Montréal
Série Scientifique
Scientific Series
2000s-58
Competition and the Reform
of Incentive Schemes in the
Regulated Sector
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© 2000 Marcel Boyer et Jean-Jacques Laffont. Tous droits réservés. All rights reserved.
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Competition and the Reform of Incentive Schemes
in the Regulated Sector
*
Marcel Boyer
†
, Jean-Jacques Laffont
‡
Résumé / Abstract
Nous analysons dans un modèle principal-agent avec sélection adverse et
contrats complets comment les incitations dans une entreprise réglementée sont
affectées par la concurrence externe à travers son effet sur l’information et la
fonction objectif du principal d’une part et les contraintes de compatibilité
incitative et de rationalité individuelle de l’agent d’autre part. Nous considérons
plus précisément les sources suivantes de pressions concurrentielles accrues : une
meilleure structure d’information, une menace plus forte de liquidation, une
concurrence plus intense pour le talent, un secteur privé plus efficace, et
l’existence de meilleurs substituts. Nous caractérisons dans chaque cas les
conditions sous lesquelles l’effet sur les incitations est positif.
We consider a regulation problem with complete contracting in a
principal-agent model with adverse selection and review within this model the
various channels by which external competition parameters affect incentives
within the regulated firm. The channels are: the principal’s information, the
principal’s objective function, the agent’s incentive constraint, the agent’s
participation constraint. We consider in particular a better information structure,
a threat of liquidation, a fight for talent, a more efficient private sector, and the
existence of better substitutes. We characterize in each case the conditions under
which the effect on incentives is positive.
Mots Clés :
Incitations secteur public, concurrence
Keywords:
Public sector incentives, competition
*
Corresponding Author: Marcel Boyer, CIRANO, 2020 University Street, 25
th
floor, Montréal, Qc, Canada
H3A 2A5
Tel.: (514) 985-4002
Fax: (514) 985-4035
email: boyerm@cirano.umontreal.ca
We thank participants in the Canadian Economic Theory Conference (Ottawa) and in the microeconomic seminar at
Rutgers University for their comments. Financial support from the SSHRC (Canada) and CNRS (France) is
gratefully acknowledged.
Theglobalizationofnationaleconomiesandthepaceoftechnologicalprogressincreasethe
over-alllevelofcompetitionlocallyandglobally. Thisinturnraisesquestionsregardingtheeconomic
role of the State as a regulator of economic activities. Demands for profound reforms of the
regulated sectorand governmentinstitutionsareexpressedinnumerouscountries. Indeedmost
countrieshave designedand begun implementingreforms aimed at increasing theperformance
of their regulated sector. The OECD public management service (PUMA) provides
numer-ous examples of national policies towards that goal. A signicant exampleamong manyis the
implementation in England of Compulsory Competitive Tendering (CCT) procedures at the
municipalandregionallevels. Thelocalauthoritiesmustcallfortendersforanincreasingarray
of government servicesbeforedeciding whetherthose serviceswillbeoeredbyadirect
munic-ipal service organization ora private companyunder dierent forms of delegated management
contracts withthelocalauthorities. 1
Theunderlyingleitmotifsareperformance, incentivesand
competition of one kindor another. 2
The preferred pathof public sector reform has generally
beento privatizeand liberalizeformer publicsectors. 3
However, manyactivities willand must
remain under government's direct control for natural monopoly or national interest reasons.
How incentives, within public rms and institutions or within regulated private rms, should
respond to thismore competitiveenvironment ? 4
In this paper, we study a regulation problem with complete contracting and a benevolent
regulator (the principal) in a principal-agent model with adverse selection. We analyze the
comparative statics eects on the power of incentives in the regulated rm or administration
(the agent)ofvariousparameterchangesusuallyassociatedwithgreatercompetitivepressures.
Thisisanecessarysteptoexplorehowgreatercompetitivepressuresmayaectincentiveswhen
1
SeeLeGallo(1998). 2
There exists also an important literature which has investigated Machlup's (1967) claim that there is no
managerialslackwhenarmoperatesinaperfectlycompetitiveoutputmarket. Leibenstein(1966)hasprovided empiricalevidencefromcasestudiessupportingthecommonviewthatincreasingcompetitionreducesslack.
3
It is often accompanied by sizable downsizing. See Jeon and Laont (1999) for a study of the incentive problemscreatedbydownsizing.
4
Onepossibility issometimestointroducecompetitionwithintheregulatedsector itself. However,thisoften requires wasteful duplications so that competition will remain limited for example to duopolies or restricted oligopolies. AuriolandLaont(1992)haveinvestigatedthechangesonincentivesbroughtaboutbythisduopoly
there is nosingle denitionofwhat increasedcompetition means.
The level of competitive pressures exerted on thepublic sector may be directly decided by
the government when activities which were monopolies are liberalized. We want to explore a
more indirectnotionofincreasedcompetitivepressures. Thelevelofcompetitionitselfisnotan
exogenous parameter asemphasized by Blissand DiTella(1997) who criticizethemodeling of
greater competition bythe increaseinthenumberofcompetitorswhich isan endogenous
vari-able at theindustry level. They suggest thatmore fundamentalparameters, such as transport
costs, cost uncertainty, distribution and overhead cost, should be used. We recognize the
im-portanceofthispointand we willfollowtheirsuggestionasmuchaspossible. However, forour
purpose,apartialequilibriumanalysiswhichconsiderschangesinparametersoftheenvironment
(which are indeed endogenous in a more general analysis) is suÆcient to understand some of
the linksbywhichincreasedcompetitionaects incentivesintheregulated sector. We consider
suchparameters,namelytheinformationstructure,thethreatof bankruptcy,theopportunities
availableto agents, thetechnologies andthequalityofsubstitutes.
Thepaperisorganizedasfollows. WediscusstherelevantliteratureinSection2followedby
the presentation of the basicmodel inSection 3. We study the informational eect in Section
4. We devoteSection5 to thethreatof liquidationeect whileSection6examines competition
in talent. Theimpactsof a more eÆcientprivatesector and ofimprovementsinsubstitutesfor
the regulated productareinvestigatedinSection7. We then concludeinSection8.
