An Equilibrium Model of Health Insurance Provision and Wage Determination
Author(s): Matthew S. Dey and Christopher J. Flinn
Source: Econometrica, Vol. 73, No. 2 (Mar., 2005), pp. 571-627
Published by: The Econometric Society
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Econometrica,Vol. 73, No. 2 (March,2005), 571-627
AN EQUILIBRIUM MODEL OF HEALTH INSURANCE PROVISION
AND WAGE DETERMINATION
BY MATTHEW S. DEY AND CHRISTOPHER J. FLINN1
We investigate the effect of employer-providedhealth insurance on job mobility
rates and economic welfare using a search, matching,and bargainingframework.In
our model, health insurancecoveragedecisions are made in a cooperativemannerthat
recognizesthe productivityeffects of health insuranceas well as its nonpecuniaryvalue
to the employee. The resultingequilibriumis one in which not all employmentmatches
are covered by health insurance,wages at jobs providinghealth insuranceare larger
(in a stochasticsense) than those at jobs without health insurance,and workersat jobs
with health insuranceare less likely to leave those jobs, even after conditioningon the
wage rate. We estimate the model using the 1996 panel of the Surveyof Income and
ProgramParticipation,and find that the employer-providedhealth insurancesystem
does not lead to any serious inefficienciesin mobilitydecisions.
Health insurance,equilibriummodels, wage bargaining,job mobility,
KEYWORDS:
simulatedmaximumlikelihood.
1. INTRODUCTION
HEALTHINSURANCEIS MOSTOFTENreceived through one's employer in the
United States.Accordingto U.S. CensusBureaustatistics,almost85 percent of
Americanswith privatehealth insuranceobtain their coverage in this manner.
This strong connection between employment decisions and health insurance
coverage has resulted in a substantialamount of research exploringthe possible explanationsfor and impacts of this linkage. One branch of the literature
has investigatedthe relationshipbetween employer-providedhealth insurance
and job mobility.In spite of a substantialamount of researchon the issue, the
relationshipbetween health insurancecoverage and mobility rates has not as
yet been satisfactorilyexplained. Basing their argumentslargely on anecdotal evidence, many proponents of health care reform claim that the present
employment-basedsystem causes some workers to remain in jobs they would
"rather"leave since they are "locked in" to their source of health insurance.
'This researchwas supportedin part by grantsfrom the Economic ResearchInitiativeon the
Uninsured at the Universityof Michiganand the C. V. StarrCenter for Applied Economics at
New York University.This paper was written while the first author was a Post-Doctoral Fellow at the Universityof Chicago. We are grateful for comments received from participantsin
the "Econometricsof Strategyand Decision Making"conference held at the Cowles Foundation in May 2000, the annual meeting of the European Society for PopulationEconomics held
in Bonn in June 2000, the meetings of the CanadianEconometricStudyGroup held in October
2002, and seminarparticipantsat Chicago, McGill, Ohio State, UCSD, Pompeu Fabra,CREST,
UCLA, Georgetown,and Wisconsin.We have also received manyvaluablecommentsfrom Jaap
Abbring,JeffreyCampbell,Donna Gilleskie,VijayKrishna,Jean-MarcRobin, and three anonymous referees. The views expressedhere are ours and do not necessarilyrepresentthe views of
the U.S. Departmentof Laboror the Bureauof LaborStatistics.We are responsiblefor all errors,
omissions,and interpretations.
571
572
M. S. DEY AND C. J. FLINN
While it is true that individualswith employer-providedhealth insurance are
less likely to change jobs than others (Mitchell (1982), Cooper and Monheit
(1993)), the claim that health insurance is the cause of this result has not
been established. Madrian (1994) estimates that health insurance leads to a
25 percent reduction in worker mobility, while Holtz-Eakin (1994) finds no
effect, even though they use an identical empiricalmethodology. Building on
their approach,Buchmuellerand Valletta (1996) and Anderson (1997) arrive
at an estimate of the negative impact of health insuranceon worker mobility
that is slightly larger (in absolute value) than Madrian's,while Kapur (1998)
concludes there is no impact of health insurance on mobility. The most recent and only paper in this literaturethat attempts to explicitlymodel worker
decisions, Gilleskie and Lutz (2002), finds that employment-basedhealth insurance leads to no reduction in mobility for married males and a relatively
small (10 percent) reductionin mobilityfor single males. Using statewidevariation in continuationof coverage mandates, Gruberand Madrian(1994) find
that an additionalyear of coverage significantlyincreases mobility,which they
claim establishesthat health insurancedoes indeed cause reductionsin mobility. While this literaturehas extensivelyexaminedhow the employment-based
health insurance system affects mobility, the more pressing welfare implications have largelybeen ignored (Gruber and Madrian(1997) and Gruberand
Hanratty(1995) are notable exceptions).
If health insurancecoverage is strictlya nonpecuniarypart of the compensation package offered by an employer, like a corner office or reservedparking
space, the theory of compensatingdifferentialswould predict a negative relationship between the cost (or provision) of health insuranceand wages conditional on the value of the employment match. Somewhat surprisingly(from
this perspective), Monheit et al. (1985) estimate a positive relationship between the two. Subsequent research has attempted to exploit potentially exogenous variationfrom a varietyof sources in order to accuratelyidentify the
"effect"of health insuranceon wages. Gruber (1994) uses statewidevariation
in mandated maternitybenefits, Gruber and Krueger (1990) employ industry
and state variationin the cost of worker'scompensationinsurance,and Eberts
and Stone (1985) rely on school districtvariationin health insurancecosts to
estimate the mannerin whichwages are affected. All three conclude that most
(more than 80 percent) of the cost of the benefit is reflected in lower wages.
In addition, Miller (1995) estimates significantwage decreases for individuals moving from a job without insurance to a job with insurance.Hence, the
research that examines to what extent health insurancecosts are passed on to
employees findsthat a majorityof the costs are borne by employees in the form
of lower wages.
These results from the two branches of the literatureseem inconsistent on
the face of it. If individualsare bearing the cost of the health insurancebeing
providedto them by their employer,why are they apparentlyless likelyto leave
HEALTHINSURANCE PROVISION
573
these jobs? In addition, the absence of a conceptual frameworkthat is consistent with many of the empiricalfindingson "joblock" and the indirectcosts of
health insuranceto workers means that few policy implicationscan be drawn
from the empiricalresults that have been obtained.
In this paper we attempt to provide such a framework by developing
and estimating an equilibriummodel of employer-providedhealth insurance
and wage determination. The model is based on a continuous-time stationary search model in which unemployed and employed agents stochastically
uncover employment opportunities characterized in terms of idiosyncratic
match values. Firms and searchers then engage in Nash bargainingto divide
the surpluses from each potential employment match. In contrast to traditional matching-bargainingmodels (e.g., Flinn and Heckman (1982), Diamond
(1982), and Pissarides (1985)), we allow employee "compensation"to vary
over both wages and health insurance coverage. In our framework, health
insurance has two potential welfare impacts on the worker-firmpair. First,
by inducing the employee to utilize health care services more frequently it
increases his productivity in the sense of reducing the frequency of negative health outcomes that lead to the termination of the match. Due to
search frictions, the preservationof an acceptable match provides a benefit
to both the employer and the employee. Second, at least some individuals
may exhibit a "private"demand for health insurance.We view this demand as
mainly arisingfrom the existence of uncovered dependents in the employee's
household.
The main novelty in our modeling frameworkis our view of health insurance as a productivefactor in an employmentmatch, in addition to any direct
utility-augmentingeffects it may have. As a result, the level of health insurance
coverageis optimallychosen given the value of the productivitymatchand idiosyncraticcharacteristicsof the worker and firm. The productivity-enhancing
nature of health insurance is modeled as follows. Since an employment contract may terminate due to the poor health of the employee, we view health
insurancecoverage as reducingthe rate of "exogenous"terminationsfrom this
source. There is some support for our assumptionin the empiricalliterature.
Levy and Meltzer (2001) provide a very thoroughsurveyof empiricalresearch
examiningthe relationshipbetween health insurancecoverage and health outcomes. With special reference to results from large-scale quasi-experimental
studies and the Rand randomizedexperiment,the authorsconclude that-there
exists solid evidence to "suggest that policies to expand insurance can also
promote health." Since we find overwhelmingsupport for our assumptionin
the course of estimatingthe model,2our results could be taken as adding fur2Wedo not impose the theoretical restrictionwhen the model is estimated, yet find that the
estimatedrate of "involuntary"
job separationsfor the uninsuredis an orderof magnitudegreater
than the rate for the insured.
574
M. S. DEY AND C. J. FLINN
ther (albeit indirect)supportto the propositionthat health insuranceimproves
health.
Beginning from this premise, we are able to derive a number of implications from the model that coincidewith both anecdotaland empiricalevidence.
Most basically,the model implies that: (i) not all jobs will providehealth insurance, (ii) workers "pay"for health insurancein the form of lower wages, and
(iii) jobs with health insurancecoverage tend to last longer than those without
(both unconditionallyand conditionallyon wage rates).
In order to explicitlyinvestigatethe "joblock" phenomenon, it is necessary
for us to allow for on-the-job searchwith resultingjob-to-job transitions.This
necessitates that we model the process of negotiation between a worker and
two potential employers,which we are able to accomplishafter making some
stringent assumptionsregarding the information sets of the agents involved
in the bargaininggame. This is the first attempt to estimate such a model in
a Nash bargaining context, though using French worker-firmmatched data
Postel-Vinayand Robin (2002) estimate an equilibriumassignmentmodel that
includes renegotiation.3They assume that firms appropriateall of the rents
from the match,which is a limitingcase of the Nash bargainingmodel we employ.
Estimates of the primitiveparameterscharacterizingthe model using data
from the 1996panel of the Surveyof Income and ProgramParticipation(SIPP)
tend to support our model specification. In particular,we find that the rate
of "involuntary"separationsfrom jobs without health insuranceis about eight
times greaterthan at jobs with health insurance.We find broadconformitywith
the implicationsof the model on a number of other dimensions as well. The
raw data suggest that jobs providinghealth insuranceare substantiallylonger
than those that do not provide it, though due to substantialamounts of rightcensoring of spell lengths the precise magnitude of the difference is difficult
to determine. Model estimates imply that the ratio is on the order of six. The
model also does a reasonablygood job of fittingthe observedconditionalwage
distributions(by health insurancestatus) and the unconditionaldistributionof
wages.
We are also able to look at the claim that nonuniversally-providedhealth
insurance leads to inefficient mobility decisions. We demonstrate that within
our model all mobility decisions are efficient in a generalized sense. We allow
for heterogeneity in the population of firms with respect to the cost of providing health insurance and within the population of searcherswith respect
to the "private"valuation of health insurance.While time-invariantsearcher
heterogeneitycannot lead to inefficientturnoverdecisions, time-invariantfirm
heterogeneitycan, at least in one specificsense. A workerwho faces the choice
3Therecentpaperby Cahuc,Postel-Vinay,and Robin (2003) uses a similarbargainingstructure
to the one employed in this paper, though that paper provides a much deeper analysisof the
bargaininggame playedby potential employersand workers.
HEALTHINSURANCE PROVISION
575
between two firmsat a point in time with two knownmatchvalues 0 and 0' may
choose to work at the lower match value firm if that firm offers lower costs of
health insurancethan does the other. While the match chosen will alwaysbe
the one providinghigher total surplusto the worker-firmpair, the fact that a
lower matchvalue is chosen may be taken to representa form of inefficiencyat least in comparisonwith a world in which all potential matches face the
same cost of providingcoverage. Given our estimates of the health insurance
cost distributionwithin the population of employers,even this limited form of
"inefficiency"is found to be virtuallynonexistent.Based on our model specification and estimates,we find little indicationthat the currentemployer-based
health insurance system causes individualsto pass up productivity-improving
job opportunities.
Due to data limitationsand also for reasons of tractabilitywe have decided
to only consider whether health insurance is provided or not and additionally assume that the employer's direct cost of purchasing health insurance
is exogenously determined. In reality the provision of health insurance involves many complicating features, both at the plan level and with respect
to the costs that employers face. In particular, plans may cover both the
worker (who supplies the productivity)and his family. Implicit in our empirical work is the assumptionthat health insurancecoverage is extended to other
family members. Moreover, for various reasons insurance contracts usually
involve cost-sharingand risk-sharingfeatures (e.g., deductibles, co-payment
rates, and annual maximums,etc.). While these features are undoubtedlyimportant factors in the decision-makingprocess, the data requirementsnecessary to consider these elements are well beyond the scope of any currently
available dataset. In a similar vein, employers can either purchase coverage
from an insuranceproviderwith whom they may be able to bargain over the
premiumsbased on employee demographicsor self-insure. By allowing firmlevel heterogeneity in the cost of health insuranceprovisionwe are capturing
(in an admittedly indirect manner) some of these features. Lastly, although
perhaps most importantly,our model ignores the relative tax advantage of
compensation in the form of health insurance benefits. While the favored
tax status of health insurance affects the wage and health insurance distribution, we argue that this cannot be the entire explanation for the role employers play in the provision of health insurance in the United States and
the inclusion of tax parameterswill not change the qualitativefeatures of our
model.
The remainderof the paper is structuredas follows. In Section 2 we develop
our search-theoreticmodel of the labor market with matching and bargaining which produces an equilibriumdistributionof wages and health insurance
status. Section 3 contains a discussionof the data used to estimate the equilibriummodel, while Section 4 develops the econometricmethodology.Section 5
presents the estimates of the primitiveparametersand the implicationsof estimates for observabledurationand wage distributions.We also develop formal
576
M. S. DEY AND C. J. FLINN
measuresof the extent of "inefficient"turnover,and use our estimates to compute them. In Section 6 we offer some concludingremarks.
2. A MODEL OF HEALTHINSURANCE PROVISIONAND WAGE DETERMINATION
In this section we describethe behavioralmodel of labor marketsearchwith
matchingand bargaining.The model is formulatedin continuous time and assumes stationarityof the labor market environment.We begin by laying out
the structureof the most general model we estimate and then proceed to characterize some importantproperties of the equilibrium.We attempt to provide
some intuition regardingempiricalimplicationsof the model throughgraphical illustrationsof equilibriumoutcomes in the specificationthat ignores heterogeneity on the supplyand demand sides of the market.
The key premise of the model is that health insurance has a positive
impact on the productivity of the match. As is standard in most searchmatching-bargainingframeworks,the instantaneous value of a worker-firm
pairing is determined by a draw (upon meeting) from a nondegenerate distribution G(O). This match value persists throughoutthe durationof the employment relationshipas long as the individualremains "healthy."A negative
health shock while employed at match value 0 reduces the value of the match
to 0 and results in what we will consider to be an exogenous dissolutionof the
employment relationship.4Thus an adverse health shock is considered to be
one, potentiallyimportant,source of job terminationsinto unemployment.To
keep the model tractablewe have assumedthat a negative health shock on one
job does not affect the labor market environmentof the individualafter that
job is terminated.Thus one should thinkof these health shocks as being largely
employeror job task-specific.
The role of health insurance in reducing the rate of separations into unemployment is presumed to result from covered employees more intensively
utilizing medical services than noncovered employees. As a result, the rate
of separations due to an inabilityto perform the job task associated with the
match 0 will be lower amongcovered employees. If all other reasonsfor leaving
a job and entering the unemployed state are independent of health insurance
status, the wage, and the match value, then r71< r70,where r7d is the flow rate
from employment into unemploymentfor those in health insurance status d,
with d = 1 for those with employer-providedhealth insurance and d = 0 for
those who are uncovered.In this mannerthe purchaseof health insurancecan
extend the expected life of the match for any given match value 0 throughit's
"direct,"but stochastic, impact on health status. We will also consider other
4Itis not strictlynecessarythat the value of the matchbe reducedto 0, since any new value of 0
that would make unemploymentmore attractivethan continued employmentat the firmwould
do. The value 0 serves as a convenientnormalization.
