________
Φ + λ( m 1 )
) − (r + π)(γ + ψ)) .
Fixed-point condition for ψ .—
The condition for ψ is that ψ is the fixed point of operator T , i.e.,
ψ + c(U(0)) = T(ψ) + c(U(0)) ≡ min
σ
c(σ) .
We obtain ψ from the first-order condition (23) for m 1 ,
ψ =
r ρ −1
_____ r + π log (
Φ + e −ρw λ( m 1 ) ____________
Φ + λ( m 1 )
) − γ .
We obtain T(ψ) from the HJB equation for the cost function at time zero
T(ψ) + c(U(0)) =
πc(E(0)) + rc(u(0)) + Φ ((r + π)U(0) − πE(0) − ru(0))
_________________________________________ r + π
= π _____ r + π (
Φ ____
g(0)
− log (
Φ ____
g(0)
) − 1) + c(U(0)) .
The fixed-point condition ψ = T(ψ) is rewritten as
(C5) (r + π)γ = r ρ −1 log (
Φ + e −ρw λ( m 1 ) ____________
Φ + λ( m 1 )
) − π (
Φ ____
g(0)
− log (
Φ ____
g(0)
) − 1) .
PROPOSITION 5: The path that satisfies (C5) exists and is unique.
PROOF:
The existence of a path that satisfies (C5) follows from the intermediate value
theorem and the fact that the right side of (C5) is either extremely large or extremely
VOL. 7 nO. 2 283fuller et al.: unemployment insurance fraud
small if we vary g(0) . To see this, note that the proof of Lemma 3 shows that
lim g(0)↓Φ
m 1 = 0 = lim g(0)↓Φ
λ( m 1 ) . Therefore,
lim
g(0)↓Φ
r ρ −1 log (
Φ + e −ρw λ( m 1 ) ____________
Φ + λ( m 1 )
) − π (
Φ ____
g(0)
− log (
Φ ____
g(0)
) − 1) = 0 .
On the other hand, the proof of part (ii) of Lemma 5 shows the existence of paths with
λ( m 1 ) approaching −Φ and g(0) ∈ (Φ, Φ e ρw ) . For these paths, log (
Φ + e −ρw λ( m 1 ) _________
Φ + λ( m 1 )
)
can be arbitrarily large, while Φ ___
g(0)
remains bounded.
The uniqueness can be shown by contradiction. Suppose there are two paths satis-
fying (C5). Associated with the two paths are two fixed points, ψ < ψ ̃ . Because the
principal facing ψ ̃ may monitor at m 1 (ψ) > 0 and adopt the optimal consumption
paths under ψ ,
T( ψ ̃ ) ≤ ψ + e −(r+π) m 1 (ψ) ( ψ ̃ − ψ) < ψ ̃ ,
which contradicts the fact that ψ ̃ is a fixed point. ∎
LEMMA 5: consider the ODE system (21), (C4) with time running backward,
that is,
(C6) λ′ = π(g − Φ − λ),
(C7) g′ = rg −
r g 2
________ Φ e ρw + λ .
Suppose the initial condition is (λ(0), g(0) = Φ + λ(0)) , −Φ < λ(0) < 0 , and m − (λ(0)) denotes the first time to hit the g -axis, i.e., m − (λ(0)) = min t {t > 0 : λ(t) = 0} :
(i ) (g − λ)′(t) > 0 for all t ∈ [0, m − (λ(0))] .
(ii ) m − (λ(0)) is finite, and lim λ(0)↓−Φ
m − (λ(0)) = ∞ .
PROOF:
(i) The path starting with (λ(0), g(0) = Φ + λ(0)) has
λ′(0) = π(g(0) − Φ − λ(0)) = 0,
g′(0) = rg(0) −
rg (0) 2
__________
Φ e ρw + λ(0)
> 0 .
Hence, it moves beyond g = Φ + λ at time zero and satisfies Φ + λ < g < Φ e ρw + λ before reaching the g -axis. If Φ + λ < g < Φ e ρw + λ , then g ′ > 0 and λ′ > 0 .