2 COMPETITION AND INCENTIVES.
Toreviewtheliteratureoncompetitionand incentives,wecan regroupthearticlesaccordingto
severalcriteria. Twosuchcriteriaarethetypeofagencyproblemconsideredandthechannelby
whichcompetitionaects incentives. According to therstcriterion,papers dieraccordingto
thetypeofmodelwhichformalizestheagencyproblemwithintherm,moralhazardoradverse
selection. According tothesecondcriterion,papersdierbythechannelthroughwhichgreater
rationalityconstraint eect. Theinformationaleect referstothefactthatgreatercompetition
mayincreasetheinformationoftheprincipalabouttheagentand thereforedecreasethecost of
asymmetric information. The incentive constraint eect isthat greater competition mayaect
directly theagent's incentive constraint forexample bymaking it easier or more diÆcultfor a
good type to mimic a bad type agent. The principal's objective function eect is that greater
competition mayaect directly thevalue of production forthe principal and consequentlythe
value forhimofcreating incentivesfortheagent: thedesirablechangeinproductionmaythen
aect or not the incentive constraint of the agent. The individual rationalityconstraint eect
conveys the idea that greater competition may increaseor reduce theoutside opportunities of
the agent and thereforeher reservation utilitylevel.
We canillustratethisclassicationinthefollowingmatrix. Ineachcell, weplacethepapers
dealing withasimilaragency problemand asimilarchannelbutdealingpossiblywithdierent
institutional contexts. As indicated in Table 1, our paper ts into four cells. In Holmstrom
(1982),relativeperformanceevaluation,orcompetitionamongagents, isusedinordertoexploit
the valuable informationconveyed bythe other agents' outputsregarding the eortof a given
agent. InNalebuandStiglitz(1983)andShleifer(1985), theprincipalcaninfer,fromobserving
the outputs of all agents, some information on the level of eort chosen by a given agent and
therefore competition enables the design of more eÆcient reward structures. 5
In Hart (1983)
andScharfstein(1988),competitionbetweenentrepreneurialrmsandmanagerialrmsmakesit
more diÆcultformanagers to shirk. Hart(1983) shows thatthemarketsystembyitselfmakes
the actions and utilities of dierent managers interdependent via prices. 6
Scharfstein (1988)
showsthattheneteectofcompetitionon managerialslackdependsonmanagerialpreferences
and thenumberof statesand therefore,incentiveproblemsarenotalwaysmitigatedbygreater
competition. Increasingthenumberofbiddersinanauctionof contractswithadverse selection
5
Theyardstickcompetitionargumentwas furtherextendedtoagentswithdierentbutcorrelated
character-istics by Cremerand McLean(1985,1988)for aprincipalwho is adiscriminatingmonopolist oranauctioneer, and byRiordanand Sappington(1988)whenregulating armwithanexpostsignalcorrelatedwiththerm's
characteristics. 6
Competitionmakestheperformanceofdierentrmsinterdependentbuttheimpactonincentivesworksin a dierent wayfromHolmstrom(1982). Whilethelatterassumedthat theperformancesofsimilarrms could
TABLE 1 Moral Hazard Adverse Selection Information structure Holmstrom(1982) Hermalin (1992)
Nalebu and Stiglitz (1983)
Shleifer(1985)
this paper (prop:1;2)
Incentive constraint Hermalin (1992) Schmidt(1997) Hart (1983) Scharfstein(1988)
Laontand Tirole (1987)
this paper (prop:3)
Principal 0 s objective function Hermalin (1994) Schmidt(1997)
this paper (prop:5;6)
Participation
constraint
this paper (prop:4)
In Hermalin (1992), the manager makes an oer to the owner-shareholder through a contract
satisfyingaparticipationconstraintfortheshareholdersandanincentiveconditionfortheagent.
More competition mayreduce theagent'sexpectedutilityorincome,therebygeneratinga
neg-ative directincome eect bywhich theconsumption of perksor slackarereduced. But amore
competitive environment means better informed shareholders regarding the actions taken by
the agent, increasing the agent's net-of-risk-premium income and generating a positive income
eect infavoroflesseort. Thecompetitioneect istherefore ambiguous. InHermalin(1994),
principal-agent hierarchies compete inthe same market. Weak incentives may be the best
re-sponse to strong incentives because the value of incentives is proportional to the rm's lower
to the rm'slowercost. Hence,more competition hasanambiguouseect. Finally,inSchmidt
(1997), competition aects incentives through a positive threat-of-liquidation eect, inducing
the manager to work harder, and a negative prot-reduction eect making it less valuable for
the principal to implement strong incentive schemes. He obtains that increased competition
mayincreaseorreduce managerialslackgiven theoppositesignsofthetwo maineects
identi-ed. Moreover, increasingcompetitionmaylowermanagerialslackwhencompetitionislowbut
increasemanagerial slackwhen competitionis already intense.
3 THE MODEL
We considera naturalmonopoly 7
which realizesa publicproject valuedS at acost of
C = e (1) where 2 n ; o
is a parameter of cost eÆciency which is privately known by the manager,
where = > 0; let =Pr
=
be the common knowledge probability that
therm isalowcost rm;wewillreferto a rmasagoodtypermand toa rmas
a bad type rm;
eis themanager'seort levelwhich hasa disutility ( e) with 0 >0, 00 >0, 000 0; let ( x)= (x) (x ) ;
Cisobservablebythepublicregulator;wetaketheaccountingconventionthatitisdirectly
paid by theregulator.
The rm's utilitylevelis then
U =t (e) (2)
7
socialopportunitycost of1+with>0: Consumers'welfareis
S ( 1+)(t+ e) (3)
and social welfareistaken to be
W =S ( 1+)(t+ e)+U =S (1+)( (e)+ e) U: (4)
Undercompleteinformationtheregulatorwouldmaximizesocialwelfareundertheindividual
rationalityconstraintoftherm,thatis,U 0. OptimalregulationwouldleadtoeÆcienteort
levels 0
( e)=1 ore=e
forbothtypes,and no rentsU =0forbothtypes. Underincomplete
information, the regulator maximizes expected social welfare under the usual incentive and
individualrationalityconstraints,that is,solves(with obviousnotations)
(MP) max h S ( 1+) e+ (e) U i +(1 ) h S (1+) e+ (e) U i subjectto U U +( e); U U ( e+); U 0; U 0
forwhichthe solutionis
0 ( e)=1; 0 (e)=1 1+ 1 0 (e )<1 (5) U =(e ); U =0 (6)
The regulatormakestheoptimaltrade-obetweeneÆciencyandrents. He decreasesthe
incen-tivesofthe badtype(e<e
) to decreasetherent ofthegoodtype(e ).
