Information Acquisition in Global Games of Regime Change (Online Appendix)

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1 Informaton Acquton n Global Game of Regme Change (Onlne Appendx) Mchal Szkup and Iabel Trevno Augut 4, 05 Introducton Th appendx contan the proof of all the ntermedate reult that have been omtted from the paper, a well a extenon of everal reult reported n the paper. The appendx dvded nto 6 ecton. In Secton we how that for any dtrbuton of precon choce, there are no non-monotonc equlbra n the econd tage of the game. We alo etablh properte of the ex-ante utlty functon that we nvoked n the proof of Theorem. In partcular, we how that the ex-ante utlty functon ncreang and that t dervatve unformly bounded from above and converge to zero. Fnally, we how that there ext a precon level (; ) uch that player never nd t optmal to chooe a precon hgher than. Secton 3 concerned wth the reult regardng the over-acquton and under-acquton of nformaton. We rt prove the techncal reult that we ued to argue that the unque equlbrum of the game genercally ne cent. We then analyze the condton under whch nvetor globally over-acqure and under-acqure nformaton and explan why a complete characterzaton of thee global e ect not attanable. In Secton 4 we prove ntermedate reult that have been nvoked n the paper when analyzng trategc complementarte n nformaton choce and the welfare mplcaton of an ncreae n the precon of the pror. We alo dcu what happen when we relax the aumpton that T. In partcular, we provde an extenon to Propoton 5 n the man text to the cae when T 6. We alo dcu why t d cult to etablh a full characterzaton of trategc complementarte and ubttutablte between prvate and publc nformaton when T 6. In Secton 5 we conder the e ect of an ncreae n the precon of the pror on the probablty of a ucceful nvetment and on welfare. In partcular, we how that for any T the probablty of a ucceful nvetment ncreae n when u cently hgh and decreae when u cently low. We alo nclude gure generated ung numercal mulaton that decompoe the ft; g pace nto regon where the probablty of ucceful nvetment ncreang or decreang n. In Secton 5: we evaluate numercally the e ect of an ncreae n on the ex-ante utlty of an nvetor for d erent parameter value. Thee numercal reult provde upport to the clam that welfare ncreang n

2 the precon of publc nformaton when the mean of the pror hgh and decreang when the mean of the pror low. Fnally, Secton 6 contan the detal of determnng the approprate lower bound for the poble precon choce,. Unquene of equlbrum and properte of the ex-ante utlty. Unquene of equlbrum Propoton For any, uppoe that nf (upp( )) >. Then the coordnaton game ha a unque equlbrum n whch all nvetor ue threhold tratege fx ( ; ) ; [0; ]g and nvetment ucceful f and only f. Proof. In the appendx of the paper we how that for any uch that nf (upp( )) > we have a unque equlbrum n monotone tratege. We how here that there are no other type of equlbra. Suppoe that n the econd tage of the game the dtrbuton of precon choce among nvetor gven by ome dtrbuton functon () wth bounded upport. The bounded upport aumpton follow from Aumpton A and A n the text. To how unquene of equlbrum for any () we ue the procedure of teratve deleton of domnated tratege, a tandard approach n the global game lterature (ee Carlon and van Damme, 993). Suppoe that nvetor expect no one to nvet, that, he expect nvetment to be ucceful f and only f > 0. Then, even though no one ele nvet, nvetor wll chooe to nvet f h gnal hgh enough,.e. he wll nvet f: Pr ( > jx ( )) > T Denote by x ( ) the value of the gnal that make nvetor nd erent between nvetng and not nvetng (x ( ) olve the above equaton wth equalty). Snce the above equaton monotone n x ( ) ; t follow that the nvetor wll nvet f x x ( ) and not nvet otherwe. Th mple that each nvetor who receve a gnal wth precon wll nd t optmal to nvet f the gnal he receve hgher than x ( ) ; where x ( ) 0 ( ) (T ) Note that all nvetor wth the ame precon level wll have the ame x ( ). Therefore, n what follow we drop the upercrpt. The fact that all nvetor follow a monotone trategy mple that nvetment wll be ucceful whenever: Z Pr (x x ( ) j) d >

3 De ne a the value of that olve the above equaton wth equalty. For a normal dtrbuton th correpond to the followng equaton: Z x ( ) d () Note that R x ( ) d > 0 when 0 and R. Moreover, Z x ( ) d Z x ( ) d < 0 8 [0; ] x ( ) d < when Snce the rght hand de of () ncreang, t follow that there a unque (0; ) that olve the above equaton. Moreover, < 0. For the nducton tep, aume that there a value n uch that n < n < ::: < < 0. n de ned uch that nvetor expect nvetment to be ucceful whenever > n. Hence, nvetor who receve a gnal wth precon wll nvet whenever h gnal hgher than x n ( ), where x n ( ) olve or Pr n jx n ( ) T x n ( ) n ( ) (T ) Note that nce n < n t follow that 8 upp ( ) ; x n ( ) < x n ( ). Conder agan the CM condton. The fact that each nvetor nvet whenever h gnal hgher than x n ( ) mple that nvetment wll be ucceful for all n, where n olve the followng equaton: Z Pr x x n ( ) j n d n or Z n ( ) (T ) n d n By the prevou argument we know that there ext n that olve the above equaton. Note that nce 8 upp ( ) x n ( ) < x n ( ), we have Pr x x n ( ) j n < Pr x x n ( ) j n Suppoe now that n n. Then the RHS of the CM condton above now greater than the LHS: But note that the RHS ncreang n n, whle the LHS decreang n n. Hence t mut be the cae, by contnuty of the LHS and RHS expreon, that n < n and n [0; ]. Th mple a new threhold gnal for each nvetor, x n ( ) uch that x n ( ) < x n ( ) < x n ( ). 3

4 By nducton we obtan decreang equence n n and fx n ( )g n for each. Snce n n bounded from below by 0, t follow that th equence convergent. Th mple that fx n ( )g n alo a convergent equence. De ne lm n n and lm n x n ( ) x ( ). Th conclude the teratve deleton of trctly domnated tratege. Conder now the cae when the nvetor expect that everyone wll nvet regardle of ther gnal and agan apply the procedure of teratve deleton of domnated tratege. Followng exactly the ame argument, we obtan ncreang and bounded equence f n g n and fx n ( )g n. De ne lm n n and lm n x n ( ) x ( ). n o By contructon ; fx ( )g upp( ) ha to olve Pr jx ( ) T,8 upp ( ) Pr x x ( ) j n o and ; fx ( )g upp( ) ha to olve Pr ( jx ( )) T,8 upp ( ) Pr (x x ( ) j) However, n the paper we how that the above ytem of equaton ha a unque oluton. Therefore and x ( ) x ( ) 8 upp ( ) mplyng that there a unque trategy pro le that urvve the teratve deleton of domnated tratege. Th trategy pro le conttute the unque equlbrum of the game n monotone tratege. Lemma Conder the bene t functon B ( ; ).. B ( ; ) trctly ncreang n,. B bounded from above, 3. lm B 0, 4. For > ; B < 0. Proof. () Condton () mple that the value of nformaton n our ettng alway potve. Recall that: B ( ; ) Z Z x Z T df (xj) dg () ( T ) dg () C ( ) Z Z x ( T ) df (xj) dg () We d erentate B ( ; ) wth repect to and note that: x Z f (x j) g () d x Z ( T ) f (x j) g () d 0 4