284 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
To show that (g − λ)′(t) > 0 for all t ∈ [0, m − (λ(0))] , suppose
to the contrary that (g − λ)′(s) ≤ 0 for some s . Let t ∗
= min s {s > 0 : (g − λ)′(s) ≤ 0} . It is easily seen that (g − λ)′( t ∗ ) = 0
and (g − λ)″( t ∗ ) ≤ 0 . Since (g − λ)′ = rg − r g
2 ______ Φ e ρw + λ − π(g − Φ − λ) ,
(g − λ)″( t ∗ ) =
(
r −
2rg(Φ e ρw + λ)
___________
(Φ e ρw + λ) 2
− π
)
g′( t ∗ ) + (
r g 2
__________
(Φ e ρw + λ) 2
+ π) λ′( t
∗ )
= (r +
r g 2 − 2rg(Φ e ρw + λ)
________________
(Φ e ρw + λ) 2
) g′( t
∗ )
= r
(Φ e ρw + λ − g) 2
_____________
(Φ e ρw + λ) 2
g′( t ∗ ) > 0 ,
where the second equality follows from g′( t ∗ ) = λ′( t ∗ ) . This contradicts that
(g − λ)″( t ∗ ) ≤ 0 .
(ii) First, we show that m − (λ(0)) is finite. We know from the proof of part (i) that
λ′ > 0 . It follows from (C6) and (g − λ)′ > 0 in part (i) that
λ″ = π(g − λ)′ > 0 .
Hence, starting from λ(0) < 0 , λ(t) accelerates and will reach zero in finite time.
Second, we show that lim λ(0)↓−Φ
m − (λ(0)) = ∞ . If λ(0) = −Φ and
g(0) = 0 , then
λ′(0) = π(g(0) − Φ − λ(0)) = 0,
g′(0) = rg(0) −
rg (0) 2
__________
Φ e ρw + λ(0)
= 0 .
Continuity of the ODE system (C6), (C7) implies that (λ, g) will stay in
a small neighborhood of (−Φ, 0) for a long duration if λ(0) is sufficiently
close to −Φ and g(0) = Φ + λ(0) . Therefore, lim λ(0)↓−Φ
m − (λ(0))
= ∞ . ∎
VOL. 7 nO. 2 285fuller et al.: unemployment insurance fraud
Appendix D. Stochastic Verification
A. construction of a contract
To prove Proposition 4, we first construct a contract σ ∗ in which E(t) > U(t)
implies p(t) = 0 and E(t) = U(t) implies p(t) > 0 . This contract has the features
described in Proposition 4, and in the next section we verify it is indeed optimal.
First, since the principal does not monitor in this contract when E > U , we still
use the ODE system (20), (21) to find a solution path in the interval [0, n] , where
n satisfies
(D1) − ∫
0
n
λ(t) (rE − r e −ρw u) dt − λ(n )( e ρϕ − 1)E(n ) + γ = 0 .
The two boundary conditions for the ODE system (20), (21) are still λ(0) = 0 and
E(n ) = − ρ −1 (Φ + λ(n )) −1 .
LEMMA 6: The n that satisfies (D1) exists and is unique.
PROOF:
For uniqueness, we show that f (n ) ≡ − ∫ 0
n λ(t) (rE − r e −ρw u) dt − λ(n )
× ( e ϕ − 1)E(n ) decreases with n . Since both λ(n ) and E(n ) are negative and
decreasing with n , −λ(n )( e ϕ − 1)E(n ) decreases with n . Moreover,
−λ (rE − r e −ρw u) = r|λ| __________
g(Φ e ρw + λ)
(g − λ − Φ e ρw ) .
For fixed t ,
r|λ|
________
g(Φ e ρw + λ)
increases with n , while (g − λ − Φ e ρw ) is more nega-
tive with higher n . Therefore, − ∫ 0
n λ (rE − r e −ρw u) dt decreases with n too.