Such a mechanism can be implemented by an incentive compatible menu of transfer-cost
pairs f(t ;C); (t;C)g: the good type rm chooses the rst pair, implying the eÆcient level of
eort (C= e
) and apositive rent U =t C =(e );thebad type rmchoosesthe second
pair, implyingan ineÆcientlevel ofeort(C = e)and no rent.
>From (5), incentives of the bad type decrease with increases in , and . When is
reduces thisexpected cost. However, if is larger than some value
, it is better to give up
productionbythebadtypeandoeracontractwithnorenttothegoodtype. Theincentivesof
theonlyactive rmarethenmaximal,asalways. Thevalueof
isdeterminedbytheequality
of expectedwelfarewithand withoutthebad type rm,thatis,using(5):
(1 ) h S (1+) e( )+ ( e( )) i = (e( ) ) (7)
We want to study the eect of competition on this optimal trade-o between eÆciency and
rents. We startbyconsideringtheinformational eect of competition.
4 THE INFORMATIONAL EFFECT OF COMPETITION
It has often been suggested that competition generates information. Competition allows
com-parisons and beingable to compare isbeing better informed. Rather than lookingdeeperinto
how exactly more competition means more information for the principal or the regulator, we
considerinthissectionhowthemenuofcontractsoeredchangeswhentheregulatorhasaccess
to abetterinformationstructure. Weprovideageneraldescriptionofthisinformationeect on
the powerof incentivesbeforelookingat specialcases.
We modelthisbetter informationstructureas thepossibilityforthe regulatorto observe a
signal correlated withthetrue type of therm beforedesigningthe contractsand determining
in particular the eort level to be induced from the bad type rm. However, contrary to the
literature reviewed inthe introduction,we assumethat thissignal isnon veriable and cannot
be used to condition the contract. It is the way yardstick competition works in practice, for
exampleinthetelecommunicationandelectricitysectors. Wethenproceedwiththecomparative
statics analysisof abetter informationorsignal.
4.1 Competition as a better information structure
Aninformationstructurefortheregulatorisasetofsignals=f 1 ; 2 ;:::; I gandconditional probabilities Pr( i j); i =1;:::;I. For each signal i
^ i =Pr( j i ); i=1;:::;I
and an associatedstrength ofincentivesdened by thelevelof eortofthe badtype
0 (e i )=1 1+ ^ i 1 ^ i 00 (e i ) (8)
when the regulatorwants to keep bothtypes of rms. Let e i = Z( ^ i 1 ^ i ) = ^ Z(^ i ) denote the solutionof (8). 8
Proposition1: If Z isconcave and^ i
<
forall i,then theregulator wishes tokeepboth
types of rmsand theexpected power ofincentives decreases whenthe regulator has access
to an informativesignal on the privateinformation of the rm.
Proof: Bydenition, E i ^ i =E i (E1 fg j i )=E1 fg = :
>From Jensen's inequality, E i e i e() i ^ Z is concave. Since 00 > 0; 000 > 0, then Z is
a decreasing function. Let h(^ i ) = ^ i 1 ^ i
, then h() is increasing and convex. Hence, ^ Z 00 = Z 00 h 02 +Z 0 h 00 . Therefore ^ Z isconcave ifZ 00 0. Q:E:D:
Weshowedabovethat ^
Z isconcaveif ()isquadraticorifissmallenough. Notealsothat
ifwedeneasignal i
asbeingfavorable[unfavorable]if^ i > i [^ i < i
],thepowerofincentives
8
Dierentiating(8)twiceandsubstituting,weobtain
^ Z 00 (^)= ^ Z 02 000 + 1+ ^ 1 ^ 0000 +2 ^ Z 0 1+ 1 (1 )^ 2 000 + 1+ 2 (1 )^ 3 00 00 + 1+ ^ 1 ^ 000 :
Therefore,if (e)isquadratic( 00
=,whereisapositiveconstant; 000 =0),weobtain ^ Z 00 (^)= 1+ 2 (1 )^ 3 <0: Similarly,if!0,weobtain ^ Z 00 (^)= 2 (1 ^) 3 00 1+ 00 + ^ 1 ^ 000 <0: Therefore, ^
theregulator,observingafavorablesignal,keepsonlythegoodtyperm,thatis,where^ i
>
for all favorable signals, it is obvious that the expected power of incentives increases since it
increaseswhen thesignalisunfavorableandsincethebadtype rmisdroppedwhenthesignal
is favorable.
Proposition2 : The expected power of incentives of the active rms increases with the
availability of a signal correlated with the true i, for all favorablesignals (^ i > i ), we have ^ i > .
Between the two extreme regimes dened by propositions 1 and 2, the expected power of
incentives mayincreaseordecrease dependingon theprobabilitiesofthe signalsleadingto the
liquidation of the bad type rm. We now illustrate these two propositions with four relevant
cases.
4.2 Four Special Cases
Case 1: A generalnon-degeneratesignal.
Werstassumethatcompetitionallowstheregulatortoobtainimperfectbutvaluable
informa-tion on the cost function of the rm. We model thisform of yardstickcompetition as follows:
the regulatorcanobserve aninformativesignal 2f; g correlatedwiththetrue typeof the
rm, thatis, Pr j =Pr j = 1 2 Pr j =Pr j =1 :
As increases, the informativeness of the signal increases, that is, the regulator's condence
in the signal increases. We want to investigate the eect of an increase in on the power of
incentives. For reasons of simplicity, let us considerthe special case where (e) = 1 2
e 2
with
=1. Aslong astheregulatorwants to keepbothtypes ofrms, we ndthat:
dEe
d
adirectillustrationofProposition1. Inasense,thecostofimplementingalargereortfromthe
bad typerm increases [decreases]with followinga favorable [anunfavorable] signal because
isthena morereliablepredictoroftheeÆciencyoftherm. Observing []impliesahigher
[lower]probabilitythataninformational rentwillbecapturedbytherm,henceanincrease[a
decrease] inincentives. Thereis avalue denedby +(1 )(1 ) = ; where
isdenedby(7), beyondwhichthebad typermisshutdownafterafavorablesignal
butremains active after an unfavorable signal . >From then on, the expected eort of the
bad typeincreaseswithand reachestheeÆcientlevelwhen=1(SeeFigure1). Proposition
1 holdsfor<
and Proposition2 holdsfor>
.