5 nce x ha to atfy: Pr ( jx ) T Evaluatng the remanng ntegral and ung the properte of the truncated normal dtrbuton we obtan: B x for all <. Th prove the rt part of the lemma. () To ee that the econd part of the lemma alo true note that: B x p p p < 4 > 0 (3) To prove the thrd part of the above lemma note that: 0 B p p 0 a (4) The lat part of the lemma le traghtforward. The econd dervatve of the bene t functon gven by: B x (x ) (x ) (x ) Thu, f we how that the term n the quare bracket potve, then the clam would hold. De ne A ( ; ) (x ) (x ) (x ) Note that: (x ) p ( ) (T ) (x ) ( ) p (T ) 5

6 and hence, A ( ; ) (x ) ( ) 3 ( ) (x ) ( ) p (T ) ( ) p (T ) ( ) p (T ) p (T ) where the lat term n general negatve. Before proceedng, we further mplfy the above expreon: p (x ) ( ) (T ) ( ) p ( ) (T ) p ( ) (T ) ( ) p (T ) p (T ) p (T ) Therefore, B 4 3 x 3 ( ) p (T ) p (T ) where we took n front of the quare bracket. At th pont we conder three cae: T <, T ; and T >. Let T <. When T < ; then: p (T ) p j j (T ) j j 0 a p (T ) p (T ) Note alo that: Therefore, t follow that: lm 3 ( ) 3 lm A ( ; ) 3 > 0 6

7 We conclude that, for a xed ; there ext a bound for uch that above th bound the econd dervatve of nvetor bene t functon negatve. Next, recall that: [ ; ] ; where 0 < < < Snce A ( ; ) contnuou and converge to zero for any a ; and nce [ ; ] a compact ubet of R, t follow that there ext a unform bound for that nte, and uch that f larger than th bound, then the econd-order dervatve of the bene t functon wth repect to negatve for all [ ; ]. We aume that greater or equal to th bound. The argument for the cae when T analogou,.e., a mlar argument etablhe the extence of a bound on that guarantee that the econd-order dervatve of the bene t functon negatve when T. We aume that greater or equal than both of thee bound. Snce each of thee bound nte, t follow that nte. Th conclude the proof. Lemma 3 There ext < uch that nvetor wll never nd t optmal to chooe >. Proof. From Lemma we know that B p 4. Moreover, by Aumpton A we have that lm C 0 ( ). Hence, there ext < uch that for all > B C 0 ( ) < 0 Snce n the neghborhood of the followng true: B C 0 ( ) > 0 then t follow that no nvetor wll ever chooe >. 3 E cency of nformaton choce In th ecton we provde proof for the reult that we ued n the e cency ecton of the paper that were omtted n the man appendx. We alo provde a partal characterzaton of global over and under acquton of nformaton and we explan why a complete characterzaton of the global reult not attanable. We tart wth a techncal reult that we nvoked when argung that equlbrum precon choce are genercally ne cent. Lemma 4 Denote by ( ) the equlbrum precon choce a a functon of. Then, for each T there ext a unque, call t E where b (T; ( ) ; ) (T ), that olve: b (T; ( ) ; ) ( ) (T ) p (T ) ( ) ( ) Moreover, f > E (T ) then > b (T; ( ) ; ) and f < E (T ) then < b (T; ( ) ; ). 7

8 Proof. We are ntereted n oluton to b (T; ( ) ; ) De ne f ( ) b (T; ( ) ; ) To prove our clam we wll how that f ( ) nterect zero only once and potve a and negatve a. Fx T and. Note that f T > then b () bounded from below by 0 and from above by p (T ), and f T < then b () bounded from below by p (T ) and from above by. If follow that, regardle of the value of T, the functon f ( ) potve for large enough and negatve for mall enough. Thu, t reman to how that f ( ) nterect zero only once. Note that, nce b (T; ( ) ; ) contnuou n ; t follow that there ext E uch that: f E E b T; E ; 0 We now how that at any uch E the dervatve of f ( ) wth repect to potve, mplyng that f ( ) croe zero only once. Takng the dervatve of b T; E ; wth repect to we get: Conder rt b (T; ( ) ; ) b (T; ( ) ; ) and recall that a xed pont of the bet repone functon,.e. t at e: ( ; ) where denote the nvetor bet repone functon, mplctly de ned by the rt-order condton. Hence, Snce ( ; ) <. Suppoe that E ( ; ) / ( ; ) and conder the CM condton: U ( ; ; ) U ( ; ; ) C 00 ( ) r p E (T ) ( ) 0 Rearrangng and ung the fact that E b T; E ; we obtan: p ( ) r r p (T ) (T ) The fact that ( ; ) < mpled by the aumpton that large enough. See Theorem and t proof n the paper. To economze on notaton we wrte below ntead of E. 8

9 mplyng that: r E (T ) E p (T ) whch n turn mple that x E ; E ; E E. But then t follow that: U x (x ) zero when E.3 Therefore, U E ; E ; E 0 and o j E 0. Snce b (T; ( ); ) alway nte, t follow that b (T; ( ); ) j E 0 and o at any E uch that f E 0, the dervatve of f () wth repect to potve. Th mple that f () croe zero only once, whch complete the proof for the cae when T >. The cae for T < follow by an analogou argument. 3. Global over-acquton and under-acquton of nformaton Now we conder the global over and under acquton of nformaton. Recall that we ay that nvetor globally over-acqure nformaton f the e cent choce of precon,, le than the equlbrum precon choce,, where whle the oluton to arg max U (; ) U (; ) C 0 () 0 C () To tudy global reult we need to make ue of the followng reult, etablhed n Szkup (05): Lemma 5 Let be the precon choce of all the other nvetor.. If b (T; ; ) then > 0: where r b (T; ; ) (T ) p (T ) Proof. Th lemma follow drectly from Propoton n Szkup (05). Lemma 6 Let be the precon choce of all the other nvetor.. Suppoe that T >, then: (a) If T then decreang for all > ; 3 In Lemma we etablh that < 0. 9

