Getting to Know your Agent: Interim Information in Long Term Contractual Relationships

Transcription

1 Getting to Know your Agent: Interim Information in Long Term Contractua Reationsips Roand Strausz Free University of Berin November 7, 2001 Abstract In a finitey repeated principa agent reationsip wit adverse seection I study (exogenous) interim information tat is reveaed during a ong term reationsip Interim information mitigates adverse seection Verifiabiity, measured by te cost of signa manipuation, and te signa s informativeness determine te use and effectiveness of interim information: Less precise and more manipuabe signas are used in a forward ooking way excusivey More precise and ess manipuabe signas are aso used in a backward ooking way and extract a information rents Higy precise signas wit a ig degree of verifiabiity yied te first best Moreover, verifiabiity and informativeness are substitutes Keywords: verifiabiity, monitoring, repeated adverse seection; JEL Cassification No: D82 Free University Berin, Dept of Economics, Botzmannstr 20, D Berin (Germany); Emai addresses: strausz@zedatfu-berinde Quatscgruppe 2 I tank Hemut Bester, Anette Boom and participants of te

2 2 1 Introduction It is widey recognized tat contracting reationsips are typicay formed under asymmetric information A bank wi question te creditwortiness of a potentia customer An empoyer wi be concerned wit te productivity of is young empoyee And a marrying coupe wi worry about te suitabiity of is or er respective partner Yet, muc of tis asymmetric information disappears over time Due to repeated interaction contracting partners get to know eac oter better and te interim information tat is reveaed during repeated interactions diutes te asymmetric information tat exists ex ante Tis paper investigates ow ong term contracts use interim information optimay Intuitivey, tere exist two different ways to empoy te information First, contracting parties may use te information in a foreward ooking way, improving te coordination of future actions and decisions But taking a more strategic perspective te partners may aso want to try to use te information to infuence decisions tat are taken before any interim information is received Tat is, agents may want to empoy te information in a backward ooking way Ceary te effectiveness of interim information depends argey on its informativeness Yet, aso te verifiabiity of te information pays an important roe Wit verifiabe information te contracting parties can condition teir contracts directy on te interim information Someting wic cannot be done wen te information is nonverifiabe Indeed, in rea ife muc of te acquired knowedge due to repeated interactions tends to be subjective and simpy refects a persona opinion about te counter party suc as its trustwortiness or ikeabiity In tis case, contracts can condition on te information ony indirecty Tis indicates tat te verifiabiity of te interim information pays a crucia roe in determining its effectiveness Hence, apart from te question ow peope use interim information, tis paper aso focuses on te roe of its verifiabiity More specificay, tis paper anayzes a repeated principa agent setting in wic te agent possesses private information ex ante Te principa and agent interact during two periods in between wic te principa receives some information about te agent Te anaysis confirms tat te verifiabiity of te interim information is crucia Wenever interim information is verifiabe, interim information resoves te repeated adverse seection probem competey Tis ods for any degree of informativeness Hence, under verifiabiity te principa uses interim information in bot a forward and backward ooking way, adjusting first and second period aocations to increase efficiency as compared to a situation in wic no interim information is avaiabe On te oter and, if interim information is not verifiabe, it cannot resove te probem In fact, if te informativeness of te information is ow, te principa uses interim information ony in a forward ooking way and does not adjust te inefficient aocation in te first period Ony if te informativeness is above a certain

3 3 tresod, does te existence of interim information affect first period aocations Moreover, nonverifiabe interim information resoves te probem of asymmetric information ony if it is perfect To investigate te roe of verifiabiity I introduce te possibiity of information manipuation and argue tat information manipuation offers a natura way to arrive at a continuous concept of verifiabiity It enabes a parametrization of verifiabiity and eps to reconcie te dicotomous caracter of te verifiabiity versus te non-verifiabiity mode More specificay, it sows ow te use of interim information differs according to te verifiabiity of te information First, a igy verifiabe signa enabes te principa to acieve te first best Second, wenever te verifiabiity of interim information is ow, te principa uses it to reduce te informationa rent of te agent In tis case, te interim information does not affect te first period aocations and te principa uses te information ony in a forward ooking way Tird, for intermediate ranges of verifiabiity te principa extracts a informationa rents and uses te interim information to reduce te distortion on te aocation of bot first and second period aocations Te reative sizes of te tree regions depend on te precision of te interim signa: For more precise signas te range of parameters signifying a ig verifiabiity is arger Wit a perfect signa, any degree of verifiabiity enabes te impementation of te first best Reated to te current paper is Cooper and Hayes (1987) wo anayze a repeated insurance mode wit adverse seection in wic accidents provide interim information about te agent s true type However, te autors do not address te issue of verifiabiity Teir setup aso does not offer a straigtforward way to anayze te roe of te informativeness of interim information, because te informativeness is directy inked to te agent s type Yet, te paper offers anoter exampe of te use of interim information in ong term reationsips and notes its common use in insurance contracts Finay, it is important to note tat te existing iterature on contracting as studied a different form of interim information Baron and Besanko (1984) anayze a muti period adverse seection setting in wic interim information reveation may occur due to te agent s beavior Specificay, if te principa offers a reveation mecanism in te first period, te agent s message may revea is type and embody interim information Te important difference to our framework of interim information is tat in Baron and Besanko (1984) te reveation of interim information ies under te fu contro of te agent and is endogenous Baron and Besanko sow tat endogenous interim information does not benefit te principa 1 Indeed, wenever te principa as fu commitment, er optima poicy is to commit not to use interim information Hence, te current paper differs from te ratceting iterature 1 In fact, tis is a direct resut of te cassica reveation principe Any information tat is reveaed by te agent s beavior coud aso be reveaed in te first period