For existence, note that li m n→0 f (n ) = 0 . Because li m n→∞ λ(n ) = −Φ and
li m n→∞ E (n ) = −∞ , we have li m n→∞ f (n ) = −∞ . ∎
Second, choose p > 0 after n so that the state vector stays on the 45-degree
line before the monitoring arrives, i.e., U(t) = E(t) for all t ≥ n . Choosing
U ̃ (n ) = U(0) = − 1 __ ρΦ and solving the equation U ′(n) = E′(n) , we have
(D2) p =
r(1 − e −ρw ) (Φ + e −ρw λ(n )) −1
______________________
e ρϕ (Φ + λ(n )) −1 − Φ −1
> 0 .
Note that p is independent of Φ . This also implies that p > 0 is time invariant after
n because U(t) = E(t) for t ≥ n .
Third, the constructed solution path defines a contract σ ∗ as follows. For each
t ∈ [0, n ] , the policy u(t) is obtained by the first-order condition (13)
(D3) u(t) = − 1 ____________
ρ(Φ + e −ρw λ(t))
.
286 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
If t ≥ n , then the state vector moves along the 45-degree line, and u(t) is always
proportional to (U(t), E(t)) . That is, for all t ≥ n ,
(D4)
u′(t)
____
u(t)
=
E ′(t)
____
E(t)
=
U ′(t)
____
U(t)
= r −
r(Φ + λ(n ))
___________
Φ + e −ρw λ(n )
+ p (1 −
Φ + λ(n)
________ Φ ) > 0 .
The contract σ ∗ is defined by (D1–D4), and the property that the continuation con-
tract after a monitoring at t ≥ n starts a new cycle in which the continuation utility
is U ̃ (t) =
Φ + λ(n )
______ Φ U(t) instead of U(0) . In this construction, σ
∗ has the features
mentioned in Proposition 4.
B. Optimality of the contract
First, using the path obtained in Lemma 6, we construct a cost function c as
(D5) (r + π)c(U(t), E(t)) = πc(E(t)) + rc(u(t))
+ Φ((r + π)U(t) − πE(t) − ru(t)) + λ(t)(rE(t) − r e −ρw u(t)) .
LEMMA 7: c U (U(t), E(t)) = Φ , and c E (U(t), E(t)) = λ(t) .
PROOF:
Differentiate (D5) with respect to t , we have
(r + π)( c U U ′(t) + c E E ′(t))
= πc′(E)E ′(t) + Φ((r + π)U ′(t) − πE ′(t)) + λ(t)rE ′(t) + λ′(t)E ′(t),
which, after substituting λ′(t) = π(Φ − c′(E) + λ) , becomes
c U U ′(t) + c E E ′(t) = ΦU ′(t) + λ(t)E ′(t) .
Homogeneity of c( ·, · ) implies that c U U(t) + c E E(t) + ρ −1 = 0 = ΦU(t) +
λ(t)E(t) + ρ −1 . Because the vectors (U ′(t), E ′(t)) and (U(t), E(t)) are linearly
independent (we have shown that (
U __
E
)
′ (t) < 0 in the proof of Proposition 2, which
is
E ′(t)
___
E(t)
>
U ′(t)
___
U(t)
) , we have c U = Φ and c E = λ(t) . ∎
VOL. 7 nO. 2 287fuller et al.: unemployment insurance fraud
Second, we verify that the cost function c satisfies the HJB equation:
(D6) (r + π)c(U, E ) = min
u, p, U ̃ , E ̃
{rc(u) + πc(E ) + p (c( U ̃ , E ̃ ) + γ − c(U, E ))
+ c U (r(U − u) − π(E − U ) − p( U ̃ − U))
+ c E (rE − r e −ρw u − p( e ρϕ − 1)E ) } ,
where ( U ̃ , E ̃ ) is the new state vector the principal chooses after the next monitoring.
LEMMA 8: The c( ·, · ) defined in (D5) satisfies (D6).
