Technical Appendix o accompany Uncerainy abou Governmen Policy and Sock Prices Ľuboš Pásor Universiy of Chicago, CEPR, and NBER Piero Veronesi Universiy of Chicago, CEPR, and NBER July 8, 0
Conens. Learning Page Opimal changes in governmen policy Page Sock prices Page 3 Exension: Endogenous iming of policy change Page 6 Exension: Invesmen adjusmen Page 3 Exension: Heerogeneous beas Page 36
Background Informaion. Firm profiabiliy evolves sochasically for all 0, T: dπ i µ + g d + σdz + σ dz i. B The prior disribuions of boh g old and g new a ime 0 are normal: g N 0, σ g. B Learning. Proposiion. Observing he coninuum of signals dπ i in equaion B across all firms i 0, is equivalen o observing a single aggregae signal abou g : ds µ + g d + σdz. B3 Under he prior in equaion B, he poserior for g a any ime 0, T is given by g N ĝ, σ. B4 For all τ, he mean and he variance of his poserior disribuion evolve as dĝ σ σ dẑ B5 σ +, B6 σ σ g where he expecaion error dẑ is given by dẑ ds E ds /σ for all 0, T. If here is no policy change a ime τ, hen he processes B5 and B6 hold also for > τ. If here is a policy change a τ, hen ĝ jumps from ĝ τ o zero righ afer he policy change, and for > τ, ĝ follows he process in equaion B5. In addiion, for > τ, σ follows σ σ g + τ. B7 σ Proof of Proposiion : Le denoe a small ime inerval. Each signal s i dπ i can be wrien as he sum of a common componen and idiosyncraic noise. The common componen is c µ + g + σε, where ε N 0,. Therefore, each signal a ime is given by s i c + σ ε i where he ε i s are cross-secionally independen and disribued as ε i N 0,. Consider he informaion from he cross-secion of signals for a given ime. Condiional on c, all firm-level signals s i are independen, as a resul of which hey reveal c perfecly. However,
hese signals canno reveal more han c for any, because c is he same across signals. I follows ha he agens informaion se includes he common componen c and nohing else. As 0, we have c ds. This common signal is equal o average profiabiliy across firms: ds 0 dπi di. Subsiuing for dπ i from B yields ds µ + g d + σdz + σ dz i. The Law of Large Numbers implies ha he las inegral is idenically zero, as 0 dzi E i dz i 0, where E i denoes an expecaion aken across i. Having esablished he equivalence in B3, he remainder of he proposiion follows from sandard resuls abou he Kalman Bucy filer see e.g. Lipser and Shiryaev 977. QED Proposiion 0 Opimal Changes in Governmen Policy. Saemen from he ex: The aggregae capial a ime T, B T 0 Bi Tdi, is given by µ+g B T B τ e σ T τ+σz T Z τ, B8 where g g old if here is no policy change and g g new if here is one. Proof: From he capial evoluion equaion db i B i dπ i, where dπ i is given in B, we immediaely obain he following expression for firm i s capial a ime T: B i T Bi τ e µ+g σ σ «T τ+σz T Z τ+σ Z i T Zi τ, B9 where g g old if here is no policy change and g g new if here is one. Aggregaing across firms, we obain B T 0 B i Tdi e The Law of Large Numbers implies ha 0 B i τe σ Z i T Zi τ di E i µ+g σ σ «T τ+σz T Z τ 0 B i τe σ Z i T Zi τ di. Bτe i σ ZT i Zi τ E i Bτ i E i e σ ZT i Zi τ, B0 B where he las sep follows from he fac ha he random variables B i τ and Z i T Z i τ are independen of each oher. The firs expecaion on he righ-hand side of B is E i Bτ i Bτdi i B τ. 0
The second expecaion is E e i σ ZT i Zi τ e σ T τ. Subsiuing boh expecaions ino B0, we obain he claim in B8. QED Saemen from he ex Proposiion. The governmen changes is policy a ime τ if and only if ĝ τ < gc, B where gc σ g σ τ γ T τ c T τγ. B3 Proof of Proposiion : Using he marke clearing condiion W T B T, we can use B8 o compue he expeced uiliy a ime T condiional on a policy change yes or no policy change no. The expecaion is condiional on he governmen s informaion se, which includes he realizaion of he poliical cos c. Recall ha if he governmen changes is policy, hen g N0, σ g; if i does no, hen g Nĝ τ, σ τ. CW γ T E γ yes W γ T E γ no B γ τ γ ec+ γµ σ T τ+ σ g γ T τ + γ σ T τ B γ τ e γµ+bgτ σ T τ+ bσ τ γ T τ + γ σ T τ γ The claim of he proposiion follows immediaely from he opimaliy condiion CW γ T W γ E γ yes T > E γ no. B4 B5 QED Proposiion Sock Prices. Proposiion 3. Each firm s sock reurn a he announcemen of a policy change is given by where Rĝ τ pĝ τ F ĝ τ Gĝ τ pĝ τ + pĝ τ F ĝ τ Gĝ τ, F ĝ τ e γbgτt τ γ T τ σ g bσ τ Gĝ τ e bgτt τ γt τ σg bσ τ γ pĝ τ N ĝ τ γt τ 3 T τ σg σ τ σ ; c, σ c B6 B7 B8, B9
and Nx; a, b denoes he c.d.f. of a normal disribuion wih mean a and variance b. The proof of Proposiion 3 comes afer he proof of he following saemen in he ex. Saemen from he ex: Righ afer ime τ, a ime τ+, he marke value of each firm i akes one of wo values: M i,yes Mτ+ i τ+ Bτ+ i e µ γσ T τ+ γ T τ σ g if policy changes B0 M i,no τ+ Bτ+ i e µ γσ +bg τt τ+ γ T τ bσ τ if policy does no change Righ before ime τ, he marke value of firm i is M i τ ωm i,yes τ+ + ω M i,no τ+, B where he weigh ω, which is always beween 0 and, is given by p τ ω p τ + p τ F ĝ τ, B using he abbreviaed noaion p τ pĝ τ. Proof: The sochasic discoun facor is π λ E B γ T. Is value righ afer he policy decision, a ime τ+, is given by π τ+ λ Bτ+E γ e γµ+g σ T τ γσz T Z τ B3 π yes τ+ λ B γ T τ σg if policy changes π no τ+ λ B γ τ+ e γµ+ γγ+σ T τ+ γ τ+ e γµ+bgτ+ γγ+σ T τ+ γ where we have used B8. Righ before he policy decision, we have π τ E τ π τ+ p τ π yes τ+ + p τ π no τ+, B4 T τ bσ τ if policy does no change B5 where p τ pĝ τ Prob ĝ τ < gc is he probabiliy of a policy change from he perspecive of invesors who know ĝ τ bu no c. I is easy o see from B3 ha his probabiliy is given by expression B9. The marke value of sock i is given by M i π E π T BT i π λ E B γ T T Bi. Righ afer a policy decision, a ime τ+, using boh B8 and B9, we obain E τ+ B γ T Bi T B γ τ+bτ+ i E τ+ e γµ+g σ T τ γσz T Z τ e µ+g σ σ «T τ+σz T Z τ+σ ZT i Zi τ Bτ+B γ τ+e i τ+ e γµ+g σ T τ+ γσz T Z τ e σ T τ+σ ZT i Zi τ Bτ+B γ τ+e i τ+ e γµ+g σ T τ+ γσz T Z τ E τ+ e σ T τ+σ ZT i Zi τ B γ τ+bτ+e i τ+ e γµ+g σ T τ+ γσz T Z τ τ+bτ+ i e γµ+ γγ σ T τ+ γ B γ Bτ+B γ τ+ i e γµ+bgτ+ γγ σ T τ+ γ 4 T τ σ g if policy changes T τ bσ τ if policy does no change