HEALTHINSURANCE PROVISION
577
indirect effects of health insuranceon match longevitythat operate through a
sortingmechanism.
Individualsare assumed to possess an instantaneous(indirect) utility function given by
u,(w, d) = w +,d,
where 5 > 0. Individualsof type 4 = 0 will then behave as classic expected
wealth maximizersand exhibit only a "deriveddemand"for health insurance
as a productivefactor in an employmentrelationship.Those individualswith
4 > 0 have some "private"demandfor health insuranceand maximizea slightly
different objective function. We assume that the population distributionof
preference types is given by H(S).
The heterogeneityon the demand side of the marketrelates to the cost to a
firmof providinghealth insurancecoverageto any one of its employees. While
modeling the cost of providingsuch coverage is an interesting issue in itself,
here we simplyassume that these costs are exogenouslydetermined.The pre? c R+. We denote the populamium paid by a firm is given by 4, where 4 E
tion distributionof firm types by F(0).
Given the natureof the data availableto us (from the supplyside of the market), firmsare treated as relativelypassive agents throughout.In particular,we
assume that the only factor of production is labor, and that the total output
of the firm is simply the sum of the productivitylevels of all of its employees.
Then if the firm "passes"on the applicant-that is, does not make an employment offer-its "disagreement"outcome is 0 (it earns no revenue but makes
no wage payment).Withthe additionalassumptionthat there are no fixedcosts
of employmentto firms,the implicationis that employmentcontractsare negotiated between workersand firmson an individualisticbasis, that is, without
reference to the composition of the firm'scurrentworkforce.
All individualsbegin their lives in the nonemploymentstate, and we assume
that it is optimal for them to search. The instantaneousutilityflow in the nonemploymentstate is b, whichcan be positive or negative.When an unemployed
searcher and a firm meet, which happens at rate A,, the productivevalue of
the match 0 is immediately observed by both the applicant and the firm as
are the firm and searchertypes, ( and 5, respectively.After both parties have
been fully informed, a division of the match value is proposed using a Nash
bargainingframework.If both parties realize a positive surplus the match is
formed, and if not the searcher continues looking for an acceptable match.
Let V/N denote the value of unemployed search to a searcher of type :, and
denote by Q (0, () the value of the match if the searcher receives all of the
surplus.Then since the disagreementvalue of the firm is 0, all matches with
QJ(0, 4) > V/Nwill be accepted by an unemployed searcher of type 4 and a
type 4 firm.5Let the value of an employmentcontractbetween a type 4worker
SThis claim is predicated on
VN
> 0, which we assume to be the case.
578
M. S. DEY AND C. J. FLINN
and a type 0 employerwith matchvalue 0 to the employee be VE(w, d; 0, 4)
and let the value of the same match to the firm be given by VFF(w, d; 0, )).
Then given an acceptable match, the wage and health insurancestatus of the
employment contract are determined by the solution to the Nash bargaining
game
(1)
(wii, d)(0,
d; 0, , VN).
4, V:) = argmaxS(w,
w,d
The Nash bargainingobjectivefunction is given by
(2)
?'(w,
d; 0, , Vf ) = (V (w, d; 0,
4) -
Vf)V/F(w,
d; 0, 4)l-,
where a e (0, 1) is the bargainingpower of the individual.
Employed agents meet new potential employers at rate Ae. To keep the
model tractable we assume that there is full information among all parties
as to the characteristicsassociated with the currentmatch (0, 0) and the potential match (0', )'), as well as the worker's type s. This means that each
firm knows the match value of the individualat the other firm with which it
is competing for the worker'sservices as well as that firm'stype. While these
are strong informationalassumptions,they are relativelyinconsequentialfor
the empiricalanalysisconducted below given the nature of the data available
to us.
We now consider the rent division problem facing a currently employed
agent who encounters a new potential employer. Assume that a currently
employed type s individual faces two potential employers with characteristics, (0, ()) and (0', 4'), where Q~(0', )') > Q(0, 0) > VN. Under our bidding mechanism, the individualwill end up accepting the match at the type
)' employer after the "last" offer by the type ) firm. The idea is that the
type 4' employer can match any feasible offer (i.e., any (w, d) such that
6)) > 0) made by the type 4)firm and will therefore "win"the
VF(w, d; 0,
worker'sservices. The value of the offer of the dominated firm serves as the
threat point of the employee in the Nash bargainingproblemfaced by the employee with the winning firm. When this is the case, the Nash bargainingobjective function is given by
(3)
((w, d; 0', ', 0, 4)
= (
(wd; -;0, ,
)) - Q (0,
d))
(
; 0',
4))
and the new equilibrium wage and health insurance pair will be given
by
(4)
d; 0',4', 0, >).
(w d), )(0', ', , )) = argmaxS'(w,
w,d
579
HEALTHINSURANCE PROVISION
The firm'svalue of the currentemploymentcontractis defined as:
(5)
VF(w, d;0,)
= (1 + p)-1
(0- w - d)e
+
+ rd X 0
Ae8
dJb(w,d, 3)
0,,0)dG()
V F(0,
(0
dF(&)
+ Ae,s G(06(w, d, d4))dF()) x V (w, d; 0, 4)
+ Ae?
G(B(O0,
o,0
+ (1 - Ae? -
))dF())
r7d?)VF(w,
x0
d; 0, ))+
o()
,
where VF(, (), ), () =, VF(w^(, (, 0 )),
; 0, )) represents
, d(0,
,
),
the equilibriumvalue to a type 0 firmof the productivematch 0 when the type
s worker's next best option has characteristics(0, 4), e is an infinitelysmall
period of time, and o(?) is a term with the propertythat lim,o(o(e)/e) = 0.
The function 06(w, d, () is defined as the maximumvalue of 0 for which the
contract (w, d) would leave a type 4 firmwith no profitgiven that the individual is type s. This value is implicitlydefined by the equation
), )=.
VF(W, d; 0(w,d,
Any encounter with a potential type ( firm in which the match value is less
than 0e(w, d, )) will not be reported by the employee; any new contact with
a match value greater than 0e(w, d, () will be reported to the current firm
and will result in either a renegotiationof the currentcontractor a separation.
When the new potential employercan match any offer extended by the current
firm such that Qe(0, 4) > Qe(0, 4)), a separationwill occur. That is, any new
match 0 > 0((0, 40,4) will induce the worker to quit where the criticalmatch
is implicitlydefined by
4) , )=Q,(,
QW(0O(0,
4)).
After rearrangingterms and takinglimits, we have
(6)
Vff(w, d; , )
(~O^w
--1'd+Ae
=P
/
d
~)dF~--1
d + Ae G(0(w, d, +)) dF(4.)
580
M. S. DEY AND C. J. FLINN
x
- w-d(
+ Ae Ja
0) dG(0)dF(4).
0,V0,
V(0,
(iv,
S d, J)
For the employee, the value of employment at a current match (0, )) and
wage and health insuranceprovisionstatus (w, d) is given by
(7)
VE(,
d; 0,)
= (1 + pe)- ' (w + )d)e + TdeVN
04(0,
r
+ oAe[J
^
E
V (0,
c)
d b~(uv,d,
+ Ae8 J(
dG(),4dF(,)
(0
4, 0, 4) dG(6) dF(O)
0, 4))
+ AeE
G(QB(w, d, 4)) dF()
+ (1 -
-e- 7Id) )V/(W, d; 0,
x VF(w, d; 0, )
)
+ o(8)
,
where VE(0, 4),0, () = VE(w(0, 0, 0, d()d(0,
6,0, ); 0, 4) represents
,
4
the equilibriumvalue to a type worker employed at a match with characteristics (0, 4) when his next best option is defined by (0, 0). Note that when
an employee encounters a firmwhere the total matchvalue is less than that at
the current employer but sufficientlygreat that it can be used to increase his
share of the match surplus,his new value of employment at the current firm
becomes VJE(w^(0, 6,0, 4), d(0,^ , 0, 4); 0, 4).
The receipt of outside offers duringan employmentmatch provides a rationale for wage growth on the job, as well as allowing for the possibilityfor a
change in health insurance coverage. In fact, while the bargainingpower parameter a is often the focus of attention when examiningthe labor share of
firm revenues, competition between firmsfor workerscan result in a very high
labor share even in the presence of low bargain power. This point is clearly
made in the empiricalresultsof Postel-Vinayand Robin (2002), where the bargaining power of searchersis assumed to be zero, and in the empiricalresults
presented below. In both cases, substantialamounts of wage growth are indicated with small or no bargainingpower as long as employersare forced to bid
againsteach other for a worker'sservices sufficientlyoften.
581
HEALTHINSURANCE PROVISION
In addition,when the potential surplusto the workerat the newly-contacted
firm exceeds that of the current firm (i.e., a new draw 0 such that 0 > 06(0,
4, 4)), mobilityresults. The value of employment at the new firm is given by
6,
0, ), d(0, ), 0, 4); 0, )-that is, the match value at the curVE(wA(0,
rent firmbecomes the determinantof the "threatpoint"faced by the new firm
and plays a role in the determinationof the new wage. Finally,when the match
value at the new firm is less than 60(w, d, 4), the contact is not reported to
the current firm since it would not result in any improvementin the current
contract. Because of this selective reporting, the value of employment contractsmust be monotonicallyincreasingboth within and acrossconsecutivejob
spells. Declines can only be observedfollowinga transitioninto the unemployment state.
After rearrangingterms and takinglimits,we have
--1
1
(8)
VE(w, d;
0, 4) = p+
d
+Ae e
x 'w+4d
G((w,
d,
))dF(4)
+ rdl/
e
+ A,
J(p J 6(w,d,<f)
+ Aef
, V/E(0,(f, 0, )) dG(0)dF())
,
Vff
,f,0
,))dG(0)
dF()
J( Je(0,(f,))
The model is closed after specifyingthe value of nonemployment,VJN,and
the total surplus of the match, Qe(0, )>).Passing directly to the steady state
representationof VN, we have
(9)
N
= [p + AG(0)]-1
x tb+
I ,'J,
Anf,
V
(0 dF(),
dG()
,
where 06 is the critical match value associated with the decision to initiate an employment contract for a type s individual and E(0, , VN) =
V(E(tB(0, 4),VN), de(0 , , VN); 0, 4) represents the equilibriumvalue to a
type ~ worker employed at a match with characteristics(0, () when coming
out of the nonemploymentstate.6
6Weare implicitlyassumingthat for every pair of worker-firmtypes (s, 4), there exist some
employmentcontractsthat do not result in the purchaseof health insurancecoverage.Without
this assumption,the criticalmatchwould depend on the type of firm the individualmeets out of
nonemployment.
582
M. S. DEY AND C. J. FLINN
The total surplusof the match is given by
Qe(0
= V)E(w,
( ) (0,, ,006,
,
,0,);
0, )),
where
(iw, dl)(0,
4, 0, 4) = (dw,d)(0, 40).
When the workerreceives the entire surplusof the matchthe equilibriumwage
function is particularlysimple to derive. Setting VF(w, d; 0, () 0= and recognizing that there is no room for renegotiation,we have
(10)
- w - d4O=O
(, 4()=e-
4d(O,0
).
Then
(11)
-Q(0
) = p+
T(d((O, 4)) + Ae
x
(-- (4
+Ae
G(d6(6,
)d(0, 4) +
L
(, 4))dF(4)
(dc(0, 4))fVN
((006, dG,()
4,
)
dF(),
wherewe haveusedthe factthat0Q(h^(0,4)),dr(0, 4), 4) = 0 (0, 4, 4).
We can now characterizethe wage and health insurancedecisions with the
following set of results.
1: Let Qf(H',
PROPOSITION
)')> Q(O, (f), where(0, <) representthe characteristicsof the next best option available to a type 5 employeeat the time a
bargain is made. The decision to acquire health insuranceis only a function
of ( e', O', ).
For the proof see AppendixA.
The decision to acquire health insurance is an efficient one in the sense
that it only depends on characteristicsof the currentmatch. The drivingforce
behind this result is the manner in which the "private"demand for health
insurance enters the individual'scontemporaneousutility function. It enters
linearly,and essentiallycombines with the firm'scost characteristic4)to produce a net health insurance cost of (>- . The linearity implies the health
insurancedecision is never revisited as a result of a new division of the match
surplus.Thus the health insurancedecision is made solely to maximizethe total
surplusfrom the match and is efficientgiven the cost of health insurance4 -.-
HEALTHINSURANCE PROVISION
583
2: Assume thereexist employmentcontractsbetweena type 0
PROPOSITION
employerand type ~ employeethat do not resultin thepurchaseof health insurance. The decision to initiatean employmentcontractcan be characterizedby a
whena typee individualworksfor a type4
uniquecriticalmatch 0*.Furthermore,
employerthe decisiontopurchasehealthinsurancecoveragecan be characterized
by a uniquecriticalmatch, 0**(>).
For the proof see Appendix B.
This result simplyestablishesthe existence of criticalvalue strategiesfor the
bargainingpair. The assumptionthat there exists some acceptable match values that do not resultin health insurancecoveragefor all pairs (0, s) simplifies
the characterizationof labor market equilibriumand the computationaltask
we face. There would be no conceptual difficultyin relaxingthis assumption.
When there exists firm heterogeneity "inefficient"mobility decisions will
occur, in general. By an "inefficient"mobility decision we mean that when
confronted with a choice between 0 and 0', where 0' > 0, the individual
(optimally) opts for 0.7 The reason for this is that, from a type s searcher's
perspective, an employment contract now is characterizedby the two values (0, 0). Since the value of the potential match is a function of both, when
confrontedwith a choice between (0', 4') and (0, 4), the decision rule will not
generallybe only a function of 6' and 0.
The potential for inefficiency,by which we really only mean that (0, 0') is
not a sufficient statistic for the mobility decision, only arises in certain cases.
Clearly,when two potential matches exist with the same type of firm8the individualwill alwayschoose the one with the highest value of 0, or
Q(O>', ) > Q(,
4)) o'> 0 vO.
Thus inefficientmobilitydecisions can never occurwhen the searcher'schoice
is between two firms of the same type. In this case the values (0, 0') are sufficient for characterizingthe mobilitydecision.
Now consider the case in which the agent faces a choice between firms of
different cost types. Say that his currentmatch is at a lower cost firm than the
potential match, 0 < 4'. Since the value of employmentis nonincreasingin the
cost of health insurance,the matchvalues at the highercost firmwill have to be
at least as large as the matchvalue at the lower cost firmfor mobilityto occur.
If the individualwould not purchase health insuranceat either firm, then the
types of the firms are irrelevant.In such a case, the match values at the two
firms are (conditionally)sufficientfor the mobilitydecision and no inefficient
7Byinefficiencyhere we mean that the highest matchvalue is not alwaystaken. In the context
of this model it is doubtfulthat this is the correctcriterionto use as is discussedmore fullybelow.
8For the likelihood to be nonzero that two firms of the same type would be bidding against
each other it is necessarythat the distributionof firm types have mass points. This is the case in
the econometricspecificationwe employ.