PROOF:
The only differences between (D5) and (D6) are the terms associated with arrival
rate p , which will be shown to be zero in this proof. Fix a t ∈ [0, n ] and con-
sider the HJB equation at (U(t), E(t)) . The first-order condition for U ̃ implies that
U ̃ = U(0) . Then we have
c( U ̃ , E ̃ ) + γ − c(U, E) − Φ ( U ̃ − U ) − c E ( e ρϕ − 1)E
= − ∫
0
t
λ(s) (rE(s) − r e −ρw u(s)) ds − λ(t)( e ϕ − 1)E(t) + γ .
The above is decreasing in t because λ(t) < 0 , and E(t) < 0 both decrease in t . Moreover, the integral − ∫ 0
t λ(s) (rE(s) − r e −ρw u(s)) ds decreases in t because
rE(t) − r e −ρw u(t) = E ′(t) = ρ E 2 (t)g′(t) < 0 .
Therefore, the definition of n in (D1) implies that
c ( U ̃ , E ̃ ) + γ − c(U, E) − Φ ( U ̃ − U ) − c E ( e ρϕ − 1)E {
> 0, if t < n,
= 0,
if t = n .
This implies that
min
p≥0
p (c ( U ̃ , E ̃ ) + γ − c(U, E) − Φ ( U ̃ − U ) − c E ( e ρϕ − 1)E ) = 0,
which finishes the proof. ∎
Finally, to complete the proof of Proposition 4, we show that the contract σ ∗ is
optimal.
288 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
PROOF OF PROPOSITION 4:
Because the technique of using the HJB equation to verify optimality is standard,
we spare the reader of detailed steps. Given the initial promised utilities (U, E) , we
need to verify that
(i) The cost of the contract σ ∗ is c(U, E) .
(ii) The costs of other I.C. contracts are weakly higher than c(U, E) .
We only verify (ii) here, since the proof for (i) can be obtained simply by replacing
the following inequalities with equalities.
To see that the cost of an I.C. contract { ( c ̃ E (t), c ̃ U (t), p ̃ (t)) ; t ≥ 0} is higher than
c(U, E) , define
h(T ) = ∫
0
T
e −(r+π)t− ∫ 0
t p ̃ (x)dx (πc ( E ̃ (t)) + r c ̃ U (t) + p ̃ (t) (c ( U ̃ (t), E ̃ (t)) + γ ) ) dt
+ e −(r+π)T− ∫ 0
T p ̃ (x)dx c(U(T ), E(T )) .
The HJB equation implies that f ′(T ) ≥ 0 . Therefore, h(T ) increases in T , and
c(U, E) = h(0) ≤ h(T ) .
Taking limit T → ∞ , we have
c(U, E) ≤ ∫
0
∞
e −(r+π)t− ∫ 0
t p ̃ (x)dx (πc ( E ̃ (t)) + r c ̃ U (t) + p ̃ (t) (c ( U ̃ (t), E ̃ (t)) + γ ) ) dt,
which can be rewritten as
c(U, E) ≤ E [ ∫ 0
τ 1
e −rt (πc ( E ̃ (t)) + r c ̃ U (t)) dt ] + E [ e −r τ 1 γ ]
+ E [ e −r τ 1 c ( U ̃ ( τ 1 ), E ̃ ( τ 1 )) ] ,
where τ 1 is the first monitoring time and ( U ̃ ( τ 1 ), E ̃ ( τ 1 )) is the state vector immedi-
ately after monitoring. Inductively, we obtain
c (U, E ) ≤ E [ ∫ 0
τ n
e −rt (πc ( E ̃ (t)) + r c ̃ U (t)) dt ] + E [ ∑ i=1
n
e −r τ i γ ]
+ E [ e −r τ n c ( U ̃ ( τ n ), E ̃ ( τ n )) ] ,
VOL. 7 nO. 2 289fuller et al.: unemployment insurance fraud
where τ n is the n th monitoring time. Without loss of generality, we may assume that
li m n→∞ τ n almost surely (otherwise the principal monitors infinitely many times in
finite time and the monitoring cost is infinity). Taking limit n → ∞ yields
c (U, E ) ≤ E [ ∫ 0
∞
e −r t (πc ( E ̃ (t)) + r c ̃ U (t)) dt ] + E [ ∑ i=1
∞
e −r τ i γ ] . ∎
Appendix E. Imperfect Detection
This section presents a version of the stochastic verification model where detec-
tion is imperfect. Specifically, there is a positive probability ϖ > 0 of monitor-
ing error. In the event of monitoring error, an unemployed worker is labeled as
employed. If an unemployed worker is monitored after reporting unemployment,
the principal observes either an unemployed signal with probability 1 − ϖ or an
employed signal with probability ϖ . On the other hand, there is no monitoring
error that labels an employed worker as being unemployed, i.e., if an employed
worker is monitored after reporting unemployment, the principal observes with
probability one.