4 4 as for exampe Laffont and Tiroe (1988, 1990) in wic, due to te principa s imited commitment, te interim information actuay urts te principa and repeated interaction soud be avoided (eg Ickes and Samueson 1987) Contrasting te current paper to tis iterature reveas tat tere exists a crucia difference between endogenous and exogenous information in a repeated adverse seection mode 2 Te Mode Consider a principa empoying an agent wo is privatey informed about is margina cost Wit probabiity α te agent s constant margina cost is θ, wit probabiity 1 α is constant margina cost is θ, were θ > θ Te agent s action a resuts into a verifiabe output v(a) Te principa empoys te agent for two periods In between te two periods te principa receives an exogenous signa s {, } about te agent s margina cost Te signa is correct wit probabiity p > 1/2 Te common discount rate between te two periods is δ As owner of te firm te principa receives te agent s output and compensates te agent by paying a wage w Hence, if te principa pays wages (w 1, w 2 ) over te two periods and te agent cooses actions (a 1, a 2 ) te principa s and agent s payoffs are V (w 1, a 1, w 2, a 2 ) = v(a 1 ) w 1 + δ(v(a 2 ) w 2 ), U i (w 1, a 1, w 2, a 2 ) = w 1 θ i a 1 + δ(w 2 θ i a 2 ), respectivey, were i {, } We assume tat v is increasing and concave, ie, v > 0 and v < 0, and, for tecnica reasons, tat v < 0 Since output is verifiabe and invertibe, an enforceabe contract γ specifies an action a, and a transfer, te wage w, for eac period, ie γ = (γ 1, γ 2 ) = (w 1, a 1, w 2, a 2 ) As is standard, te contract may depend on a message m of te agent about is type If te signa is verifiabe, te contract conditions, in addition, directy on te signa s In tis case, te contract as te form γ ms = (γ 1 (m), γ 2 (m, s)) Wen s is nonverifiabe, te contract can depend ony indirecty on te signa, in te sense tat te principa may report it by sending some message r Tat is, a genera contract as te form γ mr = (γ 1 (m), γ 2 (m, r)) 3 Two Bencmarks In tis section I anayze two bencmarks to wic I wi ater reate te existence of interim information First consider a setting wit fu information in wic te agent s type is observabe In tis case te principa can prescribe eac type of agent to work efficienty

5 5 and appropriate te entire surpus Efficient effort eves, a fb i, are defined by 2 v (a fb i ) = θ i, wit i {, } Hence, wit fu information te optima contract γ is a first best contract tat impements in eac period te respective first best action eves a fb i at first best costs θ i a fb i + δθ i a fb i Note tat te prescribed actions eves are timeinvariant and tat te optima transfers are partiay undetermined, because, due to te common discount factor δ, te principa and agent can freey aocate transfers over te periods witout affecting utiities To circumvent tis indeterminacy I assume, witout oss of generaity, tat a transfers in te first period are zero Tat is, te first best contract specifies a first period transfer w 1i = 0 and a second period transfer w 2i = θ i a fb i + δθ i a fb i Now suppose te agent s type is private information and te signa s is not avaiabe, ten, as sown by Baron and Besanko (1984), te optima ong term contract is time-invariant 3 It is a twice repeated version of te optima contract in te static, singe period mode Tat is, a standard principa agent mode wit adverse seection in wic te principa faces te famiiar trade off between efficiency and rent appropriation By te cassica reveation principe te optima contract is a direct mecanism and te soution to te foowing maximization probem: 4 P0: max V = α(v(a ) w ) + (1 α)(v(a ) w ) st w θ a 0 (1) w θ a 0 (2) w θ a w θ a (3) w θ a w θ a, (4) were (1) and (2) are te participation constraints and (3) and (4) are te incentive constraints tat ensure trutfu reveation Let V sb represent te soution to P0 By standard arguments ony te incentive constraint of te efficient type θ and te participation constraint of te inefficient type θ are binding Te soution to tis probem is a second best contract tat impements te second best action eves (a sb, a sb ) As is standard, te efficient type receives a strict positive information rent U sb > 0, wie is action is efficient (a sb not receive a rent U sb (a sb < afb ), were = a fb ) Te inefficient type does = 0, but uses an action wic is beow is efficient eve, v (a sb ) = θ + α 1 α (θ θ ) > θ (5) 2 Trougout te paper we assume tat te soution of te probem is interior Sufficient conditions are v(0) = and v ( ) = 0 3 See aso Laffont and Tiroe (1993, p 104) 4 To ave a non-trivia probem, I assume trougout tis paper tat te principa wants to empoy bot agents, wic is te case if α is sma enoug