With more signals, we have the following situation, illustrated in Figure 2. For < 1
,
Proposition 1 holds and expected incentives decrease. When reaches 1
, the bad type rm
is dropped after signal 4
. When reaches 2
, the bad type rm is dropped after signal 3 . Between 1 and 2
theexpectedincentivescandecreaseorincreasedependingontheprobability
of 4
. For> 2
,Proposition2holdsandexpectedincentivesincrease. Thetransitionbetween
the two extremecases dependsnelyon thespecics oftheproblem.
9
Letandbetheposteriorprobabilitiesthatthermisofthegoodtypeafterobservingandrespectively. Wehave 1 = 1 1 and 1 = 1 1
Theeortlevelofabadtypermafterafavorablesignalisthen
e()= 1 1+ 1 1 ; de() d <0; d 2 e() d 2 <0
Similarlyafteranunfavorablesignal
e()= 1 1+ 1 1 ; de() d >0; d 2 e() d 2 <0
SincePr()=(1 )+(1 ),Pr ()=+(1 )(1 )andEe=e()Pr()+e()Pr(),weobtain
dEe d <0; d 2 Ee d 2 <0:
-FIGURE 1 0.5 e 1 e () 1 1+ 1 Ee e () 1 -FIGURE 2 6 0.5 e 1 1 1+ 1 1 2 Ee * 4 3 1 1 2 3
We nowconsider anotherexample wherethe regulatordiscovers thetrue type of therm with
probability and nothing with probability 1 . 10
This case refers to the possibility that a
benchmarking studythe regulatorcan conduct willreveal either thetrue eÆciency of the rm
(with probability)ornothinguseful(with probability1 ). The uninformative signalislike
a signal 1 such that Pr( 1 j ) = Pr( 1 j ) = 1 . Observing 1 gives no informationto
the regulator and the power of incentives is unchanged. The favorable signal 2 is such that Pr( 2 j)= and Pr( 2 j)=0. Observing 2
,we knowforsure that therm isof thegood
type: eort is eÆcient and no rent is captured bythe rm. The unfavorablesignal 3 is such that Pr( 3 j) =0 and Pr( 3 j) =. Observing 3
, theregulator knows forsure thathe is
facing abad type rm: eort is eÆcient and no rent is leftto therm. Globally,theexpected
eort of thebad type(conditionally onbeing used)increases with: eitherthere is no change,
if 1
is observed, ore=1 if 3
isobserved. We are indeedin thespecialcase ofProposition2.
Case 3: Onlygoodtype rmsarediscovered.
If onlygood rmscan be identiedfollowingthebenchmarkingstudy,thenPr( 1 j)=1 , Pr( 1 j) =1, Pr( 2 j) =, Pr( 2 j) =0, and Pr( 3 j) = Pr( 3 j) =0. If the study
reveals no new information,thenitis an unfavorablesignalsince
^ =
(1 )
1 < ;
and incentivesincrease.
Case 4: Onlybad type rmsarediscovered.
If onlybad type rms can bediscovered, thenPr( 1 j)=1,Pr( 1 j) =1 . Pr( 2 j)= Pr( 2 j) = 0, Pr( 3 j) =0, Pr( 3
j ) = , The power of incentivesincreases when a bad
type rm is observed. When the type of the rm is not observed, it is a favorable signal. As
longas^=
1 (1 )
<
,bothtypesofrmarekeptand weareinthecaseofProposition
1 andincentivesdecrease. If^>
,weareinthecase ofProposition2 andincentivesincrease.
10
To thispoint, we consideredthe valueof as given. But thisvalue is theshadow priceof the
government budgetconstraintand therefore willingeneralvary withthelevel ofexpenditures.
We considerinthis section thegeneral equilibriumeect of an improvement of informationon
thepowerofincentivestakingintoaccounttheinducedchangein. Asincreases,theregulator
has access to a better informationstructureand therefore can leave a smaller rent to therm,
thus reducing thepressures on the budget constraint and decreasing its shadow value. 11
This
enables him to provide more incentives. The interesting questionis then whether this positive
eect on incentivesdominatesthenegative eect identiedinProposition1. Thefollowingcase
showsthatindeedthegeneralequilibriumeect,throughthechange intheshadowpriceofthe
budget constraint may be strongenoughto changethesign of thepartialequilibriumeect.
Let us considercase 4 above where Pr( = j) = and =; otherwise; we know that
in thiscaseincentives decrease withan increasein when thetype of rmis notdiscovered, a
favorablesignal leadingto ^==[1 (1 )]>. Thegeneral welfareproblemis
max E[ (S t)+( t ( e) (e))]=S e+ ( e) (1 ) e+ (e ) subject to Et= e+ (e) +( 1 ) e+ ( e) +(e )K : We obtain 0 (e)=1 1+ 1 1 1 0 (e ) (10)
that is,inthequadraticcase where (e )= 1 2 e 2 and (e )=(e 1 2 ) e= 1 1+ 1 1 1 (11)
where isnowthe (endogenous)multiplierofthe budgetconstraint. The budget constraintis
[ e + (e )]+(e ) +(1 )[ e + (e )]+(1 )(1 )[ e+ (e)]=K (12) 11
The general equilibrium eectis related to the government global nancialneeds and becomes signicant
when someformofbenchmarkingorCCTisavailableandimplementedfor mostifnotallgovernmentactivities orexpenditures.
E+[(+(1 ))( 1 2 e 2 e )]+(e 1 2 )+(1 )(1 )( 1 2 e 2 e )=K : (13) We obtain de d = (1 )( 1 2 e 2 e ) (1 )( 1 2 e 2 e ) +(1 )(1 )(e 1) >0 (14) d d <0 (15)
Thepartialanalysisofcase4above(consideringasxed)ledto de d
<0;wethereforeconclude
that the general equilibrium eect of d (through d d
< 0) is, in this case, of a sign dierent
from that of the partialequilibrium eect of d. The power of incentives increases because a
larger implieson average a less stringent budget constraint. However, it is not always true
that the general equilibrium eect will overcome the partial equilibrium eect. We illustrate
thisin Appendix1.
Abetterinformationstructureaectstheregulator'swelfaremaximizationproblemandthe
levelofincentivesinthreeways. Itaectstheexpectedcostoftheproject, 12
theleveloftherent
(e ) throughits eect on the eortlevele, and theprobabilitythat thisrent willbe captured,
that is, the posterior probability that the rm is a good type rm. Those eects modify the
budget constraint and thereforeits shadowvalue.