The timing of the problem is similar to the stochastic verification case in
Section VII. The planner still chooses the arrival rate of monitoring, p(t) , conditional
on the report of unemployment in period t . There are, however, two differences in
the case of imperfect detection. First, the planner assigns continuation utilities based
not only on whether or not monitoring occurs (as above) but also on the signal from
monitoring. Let U (t) and U (t) be the continuation utilities of a monitored unem-
ployed worker with signals and at t , respectively. Let E (t) be the continuation
utility of a monitored employed worker (whose signal can only be ) at t . Finally,
E (t) is the continuation utility of a monitored unemployed worker with signal
who transited to employment immediately after being monitored. Second, the pen-
alty is exogenous in the case of perfect detection above, but is endogenous with
imperfect detection.
Similar to (24) and (25), the promise-keeping constraint and incentive constraint
are
(E1) U ′ = r(U − u) − π(E − U) − p [(1 − ϖ) U + ϖ U − U ] ,
(E2) E′ ≤ rE − r e −ρw u − p( E − E) .
There are two differences between these two equations and (24) and (25). First, the
promise-keeping constraint (E1) incorporates the possibility that an unemployed
worker may be labeled as employed after monitoring. Second, in (25) the last term
on the right-hand side results from the exogenous and finite penalty, ϕ , whereas in
(E2) the last term allows the penalty E to be endogenous.
The main results from the perfection detection case and stochastic monitoring
still hold here. That is, the optimal monitoring mechanism consists of cycles. Within
each cycle, there exists some n such that the planner sets p = 0 before n , and then
monitors at rate p thereafter. Formally we state the following proposition.
290 AMEricAn EcOnOMic JOUrnAL: MAcrOEcOnOMicS ApriL 2015
PROPOSITION 6: There exists an n > 0 such that the principal monitors the
unemployed with a constant arrival rate p > 0 if and only if t ≥ n . Before n ,
the time path (U( · ), E( · )) converges to the 45-degree line; after n , the utility pair
(U(t), E(t)) remains stationary (i.e., U(t) = E(t) = U(n ) = E(n ) for all t ≥ n )
until the worker is randomly drawn to be monitored. if the observed signal from
monitoring is , the worker is punished, U = E < U(n ) . if the signal is , the
worker is rewarded, U > U(n ) , and the contract enters a new cycle.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without
permission.
Unemployment Insurance Fraud and Optimal Monitoring
I. Unemployment Insurance Fraud Data
II. Model
III. A Simplification of the Optimal Contract
A. Scaling
B. Periodicity
IV. Optimal Unemployment Insurance with Exogenous Monitoring
A. Optimal Control
B. Continuation Utilities
C. Employment Tax
D. Unemployment Benefits
E. Pontryagin Minimum Principle
V. Optimal Monitoring
VI. Quits
VII. Stochastic Verification
A. Comparison of Monitoring with the Deterministic Case
B. Comparison of Tax/Subsidy with the Deterministic Case
C. Quantitative Analysis
VIII. Conclusion
Appendix A. Data
Appendix B. Microfoundations for E(t) ≥ U(t)
Appendix C. Proofs
Appendix D. Stochastic Verification
A. Construction of a Contract
B. Optimality of the Contract
Appendix E. Imperfect Detection
REFERENCES