These expressions can be subsiued ino M i,yes τ+ λ E τ+ B γ T Bi T yes /π yes τ+ and M i,no λ E τ+ B γ T Bi T no /πτ+ no o yield equaion B0. M i τ Finally, he sock price righ before he policy decision announcemen is equal o E τ Eτ+ λ B γ T Bi T p τe τ+ λ B γ T Bi T yes + p τ E τ+ λ B γ π τ p τ π yes τ+ + p τ πτ+ no p τπτ+m yes i,yes τ+ + p τ πτ+m no i,no τ+ p τ π yes τ+ + p τ πτ+ no which is equivalen o B when we define ω p τ π yes τ+ p τ π yes τ+ + p τ πτ+ no I follows immediaely from B4 ha Fĝ τ πno τ+ QED Saemen from he ex π yes τ+ p τ p τ + p τ πno τ+ π yes τ+. τ+ T Bi T no Proof of Proposiion 3: From he definiion of he announcemen reurn, we have Rĝ τ Mi,yes τ+ M i τ M i τ ωmi,yes τ+ M i,no τ+ ωm i,yes τ+ + ωm i,no τ+ ω Mi,no τ+ /M i,yes τ+ ω + ωm i,no τ+ /M i,yes τ+ B6 I is easy o see from B0 ha Gĝ τ M i,no τ+ /M i,yes τ +. Subsiuing his expression and ω from B proves he claim. QED Proposiion 3 Corollary. As risk aversion γ, he announcemen reurn Rĝ τ for any ĝ τ. Proof of Corollary : For any given ĝ τ, expression B9 implies ha p τ 0 as γ, as he erm involving γ dominaes he limi. In addiion, an applicaion of l Hospial s rule shows ha p τ /Fĝ τ Gĝ τ 0. Finally, Gĝ τ rivially diverges o infiniy. Therefore, R ĝ τ p τ F ĝ τ Gĝ τ p τ + p τ F ĝ τ Gĝ τ p τ /Gĝ τ p τ /F ĝ τ Gĝ τ + p τ. QED Corollary Corollary. As risk aversion γ, he expeced value of he announcemen reurn goes o zero E {Rĝ τ } 0, where he expecaion is compued wih respec o ĝ τ as of ime 0. Proof of Corollary : From expression B9, we have p τ p N0, σ c, σ c as γ. In addiion, Fĝ τ /Gĝ τ. Therefore, R ĝ τ p τ F ĝ τ Gĝ τ p τ + p τ F ĝ τ Gĝ τ p τ F ĝ τ F ĝ τ Gĝ τ p τ + p τ F ĝ τ Gĝ τ p e bgτt τ T τ σg bσ τ 5
Recalling ha ĝ τ N0, σ g σ τ, i is easy o compue he expecaion: E R ĝ τ p e + T τ σg bσ τ T τ σg bσ τ 0. QED Corollary Proposiion 4. The marke value of each firm drops a he announcemen of a policy change i.e., Rĝ τ < 0 if and only if ĝ τ > g, B7 where g σ g σ τ T τ γ. B8 Proof of Proposiion 4: From expression B6, we see ha Rĝ τ < 0 if and only if Gĝ τ >. From he formula for Gĝ τ in B8, we see ha he laer condiion is saisfied if and only if condiion B7 is. QED Proposiion 4 Proposiion 5. The expeced value of he announcemen reurn condiional on a policy change is negaive: E Rĝ τ Policy Change < 0. Proof of Proposiion 5: The proof proceeds hrough four Lemmas, A A4. Lemma A: The announcemen reurn is given by N c xe x R x e x N c x + N c xe x γ γt τ σg bσ τ, B9 where N c x N x, σ c, σc Prc < x is he cumulaive normal densiy, and he random variable x has he normal disribuion where x N σ x, σ x, σ x γ T τ σ g σ τ. B30 Moreover, a policy change occurs if and only if he poliical cos is sufficienly low, i.e. c < x. 6
Proof of Lemma A: We can rewrie he formula for Rĝ τ in Proposiion 3 as where R ĝ τ pĝ τ F ĝ τ Gĝ τ G ĝ τ pĝ τ + pĝ τ F ĝ τ Gĝ τ pĝ τ V ĝ τ G ĝ τ pĝ τ + pĝ τ V ĝ τ V ĝ τ F ĝ τ Gĝ τ e γbgτt τ γ T τ σ g bσ τ e bg τt τ γt τ σ g bσ τ e bgτ γt τ γ T τ σ g bσ τ Noe ha he exponen in V ĝ τ is idenical o he argumen in pĝ in B9. Denoing his exponen by x, x ĝ τ γ T τ γ T τ σ g σ τ, we can rewrie he announcemen reurn as wih a sligh abuse of noaion: N x, σ c, σc e x G ĝ τ R ĝ τ R x N x, σ c, σc + N x, σ c, σc e x In addiion, expressing ĝ τ in erms of x, ĝ τ x γt τ + γt τ σg σ τ we also obain G ĝ τ e bgτt τ+ γt τ σ g bσ τ e x γ γt τ σg bσ τ+ γt τ σg bσ τ e x γ γt τ σg bσ τ which leads o B9. From is definiion above, x is normally disribued as of ime 0, x Nµ x, σx, where σx γ T τ σg σ τ µ x γ T τ σg σ τ σ x. Finally, he condiion for a policy change can also be expressed in erms of x. Recall ha a policy change occurs if and only if σ g σ τ γ T τ c ĝ τ < g c T τγ. 7
This inequaliy is equivalen o Q.E.D. Lemma A c < ĝ τ T τγ x. σ g σ τ γ T τ Lemma A: The Expeced Annoucemen Reurn EAR can be wrien as E R x Policy Change φx xn c xdx k x e x γ γt τ σg bσ τ φ x xdx, where φ x x denoes he normal probabiliy densiy funcion of x and k x N c x N c xe x N c x + N c xe x. Proof of Lemma A: We can wrie E R x Policy Change R xφ x x Policy Change dx, where φ x x Policy Change denoes he densiy of x condiional on a policy change a τ. We know ha a policy change occurs iff x > c, where c has a normal densiy ha is independen of x. Tha is, φ x x Policy Change φ x x yes, cφ c cdc φ x x x > c, cφ c cdc x φ x x {x>c} φ c cdc φ x x φ c cdc φ x xn c x. Dividing by he inegraion consan, we obain φ x x Policy Change φ x xn c x φx xn c xdx. Therefore, we can subsiue E R x Policy Change R xφ x x Policy Change dx φx xn c xdx N c xe x e x N c x + N c xe x γ γt τ σg bσ τ φ x xn c xdx φx xn c xdx N c xn c xe x e x N c x + N c xe x γ γt τ σg bσ τ φ x xdx. 8
Defining k x as in he claim yields he lemma. Q.E.D. Lemma A The denominaor in E R x Policy Change, namely φ x xn c xdx, is always posiive. Therefore, we only need o consider he numeraor, which we denoe as S k x e x γ γt τ σg bσ τ φ x xdx B3 I is convenien o express S as a funcion of σ x and a consan l. In paricular, from he definiion of σ x in B30, we obain γ σ x T τ σ g σ τ. Subsiuing for γ in B3, we obain he equivalen expression x σ «x l σx S k x e l φ x xdx, B3 where l T τ σg σ τ. We now rewrie S in an equivalen form, swapping he exponenial erm for he funcion k evaluaed a a differen value. Lemma A3: An equivalen expression for S is S E e m k x + m E k x, where x N σ x, σ x and m σ x l. Proof of Lemma A3: Rewrie x σ «x l σx S k x e l φ x xdx e l E k xe x σ x σx «l E k x We now rewrie he firs expecaion as follows E k xe x «σ x l σx k xe x «σ x le σx x+ σ x σ x πσx dx. 9
The produc of he exponenials can be wrien as e x σ x σx «x+ σ x σ le x πσx x+ e σ x +x σ xlσx σx πσx Now, we can rewrie he exponen in he exponenial funcion as follows Exponen x + σ x Thus, we obain he ideniy + x σ x lσx σ x x + σ x + xσ x σ x x + σ x + xσx xlσ x σ x + x σ x lσx x + x σ + x σx lσ x σ x σ xlσ x σ 3 xl x + σ x lσ x + lσx σxl 3 σxl 3 σx x + σ x lσ x + lσ x σxl 3 σx x + σ x lσ x + σx l σ x l. e x «x+ σ σ x x le σx σx πσx x+ e e + l σ xl σx lσx σx + l σ xl πσx x+ σx lσx e σx πσx Noe ha he expression in he parenhesis is he densiy of a normal wih mean σ x lσ x σ x + lσ x and variance σ x. Subsiuing in he expression for S, we obain S e l E k xe x σx σx «l E k x e l k xe x «σ x le σx x+ σ x σ x πσx dx E k x 0
x+ e l k xe + l σ e σx lσx σx xl dx E k x πσx e l E k xe l σ xl E k x e σxl E k x E k x, where x N σ x + lσ x, σx Because x and x differ from each oher only because of he mean m lσ x, we have x x + m formally, we have defined x from x in he derivaion above. This implies ha we can rewrie S as S E e m k x + m E k x. Q.E.D. Lemma A3 The following lemma yields one las ransformaion of S. Lemma A4: S can be equivalenly expressed as S E k x + m k x, B33 where and k x x N σ x, σx N c xn c x N c x + N c xe x. B34 Noe ha we have redefined x so he mean of x changes he sign compared o he x defined in Lemma A. Proof of Lemma A4: From Lemma A3, we have S E e m k x + m E k x. Subsiuing for k x + m and k x, S E e m N c x + mn c x + me x+m Nc xn c xe x E N c x + m + N c x + m e x+m N c x + N c xe x N c x + mn c x + me x Nc xn c xe x E E N c x + m + N c x + me x+m N c x + N c x e x E k x + me x E k xe x E k x + m k x e x k x + m k x e x e x+ σ x σ x πσx dx,