584
M. S. DEY AND C. J. FLINN
mobilitycan result (and in this case the value of workingat either firm at the
same matchvalue is equal). Now considerthe case in which the matchvalue at
the lower cost firmresults in the purchaseof health insurance.Let the current
employment match be characterizedby (0, 0) and the potential employment
matchbe characterizedby (0', )').If both matcheswould resultin health insurance being purchased,the agent must be compensatedfor the increasedcost of
health insurance.In this case, there exists a criticalvalue 0(0, 0, )') > 0 such
that Q (0 (0, 0, 0'), )') = Q (0, C))for which the agent will be indifferentbetween the two firms.It is also possible that there could exist a drawof 0' at the
higher cost firmthat resulted in mobilitybut did not result in health insurance.
In such an instance it is also necessary to compensate the individualfor the
loss of health insurance (whatever the value of s), and this also implies that
the criticalmatch value 0Q(0,0, 4') > 0. Thus there will alwaysbe a "wedge"
between the criticalmatchvalue requiredfor mobilityto a higher cost firmand
the current match value 0 at the lower cost firmwhenever the individualhas
health insurance.This wedge generates somethinganalogousto what is known
as "joblock" in the empiricalliteraturethat studies mobility,wage, and health
insuranceoutcomes. This occurs when an individualpasses on a higher match
at a higher cost firm to keep a lower match at a lower cost firm.
The other possibilityfor inefficientmobilitydecisions occurswhen the agent
is currentlyemployed at a higher cost firm (0, )) and meets a low cost firm
(0', )'), where O' < 0. If the agent would have no health insuranceat either
firm, then the mobility decision is made on the basis of the 0 and 0' draws
exclusively and is consistent with efficiency. When the current match at the
highercost firmprovideshealth insurance,then there once againexists a wedge
between the current value of the match at the higher cost firm and that required for mobility to the lower cost firm.As before, define 0(0, 04, )') such
thatQe(0e(0, 04, )'), O')= Q0(0,0) andnote that0((0, 4, 4') < 0. Whenthe
match at the higher cost firm does not result in health insurance, there still
exists a wedge when the match at the lower cost firm does. When an individual leaves a higher cost firm match for a lower valued match at a lower cost
firmwe may term this as "jobpush" as in the empiricalliteratureon the subject. Clearly "job lock" and "job push" are two sides of the same coin in our
framework,with the distinction between the two solely arising from whether
the currentmatch is with a lower cost or higher cost firm.
The model is sufficientlycomplex that comparativestatics results are not
readily available. In light of this we will only graphicallydisplay some of the
implications of the model. Due to the relatively complicated renegotiation
process it is difficult to solve for the steady state wage distribution,which is
the cross-sectional distributionthat would be observed after the labor market had been runningfor a sufficientlylong period of time. The distributions
that are plotted are all based on some of the model estimates obtained below;
the specific model utilized for these illustrativepurposes is the simplest one
585
HEALTH INSURANCE PROVISION
in which there is no heterogeneity on either side of the market and there is
symmetricbargaining,that is, a = 0.5.
Figure 1 plots the estimated probabilitydensity function of job matches in
the population,which is assumedto belong to the lognormalfamily.The lower
dotted line represents the critical match value for leaving unemployment,0*,
which is estimated to be approximately7.76. The dotted line to the right represents 0**,which is the critical match value for the match to provide health
insurance coverage (its estimated value is 13.95). The likelihood that an unemployed searcher who encounters a potential employer will accept a job is
the measure of the area to the right of 0*.The probabilitythat an unemployed
searcher accepts a job that does not provide health insuranceis then given by
the probabilitymass in the area between the two criticalvalues dividedby the
probabilityof findingan acceptablematch,which in this case is 0.43.
The wage densities (associated with the first job after leaving unemployment) conditional on health insurance status are displayed in Figure 2. We
can clearly discern the area of overlap between these two densities. Since
both of these p.d.f.'s are derived from slightlydifferent mappingsof the same
p.d.f. g(0), it is not surprisingthat they share general features in terms of
shape. Note that the wage density conditional on not having health insurance
is always defined on a finite interval [w(), w(0'), while the range of wages
conditional on having health insurance is unbounded as long as the matching distributionG has unbounded support.These general characteristicsalso
characterizethe conditionalsteady state wage distributions.
0.09 0.08 0.07 0.06 Z.
0.05
Q
0.04 -
\
0.03 0.02 -
0.01 -
0
5
10
15
20
25
30
35
40
Match
Productivity
FIGURE 1.-Productivity density. Based on the parameter estimates for the specification with
no heterogeneity presented in column 3 of Table II. The dotted vertical lines represent the critical
matches for transitions out of unemployment and for the provision of health insurance, respectively.
586
M. S. DEY AND C. J. FLINN
0.2 -
0.2 -
0.16 -
Uninsured
-
0.12-
Jobs
0.16 -
Insured Jobs
0.12-
,
~~~~~~~~~~~~~~~~~~~~~~~,
'Ci
*:
0.08-
0.08-
\
0.04-
0.04 -
\
0
0
0
5
10
15
Wage
20
25
30
0
5
10
15
20
25
30
Wage
FIGURE2.-Conditional (on health insurancestatus) wage densities and marginalwage density. The panel on the left representsthe conditionalwage densities and the panel on the right
depictsthe marginalwage density.Based on the parameterestimatesfor the specificationwith no
heterogeneitypresented in column 3 of TableII. Wagesrepresentthe firstwage reported in the
firstjob directlyfollowing an unemploymentspell. The dotted vertical lines representthe minimumwage in an uninsuredjob, the minimumwage in an insuredjob, and the maximumwage in
an uninsuredjob, respectively.See text for details.
The rightmost graph in Figure 2 exhibits the marginalwage density associated with the first wage observed following unemployment.The interval of
overlap in the wage distributionadds a "bulge"to a density that otherwise resembles the parent lognormaldensity. Recall that wages in the intervalof the
bulge are the only ones that can eitherbe associated with health insuranceor
not. Wagesin the right tail are alwaysassociated with jobs that provide health
insurancewhile those to the left of the bulge are associated with jobs that do
not provide health insurance.
Figure 3 containsgraphsof the simulatedsteadystate conditional(on health
insurance status) and unconditional wage distributions.The simulation on
which these histogramsare based is for one million labor marketcareers. The
figuresplot the equilibriumwage distributionsfrom the model, that is, they do
not include measurementerror.
The leftmost graphcontains the steady state conditionalwage distributions.
For individualsin jobs covered by health insurancethe shape of the distribution is ratherunremarkable.The lowest value of the wage in this case is $9.12
using point estimates of the model parameters.The steady state wage distribution for individualsholding jobs not providinghealth insuranceis more unusual. We know that this distributionis bounded, and we note a precipitous
drop in the "density"at relativelyhigh wages in the support of the distribu-
J,-,m
HEALTHINSURANCE PROVISION
0.35 -
0.10.09-
0.3 -
f~~1ki
0.25 -
~~\
~0.08UninsuredJobs
-
0.0
Insured Jobs
0.06
0.0
0.2'i
p
.
-
0.05 -
'a
'
0.15-
557
0.04 0.03 -
0.1 -
0.02 0.05 0.010
1;
0
10
,
0I
20
Wage
'-30
_
0O ?, ,
40
0
I
10
20
30
40
Wage
FIGURE3.-Steady state conditional (on health insurance status) wage densities and marginal wage density. The panel on the left representsthe steady state conditionalwage densities
and the panel on the right depicts the marginalwage density.Based on the parameterestimates
for the specificationwith no heterogeneitypresented in Column 3 of TableII. The distributions
are based on the simulatedlabor markethistoriesof 1,000,000individualswho begin their working lives in the unemploymentstate. The dotted vertical lines represent the minimumwage for
an uninsuredjob, the minimumwage for an insuredjob, the maximumwage for an uninsured
job directly following an unemploymentspell, and the maximumwage for all uninsuredjobs,
respectively.
tion. This drop is due to the small proportion of histories that could lead to
such a high wage rate. For an individualto have a high wage without health
insuranceimplies that he is workingat a firmwith a relativelyhigh value of 0
but one that is less than 0**.If he is getting a high share of the surplus at
this match, this firm has to bid against other firm(s) with match values less
than 0 but sufficientlyclose to it. In our case, since the critical match value
is 13.95, the highestwage that could possiblybe observedwithout health insurance is $13.95.
In terms of the unconditionalsteady state wage distribution,the upper tail
of the density has a shape solely inherited from the relevant part of the conditional (on health insurance) wage density. Overall, the density is not very
much at odds with what we typicallyobserve in cross-sectionalrepresentative
samples.The one possible exception to this claim pertainsto the smallbut perceptible notches above and below the intervalof overlap in the supportof the
two conditionalwage distributions.These discontinuitiesin the densitywould
be hidden if any amount of measurementerrorwas added to the model, as it
is when constructingthe econometric specificationbelow.
Though the model tautologicallyimplies higher involuntaryexits from jobs
without health insurance,there are two possible routes by which anyjob spell
588
M. S. DEY AND C. J. FLINN
can end. Given the efficient separationsimplied by the search and bargaining
process in the absence of firmheterogeneity,we know that voluntaryexits from
a job spell occur whenever a job with a higher match value is located (independent of whether the currentjob provides health insuranceor not). For the
model with no worker or firm heterogeneity, the instantaneousexit rate from
a job with matchvalue 0 (0 > 0*) is given by
(12)
r(0) = r7oX[* < 0 < 0**]+ 7iX[O > 0**]+ AeG(0),
so that the durationof time that individualsspend in a job spell conditionalon
the currentmatchvalue is
fe(teIO) = r(0) exp(-r(O)t,),
te > 0.
Then the density of durationsin a given job spell conditional upon health insurancestatus is
fe(teld)=
fe(telO)dG(0ld).
The corresponding conditional hazard, he(teld) = fe(teld)/Fe(teld),
exhibits
negative duration dependence for both d = 0 and d = 1. However, because
> rl] and because the lowest value of 0 for d = 1 exceeds the greatest value
mr0
of 0 for d = 0, the hazard out of jobs covered by health insurance exceeds
the hazardout of jobs with health insurancefor any value of te. Note that the
limiting value (as te -> oo) for the hazard in jobs without health insurance is
limt,o he(teld = 0) = 770+ AeG(0**), while the corresponding limiting value
for jobs with health insurance is lim,,, he(t Id = 1) = ri.
The job exit rates conditional on the currentwage as well as health insurance status also have interesting properties. For simplicity,consider the exit
rate from the first job following an unemploymentspell (outside options are
identical for these types of spells in a model with no individualor firmheterogeneity). For a given (w, d) the hazardout of the job will be constant. We can
write the hazardas
(13)
r(w, d) =
r7d + AeG(0(w,
d)).9
If it was the case that there was no overlap in the supportsof the conditional
wage distributionsby health insurance,then w would be a sufficientstatisticfor
9Weare assumingthat (w, d) are the wage and health insurancepairnegotiatedwhen entering
the job from the unemploymentstate. The individualcan meet other potentialemployersthatwill
not lead to a move but will resultin a renegotiationin the wage. Allowingfor renegotiationsleads
to a more complex mappingbetween the current(w, d) pair and 0. In this case, the mappingwill
be randomratherthan deterministic.
589
HEALTHINSURANCE PROVISION
0.03
0.025 -
-
"
InsuredJobs
0.0 2
0.02 -
Jobs
Uninsured
0.0150.01 0.005 -
,
0
0
5
,
10
.
15
.
,
20
25
30
Wage
FIGURE4.-Hazard rate comparisons.Based on the parameterestimatesfor the specification
with no heterogeneitypresented in column 3 of Table II. Wages representthe first wage in the
firstjob following an unemploymentspell. The hazardrate equals the weekly rate of exiting the
current employment state, either through a dismissalto unemploymentor a job change. The
dotted verticallines representthe minimumwage for an insuredjob and the maximumwage for
an uninsuredjob, respectively.
the rate of leavingthe job since we could write d(w) =X r(w, d) - r(w, d(w))
r(w). However, with overlap in the support of the conditional wage distributions this is no longer the case and the pair (w, d) are requiredto completely
characterizethe job exit rate. We illustratethis point in Figure 4. For the set
of wages consistentwith either health insurancestate, the value of d provides
informationon the value of 0 associatedwith the match. For example,individuals paid $10 an hour could be receiving health insuranceor not. Those with
health insuranceare less likelyto leave the job both because they are less likely
to receive a negative health shock but also because the 0 value associatedwith
a wage of 10 and health insurance is greater than the 0 value associatedwith
a wage of 10 and no health insurance.In contrastwith claims in the empirical
literatureon job lock, the fact that the hazardrate out of a job that is covered
by health insuranceis lower than the exit rate from one that is not (conditional
on the wage or not) does not necessarilyindicate that health insurancestatus
distortsthe mobilitydecision.
3. DATAAND DESCRIPTIVESTATISTICS
Data from the 1996panel of the Surveyof Income and ProgramParticipation
(SIPP) are used to estimate the model. The SIPP interviewsindividualsevery
590
M. S. DEY AND C. J. FLINN
four monthsfor up to twelve times, so that at most an individualwill have been
interviewed(relativelyfrequently) over a four year period. The SIPP collects
detailed monthly information regardingindividuals'demographiccharacteristics and labor force activity, including earnings, number of weeks worked,
average hours worked, as well as whether the individual changed jobs during each month included in the survey period. In addition, at each interview
date the SIPP gathers data for a variety of health insurancevariables,including whether an individual'sprivate health insurance is employer-provided."
With the exception of the private demand for health insurance,the primitive
parametersof the model developed in this paper are assumed to be independent of observableindividualcharacteristics.Though in principleit would not
be difficultto allow the primitiveparametersto depend on observables,we instead have attempted to define a sample that is relativelyhomogeneous with
respect to a number of demographiccharacteristics.In particular,only white
males between the ages of 25 and 54 with at least a high school education have
been included in our subsample.In addition,any individualwho reportsattendance in school, self-employment,militaryservice, or participationin any governmentwelfare program(i.e., AFDC, WIC, or Food Stamps)over the sample
period is excluded."1Although the format of the SIPP data makes the task of
definingjob changes fairlydifficult,in other respects the surveyinformationis
well-suited to the requirementsof this analysissince it follows individualsfor
up to four years and includes data on both wages and health insuranceat each
job held duringthe observationperiod.
Table I contains some descriptivestatistics from the sample of individuals
used in our empirical analysis.We see that the sample consists of 10,121 individualswho meet the inclusion criteria discussed above. So as to minimize
difficultinitial conditions problems, the unit of analysisis a labor market "cycle," which begins with an unemploymentspell, which could be right-censored
(i.e., may not end before the observationperiod is completed), and ends with
the following right-censoredor complete employment spell, if there is one.12
We partitionthe full sample into two subsamples,one consistingof individuals
who experienced unemploymentat some point duringthe observationperiod
"?Thereare several issues involved in constructinga meaningful employer-providedhealth
insurancevariable.First,there is a timingproblemsince the insurancevariablecan changevalues
only at the interviewmonths,while a job change can occur at any time over a four month period.
Second, there arejob spells in which the individualreportsemployer-providedcoveragefor some
partof the spell and no coveragefor the remainderof the spell. We have made a good faith effort
to accuratelymatch job and health insurancespells, but we have been forced to make several
judgmentcalls while constructingthe event historydataset.
" Some individuals,about three
percent of the sample, had missingdata at some point during
the panel. Since estimationdepends criticallyon havingcomplete labormarkethistorieswe have
excludedthese cases as well.
12Thisapproachwas also taken by Wolpin(1992) and Flinn (2002).