10 (b) If (T; b (T; ; )) then ntally decreang and then ncreang n ; (c) If b (T; ; ) then ncreang for all >.. Suppoe that T, then: (a) f < then decreang n for all > ; (b) f then contant n ; (c) f > then ncreang n for all > : 3. Suppoe that T <, then: (a) If ; (b) If (b (T; ; ) ; T ) then ntally ncreang and then decreang n ; (c) T then ncreang n for all >. and where r b (T; ; ) (T ) p (T ) f T > then for all, b (T; ; ) > T f T < then for all, b (T; ; ) < T Proof. The reult follow from Propoton 4 n Szkup (05). Next, we de ne our noton of global pllover e ect and ue Lemma 6 to etablh condton under whch there are potve or negatve pllover e ect n nformaton choce. De nton 7 Conder nvetor and let be the precon choce of all the other nvetor.. If 8 > we have B. If 8 > we have B > 0 then we ay that the game exhbt global potve pllover. < 0 then we ay that the game exhbt global negatve pllover. Corollary 8 Let be the precon of prvate nformaton that nvetor are ntally endowed wth.. Suppoe that T >. (a) If T then there are global potve pllover e ect n nformaton choce. (b) If (T; b (T; ; )) then there ext b ( ) uch that for all b ( ) there are global negatve pllover n nformaton choce. (c) If b (T; ; ) then there are global negatve pllover e ect n nformaton choce. 0

11 . Suppoe that T. (a) f < (b) f (c) f > then there are global potve pllover e ect n nformaton choce. then there are no pllover e ect n nformaton choce. then there are global negatve pllover e ect n nformaton choce. 3. Suppoe that T <. (a) If b ( ) there are global potve pllover n nformaton choce. (c) If T then there are global negatve pllover e ect n nformaton choce. Proof. Th reult follow from Lemma 6 and the obervaton that the gn of U the e ect that an ncreae n the precon of all nvetor ha on the threhold, nce U x depend only on Lemma 6 and Corollary 8 are key for etablhng the comparon between, the equlbrum choce of precon, and, the e cent precon. In the next propoton we utlze the above lemma to provde an almot complete characterzaton of global over and under acquton of nformaton (except for the cae when T < and (b (T; ; ) ; T )). 4 Propoton 9 nothng. Suppoe that T >. (a) If E (b) If E (c) If E (T ) then nvetor under-acqure nformaton. (T ) then nvetor acqure the e cent level of nformaton. (T ) then nvetor over-acqure nformaton. where E (T ) the unque oluton to b (T; ( ) ; ).. Suppoe that T. (a) If > (b) If (c) If < then nvetor under-acqure nformaton. then nvetor acqure the e cent level of nformaton. then nvetor over-acqure nformaton. 4 See Lemma 4 for the prece de nton of E (T ).

12 3. Suppoe that T <. (a) If b (T; ; ) then nvetor under-acqure nformaton. (b) If T then nvetor over-acqure nformaton. Proof. We dvde the proof nto three eparate part. Frt, we provde the proof for the cae when T. Next, we conder the cae when T >. Fnally, we conder the cae when T <. Suppoe rt that T. In th cae, the clam follow almot mmedately from Corollary 8. To ee th, conder rt a tuaton where > and note that Corollary 8 mple that n th cae there are global negatve pllover,.e. for all > and all > we have B ( ; ) < 0. Moreover, we know that B ( ; ) decreang n (part (v) of Lemma ). planner objectve functon, gven by B (; ) we have B (; ) B (; ). Moreover, at we have It follow that the C (), decreang for all,.e. for all C () < 0. It follow that the planner wll chooe the precon level B ( ; ) B ( ; ) C ( ) B ( ; ) < 0 and, thu, the planner would chooe <. Hence, when T and > nvetor over-acqure nformaton. Ung a mlar argument one can how that nvetor under-acqure nformaton when T and < and that they acqure the e cent level of nformaton when T and. Next, uppoe that T >. From Corollary 8 we know that when T there are global potve pllover n nformaton choce that are gnored by the nvetor at the tme they chooe the precon of ther gnal. Hence, n th cae B (; ) C 0 () 0 mple that B (; ) B (; ) C 0 () > 0. It follow that. Moreover, nce B (; ) trctly potve, t follow that at we have B ( ; ) B ( ; ) C 0 ( ) > 0, o that >. Ung an analogou argument we can etablh that f T > then for all b (T; ; ) we have <. It reman to conder the cae where T > and (T; b (T; ; )). Snce T >, from Lemma 6 we know that n th cae the threhold rt decreang and then ncreang n. Fx T, then for each (T; b (T; ; )) there ext a unque precon level, call t b ( ), uch that at th precon d level d 0. From the proof of Lemma 4 we know that b ( ) f and only f E (T ). From Lemma 4 we alo know that > E (T ) mple > b (T; ( ) ; ) and < E (T ) mple E d (T ) then d < 0; () f E d (T ) then d 0, and () f < E d (T ) then d > 0. From the above dcuon we conclude that f E (T ) ; b (T; ; ) then the equlbrum choce of uch that ncreang n at, o le at the ncreang porton of (). Th mean that for all [ ; ) we have negatve pllover e ect (Corollary 8). Thu, the optmal choce of precon mut atfy <, mplyng that nvetor over-acqure nformaton. Smlarly, f T; E (T ) then the choce of uch that decreang n, o le at the decreang porton of (). Th mean that for all [; ] we have potve pllover e ect and, thu, t ha to be the cae that >. Fnally, conder the cae when E (T ). We know that for

13 all <, decreang n (there are potve pllover e ect n nformaton choce), o that. Smlarly, we know that for all >, ncreang n (there are negatve pllover e ect n nformaton choce), mplyng that. Therefore, t ha to be the cae that. We conclude that when E (T ) nvetor chooe the e cent level of nformaton. Fnally conder the cae when T <. From Corollary 8 we know that f T then there are global negatve pllover and nvetor over-acqure nformaton. 3. Dcuon Propoton 9 allow u to determne when nvetor globally over-acqure or under-acqure nformaton. The logc of the proof mght be unclear and the reader mght wonder why we have not ued a more drect approach baed on the analy of rt-order and econd-order neceary condton for local and global maxma. In th ubecton we explan why uch an approach not feable. We alo provde a detaled ntuton behnd our approach and explan why t doe not work when T < and (b (T; ; ) ; T ). The optmal level of precon olve max B (; ) C ().e. t ha to atfy the followng rt order condton: B (; ) C 0 () B (; ) 0 wth equalty f >. The uual brute force approach would be to verfy that the objectve functon quaconcave, o that the rt-order condton u cent and then nd the value of that at e th condton. Then we would compare the oluton we found th way to the equlbrum precon choce. Th tandard approach, however, not applcable n the model. The rt d culty tem from the fact that the rt and econd order condton are complex object. In partcular, we are not able to verfy that the objectve functon quaconcave or even f the oluton to the rt-order condton, f t ext, a local maxmum. Indeed, our ntuton together wth numercal example tend to ugget that both need not be true,.e. that the utlty functon, dependng on parameter, can take many d erent hape and may have a local nteror mnmum. Moreover, t poble that the e cent precon choce a corner oluton, but gven the lack of a cloed-form oluton to the model we do not have the tool to determne when th the cae. In order to crcumvent thee problem we ued an ndrect approach. Th approach baed on nvetgatng the local and global behavor of a a functon of, and buld on the comparatve tatc reult for global game etablhed n Szkup (05). Th approach follow two tep. Frt, we etablh whether a mall ncreae or decreae n the precon from t equlbrum level would ncreae or decreae welfare. We refer to th a a local over-acquton or local under-acquton of nformaton. Th, of coure, a much mpler tak nce we know that at the equlbrum precon we 3