where x is sill defined as x N σ x, σ x. We now carry ou a ransformaion o swap he exponenial e x for a change in he mean of he disribuion of x. The produc of he exponenials is e x+ σ x σx xe πσx x+ e σ x +xσx σx πσx e x + σ x +xσ x σ x πσx «+xσ x e x + σ x xσ x σ x πσx «x e σ x σx, πσx which is he densiy of he normal disribuion wih mean σ x and variance σ x noe he change in he sign of he mean compared o he definiion of x in Lemma A. Subsiuing his ideniy, we obain S k x + m k x e x e x+ σ x σ x πσx dx x σ x k σ e x x + m k x dx πσx E k x + m k x, where x is now redefined as Q.E.D. Lemma A4 x N σ x, σ x. To summarize Lemmas A hrough A4, we have proved ha E Rx Policy Change < 0 if and only if S < 0, where S is given in B33. To prove ha S < 0, we now prove ha he funcion E k x + m is decreasing in m for all m > 0. Firs, we documen some properies of he kx funcion, which is defined in B34. This funcion is always posiive, and i converges o zero for boh x and x. In oher words, he funcion kx is hump-shaped. Second, we prove ha kx peaks below zero; ha is, kx is monoonically decreasing for all x > 0. To show his resul, we show ha he firs derivaive of k is negaive for all x > 0. We have dk < 0 if and only if dx or, equivalenly, if and only if N c xn c x + N c x N c x N c xe x < 0, N c x < e x Nc x. N c x N c x Direc compuaion shows ha his condiion is always saisfied for any x > 0 and any σ c. Third, we noe ha if he funcion kx were symmeric, he claim S < 0 would follow immediaely. This is because he expecaion E k x is compued wih respec o he normal densiy of x whose mean µ x σ x is posiive. Since he funcion kx peaks
below zero, increasing he mean of x from µ x > 0 o µ x + m would decrease he expeced value of a symmeric kx for all m > 0. The funcion kx is almos symmeric: he numeraor of kx in B34 is symmeric abou /σc, while he denominaor is given by Denx N c x + N c xfx, where fx e x. If we had fx for all x, hen i would be he case ha Denx and kx would be symmeric. Insead, fx e x > for all x > 0, which implies Denx >, so ha kx is lower han he corresponding symmeric value. For x < 0, we have fx e x <, which implies Denx < and hus kx is above he corresponding symmeric value. This characerizaion implies ha kx is negaively skewed he righ ail of kx for x > 0 is hinner han i would be under he symmeric disribuion. Since increasing he mean of x from µ x o µ x + m shifs he whole densiy o he righ, he hinner righ ail of kx reinforces he argumen for S < 0 under symmery. We conclude ha Ekx + m < Ekx for every m > 0. QED Proposiion 5 Proposiion A. In he benchmark model for τ, he sae price densiy is given by where π B γ Ωĝ,, B35 and p yes p no N N Ωĝ, p yes G yes G yes G no e e + p no G no γ γµt γbgτ + T τ bσ g+τ bσ +γ+γ σ T γ γµ+bgt + T bσ g0; ĝ γ σ τ + σ c/ g0; ĝ γ σ T T τ σ τ σ +γ+γ T T τ γ, σ σ τ + σc T τ γ σ + c/ T τ γ, σ στ + σ c T τ γ Proof of Proposiion A: We have π E π τ+ E π τ+ c φ c cdc, where φ c c is he densiy of a normal disribuion wih mean σ c and variance σc. Consider E π τ+ c p ce πτ+ c, ĝ τ < g c + p ce πτ+ c, ĝ τ > g c, where he cuoff g c is given in B3, and p c Pr ĝ τ < g c N g c,ĝ, σ σ τ. We proceed in wo seps. In he firs sep, we compue he inner expecaion condiional on a paricular poliical cos c. In he second sep, we inegrae he cos ou. 3
STEP : We compue he wo condiional expecaions separaely, firs compuing E πτ+ c, ĝ τ < g c and hen E πτ+ c, ĝ τ > g c. CASE. POLICY CHANGE: The sae price densiy a ime τ righ afer a policy change i.e., a ime τ+ is given in B4 as π τ+ λ B γ τ+e γ µ σ T τ+ γ T τ σ g+σ T τ. In wha follows, we omi he consan λ which drops ou laer, and we denoe B τ+ simply as B τ his is innocuous because B τ B τ+ by coninuiy; here are no jumps in capial a ime τ. Consider he condiional expecaion E πτ+ ĝ τ < gc, c µ E B γ e γ σ T τ+ γ T τ σg+σ T τ ĝτ < gc, c e γ τ µ σ T τ+ γ T τ σ g+σ T τ E B γ τ ĝ τ < gc, c. To compue he laer expecaion, we firs find he join disribuion of b τ log B τ and ĝ τ. Aggregae capial a τ is given by B R τ τ σ µτ + e bgd τ +σ Z b τ Z b. B Io s Lemma implies he join process for b and ĝ as db µ + ĝ σ d + σdẑ ĝ τ dĝ σ σ dẑ. Inegraing from o τ: τ b τ b + µτ + ĝ u τ σ du + σ dẑ u τ g + σ σ udẑu. Lemma A5: The join disribuion of b τ, ĝ τ condiional on he informaion available a ime < τ is given by bτ E b N τ V bτ C b, τ, ĝ τ ĝ τ E ĝ τ C b τ, ĝ τ V g τ where E b τ b + µ + ĝ σ τ E ĝ τ ĝ V b τ τ σ + σ τ V ĝ τ σ σ τ C b τ, ĝ τ σ τ. 4
Proof of Lemma A5: We already know ha ĝ τ Nĝ, σ σ τ. Clearly, b τ, ĝ τ are also joinly normally disribued. We only need o find he firs wo momens of b τ and is covariance wih ĝ τ. The mean is : τ E b τ b + µτ + E ĝ u σ du b + µ + ĝ σ τ. The condiional variance of b τ as of ime is given by τ V b τ V ar µ + ĝ u σ du + σ Using Girsanov Theorem, we reexpress everyhing in erms of he original processes, τ µ + ĝ u τ τ σ du + σ dẑ u µ + g old τ σ du + σ dz u, where g old Nĝ, σ, given he informaion a ime. We hen immediaely obain V b τ τ σ + σ τ. τ dẑu. Finally, we compue he condiional covariance of b τ wih ĝ τ as of ime : C b τ, ĝ τ E b τ ĝ τ E b τ E ĝ τ. Consider he variable Io s Lemma implies h b ĝ. dh db ĝ + b dĝ + db dĝ µ + ĝ σ ĝ + σ d + ĝ σ + b σ σ dẑ. Taking he inegral on boh sides: τ h τ h + ĝu + µ σ ĝ u + σ u du + I follows ha E h τ h + h + h + τ τ τ τ ĝu σ + b u σ uσ dẑu. ĝu E + µ σ E ĝ u + σ u du ĝ + σ σ u + µ σ ĝ + σ u ĝ + σ + µ σ ĝ du h + ĝ τ + σ τ + du µ σ ĝ τ, 5
where we used he earlier finding E ĝ u ĝ + σ σ u. The condiional covariance as of ime is given by C b τ, ĝ τ E b τ ĝ τ E b τ E ĝ τ h + ĝ τ + σ τ + µ σ ĝ τ b + µ + ĝ σ τ σ τ. ĝ Q.E.D. Lemma A5 Using Lemma A5, we now finally compue E e γb τ ĝ τ < gc. Using he properies of condiional Gaussian disribuions, we know ha b τ bgτx N E b τ + C b τ, ĝ τ x ĝ,v b τ C b τ, ĝ τ. V ĝ τ V ĝ τ Thus E e γb τ ĝ τ < gc, c gc gc E e γb τ ĝ τ x φ x x ĝτ < gc, c dx e γebτ+ Cbτ,bgτ x bg V bgτ + γ V b τ «Cbτ,bgτ V bgτ φ x ĝ τ < gc, c dx, where Noe ha we can also wrie φ x ĝ τ < gc, c E e γb τ ĝ τ < gc, c e γebτ+ e γe b τ+ e γe b τ+ e x bg V bgτ πv bgτ N gc; ĝ, σ σ τ «γ Cbτ,bgτ V b τ V bgτ N gc; ĝ, σ σ τ γ V b τ Cbτ,bgτ V bgτ N gc; ĝ, σ σ τ γ V b τ Cbτ,bgτ V bgτ N gc; ĝ, σ σ τ gc «gc «e γ Cbτ,bgτ V bgτ N gc; ĝ γcb τ, ĝ τ, σ σ τ N e gc; ĝ, σ σ τ. Cbτ,bgτ γ x bg e V bgτ πv ĝτ e x bg V bgτ dx πv ĝτ e x bg γcbτ,bgτ x bg V bgτ dx gc πv ĝτ e x bg γ γebτ+ V bτ. γcbτ,bgτ V bgτ dx 6