591
HEALTH INSURANCE PROVISION
TABLE I
DESCRIPTIVESTATISTICSa
Type of History
Characteristics of Individual Employment Histories
Marital
Sample
Children
Window
Status
Number
10,121
0.642
0.443
Withoutan unemploymentspell
7,307
0.675
0.460
With an unemploymentspell
2,814
0.558
0.397
Full sample
Type of Transition
Characteristicsof Unemployment Spells (2,814 Observations)
Initial
Accepted
Spell
Number
Duration
Wage
Wage
755
Right-censored
To a job with health insurance
1,044
To a job without health insurance
1,015
Type of Transition
17.26
(30.28)
9.55
(11.18)
12.27
(14.64)
715
To unemployment
162
To a job with health insurance
144
To a job without health insurance
23
77.53
(57.31)
47.58
(34.97)
48.80
(37.92)
54.04
(48.34)
-
16.71
(10.29)
14.26
(10.62)
14.89
(10.19)
11.41
(4.78)
478
To unemployment
314
To a job with health insurance
To a job without health insurance
73
150
44.83
(47.32)
22.14
(22.24)
29.99
(25.17)
27.97
(27.10)
11.75
(8.97)
10.99
(9.01)
10.74
(7.32)
10.78
(7.48)
Children
0.336
15.96
(10.30)
11.30
(8.67)
16.98
(9.56)
16.14
(18.18)
Characteristicsof Jobs without Health Insurance (1,015 Observations)
Initial
Accepted
Spell
Number
Duration
Wage
Wage
Right-censored
Marital
Status
-0.466
Characteristicsof Jobs with Health Insurance (1,044 Observations)
Initial
Accepted
Spell
Number
Duration
Wage
Wage
Right-censored
Type of Transition
145.17
(75.14)
143.06
(77.05)
150.64
(69.63)
11.96
(5.76)
12.67
(9.91)
0.647
0.440
0.529
0.397
Marital
Status
Children
0.649
0.446
0.630
0.401
0.660
0.472
0.609
0.304
Marital
Status
Children
0.552
0.379
0.446
0.373
0.562
0.384
0.613
0.513
aBased on the 1996 Survey of Income and Program Participation. The sample includes white males aged 25-54
with at least a high school education. See text for details. Wages are measured in dollars per hour and durations are
reported in weeks. Standard deviations are in parentheses. The sample window measures the length of time (in weeks)
an individual responds to the survey.
592
M. S. DEY AND C. J. FLINN
and the other consisting of those who did not.13Approximatelytwenty-eight
percent of the full sample, or 2,814 individuals,fall into the formergroup. For
these individuals,we use informationregardingthe durationof time spent in
the initial unemploymentspell, the durationof time spent in the firstjob spell
after the unemploymentspell, and the wage and health insurancestatus of the
first two jobs in the employment spell following unemployment.In addition,
we use marital status and children dummies as observable factors that influence the probabilityof having a high "private"demand for health insurance,
and we use the length of the sample window as an exogenous factor that affects the probabilityof being observed in the nonemploymentstate sometime
during the panel. For the 7,307 individualsin the full sample whom we never
observein the unemploymentstate, we do not considerany labor marketinformation but do include maritalstatus and children dummies and the length of
their sample windowswhen conductingthe empiricalanalysis.It is interesting
to note the differences among individualsin the two groups. Sample members without an unemploymentspell over their samplewindoware much more
likely to be marriedand to have children.The lengths of the sample windows
are not very differentfor the two subsamples.
The labor market data from the sample members with an unemployment
spell provide a wealth of information regarding the relationship between
health insurance coverage, wages, and job mobility. We see that a slight majority (50.7%) of unemployment spells end with a transition into a job that
provides health insurance.Perhapsthe most strikingfeature of the data is the
difference in the averagewages of jobs conditionalon health insuranceprovision. Jobs with health insurancehave a mean wage 41 percent higherthanjobs
without health insurance.In addition,we see that individualswho exit unemployment for a job with insuranceare more likely to be marriedwith children
than individualswho take a job without health insurance.
From the informationon the firstjob following an unemploymentspell, it is
quite clear that jobs with health insurancetend to last longer on averagethan
jobs without health insurance.Another feature of the data that is interestingto
note is the differencein initialwages for the varioustransitionsout of jobs with
health insurance.In particular,while the mean initialwage for all insuredjobs
is almost $16, individualswho subsequentlymove into a job without insurance
are earning $11.41 on average. In addition, the mean wage in the subsequent
uninsuredjob is $16.14,well above the mean wage for uninsuredjobs accepted
directlyout of unemployment.
Finally, note the difference in the averagewages of insured and uninsured
jobs that follow a job without insurance.Jobs without insurancehave a mean
'3Thesample window is the length of time an individualremainsin the SIPP.While the maximum length of the sample window is four years (or 208 weeks), a majority(52%) of sample
membersdo not participatein all 12 waves of the survey.We measure the sample windowfrom
the initiationof the surveyuntil the individualfirstfails to complete the survey.
HEALTHINSURANCE PROVISION
593
wage that is almost six percent larger than jobs with insurance. This is in
marked contrast to the relationshipbetween the average wages by health insurancestatusthat are observeddirectlyfollowingan unemploymentspell. The
model constructed above is, on the face of it, consistent with all of these descriptivestatistics,and in the following section we describe our attempt to recover the primitiveparametersof the model from these data.
4. ECONOMETRICSPECIFICATION
As noted above, the informationused in the estimation process is best defined in terms of what we will refer to as an employment cycle. Such a cycle
begins with an unemploymentspell and is followed by an employment spell,
which itself consists of one or more job spells (defined as continuous employment with a specific employer). Under our model specificationwe know that
wages will generallychange at each change in employer and can also change
duringa job spell at the time an alternativeoffer arrivesthat does not result in
mobility but that does result in renegotiation of the employment contract. In
terms of our model, an employmentcycle is defined in terms of the following
randomvariables:
tu I
tU
eSS I
ItkJJh=P
k
{.klk-7,
WtmM
tWm,
ti}m=1, M=l=
(d
idIQ
=1
{dq,9 tq},
where tu is the length of the unemploymentspell, te is the length of the job
spell with the kth employer duringthe employmentspell, w, is the mth wage
observationof the M that are observed duringthe employmentspell, t, is the
time that the mth wage came into effect, dq is the qth health insurancestatus
observed during the employment "cycle"of the Q distinct changes in status,
and td is the time that the qth health insurancestatus came into effect. Note
that the total length of the employment spell (i.e., which is the length of the
consecutivejob spells) is te= Zk tk and the numberof observedwages during
the employmentspell is at least as great as the numberof jobs, or M > S. The
{dq}q is an alternatingsequence of l's and O's.Since we are assumingthat no
unemployedsearcherwill purchasehealth insurance,the process alwaysbegins
with a 0 (since an employmentcycle begins in the unemploymentstate). Other
restrictionson the wage and health insuranceprocesses will apply depending
on the specificationof searchers'utility functions and the form of population
heterogeneity.
Because of the unreliabilityof wage change informationover the course of
a job spell, in our estimation procedure we only employ wages observed at
the beginning of a job spell and in terms of durationinformationwe only use
informationon the duration of unemploymentspells and the duration of job
spells. Furthermore,to reduce the computationalburdenwe consider(at most)
the first two jobs in a given employmentspell.
As is often the case when attemptingto estimate dynamicmodels, we face
difficult initial conditions problems. In our framework,and common to most
594
M. S. DEY AND C. J. FLINN
stationarysearch models, entry into the unemploymentstate essentially "resets" the process. While we will utilize all cases in the data in estimating the
model, our focus will be on those cases that contain an unemploymentspell.
The likelihood function is written in terms of the employmentcycles to which
we referredabove, so that only those cases that contain an unemploymentspell
are "directly"utilized. Let I take the value "1" if a sample member experiences an unemploymentspell at some point during their observationperiod
and let it take the value "0"when this is not the case. At the conclusion of our
discussionof the likelihood contributionsfor sample memberswith t = 1 we
will derive the likelihood of this event. For present purposes we simply state
that it is a function of the length of the sample period, whichwe will denote T,
and the individual'stype s. Then let us denote P(T = 11|, T) by w6(T). It
is assumed that the length of the sample window is independentlydistributed
with respect to all of the outcomes determinedwithin the model.
For the sample cases in which t = 1 the data utilized in our estimationprocedure is given by {t",t[, wl, w2,dl, d2), where the two wage and health insurance status observationsare those in effect at the beginningof the relevantjob
spell. The likelihood for these observationsis constructedusing simulationsof
the equilibriumwage and health insurance process in conjunctionwith classical measurement error assumptions regardingobserved beginning of spell
wage rates and health insurancestatuses. In particular,correspondingto any
"true"wage w that is in existence at any point in time we assume that there is
an observedwage given by
(14)
i = wexp(e),
where e is an independentlyand identically(continuously)distributedrandom
variable.Our econometricspecificationwill posit that e is normallydistributed
with mean 0, so that
(15)
Inw= Inw + e,
and E(ln w) = Inw. In terms of the observationof health insurancestatus,we
will assume that the reported health insurance status at any point in time, d,
is reported correctlywith probabilityy and incorrectlywith probability1 - y,
independently of the actual state. Thus y = p(d = lid = 1) = p(d = Od = 0).
Measurement error essentially serves three purposes in our estimation
framework.First, it reflects the reality that there is a considerableamount of
mismeasurementand misreportingin all survey data (though admittedlyit is
not likely to be exactly of the form we assume). Second, it serves to smooth
over incoherenciesbetween the model and the qualitativefeatures of the data.
For example, under certain specificationsof the instantaneousutility function
the model implies that the probabilityof moving directlyfrom a job covered
by health insuranceto a job without insuranceis a probabilityzero event. Data
595
HEALTHINSURANCE PROVISION
exhibitingsuch patternswill produce a likelihood value of 0 at all points in the
parameterspace. Measurementerror makes such observationspossible at all
points in the parameterspace.
The third usage is related to the simulation method of estimation. This
method is most effective when based on a latent variablestructure.In our case,
the latent variablescorrespondto the simulatedvalues of the variablesthat appear in the likelihood function,which themselves have a simple mappinginto
the observed values as a result of our i.i.d. measurement error assumptions.
Thus any simulatedvalue will have positive likelihood no matterwhat the observed value. In this sense, measurementerror serves as a "smoother"of the
likelihood. Because of its particularproperties, measurementerror is not introduced into the durationmeasures. By the structureof the model, it is not
necessaryto smooth the likelihood with respect to this information.
Throughoutthe empirical section we assume that the distributionsof individual demands and firm costs are both discrete. Firm cost types assume the
values 0 < ( 1 < 02, with the probabilitythat a randomlyselected firm has a
high cost of providing health insurance given by PF(-
=
02)
= 'T.
On the in-
dividuals'side of the market, we assume that there are two levels of private
demand, with 0 = 1, < s2, and with the probability that the individual has a
positive "private" demand for health insurance given by PH(- = s2) = 8.
The unit of analysis in our likelihood function is the individual.Individuals may be heterogeneous with respect to their private demand for insurance,
though we do assume that their type does not change over the course of the
sample period. This implies that the decision rules used by anygiven agent will
be time invariant.Recall that for an individualof type 5 we denote the value of
employment at a firm with match value 0 and cost type 0 when the next best
alternativeis at matchvalue of 0' at a firm of cost type (' by
VE(0, , ',1c').
The characteristics(0, 4) correspondto those of the highervalue employment
contract (from the point of view of the individual). In the presence of firm
heterogeneityit need not be the case that 0 > 0', as is true when 4 = -'.
Correspondingto each set of state variables (0, X, 0', 4') for an individual
of type s is a unique wage and health insurancepair (we, da)(0, 4, 0', ('). For
an individualof type s at a currentjob with characteristics(0, 4) let the set
of alternativematches that would dominate the currentmatch be denoted by
Q2((0,(). As we have demonstratedabove, this set is alwaysconnected and can
be parsimoniouslycharacterizedas follows. For any type 4 agentwith a current
match (0, >),a potential match (a, b) dominateswhen
a > O(0, , b).
596
M. S. DEY AND C. J. FLINN
When ) = b, so that the individualmeets a potential employer of the same
type as her currentemployer,then
0= 0(0,(,
4)
for any type s. On the other hand, when 4 -7 b we have
0(0, 42,
1) <,
0 (0,
1, 2)> 0.
The individual'stype will affect the size of the match differentialrequiredfor
a move to take place in these cases, though even individualswith no private
demand for health insurance (s = 0) generallydemand some differential.
Among the sample members for whom T- = 1 we will discuss three qualitatively distinctcases. The first case, in which the observationperiod ends while
the individualis still in an on-going unemployment spell, is the simplest. In
this situation,the only contributionto the likelihood is the densityof the rightcensored unemploymentspell. We will then discuss the second case, in which
the individualhas one job spell in the employmentcycle, either due to the fact
that he moves into unemploymentat the conclusionof the firstjob spell or due
to the fact that the firstjob spell is right-censored.In this case the likelihood
contributionis defined with respect to the densityof the completed unemployment spell, the observed wage and health insurancestatus at the initiation of
the firstjob, and the length of the firstjob (be it censored or not). The finalcase
is that in which the individualhas two consecutivejob spells followingthe completion of an unemploymentspell. In this case, the likelihood contributionis
defined with respect to the durationof the unemploymentspell, the duration
of the first job spell, and the wages and health insurance statuses associated
with the firsttwo jobs (at their onset). We shall now considerthese cases in the
order of their complexity.
4.1. UnemploymentOnly
Recall that as long as an individualof type s would accept some matchvalues
at each type of firm (differentiatedin terms of 4>)that would not result in the
purchaseof health insurance,the criticaljob acceptancematchvalue for a type
4person is independentof 40.This value is denoted by 0*.Then the hazardrate
associatedwith unemploymentfor an individualof type : is given by
(16)
h = A,nG(0,),
and the density of unemploymentspell durationsfor a type 5 individualis
(17)
f' (t"))= h exp(- h t),
597
HEALTH INSURANCE PROVISION
where t" is the durationof the unemploymentspell in the observationperiod.
Since the hazardfunction out of unemploymentis constant given the individual's type, it is irrelevantwhether or not we observethe beginningof the unemploymentspell.14Then the probabilitythat an unemploymentspell of duration
tu is on-going at the end of the sample period (e.g., is right-censored)given the
individual'stype is
(18)
L1)(tu, T- = IT) = w(T) exp(-h
tu).
Let the probabilitythat the individualis a "high demand type" be denoted 8.
Then the empiricallikelihood in this case is given by
(19)
L(1)(tu,
= lIT) = 8L}2'(t", t = 1T) + (1 - 8)L(l)(tu,
= lT).
4.2. OneJob Spell Only
For all likelihood contributionsthat involve job spells we utilize simulation
methods. We will describe the process by which we generate one sample path
for an employmentspell;for each individualin the samplewe constructR such
paths. If the minimumacceptablewage with each type of employeris the same,
then the distributionof match drawsin the firstjob spell is independentof the
type of firm at which the individualfinds employment.We simulate the match
draw at the first firm by first drawinga value (1 from a uniform distribution
defined on [0, 1], which we denote by U(0, 1). The match draw itself comes
from a truncatedlognormal distributionwith lower truncationpoint given by
the common reservationwage 0}. We have
(20)
0(4,) = exp(L + o-P- 1 - (
6
- 1(
)))
The rate of leavingthe unemploymentspell is hu= AG(0*), so the likelihood
of the completed unemploymentdurationof tu is
(21)
h exp(-hutu).15
Given that the firm is a high cost firm,the wage and health insurancedecision
are given by
(22)
(w,d
d)
)(0(s1,),
42,
V/ ),
14Ina stationarymodel such as this one, the distributionof forwardrecurrencetimes of lengthbiased spells (i.e., those in progress at the time the sample window begins) is the same as the
populationdistributionof completed spells (that are not length-biased).In our case, both distributions are negativeexponentialwith parameterhu.