14 have B (; ) C 0 () 0 and, thu, the local reult are fully determned by the gn of. In the econd tep we ue the global properte of n order to extend th local reult to the planner precon choce. For example, f monotone we know that th local reult wll automatcally mply a global reult. Unfortunately, th approach fal when T < q and take value between T p (T ) and T. The ue that when T < then the threhold rt ncreang and then decreang n (Lemma 6). Whle we can etablh whether the equlbrum precon choce le on the ncreang or decreang porton of (when we conder a a functon of ) and we tll can obtan our local reult, thee local reult do not necearly decrbe the e cent choce of nformaton. In partcular, when le on the decreang porton of the local behavor of the objectve functon ndcate that a mall ncreae n precon would mprove welfare compared to the equlbrum. However, t poble that the actual e cent choce of precon maller than equlbrum. mght be ntally ncreang very fat, and then decreang very gradually, n whch cae the planner would chooe to acqure no nformaton. Wthout cloed-form oluton we are unable to determne when uch cae are. Note that th not a problem when T > becaue n th cae, f non-monotonc, rt decreang and then ncreang. 3.3 Numercal reult In th ubecton we provde a concrete numercal example whch how why the cae T < o problematc. Our goal to how that the d culte decrbed above not only are a theoretcal poblty that cannot be excluded, but do actually are for ome parameter value. The ocal planner objectve functon can be expreed a U (; ) Z Z x T df (xj) dg () Z Z x Z ( T ) dg () C () ( T ) df (xj) dg ().e., the ocal planner objectve functon the utlty of nvetor gven that all nvetor chooe the ame precon level. Thu, the planner objectve functon can be decompoed nto three term: () the expected cot of mtake (the rt two term), () the expected gan from nvetment (the thrd term), and (3) the cot of precon (the lat term). We et T 0:5 and 0:0755. The remanng parameter are 3, C () 00 ( ):0, and 3. 5 Fgure depct, a a functon of precon choce, welfare (top left panel), the expected gan from nvetment (top rght panel), the expected cot of mtake (bottom left panel), and the cot of precon (bottom rght panel). From Fgure we can ee that for thee parameter the welfare functon not only not concave, but alo not even quaconcave (top left panel). Th mple that the tandard approach baed on rt-order and econd-order condton not applcable. 5 The welfare functon non-quaconcave alo for value of parameter n the neghborhood of the parameter condered n th example. The reaon for th reult that the cot functon u cently at and the expected gan from nvetment ntally decreae harply a ncreae. 4

15 Mtake Cot Welfare Invetment Precon τ Precon τ Precon τ Precon τ Fgure : Welfare a a functon of The remanng panel provde an explanaton for why the welfare functon not quaconcave. 6 In partcular, we ee that an ntal ncreae n lead to a harp decreae n the expected gan from nvetment (whch drven by a harp ncreae n ). Th mple that welfare ntally decreang a ncreae from. However, a ncreae further, the negatve e ect of a hgher on nvetment decreae harply a approache t global maxmum. On the other hand, a hgher precon help the nvetor to avod cotly mtake (the bottom left panel). A ncreae, th bene t, due to a lower expected cot of mtake, domnate and the welfare functon become ncreang n. A ncreae further, the reducton n the expected cot of mtake become maller and maller. A uch, the welfare functon become agan decreang n, drven by the ncreang cot of a hgher precon (bottom rght panel). It mportant to note that for the welfare functon to be non-quaconcave t crucal that acheve a global maxmum at a nte. Only then the rate at whch ncreae fall u cently fat that the potve e ect of a decreae n the expected cot of mtake become a domnant force for ome value of. 6 Note that, for each, by ubtractng from the value of nvetment the value of mtake and the cot, we obtan the value of welfare correpondng to th precon level. 5

16 4 Strategc complementarte and welfare Lemma 0 A the threhold T. Proof. Recall that the unque oluton to the CM condton r ( ) (T ) ( ) 0 or ( ) r ( ) (T ) Takng the lmt a on both de of th equaton, we obtan lm ( ) (T ) or, lm T Lemma a decreang functon of. Proof. Conder the CM condton: Recall alo that ( ) r (T ) ( ) 0 > ( ( )) for all [; ). Applyng the Implct Functon Theorem to the CM condton we obtan: nce > p. ( ( )) < 0 Lemma Let T and conder.. If < then >,. If then, 3. If > then <, > 0 and < 0. 0 and 0. < 0 and > 0. 6

17 Proof. When T the CM condton become: ( ) ( ) 0 Snce > p, the left hand de of the above equaton decreang n. Moreover, t eay to ee that when the above equaton mple that. It then follow that when > t ha to be the cae that < and when < then >. Ung the above obervaton and applyng the Implct Functon Theorem to the CM condton we obtan the remanng reult regardng and. Lemma 3 Let T and conder.. If <. If 3. If > then > 0: then 0: then < 0: Proof. Note that d d where (; ) the optmal precon choce of nvetor when the mean of the pror equal to and all other nvetor chooe precon. The above expreon tell u that the dervatve of the equlbrum precon choce wth repect to equal to the change n nvetor equlbrum precon choce due to a change n, captured by, tme the multpler e ect that capture the fact that a ect equlbrum precon choce of the remanng nvetor, leadng to a further adjutment n. Fnally, nce we are ntereted n tudyng a change from an ntal ymmetrc equlbrum precon, thee e ect have to be evaluated at and. A hown n the proof of Theorem n the appendx of the paper, the equlbrum multpler e ect alway potve. Thu, to etablh whether an ncreae n lead to an ncreae or decreae n we focu on the partal e ect of a change n ha on,.e.. Applyng the Implct Functon Theorem to nvetor rt order condton we obtan: U U where the denomnator alway negatve (ee Lemma above). U x (x ) Snce < 0, t follow that: U gn gn (x ) 7

18 Next, note that when T then x ( ) It follow then from Lemma that f < then x > 0, f then x 0 and f > then x < 0. The reult follow mmedately from th obervaton. Lemma 4 Let T and conder. For all ( ; ) we have < < are the two oluton to the equaton: < 0 where ( ) ( ) Proof. The dervatve of wth repect to equal to: ( ( )) D erentatng agan, th tme wth repect to, we get: h ( ( )) h ( ) (( ( ))) h ( ( )) ( ( )) ( ) (( ( ))) h ( ( )) Snce the denomnator alway potve, we focu on the numerator of the above expreon: ( ( )) ( ) ( ( ( ))) ( ) ( ( )) ( ( ( ))) When T, the CM condton mple that ( ) ( ( ) ( ( )) Therefore, the numerator can be wrtten a: " ( ( ( ) )) ( ( ( ))) We argue below that the term n the quare bracket alway negatve. ). Moreover, # ( ( )) Note that f and only f and decreang n, t follow that for all ( ; ) : ( ) ( ) 8