6 4 Verifiabe Information 6 Now consider te signa s and suppose tat it is verifiabe suc tat a genera contract as te form γ ms = (γ 1 (m), γ 2 (m, s)) In tis case, te principa can, for any p > 1/2, attain te first best Tat is, induce eiter type to work efficienty for bot periods and appropriate te entire surpus from production Let w is represent te agent s payment in period 2 if e caims to be type θ i and a signa s is received As sown in te second bencmark, witout te signa s te principa s probem is to pick a menu tat does not give te efficient type θ an incentive to caim e is inefficient Wit te signa s tis probem can be soved costessy Indeed, by coosing w > w te principa is abe to make te contract tat is meant for type θ reativey ess attractive to type θ, since te efficient type θ woud receive te iger wage w ony wit probabiity 1 p < 1/2, wereas type θ receives it wit probabiity p > 1/2 Terefore, te expected payment associated wit te payments (w, w ) is ess for type θ tan for type θ and te contract γ = (γ 1 (, s), γ 2 (, s)) wic is meant for te type θ becomes ess attractive to type θ In fact, by prescribing eac type is efficient effort eve, ie a = a = a = a fb and a = a = a = a fb, and coosing w and w suc tat δ(pw + (1 p)w ) + (1 + δ)θ a fb 0, type θ as no incentive to misreport is type 5 Proposition 1 Suppose te signa s is verifiabe, ten for any p > 1/2 te principa can impement te first best actions at first best costs troug a direct mecanism γ ms wit first period wages w = w = 0, and second period wages w = w = (1 + δ)θ a fb, w = (1 + δ)afb ((1 p)θ pθ ) ; w = δ(2p 1) (1 + δ)afb ((1 p)θ pθ ) δ(2p 1) Wit a verifiabe signa te principa is abe to attain te first best, by conditioning er wages on te signa s and coosing w > w Te intuition is straigtforward: Te principa rewards te agent if te signa s confirms te agent s message tat e is type θ and punises im oterwise Because te agent is risk neutra wit respect to money ony is expected payment interests im and te reward-punisment as expressed by te difference w w w = w (1 + δ)(θ θ )a fb δ(2p 1) (6) is sufficient for te principa to attain te first best 5 A forma proofs are reegated to te appendix

7 7 Te positive wedge between w and w of at east w is a necessary condition for contracts tat attains te first best Indeed, by te reveation principe a soution to te principa s probem is a direct mecanism γ = (γ rs ) wit r {, } tat ensures trutfu reveation of te agent s type Te principa utiity from suc a direct mecanism is V α(pv (γ ) + (1 p)v (γ )) + (1 α)(pv (γ ) + (1 p)v (γ )) (7) According to te reveation principe te direct mecanism (γ mr ) as to induce te agent to revea is type trutfuy Ie, te foowing two incentive compatibiity conditions ave to od pu (γ ) + (1 p)u (γ ) pu (γ ) + (1 p)u (γ ); (8) pu (γ ) + (1 p)u (γ ) pu (γ ) + (1 p)u (γ ), (9) were condition (8) ensures tat te agent of type reveas imsef trutfuy and (9) induces type θ to report imsef trutfuy Finay, te contract must ensure acceptance of bot types of agents, ie, pu (γ ) + (1 p)u (γ ) 0; (10) pu (γ ) + (1 p)u (γ ) 0 (11) Hence, a soution to te foowing maximization probem soves te principa s probem: P1: max V st (8), (9), (10), and (11) Te foowing proposition sows tat te wedge is a direct impication of te incentive compatibiity constraint (8) Proposition 2 Te principa can attain te first best ony wit a wedge w of at east w Note tat te necessary punisment-reward structure w increases as te signa becomes ess informative In fact, wen p approaces 1/2 te wedge w goes to infinity A direct comparison wit te second best soution obtained in te second bencmark sows tat te signa affects bot te first and second period action eves of te inefficient type θ Tis impies tat te principa uses te signa s in a forward and backward ooking way Hence, even toug te signa is received after te first period, te signa infuences te first period contract In fact, te principa is abe to use te interim information to impement te first best action eve