5 THREAT OF LIQUIDATION AND INCENTIVES
Adierentchannelthroughwhichmoreintensecompetitionisthoughttoaectincentivesisthe
threat of bankruptcy or liquidation. For private non regulated rms, liquidationwould follow
fromthedecisionofthenancierstorefusetorenancearmwithliquidityconstraintsbecause
of its low expected payo. For regulated or publicly owned rms or projects, \liquidation"
would follow from a political decision to stop subsidizing therm or project orto change the
management team. Forinstance,it isreasonableto assumethat politicianscannotcommitnot
to useamuchbetteralternativetechnologyifandwhenitbecomesavailable. Henceatthetime
12
Inour context,thereis noquantityeectsince weare consideringthe realizationofagivenproject. More
of the contract, the current rm faces a probabilityof being dropped or liquidated. We are
discussing a kind of auctionwhich reduces the rent of therm and induces higher eortas in
Laont and Tirole (1987). The lack of correlation betweenthe rm's type and thealternative
technology explains why the full extraction of the surplus, as in Cremer and McLean (1985),
does not occur. We can suppose for instance that the alternative technology would allow the
realization of the public project at some cost C, a random variable with prior distribution
function G(C) and density g(C). If C < (e ) + e , then the switch to the alternative
technologywouldtakeplace,thatis,anewrmormanagement teamwouldbechosentorealize
theprojectand thecurrentone wouldbe\liquidated". Hence,theprobabilityofno liquidation
is given by Pr h C > (e )+ e i = 1 G (e )+ e an increasing function of e. 14 We
wantto characterize theimpact ofsucha liquidationruleon theintensityofincentives.
For reasons of simplicity and specicity, we willassume that the liquidation rule takes the
followingreduced form: when the rm is of type , there is a probabilityof liquidationwhich
increases withthecostof thermand therefore decreaseswiththeeortlevele. Let1 kÆ(e)
denote thisprobabilityof liquidation whereÆ 0
(e ) >0 and k is a positive parameter. Incentive
compatibilityimpliesthattherentoftheeÆcientrmnowtakestheformU =kÆ(e)(e ). 15
Ifwe
denote byS thesocialwelfarewhen liquidationoccurs,that iswhen thealternative technology
is usedto pursuetheproject, expected socialwelfare is
h S (1+) ( e)+ e kÆ(e )(e) i +(1 )kÆ(e) h S (1+) ( e)+ e i +(1 )(1 kÆ(e))S: 13
Strictosensu,onedoesnotexpectthatapublicquasi-monopolyrmsuchasastate-ownedpowergenerating rm,telecomsorrailormarinetransportationrmswouldsimplybeshutdown. Butthemanagementteamcould
bereplaced ifit wereperceivedas ineÆcientand the rmcould facesignicant restructuring, withdownsizing andoutsourcing,ifitsoperatingcostsweredeemedtoohigh.
14
Theexperienceof theCompulsoryCompetitiveTendering (CCT)process inthe U.K.isaninteresting real example. Beforeoeringapublicservicedirectlytothecitizens,throughapubliclyownedDirect Service Orga-nization(DSO),amunicipalgovernmentmustinviteprivatermstobidfortheprojectorservicetogetherwith
the DSOwhomustact thenasaseparateentity. If aprivatermoutbidsthemunicipalDSO,thenitwinsthe contractandtheDSOis\liquidated". Ifnot,theDSOisawardedanincentivecontractfortheserviceorproject.
The U.K.experience shows that DSOswonabout70% ofthecontractsandweretherefore\liquidated" witha probabilityof0.3.
15
Choosingthecontract(t;C),theeÆcientrmfacestheriskoflosingthecontracttothealternativetechnology;
0 (e ) = 1 0 (e ) = 1 1+ 1 0 (e) 1+ Æ 0 (e )(e ) Æ(e) + Æ 0 (e ) (1+)Æ(e) S (1+)( (e)+ e) S : (16)
A marginalincreaseinthethreat of liquidation(increaseof k) hasno eect, because ithas
no eect ontherent extraction-eÆciencytrade-o (sincetheinformationrentdecreasesexactly
asthe gainfromhavingtheineÆcientrmactive decreases), andit hasnoeect on therate of
increaseof theprobabilityof liquidationÆ 0
(e )=Æ(e).
Proposition3: The power of incentives is insensitive to a marginal increaseof the
proba-bility of liquidation, as long as the regulator's optimization program remains concave.
Note howeverthat with amore general formulationÆ(k;e), we would getresults depending
on thecross derivative @ 2 Æ(k;e) @e@k . 6 COMPETITION IN TALENT
Analternativechannelthroughwhichcompetitionoperatesisthatitincreasesthe`market'value
ofgoodagentsintheprivatesector. Itwillthenbenecessarytooerhigherpayostogoodtype
agents in thepublicsector inorder to meet theirparticipation constraint. Thischannelbrings
into focustherelationshipbetweenthepublicsector wage structureand theprivatesectorone.
There aregoodreasons whyone may expectthat thewage structureis more egalitarianinthe
publicsector. Onesuchreasonistheredistributionobjectiveofthegovernment. Inthiscontext,
public authoritiesact asif they were usinga modied utilitariansocial welfarefunction where
a weight
<1 isassociatedwiththe protorinformational rent of thegoodagent. Thiswill
lead tolargeroveralleÆciencydistortionsinorder tolowerthelessvaluableinformationalrent.
Using thesocialwelfarefunction
S (1+)(t+ e)+
S (1+) e+ (e ) (1+ )(e ) +(1 ) S (1+) e+ (e)
we obtaintherst-ordercondition
(1+ ) 0 (e ) (1 )(1+) 0 (e ) 1 =0
that is,for=1 , 0 (e )=1 1 + 1+ 00 (e)
implying, since>0,areductionine and(e), thatis,lowerincentivesand a smallerrent for
goodtypeagents.
The trade-o between eÆciencyand rent minimizationis in a sense solved infavor of rent
minimization: eÆciency is lower, that is, incentives are weaker but the good types capture
a smaller informational rent. In the private sector, the absence of a redistribution objective
together withcompetitivepressuresmake thermsbehave asif, inorderto avoidbeing
cream-skimmed on the labor market, a utilitarian objective function with
>
were used and
therefore bringouta solutionmore inlinewith eÆciency: stronger incentivesareimplemented
and larger rentsare capturedby thegood type. Ascompetition intheprivatesectorincreases,
increases, generatingmore intense competition for thepublic sectorwhich isimplicitly led
to increase the value of
in order to avoid the loss of the more capable, good type agents.