19 Thu, < < ( ) ( ( ( ))) " ( ( )) ( ( )) ( ( )) ( ) ( ( ( ))) ( ( ( ) )) ( ( )) ( ( )) ( ( )) p # ( ) ( ) where the rt nequalty follow from the fact that ( ) < ( ) then for all ( ; ) we have. Therefore, a long a at e: < 0. ( ( )) ( ) > p < 0 and the obervaton that Lemma 5 Conder.. If < then < 0:. If then 0: 3. If > then > 0: Proof. The dervatve of wth repect to gven by: ( 3 ) ( ( )) D erentatng the above equaton wth repect to we get: h ( 3 ) 3 Note that when then, 3 ( ) h h h ( ( )) ( ( )) ( ) (( ( ))) h ( ( )) 0. It follow that 0. Next, conder the cae when <. In that cae, > > and > 0. Snce ( ( )) < 0 t follow that < 0. Smlarly, when > then < < and thu > 0. < 0, and, 9

20 4. The trade-o between publc and prvate nformaton when T 6 When analyzng the e ect of an ncreae n the precon of publc nformaton on prvate nformaton acquton we aumed that T. Th aumpton mpl ed the analy and allowed u to characterze fully when an ncreae n the precon of publc nformaton encourage and when t dcourage prvate nformaton acquton. Here we conder the trade-o between publc and prvate nformaton when T 6. In partcular, we provde u cent condton for prvate and publc nformaton to be complement and ubttute. We complement our analytcal reult wth reult obtaned from numercal mulaton. Numercal mulaton con rm our ntuton that the pave nformaton e ect the key for determnng whether prvate and publc nformaton are complement or ubttute. Propoton 6 There ext T L and T H, 0 < T L < < T H < uch that for each T (T L ; T H ) there ext a non-empty nterval of, call t I (T ), uch that for all I (T ) publc and prvate nformaton are complement. Th reult extend Propoton 5 n the paper that characterze complementarty between publc and prvate nformaton for the pecal cae when T. The proof cont of three tep and mlar to the proof of Propoton 5. We rt x T and how that whether prvate and publc nformaton are complement depend on the relatve trength of three e ect: the pave nformaton e ect, the actve nformaton e ect and the coordnaton e ect. Next, we nvetgate eparately when each of thee e ect potve (encouragng nformaton acquton) and when they are negatve (dcouragng nformaton acquton). Fnally, we how that there ext an nterval (T L ; T H ) uch that for each T (T L ; T H ) there an nterval of value for where all thee e ect are potve. Proof. A hown n the proof of Propoton 5 n the paper, whether prvate and publc nformaton are complement depend on the gn of U U, whch gven by: x 6 4 ( ) (x ) x {z } (x ) {z 7 5 } {z } the pave nformaton e ect the actve nformaton e ect the coordnaton e ect We conder rt the pave nformaton e ect. Followng an analogou approach a n the proof of Propoton 5 one can how that for each T (0; ) there ext b (T ) and b (T ), wth b (T ) < b (T ) uch that for all b (T ) ; b (T ) the pave nformaton e ect potve. 7 The threhold b (T ) 7 The d erence wth the cae of T that now we cannot clam that for all b (T ) ; b (T ) the pave nformaton e ect negatve. Intead, we can only how that there ext b (T ) and b (T ), wth b (T ) > b (T ) and b (T ) < b (T ), uch that for all ; b (T ) [ hb (T ) ; the pave nformaton e ect negatve. See Propoton

21 de ned mplctly a the larget oluton to the equaton: 8 ( ) ( ) ( ( ) ) ( ) ( (T ) ) ( ) ( ( ) ) whle b (T ) de ned a the mallet oluton to the equaton: ( ) ( ) ( ( ) ) ( ) (T ) ( ) For each T (0; ) let I P (T ) ( ) ( ( ) ) b (T ) ; b (T ). Thu, I P (T ) the et of value for for a xed T uch that the pave nformaton e ect potve (.e., t encourage nformaton acquton). Next, conder the actve nformaton e ect and note that the actve nformaton e ect encourage prvate nformaton acquton f and only f (x ) x < 0 where 9 o that x ( ) p (T ) x < 0 ( ) ( ) ( ) (T ) ( ) p (T ) ( ) From npectng the left hand de and the rght hand de of the above nequalty, we conclude that t at ed for a u cently hgh, whle for a u cently low the above nequalty revered. Moreover, nce both de are contnuou n, t follow that there ext at leat one value of uch that: ( ) ( ) ( ) (T ) ( ) p (T ) ( ) Let e (T ) denote the larget value of that at e the above equalty when T > and the mallet oluton when T <. Then, for T > ; whenever > e (T ) ; then x j < 0; whle for T < we know that whenever < e (T ) ; then x j > 0 8 To ee th we olve ( ) 0 for. Th lead to two oluton for, one for the cae when > and one for the cae when <. Subttutng the rt oluton nto the CM condton and rearrangng yeld the equaton that de ne b. Followng the ame tep for the cae when < yeld the equaton that de ne b. 9 Here we gnore the e ect of a change n on. Th captured by the coordnaton e ect.

22 Next, conder the equaton: p ( ) (T ) Agan, t mmedate that the above equaton ha at leat one oluton. Moreover, t can be hown that th oluton unque. Call the unque oluton of the above equaton (T ). Ung the CM and PI condton one can etablh that x > 0 f > (T ), x 0 f (T ) ; and x < 0 f < (T ). We clam next that for all T > we have (T ) > e (T ) ; whle for all T < we have (T ) < e (T ). To ee th x and note that at T we have: p " r (T ) (T ) # p (T ) 0 Moreover, T " p (T ) r (T ) # p (T ) p ( (T )) p ( (T )) p ( (T )) p p ( (T )) r (T ) " r p p p > 0 r # p ( (T )) nce we aumed that p > p (ee Aumpton A n the paper). It follow that, for an exogenouly gven, f T > then p (T ) > r (T ) p (T ) whle the oppote nequalty hold when T <. Thu, f wa exogenou then (T ) > e (T ) for T > and (T ) < e (T ) f T <. Next, recall that p ( ) (T ) ha a unque oluton and note that p ( ) (T ) tend to a T 0 and tend to a T. Puttng together the above obervaton we conclude that (T ) larger than e (T ) for T > and maller than e (T ) for T <.0 0 More precely, de ne F ( ) p ( ) (T )