8 8 aso in te first period Tis resut is not new From te perspective of te first period te interim information may be seen as ex post information As Riordan and Sappington (1988) sow verifiabe ex post information is sufficient to eiminate inefficiencies due to ex ante asymmetric information As becomes cear in te next section tis resut depends cruciay on te verifiabiity of te signa 5 Nonverifiabe Information If te signa s is privatey observed by te principa, a verifiabe contract γ cannot depend directy on te signa s Instead, te second period contract may ony depend on te principa s announcement r about er signa s Invoking te reveation principe te optima contract is a direct mecanism of te form γ mr = (γ 1 (m), γ 2 (m, r)) wic ensures trutfu reveation of te private information of te agent and principa 6 Te second period contract, γ 2 (m, r) = (a mr, w mr ), must terefore be structured in suc a way tat te principa as an incentive to revea er signa trutfuy Given tat te agent sent a message m, te principa announces te signa s = trutfuy if v(a m ) w m v(a m ) w m On te oter and, a trutfu reveation of a signa s = requires v(a m ) w m v(a m ) w m Since bot inequaities ave to od at te same time, tey must be satisfied in equaity Hence, for eac m {, } it must od tat v(a m ) w m = v(a m ) w m (12) Equaity (12) as an important impication and refects te imitations due to a nonverifiabiity of te signa s It sows tat te contract γ mr must be structured suc tat after te agent as sent is message m, te reaization of te signa does not affect te principa s payoffs Indeed, substitution of (12) into (7) yieds V = α(v(a ) w ) + δ(v(a w )) + (1 α)(v(a ) w ) + δ(v(a w )) (13) and sows tat te principa s objective function is independent of probabiity p, ie, independent of te signa s reaization Hence, nonverifiabiity imits te effectiveness of te signa s drasticay In fact, wen te signa s is nonverifiabe, te principa is unabe to gain directy from it Yet, te remainder of tis section sows 6 Note tat because te principa cannot commit to some reporting strategy ex ante, se as imperfect commitment and te cassica reveation principe may fai to od (See Bester and Strausz 2001 for more detais) Yet, because te principa s announcement decision is independent of er beief concerning te agent s type, te standard proof and terefore te cassica reveation principe itsef neverteess ods

9 9 V fb U sb V U V sb 1 2 ˆp Figure 1: Utiities wit nonverifiabiity 1 p tat, due to an indirect incentive effect on te agent, te principa neverteess gains from te signa s By te reveation principe te agent as to revea is information, given tat te principa reveas er information trutfuy Tis is expressed by te earier conditions (8) and (9) In addition, bot types of agents soud accept te contract A requirement tat is expressed by (10) and (11) Summarizing, te principa must sove te foowing maximization probem P2: max V st (8), (9), (10), (11), (12) Observe tat te ony difference between te principa s maximization probem P1 and te new probem P2 is te principa s incentive constraint (12) It signifies tat wit nonverifiabiity te principa s use of te interim information is more restricted Indeed, te soution of Proposition 1 vioates te constraint and, due to te new constraint, te first best is no onger attainabe for p < 1 For any p [1/2, 1) te first best impies a = a = a fb Hence, to satisfy (12) it must od w = w But according to Proposition 2 te first best requires w > w Proposition 3 Tere exists a ˆp < 1/(1 + α) suc tat for a p < ˆp te optima contract eaves te efficient agent a positive rent and prescribes an action eve for te first period tat is identica to a situation in wic no signa is avaiabe For p > ˆp te optima contract extracts a rents and te first period action eve is more efficient tan in a situation witout te signa s Moreover, for p = 1/2 te soution coincides wit te second best For p = 1 te soution coincides wit te first best Proposition 3 sows tat, depending on te informativeness of te signa s, te optima contract is of two types If te signa s informativeness is ow, p < ˆp, te principa uses signa s to reduce te informationa rent of te efficient type excusivey Tat is, se does not use te signa to improve te aocative efficiency of te first period contract In contrast, for more informative signas, p > ˆp, te

10 10 principa extracts a rents and aso adjusts te action eve of te first period coser to te first best Moreover, te first best is ony attainabe wit a perfect signa, ie, wit p = 1 Figure 1 iustrates ow utiity of te principa and te efficient type θ depends on te informativeness of te signa s For p = 1/2 te signa is uninformative and te principa s utiity coincides wit te second best As p increases te signa s becomes more informative and may be used to reax te agent s incentive constraint In te range of 1/2 to ˆp te principa s uses te signa to reduce te informationa rent to te agent At p = ˆp te informationa rent is competey depeted and te individua rationaity constraint of te efficient agent becomes binding From ˆp onwards, te principa reduces te aocative distortion At p = 1 a distortions disappear and te principa acieves te first best 6 Manipuative Signas Te previous two sections sow tat te use and effectiveness of interim information depends on its verifiabiity Wit verifiabiity te principa acieves te first best for any informative signa, wereas wit an nonverifiabe signa te principa can ony attain te first best if te signa is perfect Verifiabiity is key and it is terefore wortwie to ave a coser ook at te roe it pays In addition, aso te discontinuity at p = 1/2 in te case of a verifiabe signa seems puzzing: For any p > 1/2 te principa attains te first best, but for p = 1/2 ony a second best obtains To address tese issues more carefuy it is epfu to generaize te binary view of verifiabiity and to arrive at a more continuous concept Starting point of te generaization is te observation tat te two cases iustrate two extremes: Wen a signa is verifiabe, te principa is unabe to infuence it Wereas, wen te signa is private information, te contract depends ony on a vountary announcement of te signa, wic effectivey impies tat te principa can freey determine its reaization It is terefore natura to view verifiabiity in terms of signa manipuation To mode te idea of manipuation I assume tat te principa can, at some commony known cost K, cange te signa into a signa 7 Tat is, after receiving a signa s = te principa can invest K and cange te signa s reaization into s = Note tat tis mode wit manipuation comprises te previous two modes If te signa is verifiabe, te cost K is infinitey arge If te signa is non-verifiabe, te manipuation cost K is zero Te view of verifiabiity as a degree of manipuation carifies te discontinuity at p = 1/2 and, in addition, provides a furter argument in favor of signa manip- 7 For simpicity I assume tat te principa can ony cange a signa into a signa Tis is te reevant direction of manipuation