15Notethat this density is the same whether the individualbegan the sample window in this
spell or whether it began after the observationperiod had commenced.
598
M. S. DEY AND C. J. FLINN
where the equilibriumvalues x4denote the value of choice x at the beginning
of job spell 1 given a firm type 4j and an individualtype 5, and x^ denotes
the equilibriummapping from these state variables into the contract, where
x = w, d. The critical value for leaving the first firm will be equal to 0(,1)
whenever another high cost employer is encountered, and otherwise is equal
when a low cost firm is met. The likelihood that the first
to 60(06(1), 02,
21)
to
a
ends
due
quit of any kind is then
job
(23)
h (,,
02) = Ae(T7r(0(0l))
+
(1 - 7)G(6(0((0(0 ), 02, 0 1))).
Since the "totalhazard"associatedwith the firstjob in the employmentspell is
simplythe sum of the hazardassociatedwith a voluntaryquit and an involuntaryone, we have
(24)
he((1, &2) = h(d,
42)
+q d.
When the first employer is a low cost firm the situation is symmetric.The
equilibriumwage and health insurancedecisions are given by
(25)
(wl, d) = (w, d)(0(,),
1 V)
The criticalvalue that will induce job acceptance at a competing low cost firm
is 0(1,), while a higher match is in general required if the individualis to
accept employmentat a high cost firm.The rate of leaving this job for another
employeris
(26)
hq/(1, 0) = Ae(1rG( (((i)
,0
02))
+ (1-
T)G((l))),
and the total rate of leaving this job is
(27)
hf(1,, 01) = h4(q,
) + rd.
If the firstjob in the employment spell is still in progress at the end of the
sample period then it is right-censoredand we have all of the informationrequired to compute the likelihood contribution.Conditioning on the individual's type for the moment, the likelihood value associatedwith this particular
simulationis given by
1, = 111, T)
L[2)(t,, wl, dl, tl, cl = It
= we(T) x h exp(-htu )
x {rr (wll|Iwx)x p(d\ld ) x exp(-hx(06(
), 02)tl)
+ (1 - r)f(wi|lw) x p(dl\ld) x exp(-h'(O((,),
l)tl),
599
HEALTHINSURANCE PROVISION
where c1 = 1 if the first job spell is right-censoredand is equal to 0 if not.
The density f(wi1lw) is generated from the measurement error assumption,
as is p(dlldl). The term exp(-hi(sl, O)tl) is the probability that the first
job spell has not ended after a duration of t1 given that it is with a firm of
type 4.
To form the likelihood contribution for the individual we have to average over a large number of simulation draws and over the possible individual types. Since both averagingoperations are linear operations it makes no
difference in which order we perform them. Then define the likelihood contribution for an individualwith observed characteristics(tU,iil, di, tl, cl = 1,
T =1, T)by
(28)
L(2)(tu,
l, dl, tl, Cl= 1,1
= lIT)
R
= R-1 {8L {(t,
wl, dl, t,
t(r),cl- = 1,
= l|
T)
r=l
+ (1 -
)L(2)(t
= 1, I = 11|1(r), T)},
, dlit,cl
i,
where , (r) is the rth drawfrom the U(O, 1) distribution.
For the case in which the firstjob spell is complete and ends in an unemployment spell, the conditional likelihood function is slightlydifferent. The likelihood that an individualof type 5 with a firstjob drawof 0 who is employed at
a firm of type 4 exits into unemploymentat time tl is the density of durations
into unemployment conditional on exiting into unemploymentmultiplied by
the probabilitythat the individualhas not found a better job by time t1. This is
simplythe survivorfunction associatedwith the "voluntaryexits"density evaluated at tl, so that the productof these two terms is
(29)
rd exp(-rqdjtl)
x exp(-h(1,
/1j)tl) =
(d')exp(-h"(
when the firstjob spell was at a firm of type 4j. Then we have
(30)
L 2)(tU, wl, dl, tl, cl = O,I = 11l|, T)
= tw(T) x h exp(-h t)
x {7r/(w1|w2) x p(d\ld2) x r(d) exp(-h(el,
02)tl)
+ (1-
7r)f(wli?Iw,)
x p(dt1ld) x (dl)exp(-h(
l
)tl)},
, j)tl)
600
M. S. DEY AND C. J. FLINN
and the empiricallikelihood contributionfor this case is
(31)
L(2)(t', wl, d1, t1, c1 = 0, I = lT)
R
= R-1
L{Lg2)(tu,
w1, dit, 12c1=0, l/ = 1l (r), T)
m=l
+ (1 - 8)L;])(tu,
w,i,dt,
c = 0, T = 11(r),
T)}.
4.3. Twoor MoreJob Spells
When there exist two or more job spells we use only the informationon the
wage and health insurancestatus of the first two job spells as well as the duration of the firstjob in the employmentspell. This simplifiesour computational
burden, and results in very little loss of informationsince only a small proportion of employmentspells contain more than two jobs in our data.
To obtain the wage and health insurance associated with the second spell
we proceed as follows. Conditionalon the firstjob being with a high cost employer, for example, and given the first random draw of Of(1,), the individual
has three ways to exit the firstjob spell. First, she may find employmentwith
anotherhigh cost employer.The rate at whichthis occursis Ae7rG(06(1 )). Second, she may find employmentwith a low cost employer,which occurs at rate
Ae(l - 7rG(0)(06(1)),
2, 0)).
Third, she may exit the spell due to a forced
termination,which occurs at rate rl(d2).Then the likelihood that an individual
of type s with match 0f ( l) at a high cost firmfinds another high cost firmjob
at firstjob spell durationt1 is
(32)
Ae,rG(0
02)tl),
((i))exp(-h(1,,
while the likelihood that she will find a job with a low cost firmis
(33)
Ae(l-
7r)G(0(0f(1),
02,
4))) exp(--he(,l,
02)t).
We drawa pseudo-randomnumber s2 from U(O, 1). If the individualfinds a
job in a high cost firm,determine her matchvalue as
(34)
02
(;,,
2)
= exp / + o-iP- 1 -
c(ln(0(;,))
(1
-
)))
where 022 (i, s2) is an acceptablematchvalue at the secondjob given that both
jobs are with type 02 employers (the first term in the superscriptcorresponds
to the employer type at the firstjob and the second term is the employer type
HEALTHINSURANCE PROVISION
601
at the second job). Then the wage and health insurance status at the second
job are given by
(35)
(W2,22d,2) = (w,
2), t2, 0(1),)
df)(0f'2(~,
2),
where the superscriptson the wage and health insuranceoutcomes denote the
types of the firmsin both periods.
If the individualfinds a job in a low cost firm,define
(36)
0
(,
2)
exp(tL + r-
(ln(0e(06e()'
(-
02,
1)) -
)(1-
))
so that the wage and health insuranceoutcomes for this case are
(37)
(w "
d<)(
de1 =),
0^(
p
2).
li
Then the likelihood contributionfor an individualwhose firstjob was at a high
cost firm with a match value of 0O(1,) and who spent a duration of t1 at that
firm is given by
(38)
( 38)
w1,dl, tl, W2, d2, i = Pll
Le3)(tu,II',i2^
11K1,C2, 4(1) =
= w(T) x hUtUexp(-hutU)
42,
0,T)
T)
x {A,eT(
G(0e(i1))exp(-h'(il, 02)tl)
x f(i
lw2) x f(w2lw,2)
+ Ae(1 - 7r)G(0e(0e(0),
x(wlw2)
x p(dl ld) x p(d2ld2 )
42)t1)
k2, 4i)) exp(-he('1,
x p(d2ldd2))
x f(w21w l1) x p(dlld2)
We construct an analogous term for the case in which the firstjob is with a
low cost employer,namely
(39)
Lf3)(tu, w1, d1, tl, w2, d2,
=- l| 1,
2,
()=
1,
T)
= w)(T) x huttUexp(-hutu)
x {Aer7T(G( (05( 1),(1, 02)) exp(-hr(lq,
x f(wllw)
x f(w2w 2)p(d\lld)
4>l)tl)
x p(d2ld'
)
+ Ae(1- r)G(6e((;))exp(-h (eq, 0i)tl)
x
f(wll
w f(p2W1
)p(dld1)
)p(d
x p(d2ld
)}.
602
M. S. DEY AND C. J. FLINN
The likelihoodcontributionfor these particulardrawsof s1 and 52 for this
type 5 individualis then
(40)
L3)(tu,/ W1, ditl,t
= rTLf3(tu,W1,
l
Wd2,dT2,
= 11,
d1,l,tl, i,
d 2,
+ (1 - rT)L3)(tU
i, d,, tl,tl2,
~2)
= 11l,
2, ((1) -= 2, T)
l=
d2,
- If
1, 2, (2)=
+1,
T).
As was the case when there was only one job in the employment spell, the
"unconditional"likelihood contributionis given by
(41)
L(3)(tu,w1, di, t, if2, d2, I= lT)
R
= R-1 {L3
(tu, i,
t, 2,d2,
d,1l(r),(r)
r=l
+ (1 - 8)Le (tu, il, dl, tl, w2, d2,
= lj(r)
(r))}.
4.4. The CompleteLikelihood
In forming the likelihood function contributionsabove, we have only used
information from individualswho experienced unemploymentat some point
duringthe sample period (i.e., cases with t = 1). We will now derive the likelihood of this event.
Let us say that the individualis randomlysampled at time r and that his labor marketexperiencesare observeduntil time r + T, where T is the length of
the samplingwindow.Thus the individualcan experience (at least one) unemployment spell over the interval [, r + T] in one of two distinct ways: (1) by
being unemployedat time r or (2) by being employed at time r and exitinginto
unemploymentprior to r + T.
We have alreadyseen that under the stationarityassumptionsof the model
the hazardrate out of unemploymentis hu. Thus the mean durationof an unemployment spell for a type s individualis simply (hu)-1. A type : individual
will utilize a set of decision rules adapted to his type, and will have a distribution of completed employmentspell durations,which is the sum of consecutive
job spells, that does not belong to the negative exponentialfamily.The distribution will be a complicatedfunction of all of the primitiveparametersof the
model, including the distributionof firm types ). While there does not exist
an analyticexpressionfor this distribution,it will be stationaryand can be approximatedto any arbitrarydegree of accuracyusing simulationmethods. Let
the distributionof completed employmentspell lengths for a type s individual
be given by F (t), with the mean of the distributiondenoted ,.
HEALTHINSURANCE PROVISION
603
The probabilitythat a type 4 individualwill be found in the unemployment
state at a random sampling time r in the steady state is given by the ratio of
the average length of an unemploymentspell to the average length of a labor
marketcycle, or
P =
(42)
+
")-I e
(hu(h~)
This representsthe probabilitythat the samplingwindowbeginswith an unemployment spell.
To compute the probabilitythat an individualenters an unemploymentspell
given that he began the samplingwindow in the employment state, it is necessary to proceed as follows. Assume that the individualis in an employment
spell of length t when the samplingperiod begins. Then the probabilitythat
the employment spell will end before the completion of the sampling window is
(43)
(43)
F(t+
T) - Fe(t)
1F
-(i)
1 - F(t)
The distributionof on-going durationsof an employmentspell in progressat a
randomlychosen date is well-knownto be given by
1-F F(t)
e
(44)
e
and the probabilitythat the individual is employed at the sampling time is
1 - pu. Then the likelihood that an unemploymentspell will start during the
period [r, r + T] given that the individualwas employed at time r is
+ T) -F(i)
(45)
e
Putting all of these elements together, we have that the probabilitythat the
individualwill be in the unemploymentstate at some time duringthe randomly
selected period [r, r + T] is
e
(hu )-i
(46
r(T) -
'+ 1
(h)-1
e
(hu()-1+
=
(h)1
/0
{ {((Fh(
+1
Fe + T)-Fet
FFe(t+ T) -Fe(t)dt
T)-F~(t))di.
604
M. S. DEY AND C. J. FLINN
Note that
(47)
l(T)im
w(r)-
h(h)r--->)1
+ lim
T) - F()
(F
d
_? T--ocj'
+
(h)l/(1-Ft))dt
(h)1
=1 v.
+.
This last result demonstratesthat all nonrandomnessin our subsampleof individualswho experience an unemploymentspell at some point in the observation period is attributableto the finiteness of the samplingwindow (given
our assumptionthat the original sample to which we have access is randomly
drawn). As the sampling window grows indefinitely large the model implies
that the set of original sample members excluded by our unemploymentspell
requirementis of measure 0 so that nonrandom samplingproblems are precluded.
The final specification of the likelihood function can now be derived. We
have already specified the likelihood contributions for the individuals for
whom T = 1. For those individualswho do not experience an employment
spell we only utilize the informationthat T = 0. This probabilityis given by
(48)
p(T = 0IT) = 8(1 - w62(T)) + (1 - 8)(1 - o (T)).
Let the set of individualswho were unemployed at some time in the sample
period and who contribute only a right-censoredunemploymentspell to the
likelihood (our Case 1 above) be given by Y1,the set of individualswith an
unemploymentspell followed by one job spell be given by Y2,and the set of individualswith unemploymentand two consecutivejob spells be denoted by Y3.
Let the set containing the remaining individuals,those who experienced no
unemploymentduringtheir sample observationperiods, be denoted Y4.Then
the likelihood of the sample is given by
(49)
L = n- L(t
iE Y1
=I
lIT)
tl,i, c ii
n L(2)(ti'wl,i,dl,,
x 7 L(3)(t, wi,,, dl, t-i, W2,i,d2,
i
icEY3
= lTi)
i Yr2
= 1,T)Ii
p(ti
=
01Ti).
icY4
Maximizationof the log of this function with respect to the primitiveparameters of the model yields estimatorswith desirable asymptoticproperties as
long as the number of simulations R is growing at an appropriaterate with
respect to the sample size. In our implementationwe have set R = 1,000 and
have located the maximumlikelihood estimatorsthroughthe use of a simplex
algorithm.To compute the standarderrors of the estimates we have utilized
HEALTHINSURANCE PROVISION
605
bootstrapmethods. We found the bootstrapapproachattractivedue to the discontinuities in the numerical likelihood that arose from the use of simulated
match draws and due to the nature of the approximationsused in solving the
decision rules (see Appendix C). Although solving the model is computer intensive, it was feasible to reestimate each of the four specificationsreported
below 50 times each, and our bootstrap estimates of the standarderrors are
based on these replications.
4.5. IdentificationIssues
While our goal is to estimate all of the primitiveparametersof the model,
from Flinn and Heckman (1982) we know that the search model (with or
without bargaining) is fundamentallyunderidentifiedusing the type of data
to which we have access. They show that the pair of parameters(p, b) are not
individuallyidentified;as is common in the literature,our response is to fix the
discount rate at a given value. We have chosen an annualizedrate of 0.08.
Those authors also show that when only accepted job informationis available, identificationof G requires that functional form assumptionsbe made.
We assume that the productivitydistribution G(O) is lognormal with parameters ,/0Land o-. Furthermore,we assume that the measurement error distribution is lognormal with parameters ,g = 0 and o-r> 0.16 These types of
functionalform assumptionsare relativelystandardin the literature(see, e.g.,
Flinn (2002)).
Given a value of the bargainingpower parameter, a, identificationof the
other primitiveparametersis relativelystandard.It is exceedingly difficultto
attaincredibleestimatesof a given access to only supplyside information.This
problem is discussed in Eckstein and Wolpin (1995), and an extensive analysis
of the issue in a context similarto the one consideredin this paper is contained
in Flinn (2003). He shows that the parameter a can be identified using only
supply side information (i.e., wage and duration informationfrom surveyrespondents) if the matching distributionhas a known scale parameter.This is
unlikely to be the case in most applications,and is certainlynot the case here
since we are attemptingto estimate the unknownscale parameter,u of the log
normal distribution.