23 We have etablhed that f T > then: () (T ) > e (T ), () for all < (T ) we have x < 0; and (3) for all > e (T ) we have x j > 0. Thu, f T > then for all (e (T ) ; (T )) we have (x ) x j < 0 and, hence, the actve nformaton e ect potve. Smlarly, f T < then for all ( (T ) ; e (T )) we have (x ) x j < 0 and, hence, the actve nformaton e ect potve. Fnally, f T then (T ) e (T ) and the actve nformaton e ect alway non-potve (a argued n the proof of Propoton 5 n the paper). For each T 6, let I A (T ) denote the nterval of uch that the actve nformaton e ect potve. Fnally conder the coordnaton e ect. The gn of the coordnaton e ect determned by the gn of (x ). In partcular, the coordnaton e ect encourage nformaton f and only f (x ) < 0 Frt conder. From Szkup (05) we know that > 0 whenever > ( ) ( ) ( ) (T ) p (T ) ( ) ( ) A above, let e (T ) be the larget oluton to ( ) ( ) ( ) (T ) p (T ) ( ) ( ) when T >, and the mallet oluton when T <. Gven th de nton of e (T ) t follow that f T > and > e (T ) then > 0; whle f T < and < e (T ) then < 0. and Next, conder x G ( ). Ung the PI condton we have: x > 0 > ( ) (T ) ( ) x 0 x < 0 < and uppoe that T >. Then, for any we have: Thu, at e (T ) we have ( ) ( ) ( ) (T ) ( ) F ( ) < G ( ). ( ) (T ) ( ) ( ) (T ) ( ) e (T ) F (e (T )) < e (T ) G (e (T )) 0: p (T ) ( ) Snce F ( ) ha a unque xed pont and nce F ( ) < 0 for u cently mall and F ( ) > 0 for u cently large, t follow that the unque xed pont of the equaton F ( ) (denoted (T )) larger than e (T ). An analogou argument etablhe that for T < we have (T ) < e (T ). Note that e (T ) contnuou n T nce at T the above equaton ha a unque oluton, e (T ). 3

24 q Ung the CM condton t can be hown that > ( ) ( ) (T ) f and only f < ( ) (T ) p (T ) ( ) ( ) and q ( ) ( ) (T ) f and only f ( ) (T ) ( ) p ( ) (T ) In Lemma A. n the paper we howed that the above equaton ha a unque oluton, whch we call E (T ), uch that f > E (T ) then > ( ) (T ) p (T ) ( ) ( ) whle the oppote true f < E (T ). Therefore, f < E (T ) then x > 0, f E (T ) then x 0, and f > E (T ) then x < 0. Next, followng analogou tep a n the cae of the actve nformaton e ect one can how that f T > then E (T ) > e (T ), f T then E (T ) e (T ) and f T < then E (T ) < e (T ). Thu, we have etablhed that f T > then: () E (T ) > e (T ), () for all > e (T ) we have j < 0, and () for all < E (T ) we have x > 0. Thu, f T > then for all e (T ) ; E (T ) we have (x ) j < 0 and, hence, the coordnaton e ect f potve. Smlarly, f T < then: () E (T ) < e (T ), () for all < e (T ) we have j > 0, and () for all > E (T ) we have x < 0. It follow that f T < then for all e (T ) ; E (T ) we have (x ) j < 0 and, hence, the coordnaton e ect potve. Fnally, f T then E (T ) e (T ) and the coordnaton e ect alway non-potve (a argued n the proof of Propoton 5 n the paper). For each T 6 denote by I C (T ) the nterval of uch that the coordnaton e ect potve. The nal tep of the proof requre u to how that there ext an nterval (T L ; T H ) uch that for each T (T L ; T H ), the et I (T ) I P (T )\I A (T )\I C (T ) ha a non-empty nteror. Frt note that for T > we have e (T ) ; E (T ) I A (T )\I C (T ) and for T < we have E (T ) ; e (T ) I A (T )\I C (T ). Thu, t reman to conder the nterecton of I P (T ) wth e (T ) ; E (T ). More precely, we need to compare b (T ) and b (T ) wth e (T ) and E (T ). Frt, conder the cae when T. When T then e (T ) E (T ) whle b (T ) < lm T 0 e (T ) that for all T T L ; we have b (T ) < e (T ). Next, de ne T L. Thu, we know a the larget T < uch Th requre howng that for each T > we have E (T ) < (T ) ; whle for each T < we have E Th can be etablhed n exactly the ame way a the prevou nequalte n th proof. (T ) > (T ). 4

25 that b (T ) E (T ). If there no uch T,.e., f for all T < we have b (T ) > E (T ) (note that at T we have E (T ) E (T ). Fnally, de ne T L max T L ; T L I P (T ) \ E (T ) ; e (T )?.. We clam that for all T (T L; ) we have To ee th uppoe that T L T L, o that T L > T L, and conder any T > T L. By de nton of T L t ha to be the cae that at T > T L we have e (T ) > b (T ). Snce b (T ) > b (T ) t follow that E (T ) ; e (T ) \ b (T ) ; b (T )?. It follow that mplyng that for all T > T L we have I P (T ) \ E (T ) ; e (T )? and by de nton of thee nterval, we know that for all I P (T )\ E (T ) ; e (T ) prvate and publc nformaton are complement. Next, conder the cae when T L T L. Th mple that T L > T L and hence there ext 0 < T < uch that E (T ) b (T ). Recall that by de nton of T L we know that for all T > T L, b (T ) > E (T ). Snce T L > T L, we know that for all T > T L we have e (T ) > b (T ). Thu, b (T ) ; b (T ) \ E (T ) ; e (T ) 6?. Th etablhe the above clam. In a mlar fahon we de ne T H >. Th complete the proof a we etablh extence of T L and T H, where T L < < T H uch that for all T (T L ; T H ) we have I (T ) 6?. Above we howed that for all T (T L ; T H ) (0; ) there are value of uch that prvate and publc nformaton are complement. One may wonder whether we can extend the above reult to how that for each T (0; ) there a et of value for uch that prvate and publc nformaton are complement. Unfortunately, the argument ued above doe not allow u to prove th more general reult. In partcular, our argument rele on the fact that for all T (T L ; T H ) we have I P (T ) \ I A (T ) \ I C (T ) 6?. However, for T cloe enough to (or cloe enough to 0) we necearly have I P (T ) \ I A (T ) \ I C (T )?. In other word, for extreme value of T there are no value of uch that all the e ect of an ncreae n the precon n publc nformaton are potve. To ee th, note that lm (T ) T lm T E (T ) whle lm T b (T ) lm b (T ) T ( ) ( ) where for notatonal convenence we wrote ntead of lm T b (T ) ; T. It follow that for T cloe enough to we have E (T ) > e (T ) > b (T ) > b (T ). Therefore, when the cot of nvetment, T, very hgh or very low one would need to compare the magntude of each of the e ect. Gven the complexty of the model, th, however, nfeable. 5