11 11 uation Proposition 2 sows tat wen p approaces 1/2 te principa can ony attain te first best if se prescribes infinite penaties and rewards Tis impies tat te stakes between te principa and agent become extremey ig Given tat te agent caims e is of type θ, te principa as an extreme incentive to prevent a signa s = from occurring Hence, te impementabiity of te first best in Section 4 depends cruciay on te assumption tat te principa does not ave a possibiity to infuence te outcome of te signa s But wit infinitey ig stakes, it seems extremey unreaistic tat contracting parties wi not find ways of manipuation In contrast, wit any some possibiity of signa manipuation, captured by a finite K, te resut of Proposition 1 breaks down for p cose to 1/2 Te remainder of tis section derives te optima contract under signa manipuation and compares te resuts to te extremes of te previous two sections Given an announcement of te agent te principa s decision weter to manipuate te signa is straigtforward If s = and te principa does not manipuate te signa se receives v(a ) w Wen manipuating into a signa s = se gets v(a ) w K Hence, a principa as no strict incentive to manipuate te signa s if K w v(a ) w + v(a ) (14) Lemma 1 Witout oss of generaity an optima contract exibits w v(a ) w + v(a ) K Te emma sows tat te principa may restrict attention to contracts wic convince te agent tat a signa wi not be manipuated An intuitive resut: A signa tat is bound to be manipuated is simpy wortess Given te emma, a soution to te principa s probem is a soution to te foowing optimization probem: P3: max V st (8), (9), (10), (11), (14) Hence, te manipuation-proofness constraint (14) repaces te principa s incentive constraint (12) in probem P2 Indeed, for K = 0 te condition (14) reduces to (12) For K = te constraint becomes inconsequentia and te maximization probem is identica to P1 In te foowing, et V (p, K) represent te soution to P3 as a function of te signa s accuracy, p and its verifiabiity, K Weter te manipuation-proofness constraint is binding at te optimum wi depend on te parameters K and p In te foowing define { p(k) min 1/2 + (1 + δ)(θ θ )a fb }, 1 2δK

12 12 p 1 ˆp no rents; FL; BL First Best; FL; BL 1/2 rents; FL ˆp(K) p(k) K Figure 2: Optima forward (FL) and backward (BL) ooking use Lemma 2 If p < p(k) te constraint w v(a ) w + v(a ) K is binding Te Lemma sows tat if te signa s informativeness is ow reative to te cost of manipuation K, te requirement tat te contract must convince te agent tat te principa does not manipuate te signa becomes a binding constraint Te emma eads to to te foowing coroary Coroary 1 Te first best is attainabe if and ony if p p(k) Hence, p(k) represents a cut-off vaue above wic te principa is abe to attain te first best Te iger te cost of manipuation K, te ower te cut-off vaue p(k) Indeed, as K goes to infinity, p(k) approaces 1/2 and te principa can attain te first best for any p 1/2, confirming te resut of Proposition 1 A parameter vaue K = 0 captures te oter extreme of non-verifiabe information and yieds p(k) = 1 It impies tat, in ine wit Proposition 5, te first best is ony attainabe wit a perfect signa Proposition 4 Tere exists a ˆp(K) [1/2, ˆp] suc tat for a p < ˆp(K) te optima contract eaves te efficient agent wit a rent For p [ ˆp(K), p(k)) te optima contract extracts a rents For p p(k) te principa impements first best action eves at first best costs Te functions ˆp(K) and p(k) are bot decreasing in K It ods tat ˆp(0) = ˆp and im K ˆp(K) = 1/2 As Figure 2 iustrates, Proposition 4 sows tat interim information can be cassified according to te degree of informativeness If te interim information contains itte information, ie p < ˆp(K), te principa uses it to reduce te rent of te efficient agent, but does not affect te first period action eves Hence, sma degrees of interim information are used in a forward ooking way ony For an intermediate degree of information, te principa extracts a rents and, in addition, reduces te distortion on te action eve in te first period Tat is, se uses interim information aso in a backward ooking way Finay, te principa may use interim information wit a ig informative content, ie p p(k), to sove er adverse probem competey and tereby acieve te first best