Insteadwe adopt the strategyof Flinn (2003) to estimate a. By using a piece
of aggregateinformationon the ratio of labor compensationto total revenues,
we can effectively reduce the dimension of the parameter space by one and
achieve credible (i.e., precise) estimates of a. Heuristicallyspeaking, such a
piece of informationserves to fix the total size of the "pie" to be divided, or
the scale of the matchingdistribution.In our application,the procedure is im16Thusthe log wage has a measurementerrorwith mean 0, but the mean measurementerror
in levels is exp(or2/2).
606
M. S. DEY AND C. J. FLINN
plemented as follows. We begin by defining total revenues in the steady state
to a type 4j firm from type 4j employees as
(50)
Rss(i, j)=
OdGss(01i,j).
Let the steady state distributionof 0 and s within the employed populationbe
given by wc(i,j), where wo(1,1) + w(1, 2) + )(2, 1) +-(2, 2) = 1. The steady
state revenues are given by
(51)
Rss = E
(i, j)Rss(i, j).
i,j
Average steady state compensation of a type (i individualat a type 4j firm is
given by
(52)
Css(i, j) =
w*(0'i, jli,j)Gss(O',0li, j) + bjGss(07, oXli, j),
where the function w*(0', 0li, j) is the equilibrium mapping from the best
match 0' and dominated match 0 into the wage for a type i, individualat a
type 4j firm.Then averagesteady state compensationis given by
(53)
Css = E
o (i, j)Css(i, j).
i,j
The steady state labor share of total revenues is then simply
(54)
Fss =
Rss
Partitionthe vector of model parametersinto (a, 0), and write the steady
state labor share in the environment (a, 0) as Fss(a, 0). Flinn (2003) shows
that Fss is monotonically increasing in its first argument for all values of 0.
Then the estimation strategyis to condition on a value of a and maximizethe
log likelihood over 0. Denote this conditional maximumlikelihood estimator
by O(a). If the observedvalue of labor share is F0,then the maximumlikelihood estimatorof the model is
(55)
ro= rs(&,
(a)).
If 0(a) is unique for all a, then (&,0(&)) is unique as well.
Since equation (55) is a significantpiece of informationin determiningthe
parameterestimates, and since the observedvalue of Fois treated as "truth,"it
HEALTHINSURANCE PROVISION
607
is importantto use a reasonablevalue for oFin obtainingthe maximumlikelihood estimates. We have based our estimates on statisticsreported in Krueger
(1999). For the year 1998, which is the "modal"year of our sample, the only
computed labor share value availableis 0.766. Given that our sample only includes prime-age white males, we felt that it would be appropriateto adjust
the share upwardand settled on a value of 0.81. While model estimates, particularlyof the bargainingpower parametera, are quite sensitive to the labor
share value used in the estimation, the substantiveimplicationsof our results
are not. This is explained in more detail in the next section in the course of
discussingthe empiricalresults.
5. EMPIRICALRESULTS
We begin by reporting and discussing estimates from two specificationsof
the model.17The initialspecificationassumeshomogeneityon both sides of the
market. The second specificationallows both worker and firm heterogeneity,
and allows for the probabilitythat the workeris a high demandtype to depend
on observablecharacteristicsin the followingmanner:
(56)
8(Z)=
exp(50+
1+exp(o
+2Z2)
+81Z1 +
2Z2)'
where Z, is an indicatorvariablethat takes the value 1 when the sample member is marriedand Z2 is an indicatorvariable that takes the value 1 if he has
children. Since it is somewhat difficultto directlyinterpretsome of the primitive parameters,we also compute estimates of a number of the moments of
the distributionsof labor market outcomes, both in the steady state and outside of it.
Table II presents the simulated maximum likelihood estimates for two
specifications of the model. The first two columns correspond to the "fully"
estimated model, that is, the one in which the bargainingpower parameter
is estimated along with the other (estimable) primitiveparameters.Columns
three and four contain estimates of the symmetricNash bargainingmodel; in
this case the bargainingpower parameteris not estimated and is fixed at the
value 0.5. We will discuss both cases, but will focus most of our discussion on
the first two columns of estimates. By comparingthe estimates in the first two
columnswith those in the last two, we will be able to isolate those parameters
that are most sensitive to variationsin a.
17Weestimatedtwo other specificationsin whichwe allowed for heterogeneityon one side of
the marketwhile imposinghomogeneityon the other. We found little differencesin the estimates
from these (omitted) specificationsand the first (that posits homogeneity on both sides of the
market)and to conserve space we have omitted them.
608
M. S. DEY AND C. J. FLINN
TABLEII
ESTIMATESa
SIMULATEDMAXIMUMLIKELIHOOD
STANDARDERRORS)
(BOOTSTRAPPED
a = 0.25
Parameter
A,
A,,
7T0
Ao
070
y
b
(1)
1.044
(0.030)
6.258
(0.142)
0.156
(0.003)
1.321
(0.042)
2.737
(0.010)
0.454
(0.011)
0.551
(0.017)
0.853
(0.005)
-1.666
(0.035)
5.825
(0.089)
02
62
8o
81
52
CO
Fss
InL
0.269
0.808
-29,393.09
a = 0.50
(2)
1.254
(0.037)
7.001
(0.149)
0.153
(0.003)
1.325
(0.039)
2.609
(0.007)
0.491
(0.005)
0.550
(0.016)
0.865
(0.004)
0.423
(0.019)
5.465
(0.060)
6.781
(0.066)
0.039
(0.001)
0.662
(0.010)
-1.874
(0.024)
1.582
(0.052)
0.532
(0.014)
0.379
0.269
0.824
-29,364.45
(1)
1.247
(0.036)
6.219
(0.184)
0.157
(0.003)
1.450
(0.038)
2.500
(0.011)
0.409
(0.010)
0.553
(0.020)
0.856
(0.004)
-1.155
(0.033)
3.598
(0.130)
0.270
0.923
-29,371.29
(2)
1.322
(0.025)
7.498
(0.250)
0.148
(0.003)
1.209
(0.048)
2.410
(0.008)
0.423
(0.005)
0.554
(0.017)
0.867
(0.005)
0.268
(0.049)
3.487
(0.111)
4.811
(0.097)
0.043
(0.001)
1.057
(0.121)
-1.974
(0.047)
1.638
(0.103)
0.588
(0.029)
0.374
0.272
0.930
-29,322.86
aThe estimates are based on the assumptions that the annual discount rate is 0.08 and the log of the measurement
error is distributed normally with mean 0. The model is estimated using the Nelder-Mead simplex algorithm and the
standard errors are computed using bootstrap methods with 50 draws of the data. Standard errors are in parentheses.
The parameters are defined in the text. Estimates of the rate parameters represent weekly rates multiplied by 100.
Specification 1 assumes homogeneous workers and firms and specification 2 allows both firm and worker heterogeneity
and considers observable factors that affect the probability an individual is a high demand type.
HEALTHINSURANCE PROVISION
609
The parametera is actuallyonly estimatedfor the firstspecification,in which
there is no firm or worker heterogeneity.In this case, we employed the procedure discussedin Section 4.5 to obtain &= 0.25. Due to the lack of smoothness
in the log likelihood, particularlywhere a was concerned, we did not attempt
to determine a standarderrorfor a. In terms of the statisticalpropertiesof the
estimates reported,all standarderrorsassociatedwith other model parameters
are only consistent under the assumptionthat a actuallyequals 0.25 (or 0.5)
in the population. The reason why we did not attempt to estimate a for the
second specificationis that firm revenue informationis not availablefor individualworkeror firmtypes, particularlysince the firmtypes are assumedto be
unobservableto us in the SIPP data. While one could restrictthe labor shares
for each firm-workerpairingto be equal to the aggregateshare,there is no theoretical basis for doing so. Furthermore,Flinn (2003) finds that imposingsuch
a restrictionresults in group-specificestimates of a that are relativelytightly
clusteredaroundthe "aggregate"estimate. On the basis of this evidence,we do
not believe that fixingthe value of a at 0.25 throughoutall of the specifications
will affect the empiricalresults to any significantdegree.
For ease of exposition we will distinguish three subsets of the primitive
parameters in addition to a: (i) those parameters that are constant across
specifications (i.e., the job offer arrival rates, A, and Ae, the job dissolution rates, r71and rm0,
the parameterscharacterizingthe productivitydistriband
the
ution, OL0 o-0, parametersthat define the measurementerrorprocesses,
o-, and y, and the unemploymentutility flow, b); (ii) parameters that characterize the distributionof health insurance costs (i.e., 01, 42, and rT); and
(iii) parametersthat define the distributionof private demands for health insurance (i.e., s2, 80, 61, 52). We will begin our discussion with the first set of
parameters.For ease of presentation,the rate parameterestimates in TableII
have all been multipliedby 100.
Perhapsthe most importantfindingin TableII is that our estimates strongly
Durations are measured in
support the premise of our model that r0o> 771.18
weeks, so that our estimates imply that, on average, a job without health insurance will exogenously dissolve after approximatelyone and a half years,
while a job with insurancewill dissolve after twelve years. The point estimate
of A,, 0.070, implies that the mean wait between contacts (when unemployed)
is 3.25 months. In contrast,the point estimate of A,, 0.012, suggeststhat a contact between a new potential employer and a currentlyemployed individual
occurs about every 19 months. The standarderror of the estimate of Aeis sufficientlysmall that it is safe to say that employed search is an importantsource
of turnover,somethingwhich is obvious from the raw data as well.
18Inour parameterizationof the model, the log likelihoodwould still be well defined even if
the orderingof the estimatedexogenousseparationrateswas not consistentwith our assumption.
The "incongruity,"if you will, would be seen in an estimatedvalue of the cost of insurancethat
was negative.
610
M. S. DEY AND C. J. FLINN
As was discussed in the previous section, for the model to fit the data requires that measurementerror be incorporated.In some sense, the degree of
measurementerrorrequiredto providean acceptabledegree of fit of the equilibriummodel to the data can be considered an index of the degree of model
misspecification.The estimate of the standarddeviationof the logarithmof the
measurementerror in log wage rates, r-, takes a value similarto that found in
most similarstudies. More interestingperhapsis the estimatedamountof error
in the measurementof health insurancecoverage. The estimate of y is found
to be about 0.87, so that, in conjunctionwith the measurementerror assumed
to be present in wage rates, the probabilityof mismeasurementof health insurancestatus is approximately13 percent.
Turningto the estimates of the parameterscharacterizingthe distribution
of health insurancecosts, we find importantsimilaritiesacross the two model
specifications.In the initial specification,in which all firmsface the same cost,
the point estimate of the cost of insurance is $5.83 per hour. On the face of
it, this estimate may appear high, but compared to the mean wage of insured
jobs in the steadystate (simulatedto be a little over $20 per hour), we find that
health insuranceaccounts for slightlymore than 22 percent of total employer
costs. When we allow for covariates in the probabilitythat an individualis a
high demand type as well as heterogeneity in the cost of insurance to firms,
the difference in the costs of insurance is estimated to be $1.33, which is a
considerabledifference,althoughthe proportionof high cost firmsis only 0.03.
As a consequence of these features of the estimated cost distribution,it seems
fair to say that there is no strong indication of firm heterogeneity in ). This
resultis mainlyresponsiblefor the small amountsof "joblock"reportedbelow.
The second specificationallowsfor a "private"demandfor health insurance,
and posits that individualswith different observable characteristicswill have
different probabilitiesof being a high demand type. We find that the direct
payoff to having health insurance for private demanders is a relativelymodest 66 cents per hour. There are large differences in the likelihood of being in
the high demandgroup acrossobservabletypes.The estimatedrelationshipbetween observablecharacteristicsand the likelihood of havinga privatedemand
for health insuranceis consistentwith conventionalwisdom.An unmarriedindividualwithout childrenhas only a probabilityof 0.13 of being a high demand
type, while a marriedman without childrenhas a probabilityof 0.43 of being a
high demand type. A marriedman with children has a probabilityof 0.560 of
havinga privatedemand for insurance.We find that, given the sample composition by household type, the averageprobabilityof being a high demand type
is 0.379.
The primitiveparametersare often difficultto interpret,so in TablesIII-V
we provide some more easily interpreted statistics computed under the estimated equilibriaassociatedwith the two specifications(for the two values of a
upon which we focus). We present the implied criticalmatches for transitions
out of unemployment,0*, and for the provisionof health insurance,0T*(4),in
611
HEALTHINSURANCE PROVISION
TABLE III
ESTIMATEDDECISIONRULESAND LABORMARKETOUTCOMESa
a =0.25
Parameter
Criticalmatchfor the acceptanceof employment
Low demandworkers, 1j
a = 0.50
(1)
(2)
(1)
(2)
8.85
9.01
7.76
8.37
9.24
-
8.74
Criticalmatchfor the provisionof health insurance
Low demandworker-high cost firm, (61, (2)
17.53
19.38
13.95
17.55
High demand worker-high cost firm, (62, 02)
Low demand worker-low cost firm, (1,, <1)
High demand worker-low cost firm, (22, <P1)
18.60
17.36
16.57
-
15.91
15.01
13.36
High demand workers, s2
-
-
-
Probabilitymatchis acceptableout of unemployment
Low demandworkers,s1
0.89
0.80
High demand workers, s2
0.78
Probabilityacceptablematch results
in health insuranceout of unemployment
Low demandworker-high cost firm,(s1, (b2)
High demand worker-high cost firm, (2, <02)
Low demand worker-low cost firm, (1, <1)
High demand worker-low cost firm, ( 02,<1 )
Mean unemploymentduration
Low demandworkers, 1j
-
0.44
-
0.43
0.75
0.72
0.19
-
0.33
0.39
0.44
-
0.28
0.32
0.47
17.96
17.89
18.59
17.79
18.22
-
18.62
High demand workers, 62
Probabilityof an unemploymentspell over samplewindow
Low demandworkers,s1
0.269
High demand workers, 2
0.29
0.87
-0.263
0.273
0.270
-
0.285
0.251
aBased on the parameter estimates presented in Table II.
Table III. We also compute the probabilitythat a match is acceptable and the
probabilitythat an acceptable match results in the provision of health insurance. In addition,we estimate the mean unemploymentspell durationand the
probabilitythat an individualof a given type is unemployedover the length of
the sample window. Beginningwith our baseline specificationwith no heterogeneity, we find that nearly 89 percent of all potential matches are accepted
out of unemploymentand that nearly44 percent of these matches result in the
provision of health insurance. The mean unemployment duration is close to
18 weeks and nearly27 percent of the populationwould be observedin the unemploymentstate sometime duringthe sample window. Movingto the second
specification,we find relativelysmall differencesin the labor marketoutcomes
of high and low demand workersand high and low cost firms. Specifically,we
find that low demandindividualsaccept 80 percent of matchesout of the unemployment state, whereas 78 percent of matches are accepted by high demand
workers.As a result,the mean unemploymentdurationis only about one-third
612
M. S. DEY AND C. J. FLINN
TABLEIV
ESTIMATED LABOR MARKET OUTCOMES DIRECTLY FOLLOWING AN UNEMPLOYMENT SPELLa
a = 0.25
Parameter
(1)
Proportionof jobs with health insurance
Low demandworkers,(l
0.44
-0.43
High demandworkers,s2
Wagesin jobs with health insurance
Low demandworkers, l
15.36
(11.18)
High demand workers, ~2
-
a = 0.50
(2)
0.43
0.32
0.46
16.03
(11.54)
15.71
(10.92)
17.03
(11.79)
15.37
11.09
(6.59)
11.26
(6.68)
Durationsof jobs with health insurance
Low demandworkers, 1
316.55
(357.98)
High demandworkers, -2
Durationsof jobs withouthealth insurance
Low demandworkers, 1
50.47
(50.82)
High demand workers,
2
(2)
0.38
(11.34)
Wagesin jobs without health insurance
Low demandworkers,(
10.83
(6.48)
High demandworkers, -2
(1)
-15.22
-(10.64)
10.75
(6.51)
-11.52
-
11.67
(7.18)
(7.02)
325.83
(372.80)
312.20
(357.09)
298.17
(347.20)
-313.97
-
362.58
(404.02)
49.93
(50.33)
45.20
(45.46)
54.43
(55.10)
-49.55
(49.96)
-53.57
-
(360.20)
(53.70)
aBasedon the parameterestimatespresentedin TableII. Wagesare measuredin dollarsper houranddurations
are measuredin weeks.Standarddeviationsare in parentheses.The wage representsthe firstwage in the firstjob
errorprocess.
the measurement
followingan unemployment
spellandincorporates
of a week different for the two types of workers.We do see larger differences
in the probabilitythat an acceptableworker-firmmatchresultsin the provision
of health insuranceacrossworker and firm types. A high demand workerwill
have health insurance at 44 percent of low cost firms, but will gain coverage
at only 33 percent of high cost firms.On the other hand, low demandworkers
will be insuredat 39 percent of low cost firmsand 29 percent of high cost firms.