Hence, although there may not be a perfect alignment of incentives between the public and
privatesectors,theemergence of a more competitiveenvironmentwouldaect both.
In our model thiscompetition intalent phenomenon can be modeled as an increasein the
reservationutilitylevelofagoodtypemanagerwhich wassofarnormalizedat zero. LetU 0
be
thisnew reservation utilitylevel. The maximizationprogram of theregulator(MP) isas given
above exceptfortheparticipation constraint ofthe goodtypewhichbecomes
U U 0
:
-U 0 0 (e) (e ) (e +) AslongasU 0
( e) ,withedenedby(6)andobtainedforU 0
=0,theoptimalregulationis
unchangedsincetheinformationrentislargeenoughtosatisfythenew participationconstraint
of thegood typeagent. However, when U 0
becomes largerthanthisinformationrent(e )but
is stillless than (e
), therelevant bindingconstraints become thetwo IR constraints and the
incentive constraint of thebad type. Eortlevel eis stilldened by 0
(e)=1 and eis simply
dened by ( e) = U 0
: So the eort level of the bad type is increased (incentives increase) in
order tocreate asuÆcientlylargerent(butnotlargerthannecessary)forthegoodtypeto stay
in thepublicorregulated rm.
When U 0
becomes larger than ( e
) but still less than ( e
+), there is no point in
increasingfurtherewhichhasreachedtheeÆcientlevele
. OnlytheIRconstraintsarebinding
and botheortlevels areeÆcient.
Finally, if U 0
becomes larger than ( e
+), the incentive constraint of the bad type
agent becomes binding. A rent must be given up to the bad type. To mitigate this rent, the
eort leveleis increasedbeyondthe eÆcient level. Itis denedby
0 ( e)=1+ 1 1+ 0 ( e+)
Competition from the unregulated sector obliges the public sector to select incentive schemes
which are even more powerfulthan the eÆcient ones to keep the good type within the public
sector.
Proposition4: Competition in talent always favorsstronger incentives.
7 A MORE COMPETITIVE ENVIRONMENT
Another meaning of increased competition is that theunregulated (private) sector which
goods which have become better substitutes. We consider the eect of those changes on the
powerof incentivesintheregulated sector.
Letusassume thatthesocialvalue ofproductionfrombothsectors isnowgiven by
S(q 1 +q 2 )+q 1 q 2 where q 1
isthe variable output of the regulated sector and q 2
is the output of the competitive
sector and where 0 is a measure of how complementary the products of the two sectors
are. As increases [decreases], the goods become more complementary [substitute] 16
and the
demandof each good increases[decreases]asusual.
Assuming thatC(q 2
) =cq 2
, that theoutput of theregulated rm is \sold"at its marginal
value level, that theproceeds go into thepublic budget and are therefore generating an extra
value ofp 1 (q 1 ;q 2 )q 1
,andthatthebenevolent regulatorcares abouttotal welfare, we canwrite
the regulator'sobjective functionas
S(q 1 +q 2 )+q 1 q 2 +U ( 1+)(t+( e)q 1 ) cq 2 +p 1 (q 1 ;q 2 )q 1 ;
where U =t ( e) is the regulated rm'sutility. Underfullinformation, theregulator wants
thermto exerttheoptimallevelofeorte characterized by 0 (e )=q 1
forbothtypesofthe
regulated rm whileoperating a transfert leavingno rent to theregulated rm,U =U =0.
We assume that 2 n
; o
is private information of the rm while the private sector
technology, that is c, is common knowledge. This simplifying assumption can be justied by
the yardstick competition in the private sector which generates a perfect informative signal
makingcknowntotheregulator. Theregulatormaximizestheexpectedsocialwelfareunderthe
usual individualrationalityand incentive compatibilityconstraints of bothtypes and balances
eÆciency and rents by determining the output and eort levels of the regulated sector as a
functionof. We obtain(seeAppendix2): 0 ( e)=q 1 and 0 (e)=q 1 1+ 1 0 (e )<q 1 :
We can modelan increaseincompetitivepressures bymaking theprivatesectormore eÆcient,
that isreducing c,orbymakingthe twogoodsbetter substitutes,that isreducing.
16
The cross derivative ofsocialwelfare withrespect toq 1 and q 2 is S 00 +;if S 00
+ >[<]0, thegoods are
Let us rst consider a reduction in c. We can show (see Appendix 2) that, whatever the
eÆciency of the regulated sector, a reduction in production cost c increases the output of the
privatesectorbutmayincreaseordecreasetheproductionoftheregulatedsector. Letusassume
that the products are demand substitutes, that is S 00
+ < 0. 17
If the products are demand
substitutes and strategic substitutes, 18
then the production of the regulated sector decreases
when c decreases, implyinga reductionin incentives (no change in eort e buta reduction in
eort e) and in therent of the eÆcient type rm. If the products are demandsubstitutesand
strategiccomplementswithlargeenough,thentheproductionoftheregulatedsectorincreases
when c decreases, implyingan increase in incentives (no change in eort e but an increase in
eort e )and intherentof theeÆcient type rm.
The intensityof incentives, measuredby 19 k = 0 ( e) C e = 0 (e) q 1 ;
remainsconstant(atk =1) fortheeÆcient typebutmayincreaseordecreasefortheineÆcient
type,dependingon thevalue of. Forsmall,theeect ofa reductionincon theintensityof
incentives is of thesame sign as theeect on the productionlevel of theregulated sector. But
for suÆcientlylarge, we can have an oppositesign: 20
ifthe products aredemandsubstitutes
but strategiccomplementsand the value of is largeenough, a reductionin themarginal cost
c increases both production and eort in the public sector, but nevertheless the intensity of
incentives decreases,asshown inAppendix2.
Proposition5: Iftheregulated productq 1
andtheprivatesectorunregulatedproductq 2
are
strategicsubstitutes,thenareductioninmarginal costcreducesincentivesfortheregulated
17
Thereadercanadapttheanalysistothecaseofdemandcomplements. 18
The products are strategic substitutes[complements]if themarginal social revenuefrom good idecreases
[increases] withtheproductionofgoodj,thatis,ifS 000
qi+S 00
+< [>]0. Ifthegoodsaredemandsubstitutes,
wehaveS 00
+<0;hencewhetherthegoodsarestrategic substitutesorstrategiccomplementsdependsonthe
signandmagnitudeofS 000
qirelativetoS 00
+. SeeAppendix2fordetails. 19
Theintensityofincentivescorrespondstothat partk oftherealizedcostwhichisborneby thermat the
margin: ifU =t kC(;e;q) (e)=t k( e)q (e),thenU e =kq 0 (e)=0impliesk= 0 (e) q 2(0;1]: 20 Unless 00
=0,asforthequadraticcase (e)= 1 2
e 2
cincreases incentives for the regulated rm.