13 13 V fb V (p, ) V fb V ( p, K) V (p, K) V (p, 0) V ( p, 0) V sb p( )= 1 2 ˆp(K) 3a p(k) ˆp(0)=ˆp p(0)=1 p V sb ˆK 3b K K Figure 3: Principa s utiity wit manipuation Figure 3a and 3b iustrate te effect of p and K on te principa s utiity Te first figure contrasts te two extremes K = 0 and K = to an intermediate vaue K In te range from 1/2 to ˆp(K) te principa s utiity is increasing, because a iger p enabes te principa to reduce te informationa rent to te efficient agent θ At ˆp(K) te principa extracts a rents A furter increase in p, neverteess, raises te principa s utiity, because se now adjusts te impemented actions coser to te first best As of p(k) te principa is abe to impement first best action eves at first best costs Te cut-off vaues ˆp(K) and p(k) are decreasing in K, suc tat, given a ess manipuative signa, te principa may acieve te first best for a ess accurate signa Figure 3b empasizes te roe of verifiabiity Given a fixed eve of accuracy p it draws te principa s utiity as a function of te verifiabiity parameter K Again one may identify tree regions Wenever te signa s verifiabiity is ow, K < ˆK, te principa uses interim information for rent extraction ony If te eve of verifiabiity is ig, K > K te interim information aows te principa to acieve te first best For intermediate eves of verifiabiity, K ( ˆK, K), te principa uses te interim information to extract a rents and to reduce te aocative distortion Note tat in ine wit Proposition 5 te principa benefits from interim information, even wen te signa is nonverifiabe, ie, V (p, 0) > V sb for any p > 1/2 Proposition 4 and te two figures sow tat te signa s precision p and its degree of verifiabiity K are substitutes: Witout affecting er utiity te principa may substitute a signa wit a ig degree of informativeness and an intermediate eve of verifiabiity for a signa wit an intermediate degree of informativeness and a ig eve of verifiabiity Tis observation points to a potentia trade-off between a signa s precision and its degree of verifiabiity More specificay, suppose tat te principa may coose from mutipe signaing tecnoogies wic differ according to teir informativeness p and manipuabiity K Formay, et T represent te set of avaiabe signaing tecnoogies (p, K) Ten we may obtain from te set T a subset of efficient tecnoogies T e tat contain tose pairs (p, K) T suc tat

14 14 tere does not exist a K > K wit (p, K ) T Wenever T e is not a singeton, te principa faces a trade-off, as tere exist mutipe efficient tecnoogies Some produce more informative signas tan oters, but are, as a drawback, easier to manipuate Utimatey, it depends on te function V (p, K) wic auditing tecnoogy (p, K ) T e is optima Tat is, (p, K ) = arg max (p,k) T e V (p, K) Ceary, te optima auditing tecnoogy may not be te most informative one Te trade-off is reinforced if one extends te mode furter by modeing interim information as a costy auditing tecnoogy of te principa Let c(p, K) represent te cost of an interim signa wit precision p and verifiabiity K Te principa s optima auditing tecnoogy may ten be cacuated as (p, K ) = arg max V (p, K) c(p, K) Wen, as seems reasonabe, te cost c(p, K) is increasing in p and K, te aforementioned trade-off between precision and verifiabiity is strengtened 7 Concusion and Extensions Tis paper studied te use of interim information, since te presence of suc information seems a natura caracteristic of repeated interactions In fact, it may be te very reason wy economic agents enter into a contractua reationsip rater tan transact on an anonymous spot market in wic suc information is not produced 8 Interim information mitigates te adverse seection probem and, depending on its informativeness and verifiabiity, may resove te probem competey Hence, in repeated contractua reationsips tat start under asymmetric information te adverse seection probem may be ess probematic tan te standard static mode suggests Specificay, te paper sowed tat ess precise and more manipuabe signas are used in a forward ooking way ony to arrive at ess distorted future actions and a reduction in te agent s information rent More precise and ess manipuabe signas are in addition used in a backward ooking way to reduce te aocative distortion in te previous period and an extraction of a information rents Finay, igy precise signas wit a ig degree of verifiabiity are abe to yied te first best Te paper sowed, moreover, tat verifiabiity is a crucia determinant of te effectiveness of interim information More specificay, a signa s verifiabiity and its informativeness are substitute Tis points to a new potentia trade-off in te coice of auditing procedures tat te iterature as iterto disregarded: a signa s precision versus its degree of verifiabiity In particuar, contracting parties may be we advised to coose a ess accurate auditing tecnoogy tat is ess prone to manipuation at te expense of a igy accurate one wic is more manipuabe Te paper s mode suggests mutipe extensions First, te contractua reation- 8 Te vaue of te interim information is te difference V (p, K) V sb