We estimate that low demand individualsare slightlymore likely than high demand workers, by 27.3 to 26.3 percent, to be observed in the unemployment
state over the sample window.
TableIV presents some summarystatisticsfor characteristicsof the firstjob
directlyfollowing an unemploymentspell. Since the theoretical predictionsof
the model (in terms of equilibriumwages and job spell durations) are clearest following an unemploymentspell and since most of our wage and health
insurancedata come from the firstjob following such a spell, these estimates
HEALTHINSURANCE PROVISION
613
providea useful comparisonbetween the theoretical(estimated) predictionsof
the model and the data. The results from the first specificationindicates that
44 percent of first jobs provide health insurance,that the mean wage in jobs
with health insurance is 42 percent higher than the mean wage in jobs without insurance,and that (first)jobs with health insurancetend to last about six
times longer thanjobs withouthealth insurance.The estimatesfromthe second
specificationpoint to some importantfeatures of the model. First,we find that
almost 43 percent of high demand individualshave health insurancecoverage
and 38 percent of low demand workers have coverage in the first job following an unemploymentspell. Second, we see that low demandworkersactually
earn 66 cents per hour more, on average, than high demand workers at jobs
with insurancecoverage. This is not only due to the direct effect of the private
demand on the wages, as capturedby s2, but also because high demandworkers have health insurance coverage at relativelyless productivematches than
do low demandworkers.Third,we see that high demandworkersearn slightly
more than low demand workers at jobs without insurance,which is solely due
to a composition effect.19Lastly,we find that low demandworkershave longer
job durationsat insuredjobs than high demandworkers.While this result may
seem paradoxical,recall that high demandindividualshave health insuranceat
relativelyless productivematches. Thus voluntaryturnoverfrom jobs covered
by health insurancewill occur at higher rates in the high demand population
than in the low demandpopulation,while the involuntaryseparationrate is the
same for the two groups (rqi).
The model does do a reasonablygood job of fittingthe conditionalwage distributions observed in the data, as Figure 5 demonstrates.20The graphs plot
the theoretical (estimated) densities of the firstwage observed after an unemployment spell conditional on health insurancestatus against the corresponding histogramof sample wage rates. Clearlythe implicationsof the model are
less satisfactoryfor the wage distributionassociated with jobs without health
insurance.While it is true that allowing for measurementerror in wages acts
to smooth away differencesbetween the predictionsof the equilibriummodel
and the data, the measurementerror assumptionsare restrictiveenough that
its presence cannot be the sole explanationof the high degree of correspondence between the predicted and observed distributions.
TableV presents some summarymeasures of the labor marketin the steady
state. The estimates are computed by simulatingthe labor market histories of
one million individuals(of each type), who begin their labor marketcareers in
the unemploymentstate, based on the parameterestimatesfrom the two specifications.Because the implicationsfrom both specificationsare similar,we will
'9Thisis due to the fact that low demandindividualshave a lower reservationmatchvalue into
jobs without health insurance.
20Wedecided not to implementformaltests of model fit due to the large amountof censoring
in the durationdata.
614
M. S. DEY AND C. J. FLINN
0.08 -
0.12-
0.07 -
0.06-
a
08-
ii
004- _
0.030.03
~~~~0.1-
I2|~~~~
-
I
i0./H
j
-
o02- -_
0.02
-0.01-JOItk
0
:i
:
0.06 -
0.04
\
-
-
* .i
Ij-ii I i
.- .
0.02
111111^?sll
10
20
Wage
30
40
0
10
20
30
40
Wage
FIGURE 5.-Actual vs. predicted wage densities. The panel on the left representsjobs with
health insurancecoverage and the panel on the right representsjobs without health insurance
coverage. Wages represent the first wage in the firstjob following an unemploymentspell. The
predicted wage distributionis based on the parameterestimates for the specificationwith no
heterogeneity presented in column 3 of Table II. The actual wage distributionis based on the
1996 panel of the SIPP.
discuss only those from the second specification(column 2). The most striking
feature is the lack of large differences between the steady state outcomes of
low and high demand individuals.This is a result of the small size of the estimate of the utility parameter s2. We see that the steady state unemployment
rates of the two types are almost identical, as is steady state health insurance
coverage. It is of interest to compare the rate of coverage in the steady state,
about 87 percent for each group, with the rate of coverage in first jobs after
an unemploymentspell, which differed for the two groupsbut was centered at
40 percent. The steady state rate, which is roughlyconsistentwith the observed
rate of employer-providedhealth insurancecoveragein this population,is produced by the systematicdifferencebetween the steady state distributionof job
matchvalues and the population distributionof these values. The formerfirstorder stochasticallydominates the later, thus producingthe large differences
in insurancecoverage and wage rates observedin TablesIV and V.
In the bottom two panels of TableV we note the large differencesbetween
the wages associated with jobs with insurance and without insurance in the
steady state. There are small differences between the average wage on jobs
without health insurancein the steady state and immediatelyfollowing unemployment due to the fact that the upper bound on match values is the same
for both distributionsand there is little difference in the distributionof match
values below this criticalvalue in the steadystate and in the population.On the
other hand, there are large differencesbetween distributionsof matches in the
HEALTHINSURANCE PROVISION
615
TABLEV
ESTIMATEDLABORMARKETOUTCOMESIN THESTEADYSTATEa
a = 0.25
Parameter
(1)
Unemploymentrate
Low demandworkers, 1
High demand workers, :2
a =0.50
(2)
0.044
0.045
(1)
0.046
-0.044
0.868
0.878
High demand workers, 2
0.877
-0.882
Health insurancecoveragerate (employedpopulation)
Low demandworkers,61
0.914
0.909
0.921
High demand workers, ~2
-0.920
0.917
Wagesin jobs with health insurance
Low demandworkers,e1
20.33
(15.04)
High demand workers, 2
-
Wagesin jobs without health insurance
Low demandworkers,e1
11.50
(7.10)
High demand workers, -2
0.052
0.042
Health insurancecoveragerate (entire population)
Low demandworkers,fl
0.874
-
(2)
20.50
(15.36)
20.35
(15.32)
11.51
(7.06)
11.68
(7.12)
19.37
(13.61)
-
11.08
(6.68)
0.846
0.893
18.97
(13.26)
18.36
(12.98)
11.52
(7.04)
11.82
-(7.13)
aBased on the parameter estimates presented in Table II. The estimates are computed using the simulated labor
market histories of 1,000,000 individuals (of each type) who begin their lives in the unemployment state. Wages are
measured in dollars per hour and incorporate the measurment error process.
steady state and populationwhen conditioningon health insuranceprovision.
This is reflected in the large differences between the mean wage in covered
jobs in the steady state, over $20 for both demand types, and in the first job
following unemployment,a bit over $11 for each demand type.
We now consider the implicationsof the estimates for degree of inefficiency
in mobility decisions. Proposition 1 establishedthat, in a model with differential costs of health insurancecoverage (withinthe populationof firms)and differentialrates of privatedemandfor health insurance(withinthe populationof
searchers),job mobilitydecisions are conditionallyefficient. Thus, given a cost
of health insurancec(= 4 - 4) associatedwith a match of productivevalue 0
and given a cost c'(= &'- ~) associated with a match of productivevalue 0',
the workerwill opt for the job in which the total surplusof the match is greatest. If we let the total surplusof the matchbe denoted by Q(0, c), then efficient
mobilityimplies
(57)
(0, c') > (0, c) X Q(o',c') > Q(o, c).
616
M. S. DEY AND C. J. FLINN
The total surplusof the match, Q(0, c), is a strictlyincreasingfunctionof 0 and
nonincreasingfunction of c. This implies the possibilityof an efficient choice
of a job (0', c') over a job (0, c) when 0' < 0. However, in this eventualityit
must be the case that c' is strictlyless than c, or more formally
(58)
(', c') > (0, c)
and
0' < 0
= c' < c
0' <,/'
when we look at job movements for the same individual,where the last line
in (58) follows from the fact that the individualcomponent of costs (4) is the
same at both jobs.
From (58) it follows that if the cost of the provision of health insurance is
the same at all firms,then
(59)
(0', c) > (0, c) O' > 0
when evaluated at the common ) shared by all firms. Thus a degenerate distribution of 0 implies that all mobility choices are efficient in the sense of
maximizingthe instantaneousgross product of the match. The claim that differential health insurance costs distort otherwise efficient mobility decision
seems to best be interpreted as a claim that selection of the largest 0 value
is not ensured. We have shown that with differential costs of health insurance, selection of the largest 0 value is, in fact, not necessarily efficient. In
this section we define and compute measures of "inefficiency,"by which we
mean the probabilityof the choice of the smallest of the two values of 0 available to a currentlyemployed searcher. The quotation marks in the previous
sentence and in the title of this subsection refer to the fact that such choices
are not, in fact, inefficient in the presence of firm heterogeneity in the cost of
providinghealth insurance.21Nonetheless, it seems of interest to know the extent to which firm heterogeneityin 0 results in the choice of jobs with smaller
values of 0. The measures defined below compute the probabilityof such an
event occurringwhen an employed searcher meets a new potential employer.
Since these events can only occur when 4 is heterogeneous, we compute our
measures for the model specificationsthat allow for this possibility (columns
2 and 4 of TableII).
In computing the indices we use as a baseline the steady state joint distri-
bution of (0, s, &).22 Although in the population these characteristics are as-
sumed to be independentlydistributed,systematicsorting will produce some
21In the remainderof this sectionwe will not
highlightthe fact that the term inefficientmobility
is actuallya misnomer,but insteadwill use it as simplyindicatingthe choice of a job opportunity
with a lower matchvalue.
22Sincespecification2 does not include heterogeneityin searchers'privatedemandsfor health
insurancethe steadystate distributionis defined over (0, 05)only.
617
HEALTHINSURANCE PROVISION
dependence in the steady state distribution.Since there is no analyticsolution
for the joint steady state distributionof these characteristics,this distribution
is approximatedusing simulationmethods. We denote thisjoint distributionby
Pss(0, ),c).
Next we use the decision rules described in Section 2 to write a critical
match value required to induce mobility from a job at a potential employer
of type )'. Define this criticalmatch value as 0(0, 4, 4'). Thus any potential
employer of type (0', 4') will successfullyrecruit the individualif and only if
o' > 06(0,
6, ('). Using this function, the conditional probabilityof a job-tojob change given a contact with anotherfirm is
G(0(0,
,O '))p(O'),
where p(4)') is the populationproportionof type 4' firms.
Only a proportionof all moveswill be inefficientin the sense of involvingthe
choice of a job with a lower 0 than hat availableat an alternativematch. In the
empiricalliteratureon health insuranceand mobilitydecisions a distinctionis
made between the case in which one stays at a firmwith a lower 0 than at an
alternativefirm, termed "job lock," and the situation in which an individual
moves to a firmwith a lower 0 value, termed "job push."While the difference
between the two is somewhat semantic,we can decompose the probabilityof
an inefficientchoice into these two sources using the followingmetric.
For an inefficient move to occur requires that the two firms currentlycompeting for the searcher'sservices be of different types. For there to exist "job
push" requires that the current employer be type 02 and the potential employer be of type 0l1. In the case of job push the critical value for leaving a
match of 0 is less than or equal to 0. The conditional probabilitythat a move
will be inefficient and be attributableto "jobpush"is then
(60)
pss(Ijp)
E
[G(0(0,
42 01)),
G()]p
=X
01))
? (Et5t1\(2}
G(0((0,(t2
> 0O
ss(O,
n
2)dO
2 > 4>1.
For "joblock" to occur requires that the currentemployer be a low cost type
and the potential employerbe a high cost type. The criticalmatchvalue in this
case is greater than or equal to 0, and the probabilityof inefficient mobility
attributableto job lock is
(61)
pss(IJL)
E-
[ G(0)-G(0(0,
1
>0
.4 2p(O42)pss().,
?E(1fi2}
>0
b2 > 01.
G(0)
0,
2))
,
01)
)dO
dO
M. S. DEY AND C. J. FLINN
618
The likelihood of inefficient mobilityis simplythe sum of these two probabilities, or
Pss (IT) = Pss (IJP) + Pss (IJL)
(62)
Before discussingthe estimatesof the degree of inefficientmobility,we show
that whenever the firmscompetingfor an individual'slabor services are of different cost types, the likelihood of passing on the higher match value is not
negligible. In Figure 6 we have plotted the probabilityof "inefficient"mobility for low demand individualsusing the estimates from the fourth column
of Table II.23Figure 6 illustrates the probabilityof moving to a lower match
value conditional on the searcher's current match value at a high cost employer. Once 0 reaches the criticalvalue at which health insurancewould have
been purchased at the employee's current (high cost) employer, the conditional probabilityof leaving for a lower value of 0 jumps to a bit less than 0.25,
rising and then declining slowly thereafter. The rightmostgraph in Figure 6
contains a plot of the conditional probabilityfunction for the case of job lock,
in which the currentfirm is a low cost provider and the potential employer is
-
03 -0.3
,
0.25 -
0.25 -
0.2-
0.2-
'oQ
&
i, 0.15 O
. 0.15 o
-
01 0.1-
-
0.0.05
0
,
0
,
10
, ,
20
I
30
0.05 -
,,
,
0
40
Match Value at High Cost Firm
50
0
i
10
,
i
20
,
.
30
40
50
Match Value at Low Cost Firm
FIGURE 6.-"Conditional" job push and job lock. The panel on the left represents "conditional" job push and the panel on the right represents "conditional" job lock. Based on the parameter estimates for the specification with both firm and worker heterogeneity presented in
column 4 of Table II. The estimates are computed using the simulated labor market histories
for 1,000,000 "low demand" individuals who begin their working lives in the unemployment state.
See text for details and definitions of the measures.
23The probabily of inefficient mobility function for high demand individuals is very similar. We
have not plotted their conditional probability function so as to avoid clutter.
HEALTHINSURANCE PROVISION
619
high cost. We see a similarpatternfor the case of job push (the leftmost graph).
Roughly speaking,the probabilitythat the employee will pass up an opportunity possessing a higher value of 0 is approximately0.23 over a large range of
values of 0.