In conclusion, subsiuing for E b τ, V ĝ τ and Cb τ, ĝ τ, we obain E e γb τ ĝ τ < gc, c B γ e γµ+bg σ τ + γ τ bσ +σ τ N gc, ĝ γ σ τ, σ σ τ N. gc; ĝ, σ σ τ Finally, puing all erms ogeher, E πτ+ ĝ τ < gc, c B γ e concluding Case. γµt γbgτ +γ σ γ T + T τ bσ g+τ bσ + γ σ T N gc, ĝ γ σ τ, σ σ τ N, gc; ĝ, σ σ τ CASE : NO POLICY CHANGE: The sae price densiy righ afer ime τ if here is no policy change is given in B4 as follows again dropping λ and using B τ B τ+ : π τ+ Bτ γ e γ µ+bg τ σ T τ+ γ T τ bσ τ+σ T τ. The main difference from Case is ha ĝ τ now also eners he exponen, as a resul of which we canno facor i ou. Consider he condiional expecaion E πτ+ ĝ τ > gc, c µ+bg E B γ e γ τ T σ τ+ γ T τ bσ τ+σ T τ ĝτ > gc, c Define ĝ τ τ σ γµt τ+γ e T τ+ γ T τ bσ τ+σ T τ E e γbg τt τ+b τ ĝ τ > gc, c y τ ĝ τ T τ + b τ. We can now use Lemma A5, designed for he join densiy of b τ, ĝ τ, o compue he join densiy of y τ, ĝ τ. We obain yτ ĝτ T τ + E N b τ V yτ C y τ, ĝ τ. E ĝ τ C y τ, ĝ τ V ĝ τ The only wo new erms o compue are V y τ and Cy τ, ĝ τ. We have and V y τ T τ V ĝ τ + V b τ + T τc b τ, ĝ τ T τ σ σ τ + τ σ + σ τ + T τ σ τ T τ σ T τ σ τ + τ σ + σ τ + T τ σ τ σ T τ + τ + T ττ T τ σ τ + σ τ σ T T τ σ τ + σ τ C y τ, ĝ τ T τv ĝ τ + C b τ, ĝ τ T τ σ σ τ + σ τ σ T T τ σ τ. 7
I follows from he properies of condiional Gaussian disribuions ha y τ bgτx N E y τ + C y τ, ĝ τ x ĝ, V y τ C y τ, ĝ τ V ĝ τ V ĝ τ. Thus E e γy τ ĝ τ > gc, c gc gc E e γyτ ĝ τ x φ x ĝ τ > gc, c dx e γeyτ+ Cyτ,bgτ x bg V bgτ + γ V y τ «Cyτ,bgτ V bgτ φ x ĝ τ > gc, c dx, where We herefore obain E e γy τ ĝ τ > gc, c e γeyτ+ The same seps as in Case yield e x bg V bgτ πv bgτ φ x ĝ τ > gc, c N. gc; ĝ, σ σ τ «γ Cyτ,bgτ V y τ V bgτ N gc; ĝ, σ σ τ gc Cyτ,bgτ γ x bg e V bgτ πv ĝτ e x bg V bgτ dx. E e γy τ ĝ τ > gc, c γ γbgt τ+ebτ+ N gc, ĝ γc y e V yτ τ, ĝ τ, σ σ τ N. gc, ĝ, σ στ Finally, puing all erms ogeher, E πτ+ ĝ τ > gc, c e σ γµt τ+γ γ γµt + T τ+ γ T τ bσ τ+σ T τ E e γy τ ĝ τ > gc, c B γ e T bσ +σ T τ e γbg T σ T N gc, ĝ γc y τ, ĝ τ, σ τ σ N gc, ĝ, σ σ τ. STEP : We now inegrae ou he cos c. The condiional expecaion is E π τ+ c p ce πτ+ ĝ τ < g c,c + p c E πτ+ ĝ τ > g c,c B γ N g c, ĝ γ σ τ, σ σ τ e σ γ γµt γbgτ +γ T + T τ σg+τ bσ + γ σ T +B γ N g c,ĝ γc y τ, ĝ τ, σ στ γ γµt + e T bσ τ +σ T τ e γbg T σ T 8
Noe ha he cos c only eners ino he wo erms N g c, ĝ γ σ τ, σ σ τ and N g c,ĝ γc y τ, ĝ τ, σ σ τ. Since he disribuion of c is independen of everyhing else, we obain E π τ+ B γ E N g c,ĝ γ σ τ, σ σ τ e σ γ γµt γbgτ +γ T + T τ σg +τ bσ + γ σ T +B γ E N g c,ĝ γc y τ, ĝ τ, σ στ γ γµt + e T bσ τ+σ T τ e γbg T σ T. Finally, using he law of ieraed expecaions, we can compue p yes E N g c,ĝ γ σ τ, σ σ τ E Pr x < g c c E E x<gc c E x<gc Pr x < g c c Pr x < g 0 γ T τ N g 0;ĝ γ σ τ σ c γ T τ ; σ σ τ + σc. γ T τ Similarly, p no E N g c, ĝ γcy τ, b τ, σ σ τ N g 0 ; ĝ γcy τ, b τ σ c γ T τ ; σ σ τ + Subsiuing in E π τ+, he claim of Proposiion A follows. QED Proposiion A Proposiion 6. The sochasic discoun facor SDF follows he process σc γ T τ dπ π σ π, dẑ + J π {τ }, B36 where dẑ is he Brownian moion from Proposiion, {τ } is an indicaor funcion equal o one for τ and zero oherwise, and he jump componen J π is given by. J π { J yes π For > τ, σ π, is given by pτ Fbgτ p τ+ p τfbg τ if policy changes J no π pτfbgτ p τ+ p τfbg τ if policy does no change. σ π, γ σ + T σ σ, and for τ, i is given by Corollary A in he paper: B37 B38 σ π, γσ Ωĝ, σ Ωĝ, ĝ σ. B39 9
Proof of Proposiion 6: For < τ, he SDF dynamics sem from an applicaion of Io s Lemma o B35, which also yields he volailiy B39. Because he sae price densiy is a maringale, π E π T, he drif of he process is zero. For > τ, he sae price densiy is given in closed form as in equaion B4, wih in place of τ in he formula. An applicaion of Io s Lemma immediaely leads o he diffusion erm B38. A he ime of he anouncemen, τ, he sae price densiy jumps from B5 o eiher of he wo expressions in B4. We obain he size of he jump by compuing he difference: π yes Jπ,τ yes τ+ π yes τ+ π τ p τ π yes τ+ + p τ πτ+ no p τ + p τ πno τ+ p τ p τ + p τ e γbgt τ γ T τ σg bσ τ p τ + p τ e γbgt τ γ T τ σg bσ τ π yes τ+ e γbgτt τ γ T τ σ g bσ τ The expression for he jump if here is no policy change follows from he maringale condiion. which implies J no π,τ p τ p τ Jyes π,τ QED Proposiion 6 E τ J π p τ J yes π,τ + p τ J no π,τ 0, p τ e γbgτt τ γ T τ σ g bσ τ p τ + p τ e γbgt τ γ T τ σ g bσ τ. Proposiion A. In he benchmark model for τ, he sock price for firm i is given by where and p yes p no N N M i B i Φĝ, p yes K yes + p no K no K yes K no e e Φĝ, Ωĝ,, γµt + γbgτ + γ T τ σg+τ bσ γγ σ T γµt + γbgt + γ bσ T γγ σ T g0; ĝ + γ σ τ + g0; ĝ + γ σ T T τ σ τ σ c/ T τ γ, σ σ τ + B40 σc T τ γ + σ c / T τ γ, σ σ τ + Proof of Proposiion A: The proof is similar o ha of Proposiion A. For < τ, marke value saisfies M i Eπτ+Mi τ+ π. We need o compue he numeraor. As in Proposiion A, 0 σ c T τ γ
we proceed in wo seps. In he firs sep, we compue he condiional expecaions, and in he second sep, we inegrae ou c: E πτ+ M i τ+ c p ce πτ+ M i τ+ ĝ τ < gc, c + p ce πτ+ M i τ+ ĝ τ > gc, c. CASE. POLICY CHANGE. Using he expressions in B4 and B0: E πτ+ Mτ+ i ĝ τ < gc, c µ E B γ e γ σ T τ+ γ T τ σg +σ T τ τ B i τ e e γ µ σ µ σ T τ+ γ T τ σg+σ T ĝτ < gc, c T τ+ γ T τ σ g+σ T τ E B γ τ B i τ ĝ τ < gc, c. Under he original probabiliy measure, B τ B e µτ +gold τ σ τ +σz τ Z, while Thus Bτ i Be i µτ +gold τ σ τ +σz τ Z σ τ +σ Zτ i Z i B i Bτ e σ τ +σ Zτ i Z i. B E B γ τ Bτ ĝ i τ < gc, c Bi E B Bτ γ Bi B E B γ τ ĝ τ < gc, c e σ τ +σ Zτ i Z i ĝτ < gc, c B4 since he Brownian moions Z i are independen of B τ. The closed-form expression for he expecaion is hen idenical o ha obained in Proposiion A when we subsiue γ for γ. Tha is: E B γ τ Subsiue o obain ĝ τ < gc, c B γ E πτ+ Mτ+ i ĝ τ < gc, c B i B γ e CASE. NO POLICY CHANGE E πτ+ M i τ+ ĝ τ > gc, c E e γµ+bg σ τ + γ τ bσ +σ τ N gc, ĝ + γ σ τ, σ σ τ N. gc; ĝ, σ σ τ σ γµt + γbgτ γγ T + γ N gc, ĝ + γ σ τ, σ σ τ N. gc; ĝ, σ σ τ e B i τe µ+bg Bτ γ e γ τ σ µ+bg τ σ T τ+ γ T τ bσ τ+σ T τ T τ σ g +τ bσ T τ+ γ T τ bσ τ+σ T τ ĝτ > gc, c σ γµt τ γ T τ+ γ σ T τ+ γ T τ bσ τ E B γ τ B i τ e γbgτt τ ĝ τ > gc, c.