26 Above we nvetgated condton under whch publc and prvate nformaton are complement for the cae T 6. In the next propoton, we provde u cent condton under whch prvate and publc nformaton are ubttute. In partcular, below we characterze regon n whch the pave, actve and coordnaton e ect are negatve. A uch, the next proof wll utlze everal reult etablhed n the proof of Propoton 6. Before we tate our reult we need to de ne everal mportant threhold for. Thee threhold alo play a crucal role n the proof of Propoton 6 where we alo addre the ue of ther extence. Let b (T ) be the mallet oluton to the equaton: ( ) ( ) ( ( ) ) ( ) ( (T ) ) ( ) ( ( ) ) Let b (T ) be the larget oluton to the equaton: ( ) ( ) ( ( ) ) ( ) (T ) ( ) ( ) ( ( ) ). Smlarly, let e be the mallet, f T >, or the larget, f T <, oluton to the equaton: ( ) ( ) ( ) (T ) p (T ) ( ) ( ) and (T ) be the unque oluton to: p ( ) (T ) Wth the above de nton, we can tate our mot general reult that provde u cent condton for the ubttutablty between prvate and publc nformaton. Propoton 7 Conder an nvetor ncentve to acqure nformaton.. Suppoe that T >. If mn ne o n ; b (T ) ; max (T ) ; b o (T ) then prvate and publc nformaton are ubttute.. Suppoe that T. If b (T ) ; b (T ) then prvate and publc nformaton are ubttute. 3. Suppoe that T <. If publc nformaton are ubttute. n o mn (T ) ; b (T ) ; max ne (T ) ; b o (T ) then prvate and Proof. To prove th reult we ue the ame approach a we ued to etablh Propoton 6 wth the d erence that now we focu on the regon n whch the pave nformaton e ect, the actve nformaton e ect and the coordnaton e ect are negatve. Note alo that part () of the above propoton follow mmedately from Propoton 5 n the paper. Therefore, we only need to prove part () and part (3). Below, we conder the cae when T >. The proof for the cae when T < analogou. 6

27 Fx T > and conder rt the pave nformaton e ect. The pave nformaton e ect dcourage nformaton acquton f and only f ( ) > 0 Note that > and ncreang n. Moreover, lm 0 whle lm. It follow that there ext b (T ) and b (T ), wth b (T ) 0, and f b (T ) or b (T ) then ( ) 0 Followng an analogou argument a n Propoton 5 n the paper t can be hown, ung the CM condton and the above obervaton, that b (T ) the mallet oluton to the equaton: ( ) ( ) ( ( ) ) ( ) ( (T ) ) ( ) ( ( ) ) whle b (T ) the larget oluton to the equaton: ( ) ( ) ( ( ) ) ( ) ( (T ) ) ( ) ( ( ) ) Thu, we conclude that whenever b (T ) ; b (T ) then the pave nformaton e ect negatve. 3 Conder now the actve nformaton e ect and note that the actve nformaton e ect dcourage nformaton acquton f and only f (x ) x > 0 A we howed n the proof of Propoton 6, for a gven T, we have x > f and only f < (T ), x f and only f (T ) and x < f and only f > (T ) where (T ) the unque oluton to: p () where (T ) depend on T through the mpact the cot of nvetment ha on the equlbrum precon choce,. Next, note that a change n x holdng contant gven by x ( ) p 3 In Propoton 6 we found bound b (T ) and b (T ) uch that for all b (T ) ; b (T ) the pave nformaton e ect encourage prvate nformaton acquton. Unle the equaton de nng b (T ) and b (T ) ha a unque oluton, we have b (T ) < b (T ). Smlarly, unle the equaton de nng b (T ) and b (T ) ha a unque oluton, we have b (T ) < b (T ). In general, t unclear whether thee condton have a unque oluton or not and, hence, t not necearly true that the pave nformaton e ect potve f b (T ) ; b (T ). 7

28 A argued n the above Propoton, x < 0 f and only < ( ) ( ) ( ) (T ) p (T ) ( ) ( ) and x > 0 f and only f > ( ) ( ) ( ) (T ) p (T ) ( ) ( ) Fnally, x 0 f and only f a oluton to the followng equaton: ( ) ( ) ( ) (T ) p (T ) ( ) ( ) Let e (T ) be the mallet oluton to the above equaton. Snce for a xed T the rght hand de of the above equaton bounded and ( ; ) ; t follow that uch e (T ) ext. Recall that n the proof of Propoton 6 we howed that any oluton to the equaton: ( ) ( ) ( ) (T ) p (T ) ( ) ( ) ha to be maller than (T ). Therefore, e (T ) < (T ). From the above dcuon t follow that f > (T ) then x < and x < 0, whle f < e (T ) then x > and x > 0. Hence, we conclude that for all ; e (T ) [( (T ) ; ) the actve nformaton e ect dcourage prvate nformaton acquton. Fnally, we conder the coordnaton e ect. The coordnaton e ect dcourage nformaton acquton f and only f (x ) > 0 In the proof of the Propoton 6 we howed that x > 0 f < E (T ), x 0 f E (T ) and x < 0 f > E (T ). Next, conder j and recall that whle analyzng the actve nformaton e ect we de ned e (T ) a the mallet oluton to ( ) ( ) ( ) (T ) p (T ) ( ) ( ) A hown n Szkup (05), whenever < ( ) ( ) ( ) (T ) p (T ) ( ) ( ) then j > 0; whle f the drecton of the above nequalty revered then j < 0. Therefore, from the de nton of e (T ) t follow that for all < e (T ) we have j > 0. 8

29 Moreover, n the proof of Propoton 6, we argued that, for T >, E (T ) larger than any oluton to: ( ) ( ) ( ) (T ) ( ) p (T ) ( ) It follow that: () E (T ) > e (T ) ; and () for all > E (T ) we have j < 0. Thu, we have etablhed that for all e (T ) ; E (T ) the coordnaton e ect negatve. Above we have etablhed that for each T > the pave nformaton e ect negatve f b (T ) ; b (T ), the actve nformaton e ect negatve f ; e (T ) [ ( (T ) ; ) and the coordnaton e ect negatve f ; e (T ) [ E (T ) ;. Snce for T > we have (T ) > E (T ) (ee the proof of Propoton 6) part () of the Propoton follow mmedately. The proof of part (3),.e., the cae when T < analogou. The man d erence, compared to the cae when T >, that when T < then E (T ) > (T ). Moreover, for T < we need to de ne e (T ) a the larget oluton to: ( ) ( ) ( ) (T ) p (T ) ( ) ( ) where t can be hown that for T < we have e (T ) > E (T ). 4. The trade-o between publc and prvate nformaton when T 6 : numercal reult In Propoton 6 and 7 we etablh u cent condton under whch prvate and publc nformaton are complement (Propoton 6) and ubttute (Propoton 7). We complement the above analytcal reult wth numercal mulaton. In partcular, Fgure provde a complete characterzaton of the regon where a margnal ncreae n lead to an ncreae n nvetor ncentve to acqure nformaton when (left panel) and when (rght panel). Fgure how that complementarty between publc and prvate nformaton a pervave phenomenon not lmted only to the cae when T. Moreover, unle T take extreme value, the regon where prvate and publc nformaton are complement fall nde the regon where the pave nformaton e ect potve, whch ugget that even when T 6 the pave nformaton e ect play an mportant role n drvng th reult. 4 Note that a ncreae, the regon n whch the two type of nformaton are complement hrnk. The man reaon that a ncreae, the regon where the pave nformaton e ect potve hrnk (th can be deduced from the expreon for the pave nformaton e ect, ee proof of Propoton 7). Th further undercore the mportance of the pave nformaton e ect. Recall that when T 6, t poble that the actve nformaton e ect potve. In order to nvetgate the role played by the actve nformaton e ect, n Fgure 3 we plot the regon where the 4 In Fgure the regon between the two dahed lne correpond to the regon where the pave nformaton e ect potve. 9

