Internet Appendix for Uncertainty about Government Policy and Stock Prices

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1 Inerne Appendix for Uncerainy abou Governmen Policy and Sock Prices ĽUBOŠ PÁSTOR and PIETRO VERONESI This Inerne Appendix provides proofs and addiional heoreical resuls in suppor of he analysis presened in he paper The Inerne Appendix is organized in six secions whose iles correspond o he respecive secions in he paper The conens are as follows: Secion I: Learning Page Secion II: Opimal Changes in Governmen Policy Page 3 Secion III: Sock Prices Page 4 Secion IV: Exension: Endogenous Timing of Policy Change Page 9 Secion V: Exension: Invesmen Adjusmen Page 34 Secion VI: Exension: Differen Policy Exposures Page 39 Two equaions from he paper are paricularly useful for his Inerne Appendix, so hey are repeaed here for convenience: Firs, firm profiabiliy evolves sochasically for all 0, T: dπ i µ + g d + σdz + σ dz i IA Second, he prior disribuions of boh g old and g new a ime 0 are normal: g N 0, σ g IA Ciaion forma: Pasor, Lubos, and Piero Veronesi, 0, Inerne Appendix for Uncerainy abou Governmen Policy and Sock Prices, Journal of Finance vol, pages Please noe: Wiley-Blackwell is no responsible for he conen or funcionaliy of any supporing informaion supplied by he auhors Any queries oher han missing maerial should be direced o he auhors of he aricle

2 I Learning PROPOSITION Observing he coninuum of signals dπ i in equaion IA across all firms i 0, is equivalen o observing a single aggregae signal abou g : ds µ + g d + σdz IA3 Under he prior in equaion IA, he poserior for g a any ime 0, T is given by g N ĝ, σ IA4 For all τ, he mean and he variance of his poserior disribuion evolve as dĝ σ σ dẑ IA5 σ +, IA6 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 IA5 and IA6 also hold 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 IA5 In addiion, for > τ, σ follows σ + τ IA7 σ σ g 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 IA yields ds µ + g d + σdz + σ dz i 0

3 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 IA3, he remainder of he proposiion follows from sandard resuls abou he Kalman Bucy filer see, for example, Lipser and Shiryaev 977 QED Proposiion II Opimal Changes in Governmen Policy STATEMENT FROM THE TEXT: 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, IA8 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 IA, we immediaely obain he following expression for firm i s capial a ime T : B i T B i τe µ+g σ σ «T τ+σz T Z τ+σ Z i T Zi τ, IA9 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 T di e µ+g σ σ «The Law of Large Numbers implies ha T τ+σz T Z τ 0 B i τ eσ Z i T Zi τ di IA0 0 B i τ eσ Z i T Zi τ di E i Bτ i eσ ZT i Zi τ E i Bτ i E i e σ ZT i Zi τ, IA where he las sep follows from he fac ha he random variables Bτ i and ZT i Zτ i are independen of each oher The firs expecaion on he righ-hand side of IA 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 IA0, we obain he claim in IA8 QED Saemen from he ex 3

4 PROPOSITION The governmen changes is policy a ime τ if and only if ĝ τ < gc, IA where gc σ g σ τ γ T τ c T τγ IA3 Proof of Proposiion : Using he marke clearing condiion W T B T, we can use IA8 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 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 Recall ha if he governmen changes is policy, hen g γ ec+ γµ B γ τ γ e γµ+bgτ σ T τ+ σ g γ T τ + γ σ T τ σ T τ+ bσ τ γ T τ + γ The claim of he proposiion follows immediaely from he opimaliy condiion E CW γ T W γ γ yes T > E γ no IA4 σ T τ IA5 QED Proposiion III Sock Prices PROPOSITION 3 Each firm s sock reurn a he announcemen of a policy change is given by R ĝ τ pĝ τf ĝ τ Gĝ τ pĝ τ + pĝ τ F ĝ τ Gĝ τ, IA6 where 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 γ T τ T τ σg σ τ σ ; c, σ c and Nx; a, b denoes he cdf of a normal disribuion wih mean a and variance b 4 IA7 IA8, IA9

5 The proof of Proposiion 3 comes afer he proof of he following saemen in he ex STATEMENT FROM THE TEXT: Righ afer ime τ, a ime τ+, he marke value of each firm i akes one of wo values: Mτ+ i M i,yes τ+ B i τ+ eµ γσ T τ+ γ T τ σ g if policy changes M i,no τ+ B i τ+ 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 IA0 M i τ ωm i,yes τ+ + ω M i,no τ+, IA where he weigh ω, which is always beween zero and one, is given by ω p τ p τ + p τ F ĝ τ, IA 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 B γ τ+e e γµ+g σ T τ γσz T Z τ π yes τ+ λ B γ 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 IA8 Righ before he policy decision, we have IA3 T τ σg if policy changes IA4 T τ bσ τ if policy does no change, π τ E τ π τ+ p τ π yes τ+ + p τ π no τ+, IA5 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 IA3 ha his probabiliy is given by expression IA9 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 IA8 and IA9, 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+ i E 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 5