15 15 sip may extend to more tan two periods and, ence, mutipe interim signas As ong as te number of periods are finite, owever, simiar resuts obtain 9 Te type of equiibrium tat resuts ten depends on te overa informativeness of a signas taken togeter Second, te agent s type may directy affect te principa s utiity suc tat te contracting environment exibits a common vaue component A compication of a common vaue environment is tat te cassica reveation principe does not od anymore Bester and Strausz (2001) sow ow tis compicates a proper anaysis Neverteess, aso in a common vaue environment te interim signa does not make te principa worse off, since se can aways write er contract independent of te signa and tereby mimic te outcome witout te interim information A common vaue environment may, owever, reduce te vaue of interim information Tird, one may introduce risk averseness on part of te agent Tis makes interim information ess beneficia, because, as may be sown in te anaysis, it tends to increase te amount of risk on te agent Yet, even wit risk averseness interim information is usefu and one may expect a simiar roe for verifiabiity 10 9 Wit infinite periods, contracting partners may use of trigger strategies to obtain fok teorem ike outcomes 10 See aso Cooper and Hayes (1987)

16 16 8 Appendix Proof of Proposition 1 It is easy to verify tat a contract wit a = a = a = a fb, a = a = a = a fb, and wages w = (1+δ)afb ((1 p)θ pθ ) δ(2p 1), w = (1+δ)afb ((1 p)θ pθ ) δ(2p 1), w = w = 0, w = w = (1 + δ)θ a fb gives eac type of agent an incentive to report is type trutfuy Moreover, eac type receives is reservation utiity suc tat te contract impements te first best QED Proof of Proposition 2: In te first best it ods, by definition, tat a = a = a = a fb, a = a = a = a fb and tat constraints (10) and (11) bind Incentive compatibiity constraint (8) may terefore be rewritten as δ[pw + (1 p)w ] (1 + δ)θ a fb Moreover, a binding (11) impies δ[pw + (1 p)w ] = (1 + δ)θ a fb Subtracting tis equation from te previous inequaity and a rearrangement of terms yieds and te proposition foows QED Proof of Proposition 3 (1 2p)(w w ) (1 + δ)(θ θ )a fb Assume tat at te optimum ony te individua rationaity constraint of te inefficient type (2), te incentive compatibiity constraints of te principa (12), and te incentive compatibiity of te efficient type are binding (3) Hence, w = a θ + a θ /δ + (a a )pθ + (1 p)(v(a ) v(a )) w = a θ + a θ /δ + (a a )pθ p(v(a ) v(a )) w = a θ + (a a )θ w + (1 2p)(v(a v(a )) p + a θ + (a a )θ δp Substitution and a rearrangement of terms yieds + (a a )(θ + θ ) + (a a )θ + w V (p) α(v(a ) a θ ) + αδ(1 p)(v(a ) a θ ) + αδp(v(a ) a θ ) +(1 α)v(a ) (θ αθ )a +δ((1 p αp)v(a ) a ((1 p)θ αpθ )) +δ((p α(1 p))v(a ) (pθ α(1 p)θ )a ) Te first order conditions wit respect to a is v (a ) = θ and impies te first best action eve a fb Te remaining first order conditions are (1 α)[v (a ) θ ] = α(θ θ ) (15) (1 p αp)[v (a ) θ ] = αp(θ θ ) (16) (p α(1 p))[v (a ) θ ] = α(1 p)(θ θ ) (17)

17 17 Equation (15) sows tat te optima vaue for a coincides wit te second best, ie a = a sb, and is independent of p Since v < 0, te second order condition is aso satisfied and (17) defines a (p) impicity For p = 1/2 te first order condition coincides wit (5) and it foows a = a sb For p > 1/2 foows tat a < a fb Differentiating wrt p and rearranging terms yieds a p = (a ) θ ) + (v (a ) θ ) α(v (p α(1 p))v (a ) (18) Te numerator of te expression is positive due to v (a ) > θ > θ Te denominator is positive, because v () < 0 and α < 1 Hence, a increases wit p Te equation (16) defines te function a (p) impicity Note tat te second order condition is satisfied for p < ˆp For p = 1/2 te first order condition coincides wit (5) and it foows a (1/2) = a (1/2) = asb rearranging terms yieds a Differentiating wrt p and p = (v (a ) θ ) + α(v (a ) θ ) (1 p αp)v (19) (a ) Te numerator is positive and, for p < ˆp, te denominator is negative It foows tat for p < ˆp te optima a is decreasing in p Moreover as p approaces ˆp te derivative a / p goes to minus infinity Since for p = 1/2 we ave a = a = a sb and since a / p > 0 and a / p < 0 it foows for p (1/2, ˆp) tat a (p) < a (p) = a sb < a (p) Now consider te individua rationaity constraint (10) of agent Since te soution for p = 1/2 coincides wit te second best soution, te individua rationaity constraint is sack, ie for p = 1/2 it ods w θa + δ[p(w θ a ) + (1 p)(w θ a )] > 0 Substitution of te reevant constraints yied U (p) = (θ θ )(a + a δ + (a a )δp) + (2p 1)δ(S (a ) S (a )), (20) were S i (a) v(a) θ i a represents te joint surpus of te action a Differentiating wrt p and using (18) and (19) yieds [S (a ) S (a ) + S (a ) S (a )] + } {{ } } {{ } >0 >0 [ ] (1 α)α(1 p)p(θ θ ) 2 1 (1 (1 + α)p) 3 v (a ) 1 (p (1 p)α) 3 v < 0 (a ) } {{ } 0 Te first term in te square brackets is negative since a < a < a fb < afb and S i (a) is increasing for a < a fb i Te second term in te square brackets is