Using B4 again, we have E B γ τ Bτ i e γbgτt τ ĝ τ > gc, c Bi E B γ τ e γbgτt τ ĝ τ > gc, c B Bi B E e γb τ+bg τt τ ĝ τ > gc, c where he idiosyncraic Brownian moion erm Z i τ Zi is independen of boh B τ and ĝ τ. Recalling he noaion y τ b τ + ĝ τ T τ, we can use he resuls in Proposiion A wih γ replaced by γ o obain E e γb τ+bg τt τ ĝ τ > gc, c γbgt τ+ebτ+ γ V y e τ N gc, ĝ + γc y τ, ĝ τ, σ στ Puing he erms ogeher, N gc, ĝ, σ σ τ γ γµt γ σ e T + γbg T + γ E πτ+ Mτ+ ĝ i τ > gc, c BB i γ N gc, ĝ + γ C y τ, ĝ τ, σ τ σ N gc, ĝ, σ σ τ.. bσ T The las sep is o inegrae ou he random cos c. Using he same seps as in he proof of Proposiion A, we obain p yes and p no as in he claim of he proposiion. QED Proposiion A Proposiion 7. The reurn process for sock i is given by dm i M i µ M, d + σ M, dẑ + σ dz i + J M {τ }, B4 where he jump componen J M is given by J M For > τ, we have { J yes M R ĝ τ if policy changes J no M Rĝ τ Gĝ τ + Gĝ τ if policy does no change. B43 µ M, γ σ + T σ σ B44 σ M, σ + T σ σ, B45 and for τ, µ M, and σ M, are given by Corollary A in he paper σ M, Φĝ, / ĝ σ + F ĝ, / ĝ σ Φĝ, F ĝ, σ B46 µ M σ π, σ M,, B47
where σ π, and σ M, are given in equaions B39 and B46, respecively. Proof of Proposiion 7: For < τ, he dynamics of he reurn dm i /Mi sem from an applicaion of Io s Lemma o B40, which also yields he volailiy B46. The expeced reurn is given by µ M Covdπ /π, dm i /M i σ π, σ M,, which yields B47. For > τ, he price is given in closed form as in equaion B0, wih in place of τ in he formula. An applicaion of Io s Lemma immediaely yields he diffusion erm B45. We obain B44 by using he closed-form soluion for he SDF diffusion in B38. A ime τ, sock prices jump. The jump size in case of a policy change, J yes M already derived in Proposiion 3. For JM no, we have J no M Mi,no τ+ M i τ M i τ ωmi,yes τ+ M i,no τ+ p τ Gĝ τ p τ + p τ Fĝ τ Gĝ τ, ωm i,yes τ+ + ωm i,no τ+ ω Mi,no τ+ /M i,yes τ+ ω + ωm i,no τ+ /M i,yes τ+ Rĝ τ, is B48 B49 where we use ω in B and Gĝ τ M i,no τ+ /M i,yes τ+. This is he same expression as in B43, as can be easily verified by subsiuing for Rĝ τ from B6. QED Proposiion 7 Corollary 3. The marke value of each firm increases a he announcemen of no policy change i.e., J no M > 0 if and only if ĝ τ > g, where g is given in equaion B8. Proof of Corollary 3: The proof follows immediaely from expression B49, which is posiive if and only if Gĝ τ >, where Gĝ τ is given in B8. QED Corollary 3 Proposiion 8. The condiional expeced jump in sock prices a ime τ, as perceived by invesors jus before ime τ, is given by E τ J M p τ p τ F ĝ τ Gĝ τ p τ + p τ F ĝ τ Gĝ τ. B50 Proof of Proposiion 8: The proof follows immediaely from E τ J M p τ J yes M + p τjm no and subsiuing J yes M Rĝ τ in B6 and JM no in B49. QED Proposiion 8 Corollary 4. We have E τ J M < 0 if and only if where g is given in equaion B8 and g < ĝ τ < g, B5 g γ T τ σ g σ τ. B5 3
Proof of Corollary 4: The expeced jump is negaive if and only Gĝ τ Fĝ τ > 0, ha is, if and only if e bgt τ γt τ σ g bσ τ e γbgτt τ γ T τ σ g bσ τ > 0. This condiion is saisfied if and only if ĝ τ γt τ σ g σ τ ĝ τ + γ T τ σ g σ τ < 0, which yields he claim. QED Corollary 4 Corollary 5. As risk aversion γ, E τ J M 0 from above for any value of ĝ τ. Proof of Corollary 5: The condiional expeced jump is E τ J M E ĝ τ p τ p τ Fĝ τ Gĝ τ p τ + p τ Fĝ τ Gĝ τ. Recall ha he probabiliy of a policy change is given by p τ Pr γ ĝ τ < g c ĝ τ Pr c < ĝ τ γt τ T τ σg σ τ. For γ, we have p τ 0 he erm ha involves γ dominaes, Gĝ τ, Fĝ τ 0, and Fĝ τ Gĝ τ 0. In addiion, an applicaion of l Hospial s rule gives us Therefore, we have p τ Fĝ τ 0 and p τ Fĝ τ Gĝ τ 0. E τ J M E ĝ τ p τ p τ FG p τ + p τ FG p τ F p τ G F + FG p τ + p FG τ p τ F 0, p τ p τ G FG + F p τ + p τ FG where we used he fac ha Fĝ τ Gĝ τ e +γbgτt τ+ γ γ T τ σ g σ τ 0. Noe ha because boh p τ > 0 and Fĝ τ > 0, he limi of Eĝ τ is from above, which implies ha for every ĝ τ, E ĝ τ > 0 for γ sufficienly large. Indeed, noe ha he range in which E ĝ τ < 0 is g, g,, so ha for any given ĝ τ, E ĝ τ mus urn posiive as γ increases. 4
QED Corollary 5 Corollary 6. As risk aversion γ, E τ J M converges o a nonnegaive value for any ĝ τ. I converges o zero if and only if ĝ τ T τ σ g σ τ. Proof of Corollary 6: The condiional expeced jump is E τ J M E ĝ τ p τ p τ Fĝ τ Gĝ τ p τ + p τ Fĝ τ Gĝ τ. From he definiion of p τ, p τ Pr ĝ < g c ĝ Pr c < ĝ τ γt τ γ T τ σ g σ τ, we have ha γ implies p τ p N 0, σ c, σ c, independen of ĝτ. In addiion, as γ, we also have Fĝ τ /Gĝ τ. This implies ha for every ĝ τ, he expeced jump converges o E ĝ τ p p /Gĝ τ Gĝ τ p p e bgτt τ T τ σg bσ τ e bgτt τ+ T τ σg bσ τ p p e x e x, where x ĝ τ T τ + T τ σ g σ τ. This expression is always sricly posiive, excep for x 0 i.e., ĝ τ T τ σ g σ τ, in which case i is equal o zero. QED Corollary 6 Corollary 7. As risk aversion γ, EJ M 0 from above. Proof of Corollary 7: The proof follows from ha of Corollary 5 and he resul ha for every ĝ τ, we have E ĝ τ 0 as γ from above. This implies ha also uncondiionally E J M,τ E ĝ τ φ ĝ τ, 0, σg τ σ dĝτ 0, where φ ĝ τ, 0, σg τ σ is he normal densiy wih mean zero and variance σ g σ τ. In addiion, because for ĝ τ, E ĝ τ > 0 for γ sufficienly large, and because φ ĝ τ, 0, σg σ τ does no depend on γ, we have ha E J M,τ converges o zero from above i.e., i is posiive for a sufficienly large γ. This implies ha here exiss γ such ha for γ > γ, we have E J M,τ > 0. QED Corollary 7 Corollary 8. As risk aversion γ, E J M converges o a posiive value. 5