6 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+bg τ+ γγ σ T τ+ γ 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 τ+ λ E τ+ B γ T Bi T no /πτ+ no o yield equaion IA0 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 T Bi T no 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 IA when we define ω p τ π yes τ+ p τ π yes τ+ + p τ πτ+ no I follows immediaely from IA4 ha Fĝ τ πno τ+ QED Saemen from he ex π yes τ+ p τ p τ + p τ πno τ+ π yes τ+ 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+ IA6 I is easy o see from IA0 ha Gĝ τ M i,no τ+ /M i,yes τ + Subsiuing his expression and ω from IA proves he claim QED Proposiion 3 COROLLARY As risk aversion γ, he announcemen reurn R ĝ τ for any ĝ τ Proof of Corollary : For any given ĝ τ, expression IA9 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 τf 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 6

7 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 IA9, we have p τ p N0, σ c, σc 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 Recalling ha ĝ τ N0, σg 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 PROPOSITION 4 The marke value of each firm drops a he announcemen of a policy change ie, Rĝ τ < 0 if and only if where g σ g σ τ ĝ τ > g, IA7 T τ γ IA8 Proof of Proposiion 4: From expression IA6, we see ha Rĝ τ < 0 if and only if Gĝ τ > From he formula for Gĝ τ in IA8, we see ha he laer condiion is saisfied if and only if condiion IA7 is QED Proposiion 4 PROPOSITION 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 o 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, IA9 where N c x N x, σ c, σc Pr c < x is he cumulaive normal densiy, and he random variable x has he normal disribuion x N σ x, σ x, 7

8 where σ x γ T τ σ g σ τ IA30 Moreover, a policy change occurs if and only if he poliical cos is sufficienly low, ha is, c < x where Proof of Lemma A: We can rewrie he formula for R ĝ τ in Proposiion 3 as 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 IA9 Denoing his exponen by x, x ĝ τ γt 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, we also obain which leads o IA9 ĝ τ x γt τ + γt τ σg σ τ, G ĝ τ e bgτt τ+ γt τ σ g bσ τ e x γ γt τ σ g bσ τ+ γt τ σ g bσ τ e x γ γt τ σ g bσ τ, 8

9 From is definiion above, x is normally disribued as of ime 0, x Nµ x, σ x, where σ x γ T τ σ g σ τ µ x γ T τ σ g 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 This inequaliy is equivalen o QED 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 Changedx, 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 if and only if 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 c dc φ x x φ c cdc φ x xn c x 9

10 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 Changedx φx xn c xdx N c xe x e x N c x + N c x e 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 x e x γ γt τ σg bσ τ φ x xdx Defining k x as in he claim yields he lemma QED 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 IA3 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 IA30, we obain γ σ x T τ σ g σ τ Subsiuing for γ in IA3, we obain he equivalen expression x σ «x l σx S k x e l φ x xdx, IA3 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 0

11 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 l σx E k x We now rewrie he firs expecaion as E k xe x «σ x l σx k xe x «σ x le σx x+ σ x σ x πσx dx 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 Exponen x + σ x + x σ x lσx σ x x + x σ + xσx σ x x + x σ + xσx xlσ x σ x + x σ x lσx x + x σ + x σx lσ x σ x σ x lσ x σ 3 x l x + σ x lσ x + lσ x σxl 3 σxl 3 σx x + σ x lσ x + lσ x σx 3l σx x + σ x lσ x + σx l σ x l

12 Thus, we obain he ideniy e x σ x σx «x+ σ x le σx πσx x+ e e + l σ xl σx lσx σx + l σ xl πσx x+ σx lσx σ e x πσx Noe ha he expression in parenheses 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 σ x σx «l E k x e l k xe x σx «x+ le σ x σx σx πσx dx E k x 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 QED 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, IA33

13 where x N σ x, σx and k x N c xn c x N c x + N c xe IA34 x 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 x+ σx e xe σx 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 E k x + m k x, σ x πσx dx x σ x k e σx x + m k x dx πσx where x is now redefined as x N σ x, σx 3

14 QED Lemma A4 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 IA33 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 IA34 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 dx or, equivalenly, if and only if < 0 if and only if N c xn c x + N c x N c x N c xe x < 0, N c x N c x < e x Nc 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 IA34 is symmeric abou /σ c, 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 PROPOSITION A In he benchmark model for τ, he sae price densiy is given by π B γ Ωĝ,, IA35 4

15 where 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 + g0; ĝ γ σ τ + T bσ g0; ĝ γ σ T T τ σ τ σ +γ+γ T σ c / T τ γ, σ σ τ + σc T τ γ + σ c/ T τ γ, σ σ τ + σ c T τ γ Proof of Proposiion A: We have π E π τ+ E π τ+ cφ 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 IA3, 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 STEP : We compue he wo condiional expecaions separaely, firs compuinge 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 ie, a ime τ + is given in IA4 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 ha 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 E e γ µ Bτ γ e γ σ µ σ T τ+ γ T τ σg+σ T τ ĝτ < gc, c T τ+ γ T τ σ g +σ T τ E B γ τ ĝ τ < gc, c 5