30 µ θ 0.5 µ θ Subttute Complement Pave T 0.75 Subttute Complement Pave T Panel A: Panel B: :5 Fgure : Relaton between prvate and publc nformaton: pave nformaton e ect actve nformaton e ect potve (the regon between the dah-dot lne) agant the regon where the pave nformaton e ect potve (the regon between the dahed lne) µ θ 0.5 µ θ Subttute Complement Pave0 Actve T 0.75 Subttute Complement Pave0 Actve T Panel A: Panel B: :5 Fgure 3: Relaton between prvate and publc nformaton: pave and actve nformaton e ect We ee that the regon where the actve nformaton e ect potve, wth excepton for the cae when T extreme, d er ubtantally from the regon where publc and prvate nformaton are complement. Thu, Fgure 3 provde addtonal upport for the mportance of the pave nformaton e ect n explanng the complementarty between publc and prvate nformaton. 5 Change n nvetment and welfare: robutne 5. Change n the probablty of a ucceful nvetment In ecton 6: of the paper we analyzed how a change n the precon of the pror a ect the probablty of a ucceful nvetment when T. Here, we nvetgate the cae when T 6. We tate rt 30

31 ntermedate techncal reult and then addre th queton. 5.. Intermedate reult Lemma 8 Suppoe that exogenouly gven. q. If > (T ) then <. q. If (T ) then. q 3. If < (T ) then >. Proof. Recall that de ned a the oluton to the CM condton: r ( ) (T ) ( ) 0 The above lemma follow from the obervaton that the left hand de of the above equaton decreang n. Next, conder the followng equaton ( ) (T ) ( ) De ne M (T ) a the larget oluton to the above equaton when T > and the mallet oluton when T <. Note that M (T ) well de ned nce the rght hand de take only value between 0 and. Next, recall from Secton 4: that we de ned e (T ) a the mallet, f T >, or the larget, f T <, oluton to equaton ( ) ( ) ( ) (T ) p (T ) ( ) ( ) Lemma 9 Conder M (T ).. If T > then M (T ) < e (T ).. If T then M (T ) e (T ). 3. If T < then M (T ) > e (T ). 3

32 Proof. Frt, note that f T then M (T ) e (T ). Next, holdng contant, take the dervatve of M (T ) e (T ) wth repect to T to obtan: h M (T ) e r r (T ) T ( (T )) (T ) r ( (T )) r (T ) p ( (T )) " r p # p p p p ( ) ( (T )) ( (T )) < 0 Hence, t follow that f T > then, for a gven, we have r r (T ) < (T ) p (T ) In partcular, at M (T ) we have M M (T ) (T ) M (T ) (T ) M < (T ) M (T ) M (T ) M (T ) (T ) q (T ) M (T ) Recall from Secton 4: that, for T >, e (T ) de ned a the value of uch that for all e (T ) we have: M > (T ) M (T ) M (T ) M (T ) (T ) q (T ) M (T ) Thu, f T > then, M (T ) < e (T ). The proof for the cae T < analogou. Fnally, de ne Inv (T ) a n Inv (T ) max (T ) ; b o (T ) f T > n o Inv (T ) max M (T ) ; b (T ) f T < and Inv (T ) a n Inv (T ) mn M (T ) ; b o (T ) f T > n o Inv (T ) mn (T ) ; b (T ) f T < It follow from the de nton of b (T ), ; b (T ), M (T ) and (T ) that both Inv (T ) and Inv (T ) are contnuou and Inv (T ) Inv (T ) when T : Below, we ue Inv (T ) and Inv (T ) to tate our man reult of th ecton. 3

33 5.. Change n the probablty of a ucceful nvetment: man reult In Secton 6: we dent ed three channel through whch a change n a ect the probablty of a ucceful nvetment. Frt, t a ect the ex-ante dtrbuton of. Second, t a ect drectly the value of the threhold through an adjutment n nvetor nvetment tratege (holdng ther precon choce contant). Thrd, t lead ndrectly to a change n by a ectng nvetor precon choce, and through th channel t further a ect ther nvetment behavor. We then proved that when T a change n ncreae the probablty of a ucceful nvetment when >, t ha no e ect f, and t decreae t when <. When T 6 uch a characterzaton not feable. The man ue here that t d cult to etablh both the gn and the magntude of the econd and thrd channel once T 6. Intutvely, however, we hould expect that the ex-ante probablty of a ucceful nvetment ncreae when hgh and decreae when low. Th becaue the drect e ect of a change n on (the econd channel dent ed above) tend to domnate the other two channel and t potve for hgh value of the pror mean and negatve otherwe. Propoton 0 how that th ntuton correct when u cently hgh or low. De ne Inv (T ) a n Inv (T ) max (T ) ; b o (T ) f T > n o Inv (T ) max M (T ) ; b (T ) f T < and Inv (T ) a n Inv (T ) mn M (T ) ; b o (T ) f T > n o Inv (T ) mn (T ) ; b (T ) f T < Propoton 0 Suppoe that the precon of the pror,, ncreae.. Let T > : (a) If > Inv (T ) then the ex-ante probablty of a ucceful nvetment ncreae. (b) f < Inv. Let T < : (T ) then the ex-ante probablty of a ucceful nvetment decreae. (a) If > Inv (T ) then the ex-ante probablty of a ucceful nvetment ncreae. (b) If < Inv (T ) then the ex-ante probablty of a ucceful nvetment decreae. Proof. Recall from the proof of Propoton 6 n the paper that the total e ect of a change n on the probablty of a ucceful nvetment gven by: " # d Pr ( > ) d d d 33 d d ( )

Additional File 1 - Detailed explanation of the expression level CPD

Additional File 1 - Detailed explanation of the expression level CPD Addtonal Fle - Detaled explanaton of the expreon level CPD A mentoned n the man text, the man CPD for the uterng model cont of two ndvdual factor: P( level gen P( level gen P ( level gen 2 (.).. CPD factor