Proof of Corollary 8: From he proof of Corollary 6, noe ha as γ, he uncondiional expecaion E J M,τ E E τ J M,τ E E ĝ τ E p p e bgτt τ T τ σg bσ τ e bgτt τ+ T τ σg bσ τ p pe e bgτt τ+ T τ σg bσ τ + e bg τt τ T τ σg bσ τ p p e T τ σg bσ τ > 0. QED Corollary 8 Corollary 9. The correlaion beween he reurns of any pair of socks for > τ is given by ρ σ + T σ σ σ + T σ σ + σ. B53 For < τ, he correlaion is given by Corollary A3 in he paper ρ σ M, σ + σ M,. B54 For τ, he insananeous correlaion is one. Proof of Corollary 9: The saemen follows from he definiion of correlaion, which in coninuous ime is given by dm i j E dm M i M j ρ E dm i M i E dm i M i σ M, σ + σ M, Subsiuing he appropriae value for σ M, from Proposiion 7, he claim of he corollary follows. QED Corollary 9. Exension: Endogenous Timing of Policy Change. Le V ĝ, B, denoe he value funcion given no policy change a or before ime : B γ T V ĝ, B, E {max {E τ No change a τ, E τ τ> γ C B γ T γ }} Change a τ. B55 6
Proposiion A3. Le he iming of he policy change be endogenous. For every τ i, τ i+, he indirec uiliy funcion V ĝ, B, from equaion B55 is given by V ĝ, B, B γ Φĝ,, B56 where Φĝ, saisfies he parial differenial equaion 0 Φĝ, + { γµ + ĝ } γ γσ Φĝ, + Φĝ, σ ĝ σ + γ Φĝ, σ ĝ. B57 The boundary condiions a ime τ i are given by { } τi+ Φĝ τi, τ i E τi max Φĝ τi, τ i, ec+ γµt γ σg T τ i γ γ σ T τ i γ, where he expecaion is aken wih respec o c jus before he policy decision a ime τ i. The final condiion a ime T is Φĝ T, T γ. Proof of Proposiion A3: We have he final condiion V ĝ T, B T, T B γ T γ. Le τ denoe a generic τ i. Because he policy change is irreversible, we can use he aggregae capial process B T B τ e µ+g τ σ T τ+σz T Z τ o obain he value of expeced uiliy a any ime τ condiional on a change a τ in closed form: B γ V B τ, c, τ e c T E τ γ Y es, ĝ τ, B τ, c e c B γ τ γ e γµt τ+σ gt τ γ γ σ T B58 τ. A policy change occurs a ime τ if V B τ, c, τ > V ĝ τ, B τ, τ. Since no policy decisions are made beween τ i and τ i+, we have for τ i, τ i+ ha V ĝ, B, E max { V ĝτi+, B τi+, τ i+, V Bτi+, c, τ i+ }. I follows ha V saisfies he maringale condiion E dv 0, ha is 0 V + V E db + V E dĝ + B ĝ V B E db V + E ĝ dĝ V + E dĝ db B ĝ wih he following final condiion a τ τ i+ we drop he subscrip for noaional convenience: { V ĝ τ, B τ, τ E τ max V ĝτ, B τ, τ,v B τ, c, τ } B59 7
where he expecaion is aken over possible values of c. Noe ha because of he convexiy of he max operaor, we canno ake he expecaion inside he parenhesis. We conjecure ha he value funcion is: Taking firs derivaives, V B V ĝ V V ĝ, B, B γ Φĝ,. γb γ Φĝ, ; V B γ B γ In addiion, we have for < τ, B Φĝ, ; V B γ Φĝ, ĝ ĝ ĝ Φĝ, V ; γb γ ĝ B γ γ B γ Φĝ, Φĝ, ĝ. E db µ + ĝ B ; E db σ B ; E dĝ 0; E dĝ σ σ ; E db dĝ σ B. Subsiuing he derivaives of he value funcion and he expecaions in he PDE, we obain B57. The final condiion a τ i+ follows from equaion B59. QED Proposiion A3 Saemen from he ex: Condiional on a policy change a a given ime τ, he value funcion a ha ime is available in closed form: V B τ, τ, c B γ τ γ ec+µ γt τ+σ gt τ γ γ σ T τ. B60 Proof: See proof of Proposiion A3. QED Saemen from he ex Proposiion A4: For every τ i, τ i+, he marke-o-book raio M/B of firm j before a policy change is given by M j Ωĝ, F ĝ,, where F ĝ, and Ωĝ, saisfy he ODEs 0 F ĝ, + { γ µ + ĝ + } γ γ + σ 0 Ωĝ {, + γµ + ĝ + + Ωĝ, ĝ B j γ γ σ σ + γ Ωĝ, σ ĝ 8 F ĝ, + F ĝ, ĝ } σ Ωĝ, σ σ γ F ĝ, σ ĝ
wih he following boundary condiions for τ τ i+ : F ĝ τ, τ p τ e γµt τ+ γ T τ bσ g+γγ+ σ T τ + p τ F ĝ τ, τ Ωĝ τ, τ p τ e γµt τ+ γ T τ σg σ γγ T τ + p τ Ωĝ τ, τ and final condiions a T, F ĝ T, T and Ωĝ T, T. Above, p τ is he probabiliy of a policy change a τ, given explicily by p τ N x ĝ τ, τ, σ c, σ c, B6 where x ĝ τ, τ log Φĝ τ, τ γ γµt τ γ σ g T τ + γ γ σ T τ. Proof of Proposiion A4: We sar by compuing he probabiliy a τ ha a policy change will occur, assuming i has no occurred ye. From he proof of Proposiion A3, we have: p τ Pr Change a τ ĝ τ, τ Pr Φĝ τ, τ < γ Pr Φĝ τ, τ γe γµt τ γ σgt τ +γ γ σ T τ > e c Pr c < log Φĝ τ, τ γ γµt τ which leads o B6. τ+ ec+ γµt γ σg T τ γ γ σ T τ γ σg T τ + γ γ σ T τ To compue asse prices, we firs consider he dynamics of he sochasic discoun facor. The analysis is similar o above, excep ha marke paricipans do no decide wheher o change policy or no. Le W B, ĝ, E B γ T τ >. For τ i, τ i+, marke paricipans know here are no policy decisions, and herefore 0 W + W E db + W E dĝ + W B ĝ B When a policy change occurs a τ τ i+, hen W B τ, τ E τ B γ T Change a τ E τ B γ τ E db W + E ĝ dĝ W + E db dĝ. ĝ B Bτ γ e γµt τ+ γ T τ σg +γ σ γ T τ+ e γµ+gnew T τ+γ σ T τ γσz T Z τ σ T τ B γ Thus, W B, ĝ, for τ i, τ i+ has he following final condiion a τ τ i+ : W B τ, ĝ τ, τ p τ W B τ, τ + p τ W B τ, ĝ τ, τ. 9 τ e γµt τ+ γ T τ σg +γγ+ σ T τ
The final condiion a T is W B T, ĝ T, T B γ T. We conjecure so ha W W B B γ F ĝ, ; W W B, ĝ, B γ F ĝ,, B γ ĝ γ γ + B γ F ĝ, ; W ĝ F ĝ, ; W ĝ B γ γb γ F ĝ, ; B F ĝ, W ; γb γ ĝ B ĝ F ĝ, ĝ. Subsiuing hese expressions and he expecaions in he PDE as in he proof of Proposiion A3, we obain he firs PDE in he claim of he proposiion. The boundary condiion also follows from he conjecured soluion. Similarly, for he numeraor of he pricing formula M j E π T B j T /π, we mus compue Q B j,b, ĝ, E B j T B γ T τ >. For τ i, τ i+, marke paricipans know here are no policy decisions, and herefore 0 Q + Q B E db + Q + Q E dĝ ĝ B j E db j Q + E dĝ + Q E db ĝ + Q ĝ B E db dĝ + When a policy change occurs a τ, hen Q B j τ, B τ, τ E τ B j T B γ T Change a τ Q B ĝ B j E dbdĝ j + + Q B i E d B i Q B j B E db db j E τ B τe j µ+gnew T τ σ T σ T +σz T Z+σ Z j T Zj B γ τ e γµ+gnew T τ+γ σ T τ γσz T Z τ E τ Bτe j σ T +σ Z j T Zj B γ B j τb γ τ B j τ B γ τ τ. e γµ+gnew T τ+γ σ T τ+ γσz T Z τ e γµt τ+ γ T τ σg+γ σ T τ+ γ σ T τ e γµt τ+ γ T τ σg γγ σ T τ. Thus, Q B, ĝ, for τ i, τ i+ has he following final condiion a τ τ i+ : Q B j τ, B τ, ĝ τ, τ p τ Q B j τ, B τ, τ + p τ Q B j τ, B τ, ĝ τ, τ. We conjecure Q B j,b, ĝ, B j B γ Ωĝ,, 30