So if greater competition means lower cost competitors, strategic complementarity and a high
socialcostof publicfundsare thekeys to greater productionof thepublicsector and therefore
to greater eortandlarger rents. Whenislargebutnottoolarge, greatereortis equivalent
to higherpowerof incentives. 21
7.2 Improved substitutes for the regulated rm's product
Let usnow consideran increasein substitutability,that is, a reduction in . We can show
(see Appendix2) forboth and thatthe eect of a change in onthe productionleveland
eort levelin thepublicsectorare of thesame sign. Iftheproductsare strategic complements
withlargeenough, thentheproductlevelsq 1
andq 2
aswellastheeortleveleareallreduced
as decreases. Indeed, a reduction of increases substitution but also reduces the value of
bothcommoditiesto consumers. Ifon theotherhandtheproductsarestrong enoughstrategic
substitutes, thesubstitutioneect maydominate and q 1
mayincreaseaswellase.
Proposition6: If the regulated product q 1
and the private sector unregulated product q 2
are strategic complements with large enough, then the incentives of the regulated rm
decrease with an increase in the degree of substitution. If the products are strong enough
strategic substitutes, then the incentives of the regulated rm increase with the degree of
substitution.
When more competition means better substitutability between the public and private sectors
products (andan increaseintheir globaldemand), then strategicsubstitutabilityis the keyto
higher production, eort and rent in the public sector (unlessthe social cost of publicfund is
low). 21 Weneed >maxf (S 00 +) S 00 ++S 000 q 1 and 00 00 + 1+ 1 00 > 0 q 1 ;
cost functionC =( e)q which yieldstherst orderconditionof thebad type 0 (e)=q 1 1+ 1 0 (e )
from which the eects of competition can be tracked. Clearly these eects work through q 1
by changingthe social protability of eort. Witha more general cost function satisfyingthe
incentivedichotomypropertyC=C((;e);q 1
)thechannelwouldremainsimilar, @C
@e
replacing
q 1
. If we drop the incentive dichotomy assumption, competition can aect more directly the
incentive constraint and the rent. The marginal rent is now given by 0
(e)E
(;e;q) where
E(;e;q) is the solutionin e of
C =C(;e;q), thatis, the eortlevel which enablesa rm of
type to produceq witha cost of
C. Furthermore, thepricingequation requiresan incentive
correction and the eect of a change in theproductionlevel, dueto increased competition, on
the eortlevelis obtainedbydierentiating thewhole system
p 1 C q1 p 1 = 1+ 1 ^ 1 +I 0 (e )= C e (;e;q) 1+ 1 0 E (;e;q): 8 CONCLUSION
We characterized theeects ofstronger competitivepressures on publicsector incentives,more
preciselyon thelevelofincentivesinan optimallyregulatedsector. Maybe contrary to
conven-tionalwisdomorarst intuition,greatercompetition doesnotalwayscallforhigherincentives.
We identied dierent channels through which competition aects the powerof incentives.
First, we looked at the information channel, namely the idea that more competition provides
theprincipal,theregulator,withinformationabouttheagent, theregulated sector. We showed
in propositions1 and 2 thatthe eect of better informationon thepower ofincentivesmaybe
positive or negative. If the quality of the competition signal is suÆciently high to induce the
regulatortodropthepublicprojectfollowingafavorablesignal,thepowerofincentivesincreases;
is quadratic and when the social cost of public funds is small. Moreover, we characterized
the general equilibrium eect of a better information structure, considering as endogenous,
and we showed that its sign may be dierent from that of the partial equilibrium eect. The
mainreasonfortheambiguousresultsisthatgreaterinformationprovidedbyamorecompetitive
environmentmayalterthedesirabletrade-obetweeneÆciencyandrentextractiontowardsless
rent extraction on average. However, when the competition signal is better, that is, becomes
veryinformative,theexpectedresult obtains.
Second, we looked at the threat of liquidation eect and we showed that the power of
incentives isinsensitive to localincreases of thisthreat. Third,we analyzedthe competition in
talent eect and weshowed thatthiseect increases also thepowerofincentives.
Fourth,weconsideredamodelofamorecompetitiveenvironmentandweshowed(i)thatthe
eectofacostreductioninthecompetitiveunregulatedsector,producingasubstituteproductto
thatoftheregulatedrm,onthepowerofincentivesisnegative[positive]whentheproductsare
strategicsubstitutes[strategiccomplementsandislarge],and(ii)thattheeectofanincrease
insubstitutabilitybetweentheproductsispositive[negative]ifthedegreeofdemandsubstitution
isstrongenough[iftheproductsarestrategiccomplementsandislarge]. The mainreasonfor
theambiguousresultisthatthedesirablechangeinthepowerofincentivesisrelatedtothelevel
ofactivityoftheregulatedrmandtotheinteractionbetweeneortandproductionlevelinthe
cost function. Dependingon thenatureand strengthofthesubstitutabilityorcomplementarity
betweentheregulatedproductandthegoodsproducedbytheunregulatedcompetitors,greater
competitive pressures from the latter may call for a shrinkage or an expansion of the public
sector, and that determinesto alargeextent the desirablechange inthepowerofincentives.