18 18 non-positive since 1 (1 + α)p p (1 p)α and v (a ) v (a ) < 0, due to v 0 Hence, starting from p = 1/2 te utiity of type θ is decreasing in p As p approaces 1/(1 + α) te first part of te second term in te square brackets approaces negative infinity Hence, tere exists some ˆp < 1/(1 + α) suc tat U (ˆp) = 0 and te individua rationaity constraint of type θ is binding for p > ˆp It remains to be sown tat for p = 1 te principa can acieve te first best Note tat if p = 1 a contract γ wit a = a = a fb and a = a = a fb, w = w = θ a fb and w = w = θ a fb yieds, if incentive compatibe, te principa te first best outcome To ensure incentive compatibiity set w = v(a ) and w = v(a ) suc tat tat te constraints (12) are fufied In order to satisfy te agent s incentive constraint (8) set a suc tat S (a ) = v(a ) θ a (θ θ )a fb /δ, wic is satisfied for a arge enoug Finay, to respect te agent s incentive constraint (9) set a suc tat S (a ) = v(a ) θ a (θ θ )a fb /δ, wic is satisfied for a arge enoug QED Proof of Lemma 1: Consider a contract tat exibits w v(a ) w +v(a ) > K Tis contract induces te principa to manipuate a signa s = into a signa s = after an announcement Hence, irrespective of te true reaization of s te reevant second period contract is (w, a ) Terefore, any contract wit w v(a ) w + v(a ) K is equivaent to a contract wit w = w and a = a But suc a contract satisfies w v(a ) w + v(a ) K QED Proof of Lemma 2: If te constraint is not binding at te optimum, it may be disregarded in te optimization probem Probem P3 terefore reduces to probem P1 By Proposition 1 te soution of probem P1 is te first best, but as sown by Proposition 2 it necessariy exibits a wedge of at east w = (1 + δ)(θ θ )a fb δ(2p 1) But tis vioates te omitted constraint wenever p < p(k) Hence, te first best action a = a fb is not impementabe and since p < p(k) 1 te first best outcome is not attainabe QED Proof of Proposition 4: Proposition 1 estabises a contract tat witout te possibiity of manipuation attains te first best Te contract does not give rise to manipuation wen K [(1 + δ)(θ θ )a fb ]/[δ(2p 1)] Rewriting tis inequaity yieds p p(k) and p(k) = 1/2 as K goes to infinity Te mode of Section 5 obtains for K = 0 Terefore, Proposition 3 ods for K = 0 and it foows ˆp(0) = ˆp

19 19 Now consider p < p(k) and assume tat at te optimum ony constraints (8), (11), and (14) are binding Soving for tese constraints yieds w = a θ + a θ /δ + (a a )pθ + (1 p)(v(a ) v(a ) + K) w = a θ + a θ /δ + (a a )pθ p(v(a ) v(a ) + K) w = a θ + (a a )θ w + (1 2p)(v(a v(a ) + K) p Substitution yieds + a θ + (a a )θ δp + (a a )(θ + θ ) + (a a )θ + w V (K, p) αδk( 1 + 2p) + V (p) Te first order conditions wit respect to a is v (a ) = θ and yieds te first best action eve a fb Te remaining first order conditions are described by equations (15), (16), and (17) Note tat for p = 1/2 action eves coincide wit te second best and te wedge w refects, due to risk neutraity, an inconsequentia randomization Hence, for p = 1/2 te equiibrium outcome coincides wit te second best and confirms tat at te optimum ony constraints (8), (11), and (14) are binding Wen constraints (8), (11), and (14) are binding, type θ s payoff is U (p K) = (θ θ )(a + a δ + (a a )δp) + (2p 1)δ(S (a ) S (a )) +δ(1 2p)K Tis is identica to expression (20) except for te ast term, wic is decreasing in p Hence, by te argument used in te proof of Proposition 3, it may be estabised tat type θ s overa payoff is decreasing in p To sow tat ˆp(K) ˆp note first tat ˆp(0) = ˆp and tat U (p K) < U (p 0) = U (p) for any K > 0 and p > 1/2 Since ˆp(K) satisfies U (ˆp(K) K) = 0 it foows U (ˆp(K) K) = 0 < U (ˆp K) and ence ˆp(K) < ˆp for any K > 0 QED References Baron, D and D Besanko (1984), Reguation and information in a continuing reationsip, Information Economics and Poicy 1, Bester, H and R Strausz (2001), Contracting wit Imperfect Commitment and te Reveation Principe: Te Singe Agent Case, Econometrica, Cremer, J and RP McLean (1988), Fu Extraction of te Surpus in Bayesian and Dominant Strategy Auctions, Econometrica 54, Cooper, R and B Hayes (1987), Muti-period Insurance Contracts, Internationa Journa of Industria Organization 5,

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