so ha Q Q B Q B j B j B γ Ωĝ, ; Q B ĝ B j γ γ γ + BB j γ Ωĝ, ; Q ĝ Q 0; B B j γb γ Ωĝ, ; Ωĝ, ; Q ĝ B j B γ Q B ĝ j B γ γb B B j γ Ωĝ, ; Q B j Ωĝ, Q ; γb ĝ B ĝ B j γ Ωĝ,. ĝ B γ Ωĝ, Ωĝ, ĝ Subsiuing in he PDE, we obain 0 BB j γ Ωĝ, + γ γ + Bj B γ γb j B γ γb j B γ Ωĝ, ĝ Ωĝ, µ + ĝ B + B γ Ωĝ, µ + ĝ B j Ωĝ, σ B + Bj B γ Ωĝ, σ B + B γ ĝ σ σ Ωĝ, σ ĝ Bj γb γ Ωĝ, σ BB j, where we also used E db j µ + ĝ B j ; E db j dĝ σ B; j E db j db σ BB j The second PDE in Proposiion A4 hen follows afer some algebraic manipulaions. The boundary condiion also follows from he conjecure soluion. Exension: Invesmen Adjusmen. Proposiion 9. In equilibrium a ime τ, a randomly-seleced fracion α τ 0, of firms coninue invesing in heir risky echnologies, while he remaining firms disinves and park heir capial in he risk-free echnology. The governmen changes is policy if and only if ĝ τ < g c, α τ, B6 where for given c, α τ he hreshold g c, α τ saisfies he equaion e c E τ ατ e εyes τ,t + α τ γ E τ ατ e εg τ,t + α τ γ B63 where ε yes N µ σ ε g N µ + gc, α τ σ T τ,σg T τ + σ T τ T τ, σ τ T τ + σ T τ B64 B65 and he equilibrium value of α τ is described below. 3
We prove Proposiion 9 ogeher wih he following saemen from he ex of he paper: Saemen from he ex: If firm i decides a ime τ o remain invesed in is risky echnology, is marke value righ afer ime τ is given by h M i,yes E Mτ+ i τ+ Bτ+ i τ+ e εyes τ,t α τe εyes τ,t + α τ γi h E τ+ ατe εyes τ,t + α τ γi if policy changes h M i,no E τ+ Bτ+ i τ+ e εno τ,t α τe εno τ,t + α τ γi h E τ+ ατe εno τ,t + α τ γi if policy does no change B66 Righ before ime τ, he marke value of firm i is given by a weighed average of M i,yes τ+ and M i,no τ+ as in equaion B, excep ha he weighs ω τ are given by p τ ω τ, B67 p τ + p τ H τ where H τ E τ { ατ e εno τ,t + α τ γ } E τ { α τ e εyes τ,t + α τ γ} and p τ is he probabiliy of a policy change as perceived jus before ime τ i.e., he probabiliy ha he condiion B6 holds. The condiion for equilibrium wih 0 < α τ < : M i yes ω τ+ M i no τ + ω Bτ+ i τ τ+ B68 Bτ+ i Proof of Proposiion 9: We proceed in a few seps. Lemma A6. Le α τ be he fracion of firms, chosen a random, ha a τ choose he invesmen in he risky echnology. In his case, aggregae capial a T is B T µ+g α τ e σ T τ+σz T Z τ + α τ, B69 B τ where g g new if a policy change occurs a τ, and g g old oherwise. Proof of Lemma A6: Wihou loss of generaliy, le he firms randomly choosing he risky echnology fall in he inerval 0, α τ. The capial evoluion of each individual firm is given by B9. If a firm chooses he riskless echnology insead, hen BT i Bi τ. I follows ha aggregae capial a ime T is B T 0 B i Tdi ατ 0 σ µ+gt τ e T τ+σz T Z τ σ T τ Bτe i σ µ+gt τ T τ+σz T Z τ σ T τ+σ ZT i Zi τ di + Bτdi i α τ ατ 0 B i τe σ Z i T Zi τ di + α τ B i τdi The same condiion is obained as a firs-order condiion in an alernaive formulaion of he problem in which a social planner chooses α τ o maximize he invesors expeced uiliy. See Lemma A8 below. 3
where g g old if here is no policy change a ime τ, bu g g new if here is one. Applying he law of large numbers, we obain ατ 0 Similarly, Bτe i σ ZT i Zi τ ατ di ατ B α τe i σ ZT i Zi τ di α τ E B iτe σ ZT i Zi τ τ 0 α τ E Bτ i E e σ ZT i Zi τ α τ E Bτ i e σ T τ. Subsiuing, we obain Bτ i di α τ α τ α τ B T E B i τ Bτ i di α τ E Bτ i. α τ σ µ+gt τ α τ e T τ+σz T Z τ + α τ Using he same logic, he law of large numbers also implies ha EB i τ B τ, concluding he proof of Lemma A6. Q.E.D. Lemma A6 Lemma A7: The governmen changes is policy if and only if where he hreshold g c, α τ is in he claim of Proposiion 9. ĝ τ < g c, α τ, B70 Proof of Lemma A7. Given he equilibrium level of firms α τ, he governmen changes policy if and only if E CB γ T Policy Change > E γ B γ T No Policy Change γ.. B7 Given he aggregae capial in Lemma A6, from he perspecive of all agens in he economy, he exponen in he firs erm on he righ-hand side of B69 has he following disribuion: µ + g σ T τ + σ Z T Z τ ε no τ, T N µ + ĝ τ σ T τ, σ τ T τ + σ T τ if no policy change ε yes τ, T N µ σ T τ, σ gt τ + σ T τ if policy change Using his disribuion, condiion B7 is hen he same as e c E τ ατ e εyes τ,t + α τ γ < E τ ατ e εno τ,t + α τ γ B7 where ε i τ, T N µ ε g i, T τ,σ ε σ i, T τ B73 33
wih µ ε g i, T τ σ ε σ i, T τ for i yes, no and g no ĝ τ, σ no σ τ, g yes 0, σ yes σ g. µ + g i σ T τ; B74 T τ σ i + σ T τ B75 For given c, α τ, wih α τ > 0, condiion B7 deermines a cuoff rule, as he lef-handside is a consan, while he righ-hand-side decreases wih ĝ τ. Le g c, α τ be he hreshold ha solves he equaion e c E τ ατ e εyes τ,t + α τ γ E τ ατ e εg τ,t + α τ γ where ε g τ, T is as in B73 bu wih g i g c, α τ and σi σ no. We hen have ha B7 is saisfied if and only if ĝ τ < g c, α τ. Q.E.D Lemma A7 Turning o prices, we firs obain he sae price densiy a τ+, an insan afer he policy decision has been made. Given he aggregae capial in Lemma A6, we have ατ π yes τ+ Bτ+E γ τ+ e εyes τ,t γ + α τ if policy change π τ+ πτ+ no Bτ+E γ ατ τ+ e εno τ,t γ B76 + α τ if no policy change where ε i τ, T are given in B73. The value of firm i ha chooses he risky echnology is given by Mτ+ i E τ+ BT i π T/π τ+. The numeraor is equal o Mτ+π i τ+ E τ+ πt BT i σ µ+gt τ α τ e T τ+σ γ Z b T Z b τ + ατ E τ+ B γ τ+ Bτ+ i σ τ eµ+gt T τ+σ Z b T bz τ σ T τ+σ ZT i Zi τ Because Z i T Z i τ is independen of all he oher random variables, upon aking expecaion, we obain E τ+ πt BT i B γ τ+bτ+ i E τ+ e εyes τ,t α τ e εyes τ,t + α τ γ if policy changes Bτ+B γ τ+ i E τ+ e εno τ,t α τ e εno τ,t + α τ γ if policy does no change B77 where ε yes τ, T and ε no τ, T are defined in B73. Equaion B66 follows from B77 and B76. Finally, we compue marke values a ime τ righ before he policy decision is aken. Given he equilibrium funcion gc, α τ, le he probabiliy of a policy change be p τ Pr ĝ τ < gc, α τ ĝ τ, α τ φ c c; σ c, σ c dc. 34 c:bg τ<gc,α τ.