A next step in the analysis would be to recognize that, for political economyreasons,
reg-ulation is notoptimized and study to which extent the increasein competition disciplinesthe
regulators. 22
22
Thiswouldrequireanexplicitpoliticaleconomymodelas inBoyerandLaont(1999)oranexplicit
A Non-Reversing General Equilibrium Eect
Weconsiderthegeneralequilibriumeect forthefollowinginformationstructureof case1. We
have e()= 1 1+ 1 1 (17) e()= 1 1+ 1 1 ; (18)
hence e()e (),and thebudget constraint
[ e + 1 2 e 2 +(e () 1 2 )]+(1 )(1 )[ e ()+ 1 2 (e ()) 2 ] +(1 )[ e + 1 2 e 2 +(e () 1 2 )]+(1 )[ e ()+ 1 2 (e ()) 2 ] = K (19)
Totallydierentiatingthe systemof equations(17), (18) and (19)with respect to e(), e(),
and , we obtain de () de() d d 1 0 B 1 C 1 0 1 B 2 = C 2 A 1 A 2 0 where =[(e () 1 2 ) (e () 1 2 )]+(1 )[( 1 2 (e ()) 2 e()) ( 1 2 (e()) 2 e())]<0,
with e () ande() given by(17)and (18), and
B 1 = 1 (1+) 2 1 1 >0; B 2 = 1 (1+) 2 1 1 >0 C 1 = 1+ 1 1 (1 ) 2 <0; C 2 = 1+ 1 1 2 >0 A 1 = 1 1+ <0; A 2 = (1 ) 1 1+ <0
with the determinant = A 2 B 2 A 1 B 1 >0. Observingthat A 1 <A 2 <0, jC 1 j>jC 2 jand B 1 >B 2 >0,weobtain d d / + C 2 A 2 C 1 A 1 < 0
d / B 1 A 2 C 1 B 2 +A 2 B 1 C 2 < 0 de() d / B 2 +A 2 C 1 B 2 A 1 B 1 C 2 > 0: Thesignof de () d
maybepositiveornegative. Itdependsontwoeects: thepartialequilibrium
eect (xed)and thegeneralequilibriumeect through d d <0,namely de () d / @e () @ d=0 + B 1 B 2 A 2 C 2
where therst termisnegative [equal to C 1
<0] andthesecond termis of thesignof
+A 2 C 2 = (1+) 2 2 1 1 2 : Indeed, for = 2 9 ; = 1 3 ; = 3 4 , we have de () d = 1:31 if = 2 3 , and de() d = +0:49 if = 3 4 . Appendix 2
Increases in Competitive Pressures from the Private Sector
We wish to maximizesocialwelfare
W = h S q 1 +q 2 +q 1 q 2 (1+) ( e)+ e q 1 cq 2 +p 1 (q 1 ;q 2 )q 1 U i +(1 ) h S(q 1 +q 2 )+q 1 q 2 ( 1+) (e )+ e q 1 cq 2 +p 1 (q 1 ;q 2 )q 1 U i subjectto IR: U 0 IR: U 0 IC: t ( e)t ( e)+( e) IC: t ( e)t ( e ) with ( e)= (e) (e )
dq 1 dc = 1 D 00 S 00 ++ h @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 i ! dq 2 dc = 1 D " 00 S 00 + h @ 2 p 1 @q 2 1 +2 @p 1 @q 1 i ! (1+) # <0 de dc = 1 D S 00 ++ @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 ! dq 1 dc = 1 D " 00 1+ 1+ 00 S 00 ++ h @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 i # dq 2 dc = 1 D " 00 1+ 1+ 00 S 00 + h @ 2 p 1 @q 2 1 +2 @p 1 @q 1 i (1+) # <0 de dc = 1 D S 00 ++ h @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 i ! leading to dq 2 dc <0 sign dq 1 dc = sign de dc = sign " S 00 ++ " @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 # !# = sign S 00 ++ @MR(q 1 ) @q 2 = sign S 00 ++ S 000 q 1 +S 00 + = sign (1+) S 00 + S 000 q 1
As fortheintensityof incentivesk,we have
dk dc = 1 q 1 00 de dc + 0 1 q 2 1 ! dq 1 dc : Since de dc 00 = dq 1 dc and de dc h 00 + 1 1+ 00 i = dq 1 dc ,we obtain dk dc = 1 q 1 " 1 0 (e ) q 1 # dq 1 dc =0 dk dc = 1 q 1 2 6 6 4 00 (e ) 00 (e )+ 1 1+ 00 (e) 0 (e) q 1 3 7 7 5 dq 1 dc :
00 6=0, sign dk dc ! = sign dq 1 dc :
Regarding theeect of a change in,we nd:
dq 1 d = 00 D " q 1 S 00 ++ h @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 i +q 2 S 00 + h @ 2 p 1 @q 2 2 q 1 i # dq 2 d = 1 D " 00 S 00 + h @ 2 p 1 @q 2 1 +2 @p 1 @q 1 i q 1 00 S 00 ++ h @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 i q 2 +( 1+)q 1 # = 1 D " q 1 h S 00 + h @ 2 p 1 @q 2 1 +2 @p 1 @q 1 i ( 00 ) (1+) i +q 2 S 00 ++ h @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 i ( 00 ) # de d = 1 D " q 1 S 00 ++ h @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 i +q 2 S 00 + h @ 2 p 1 @q 2 2 q 1 i # dq 1 d = 00 + 1+ 1+ 00 D " q 1 S 00 ++ h @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 i +q 2 S 00 + h @ 2 p 1 @q 2 2 q 1 i # dq 2 d = 1 D " q 1 h S 00 + h @ 2 p 1 @q 2 1 +2 @p 1 @q 1 i 00 1+ 1+ 00 (1+) i +q 2 S 00 ++ h @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 i 00 1+ 1+ 00 # de d = 1 D " q 1 S 00 ++ h @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 i +q 2 S 00 + h @ 2 p 1 @q 2 2 q 1 i # where S 00 + h @ 2 p 1 @q 2 2 q 1 i
<0 bythe secondorder conditions,leadingto:
sign dq 1 d =sign de d : Therefore, { ifS 00 ++ h @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 i >0,then dq 2 d >0; dq 1 d >0; de d >0 { otherwise, if jS 00 ++ h @ 2 p 1 @q 1 @q 2 q 1 + @p 1 @q 2 i
j is large enough relative to j S 00 + h @ 2 p 1 @q 2 2 q 1 i j ,
the substitutioneect dominatesand
dq 2 d <0; dq 1 d <0; de d <0
have dk d = 1 q 1 00 de d + 0 1 q 2 1 ! dq 1 d : Since de d 00 = dq 1 d and de d h 00 + 1 1+ 00 i = dq 1 d ,we obtain dk d = 1 q 1 " 1 0 (e ) q 1 # dq 1 d =0 dk d = 1 q 1 2 6 6 4 00 (e ) 00 (e )+ 1 1+ 00 (e) 0 (e) q 1 3 7 7 5 dq 1 d :
The intensity of incentivesfor thegood type remainsconstant but, for suÆcientlylargeand
00 6=0, sign dk d ! = sign dq 1 d :
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