While the "joblock" and "jobpush"phenomenon are not negligible conditional on firmsof differentcost types meeting, from our estimates of the distribution of b we know that this event occurs very infrequently.From column 2
of TableII, the point estimate of the probabilitythat an employee of a high cost
firm meets a low cost firm is 0.961. However, the steady state (unconditional)
probabilitythat an employer is high cost is only 0.059, so that the probability
of such a comparisonin the steady state is only 0.055. The probabilityof an
employee of a low cost firm meeting a high cost potential employer is approximately equal, so that the total probabilityof these events is about 0.110. The
infrequencyof these types of meetings accountsfor the extremelylow levels of
"inefficient"mobilitychoices we observe.
Table VI contains estimates of "inefficient"turnoverprobabilitiesthat are
computed using point estimates of the model that includes both firm and
workerheterogeneity(i.e., the second specificationin TableII) for the two values of a that we consider.The amountsof inefficientturnoverare minisculein
both columnsof the table. The total amountof inefficientturnoverfor a = 0.25
was about 1 percent, and "job lock" accounted for slightlymore of the inefficient turnoverthan "jobpush."The breakdownswere virtuallyidenticalwhen
we distinguishbetween high and low privatedemandtype searchers.Inefficient
turnoverwas slightlymore prevalentunder the symmetricNash bargainingasTABLEVI
MEASURESa
"INEFFICIENCY"
Parameter
High demandtypes
Percentof "jobpush"
Percentof "joblock"
Index of inefficiency("jobpush"+ "joblock")
Low demand types
Percentof "jobpush"
Percentof "joblock"
Index of inefficiency("jobpush"+ "joblock")
All workers
Percentof "jobpush"
Percentof "joblock"
Index of inefficiency("jobpush"+ "joblock")
a = 0.25
a = 0.50
(2)
(2)
0.48
0.57
1.05
0.68
0.92
1.60
0.47
0.56
1.03
0.65
0.86
1.51
0.47
0.57
1.04
0.67
0.89
1.56
aBased on the parameter estimates presented in Table II. The estimates are computed using the simulated labor
market histories for 1,000,000 individuals (of each type) who begin their working lives in the unemployment state. See
text for details and definitions of the measures.
620
M. S. DEY AND C. J. FLINN
sumption (a = 0.5), where the estimated value of total inefficient turnoveris
1.56 percent. Even in this case, it is difficult to argue that the heterogeneity
in firm health care costs produced significantamounts of "bad"turnoverdecisions defined by the choice of the smallest 0 value in the employee's choice
set.
6. CONCLUSION
Researchers investigating the relationship between employer-provided
health insurance,wages, and turnoverhave uncovered a number of empirical
findings,not all of which are mutuallyconsistent, that to date have not been
explicablewithin an estimabledynamicmodel of labormarketequilibrium.We
propose such a model and show that it has implicationsfor labor market careers broadlyconsistentwith empiricalevidence. Using SIPP data we estimate
the model and, for the most part, obtain plausibleresults.The model is able to
capturesome of the most salient features of the event historydata. In particular, our model is based on the premise that health insurancereduces the rate
of arrivalof negative health shocks that lead to job separationsinto the nonemployment state. In the data we find this prediction to be overwhelmingly
supported. Furthermore,the model offers a cogent rationale for the necessity of conditioning on health insurance status as well as the wage rate when
estimatingjob separationhazards,and predicts that voluntaryas well as nonvoluntaryseparationswill be lower for those with employer-providedhealth
insurancethan for those without it. The wage distributionsassociatedwith the
jobs covered by health insuranceand those not covered are also broadlyconsistent with the predictionsof the model. No undue reliance on measurement
error is required to make this relatively involved dynamic model consistent
with the data.
We view one of the accomplishmentsof this paper as demonstratingtheoretically and empiricallythat what may appear to be "job lock" is consistent
with an equilibriummodel in which all turnoveris efficient. The model estimated here is innovativeon at least two dimensions. First,we have estimated
an equilibriummodel in which jobs are (endogenously) differentiated along
two dimensions:wages and health insuranceprovision. Second, the model allows for wage renegotiationswith the employee's currentfirm. While empirical implementations of matching models (e.g., Miller (1984), Flinn (1986))
are consistent with wage changes during an employment spell, they imply no
dependence between the wages paid at successive employers.The bargaining
models formulated and estimated here and in Postel-Vinayand Robin (2002)
and Cahuc,Postel-Vinay,and Robin (2003) offer a more complete view of wage
dynamicsthan other search-basedmodels that are currentlyavailable.
To assess the quantitativesignificanceof incomplete health insurance coverage for "inefficient"mobility outcomes, we introduced time-invariantheterogeneity within the population of firms and searchers. We showed that
HEALTHINSURANCE PROVISION
621
(time-invariant)worker differences in private demand for health insurance
could not produce situations in which lower match values were preferred to
higher ones; rather,firm heterogeneity in the costs of providinginsurancewas
crucial. Our estimate of the distributionof firm costs implied that the distribution was almost degenerate. As a result, our measuresof the degree to which
employees pass on match-improvingopportunitiesindicatethat less than 2 percent of mobility decisions exhibit this property. Furthermore,it must be remembered that in the presence of firm heterogeneityin the costs of providing
health insurance,all mobilitydecisions are efficientwhetheror not they involve
the choice of an option with a lower matchvalue than the alternative.
The model we have estimated and evaluated allows for rich forms of heterogeneity that are match-specific,firm-specific,and worker-specific.Nevertheless, it may be objected that all sources of heterogeneityare assumedto be
time-invariant,and thus that the model fails to capture the impact of changing patterns of employee health insurancedemand on mobilitydecisions. The
argumentusuallyput forth is that employees covered by health insuranceand
who experience an increase in their demand for health insuranceface a wedge
in the price of health insurance offered by their current employer and that
which would be offered by any future potential employer. These price differences are the forces behind the phenomenon of "job lock," though the existence of such a wedge clearly requires an assumption of no renegotiation
of employmentcontractswhen employee or employer characteristicschange.
Since such renegotiation is the foundation of our model, and for reasons of
model tractabilityand the identification of model parameters,we have neglected time-varyingheterogeneity in the current analysis.If this type of heterogeneity were to be introduced into a fully-articulatedmodel of the labor
market, an explicit rationale for precludingthe renegotiationof labor market
contractswould have to be added as well.
US. Bureau of Labor Statistics,2 MassachusettsAve., NE, Washington,DC
20212-0001, U.S.A.;[email protected]
and
New
York
269 MercerSt.,New York,NY10003,
University,
Dept. of Economics,
econ.nyu.edu/user/flinnc.
U.S.A.; christopher.flinn
@nyu.edu;http://www.
ManuscriptreceivedNovember,2000;final revisionreceivedJune,2004.
1
APPENDIX A: PROOFOF PROPOSITION
Define the total surplusof the match as
(63)
Te(w, d; 0, 4) = VE(w, d; 0, ) + VfF(w, d; 0, 4)
622
M. S. DEY AND C. J. FLINN
-r
= p+
x ' 0- ( -
+ Ae
JJ
- ---1
G( 6(w, d, ))dF(0)
d+ Ae
)d+
/
TndN
e^(0,0~,)
T(0e, , 0, ))dG(O)dF(4)
O5(w,d,k)
f+Aef
V,(E(f,o0,4 )) dG(0)dF(j) ),
where T7(0,4, 0, ()) = T7(&j(0,4, 0, 4), d(0, , 0, ); 0, 4) is the equilibrium total surplusof the match (0, 4))when a type s individualhas next best
option (0, 4). It is straightforwardto show that the total surplusof the match
is independentof the wage, or
8T(w,d; o, 4)
=0
8w
for all (w, d; 0, ), ~)
Tr(w, d; 0, )-Q (d; 0, b).
Therefore, we can redefine the Nash bargainingobjective function as a function of the health insurance status and the share of the total surplus of the
match earned by the worker,/. We then choose d and /3 to maximizethe Nash
bargainingobjectivefunction
(64)
(/,
d; 0', ',,
= [PfQ(d; 0',
)
)')- Qe(0, ()]"[(1 - 3)Qe(d; 0', ')]1-"
Conditional on health insurance status d, the equilibriumshare of the total
surplusreceived by the employee satisfiesthe first-ordercondition
aQ6(d; 0', 4')
[/3*0(d; 0', 4') - Q6(0, 4)]
=
(1 - a)Q(d;
', ')
(1 -3*)0Q(d; 0', 4')
a(1 - /*)Q(d; 0', 4') = (1 - a)[/3*Qf(d; 0', 4') - Q(0, 4)]
= 3*3((d; 0', 4') = aQ(d; 0', )') + (1 - a)Qe(0, 4).
Therefore, conditional on health insurancestatus d, the equilibriumvalue to
the workeris given by
(65)
VE(d; 0', 4)',0, 4)) = aQe(d; 0', 4') + (1 - a)Qe(0, 0)
623
HEALTHINSURANCE PROVISION
which implies that
(66)
JVF(d;0',
)', 0, 4) = (1- a(Qa)(Q(d;')- 0',
(,
))).
The conditionaltotal surplusis given by
(67)
T (d; 0', ', , ) = VE(d; 0',a,
, )) +VL(d; 06,', 0, ))
= Q6(d; 0', 4').
Therefore, in equilibrium,the total surplusof the currentmatch (0', )') is independent of the characteristicsof the worker's next best option (0, )) and
hence, the health insurance decision only depends on the "winning"firm's
characteristics.
APPENDIX B: PROOFOF PROPOSITION
2
Define the total surplus of the match conditional on health insurance status d as
(68)
Q 0(d; , 4)
--1
1r
= p+
d
+ Ae
/
x {0-(-(+ Ae
G(O(d, 0, ( , ))dF(4)
)d + rdV
Vf (0n
d, 0, 0)dG(5)dF(0)
0
(
+Je(d,0e,,I
)
,
where the criticalmatch 06(d, 0, 4, () is implicitlydefined by the equation
Qe(0e(d;0, , ), )=QO(d; 0, ))
and VE(0, ), d, 0, 4))representsthe value to the employee at the match (0, ))
when his threat point is given by QO(d;0, 4). Given Proposition 1, we can
rewritethe total surplusas
'
(69)
(d; 0, ) =
p+
x
7d
+ ae
0-( ) -
--1
f G(0(d,
d+
0,,, >)) dF(4)
rdVN
+,ae Af
, J6(d,O,&,d3)
Q(0,
)dG(0)dF()
).
624
M. S. DEY AND C. J. FLINN
Using the definitionof the criticalmatch b0(d, 0, 4, 0) we can show that
(70)
dQ6(d; , 4)
d
P + d+aA, e
G((d, O,
0,
))dF())
-1
>0
for all (d, 0, , ).
Since Q0(d; 0, 0) is strictlyincreasingin 0 for all (d, 0, 4, s) we can characterize the decision to initiate an employmentcontractwith health insurance
status d between a type 5 worker and a type 4 firm by unique critical match
0*(d, 4) that satisfiesthe implicitequation
(71)
Q ;(d;0(d, 4), 4))=
.
More explicitly,we can show that
(72)
o0(d,
d + pV
) =(-
-aA, f
J Je(d,&)
[Q,(0, 4)- VN] dG(0) dF(4),
which implies that the critical match does not depend on the type of firm )
when health insuranceis not provided and that 4 - 4 > 0 is a necessarycondition for the existence of jobs without health insurance at the worker-firm
pair (4, )). The intuitionbehind this condition should be clear. When the private demand s is greater than the cost ), health insuranceis effectively free
and, given that it has beneficial effects on the value of any match, will always
be purchased.
Assume that every ( , >) pair accepts some employment contractswithout
health insuranceso that 0 - 0(0, () < 0e(1, 4). Any match 06 [0 , 0j(1, 4)]
will not result in the provisionof health insurance.In addition, since rio > r71
and f, G(0(0, 06, 4, )) dF(4) = GG(06(1,0((1, ), ), 4)) dF(), Q(1;
0, >)increasesat a faster rate than Q(0; 0,4) for all 0 > 0*(1, 0). Therefore,
there exists a unique criticalmatch 0T*(4>)> 0* such that
(73)
Qo(0; 0**(), &) = Q(1; 0**(&),&).
APPENDIX C: APPROXIMATION
OF DECISIONRULES
In this appendix we present our strategy for deriving the equilibrium of
the Nash bargainingmodel. We will consider the most general specification
with both worker and firm heterogeneity. The computationalburden of solving the model is quite substantialand since estimation of the model requires
knowledge of the conditional (on health insurancestatus d) equilibriumwage
625
HEALTHINSURANCE PROVISION
functions,wth(d;0', 0', 0, ), and the criticalmatches for acceptanceof an employment contract and the provision of health insurance, 06 and 0**(4), we
are forced to approximatethe system of value functions.After doing so we are
able to efficiently solve for the equilibriumof the model without an excessive
computationalburden.
To begin, recall the value of an employmentcontract (w, d) at a matchwith
characteristics(0, 0) to a workerof type s is given by
p + ld +e
+,
VJE(w,d; 0,)
"--1
-
r(
(74)
G(0O(w, d, 4)) dF(()
x w + d + ndV
+
Ae/X
Jdje(w,d,3)
P+Ae| A(0(0
, 0,
(,
vE( 0
,
) dG(0)dF())
0n0 ) dG ) dF
and that the criticalmatches are implicitlydefined by the equations
(75)
VF(w, d; 06(w, d, 4),
4) = 0 and
Q(0((0,), 4), ),) = Q(0, 0).
Next, note that the maximumvalue of 0 for which the contract (w, d) would
leave a O-typefirmwith no profit equals 0((w, d, 4) = w + d4) and the equilibrium wage associated with the worker receiving all the rents from 0 is
wi(0, ), 0, )) = 0 - do. Given these two implicationsof the model we then
know that
(76)
V E(w =
- d,
d; 0, O) = Q6(d; 0, 0).
Taking the first-orderTaylor series approximationto VE with respect to w
(around w = 0 - d4), we have
(77)
VE(w, d; 0,0)
, Q(d;
0, 4)+ (w-0+d4)d 4
,
, 4))
w
dW
V(wE(w, d;
w=O-do
Using Leibniz'srule, the derivativeevaluatedat w = 0 - dOfcan be shownto be
(78)
dVE(w, d; 0, )
dOW
1
6w=e-dO
P + -'ld
+ e ,i=l p(i)G(06C(0,
1(d;
/3( d; 0 4)'
,
i))
626
M. S. DEY AND C. J. FLINN
Therefore,
(79)
VE(w, ,d; , &)
Q6e(d;0, ) +
+
/3(d; 0, 0)
Using the fact that
VfE(lw(d; 0',
', 0, 4), d; 0', 4) = Vf(d; 0', 0', ), )
= aQ3(d; 0', )') + (1 - a)Qe(0, 4)),
it follows that equilibriumwages, conditionalon health insuranced, are given
by the equation
(80)
w(d; ', ', 0, 4))
= 6' - do' - (1 - a)f38(d; 0', 4')(Q6(d; 0', )') - Q(0, 4))).
Given the close relationshipbetween the equilibriumoutcome (both wages
and the provisionof health insurance)and the function Qe(d; 0, 4), the computation of the equilibriumfollows a rather simple three-stage process. First,
we solve the fixed point equations for Q d(d;0, 4) and VfNfor all d, 0, 4, s.
Given these values, we determine the criticalmatches for the acceptanceof an
employment contract and the provision of health insurance accordingto the
system of equations
(81)
Q6(0, 0,
e
1)
0
Q(0,, 0,
2)
=
vN
and
Q(1, 0e*(4), 4) = Qe(0, 06*()), 4)
for s E {s1, ~2} and E {i01, 42). Finally,we compute the equilibriumwages
accordingto equation (80) above.
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