Using he law of ieraed expecaions, we obain π τ E τ πt BT i p τ π yes τ+ + p τ πτ+ no B78 p τ E τ+ πt BT i yes + p τ E τ+ πt BT i no, B79 where E τ+ π T BT i yes and E τ+ π T BT i no are given in B77. We hen obain he marke value righ before he decision a ime τ: M i τ E τ π T B i T π τ p τ E τ+π T B i T yes π yes τ+ where ω is in B67 wih H πno τ+ p τe τ+ π T BT i yes + p τ E τ+ π T BT i no p τ π yes τ+ + p τ πτ+ no + p τ πno τ+ π yes τ+ p τ + p τ πno τ+ π yes τ+ ωm yes τ+ + ωmτ+ no, π yes τ+. E τ+π T B i T no π no τ+ We finally obain he equilibrium condiion for a Nash Equilibrium. A firm ha chooses he riskless echnology has BT i Bτ i and hus a marke price Mτ i Eτπ T BT i π τ Bτ i E τπ T π τ Bτ i. Thus, a value-maximizing manager sricly prefers he risky echnology if and only if Mτ i > Bi τ. For here o be an equilibrium wih a fracion α τ of firms chosing he risky echnology, we mus have he indifference condiion Mi τ, oherwise all firms would op Bτ i for eiher he risky or he riskless echnology. From he pricing formula of socks B66, and because B τ+ B τ capial does no jump a he announcemen, his indifference condiion implies condiion B68. To summarize, he Nash equilibrium is as follows: Given ĝ τ, le here be α τ such ha condiion B68 is saisfied. Given ha he fracion α τ of firms choose he risky echnology, le gc, α τ be he soluion o equaion B63. The governmen changes is policy if and only if ĝ τ < gc, α τ. Given his policy funcion of he governmen, he equilibrium α τ mus be consisen wih p τ Pr ĝ τ < gc, α τ, which affecs M i τ/b i τ and hus also he equilibrium condiion B68. QED Proposiion 9 Lemma A8: The equilibrium condiion B68, obained in a seing in which firms maximize heir marke values, is equivalen o a firs-order condiion in an alernaive seing in which a social planner chooses α τ o maximize he invesors expeced uiliy. Proof of Lemma A8. Consider a social planner/represenaive manager who can choose a ime τ how much invesmen in physical capial o make. Specifically, his manager chooses he fracion α τ of firms ha will inves in he risky echnology. Physical invesmens are irreversible and he decision is made wihou knowing wheher he governmen will change is policy as in he paper. The represenaive manager decides according o B γ T B γ T B γ max E τ max p τ E τ α τ γ α τ γ yes T + p τ E τ γ no, 35
where B T yes B τ ατ e εyes,t + α τ B T no B τ ατ e εno,t + α τ and where ε yes, T and ε no, T are defined in he paper. Subsiue B γ T max E τ α τ γ B γ τ γ p τ E τ ατ e εyes,t + α τ γ + p τ E τ ατ e εno,t + α τ γ Assuming an inerior soluion 0 < α τ <, he FOC wih respec o α τ is or, equivalenly, 0 p τ E τ ατ e εyes,t + α τ γ e ε yes,t + p τ E τ ατ e εno,t + α τ γ e ε no,t ατ p τ E τ e εyes,t + α τ γ ατ + p τ E τ e εno,t + α τ γ ατ p τ E τ e εyes,t + α τ γ e ε yes ατ,t + p τ E τ e εno,t + α τ γ e ε no,t or, equivalenly, ατ p τ E τ e εyes,t + α τ γ e ε yes,t. ατ + p τ E τ e εno,t + α τ γ e ε no,t p τ E τ α τ e εyes,t + α τ γ + p τ E τ α τ e εno,t + α τ γ. This is he same condiion as in he paper, where α τ is chosen as a Nash equilibrium in which all firms have Mi τ. Bτ i QED Lemma A8 Exension: Heerogeneous Beas. Lemma A9 Learning: Le g N0, σg, le β denoe he N vecor of governmen beas, and le ds denoe he N vecor of signals ha invesors receive a, given by ds µ N + βgd + σdz Then, for 0, τ he poserior is g Nĝ, σ, wih dĝ σ σ β dẑ and σ 36 σ g + β β σ
Above, dẑ is a N vecor of Brownian moions defined by dẑ σ ds E ds The profiabiliy process for firm i in secor n under he filered probabiliy measure is hen dπ i µ + βn ĝ d + σdẑ n, + σ dz i B80 Proof of Lemma A9. The proof of he learning dynamics follows from an applicaion of he Kalman Bucy filer see e.g. Lipser and Shiryaev 977. From he definiion of he new Brownian moions dẑ, we can wrie he signals under he filered probabiliy measure: ds µ N + βĝ d + σdẑ This implies ha for every n,..., N, we have he following ideniy: β n gd + σdz n β n ĝ d + σdẑn B8 The dynamics of he profiabiliy process under he filered measure follow immediaely. QED Lemma A9 Lemma A0. Aggregae capial a T is given by B T B τ e µt τ σ N T τ n w n τ e βn gt τ+σz nt Z nτ where w n τ Bn τ B τ Proof of Lemma A0. There are N secors of firms, each wih mass λ n, where N n λn. Denoe by Λ n 0, he se of firms in secor n. The aggregae capial a T is hen B T B i Tdi N n Λ n B i Tdi where BT n is he aggregae capial in secor n. Using he same seps as in he proof of Lemma A6, we can express he laer as follows: BT n Bτe i µ+βngt σ τ T τ+σz n,t Z n,τ σ T τ+σ Z i,t Z i,τ di Λ n e µ+βn gt τ σ T τ+σz nt Z nτ σ T τ Bτ i eσ Z i,t Z i,τ di Λ n B n τ e µ+βn gt σ T +σz nt Z n 37 N n B n T B8
Thus or where B T N BT n n N n B T σ µt τ e B τ For laer reference, noe ha he vecor B n τ e µ+βn gt τ σ T τ+σz nt Z nτ N T τ wτ n gt τ+σz nt Z nτ eβn n w n τ Bn τ B τ β T τg + σ Z T Z τ N β T τĝ τ, σ τ T τ ββ + σ T τ B83 B84 QED Lemma A0 Proposiion 0. The governmen changes is policy a ime τ if and only if ĝ τ < g c,w τ, B85 where he hreshold g c,w τ is he soluion o N γ e c wτ n e xn φ x 0, σg, τ N γ dx wτ n e xn φ x g c,w, σ τ, τ dx R n n R n n and φx a, b, denoes he mulivariae normal densiy N a T β, b T ββ + σ T B86 Proof of Proposiion 0. The governmen chooses a new policy if and only if CB γ T B γ E τ γ yes T > E τ γ no Subsiuing aggregae capial in B83, we obain he condiion N γ N γ e c E τ w n e βn gt τ+σz nt Z nτ yes < E τ w n e βn gt τ+σz nt Z nτ no n Using B84, we can express he expecaion as e c R n N γ wτ n e xn φ x 0, σg, τ dx < n 38 R n n N γ wτ n e xn φ x ĝ τ, σ τ, τ dx B87 n
where φx a, b, denoes he mulivariae normal disribuion as in B86. Given w τ, he lef-hand side of B87 is independen of ĝ τ. Assuming β > 0, he righ-hand side is monoonically decreasing in ĝ τ as he mean of each x n increases, bu he variance covariance marix is independen of ĝ τ. I follows ha here is a cuoff gw τ, c such ha a change in policy occurs if and only if ĝ τ < g w τ, c QED Proposiion 0 Proposiion A5. a For τ+, he sochasic discoun facor is given by where π B γ e Ω w, ĝ, σ, σ γµt +γ R N T Ω w, ĝ, σ, B88 γ w n e xn φ x ĝ, σ, dx and φx ĝ, σ is he mulivariae normal disribuion as in B86. n b A τ, righ before he announcemen, he sochasic discoun facor is given by π τ Bτ γ e σ γµt τ+γ T τ p τ Ω w τ, 0, σ g, τ+ + p τ Ω w τ, ĝ τ, σ τ, τ+ B89 where p τ is he probabiliy of a policy change a τ, p τ Pr ĝ τ < gw τ, c. This probabiliy can also be expressed as p τ N c w τ, ĝ τ, σ τ, σ c, σc B90 where N., b, c denoes he cumulaive normal disribuion wih mean b and variance c, and c w τ, ĝ τ, σ τ log R w n γ N τ e xn φx ĝ τ, σ τ, τdx w R n N τ e xn γ φ x 0, σg, τ dx c For < τ, he sochasic discoun facor can be compued as π B γ e σ γµt τ+γ T τ Ω w, ĝ, σ, where, wih sligh abuse of noaion, we denoe for < τ Ω w, ĝ, σ, { Bτ γ E pτ Ω w τ, 0, σg, τ+ + p τ Ω w τ, ĝ τ, σ τ, τ+ } B d The dynamics of he SDF are given by dπ π σ π,dẑ For noaional convenience, we suppress he dependence of he SDF on a consan Lagrange muliplier. 39