16 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 µτ + τ σ bgd e τ +σ Z b τ bz 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 dẑu 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 ĝ τ N E b τ E ĝ τ V bτ C b, τ, 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 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 6

17 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 dẑu 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 h b ĝ Io s Lemma implies 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 + ĝ µ u + σ ĝ u + σ u du + I follows ha E h τ h + h + h + τ τ τ τ ĝu σ + b u σ u σ dẑ u ĝu E + µ σ E ĝ u + σ u du ĝ + σ σ u + µ σ ĝ + σ u ĝ µ + σ + σ ĝ du du h + ĝ τ + σ τ + µ σ ĝ τ, where we use 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 7

18 h + ĝ τ + σ τ + µ σ ĝ τ b + µ + ĝ σ τ σ τ ĝ QED 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 where Noe ha we can also wrie 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 φ x ĝ τ < gc, c x bg πv bg e V bgτ τ N gc; ĝ, σ σ τ «γ Cbτ,bgτ V b τ V bgτ E e γb τ ĝ τ < gc, c e γe b τ+ N gc; ĝ, σ σ τ e γebτ+ e γebτ+ 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 V bgτ φ x ĝ τ < gc, c dx, 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 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 8 τ bσ +σ τ N gc, ĝ γ σ τ, σ σ τ N gc; ĝ, σ σ τ

19 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 IA4 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 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 Define 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 N ĝ τ ĝτ T τ + E b τ E ĝ τ V yτ C y τ, 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 τ σ + 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 9

20 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 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 E e γy τ ĝ τ > gc, c e γeyτ+ N gc; ĝ, σ σ τ The same seps as in Case yield 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 στ 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 στ 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 γµt γbgτ +γ +B γ T + γ T τ σ g+τ bσ + γ σ T 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 0

21 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 σ τ 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 E x<gc c E x<gc Pr x < g c Pr x < g 0 Similarly, p no N c γ T τ g 0;ĝ γ σ τ E N g c, ĝ γcy τ, b τ, σ σ τ N g 0 ; ĝ γcy τ, b τ E Pr x < g c c σ c γ T τ ; σ σ τ + σ c γ T τ ; σ σ τ + Subsiuing in E π τ+, he claim of Proposiion A follows QED Proposiion A PROPOSITION 6 The sochasic discoun facor SDF follows he process σc γ T τ σc γ T τ dπ π σ π, dẑ + J π {τ }, IA36 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 pτ Fbgτ p τ+ p τfbg τ if policy changes J no π pτfbgτ p τ+ p τfbg τ if policy does no change IA37 For > τ, σ π, is given by σ π, γ σ + T σ σ, IA38

22 and for τ, i is given by Corollary A in he paper σ π, γσ Ωĝ, σ Ωĝ, ĝ σ IA39 Proof of Proposiion 6: For < τ, he SDF dynamics sem from an applicaion of Io s Lemma o IA35, which also yields he volailiy IA39 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 IA4, wih in place of τ in he formula An applicaion of Io s Lemma immediaely leads o he diffusion erm IA38 A he ime of he anouncemen, τ, he sae price densiy jumps from IA5 o eiher of he wo expressions in IA4 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 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 QED Proposiion 6 PROPOSITION A In he benchmark model for τ, he sock price for firm i is given by M i B i Φĝ, Ωĝ,, IA40 where Φĝ, p yes K yes + p no K no K yes K no e e γµt + γbgτ + γ T τ σg+τ bσ γγ σ T γµt + γbgt + γ bσ T γγ σ T

23 and p yes p no N N g0; ĝ + γ σ τ + g0; ĝ + γ σ T T τ σ τ σ c/ T τ γ, σ σ τ + σc T τ γ + σ c / T τ γ, σ σ τ + σ c T τ γ 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, 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 IA4 and IA0, E πτ+ Mτ+ ĝ i 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 Bτ i B i eµτ +gold τ σ τ +σz τ Z σ τ +σ Zτ i Zi B i Bτ e σ τ +σ Zτ i Z i B IA4 Thus, E B γ τ Bτ i ĝ τ < gc, c Bi E B Bτ γ e σ τ +σ Zτ i Zi ĝτ < gc, c Bi B E B γ τ ĝ τ < gc, c, 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 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 3

24 Subsiue o obain E πτ+ Mτ+ ĝ i τ < gc, c BB i γ e σ γµt + γbgτ γγ T + γ N gc, ĝ + γ σ τ, σ σ τ N gc; ĝ, σ σ τ T τ σ g+τ bσ CASE NO POLICY CHANGE E πτ+ M i τ+ ĝ τ > gc, c E Using IA4 again, we have e B i τ e µ+bg Bτ γ e γ τ σ µ+bg τ σ T τ+ γ T τ bσ τ +σ T τ 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 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 Z i 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 τ σ Puing he erms ogeher, N gc, ĝ, σ στ γ γµt γ σ e T + γbg T + γ E πτ+ Mτ+ i ĝ τ > gc, c B i B γ N gc, ĝ + γ C y τ, ĝ τ, σ στ 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 4

25 PROPOSITION 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 }, IA4 where he jump componen J M is given by J M { J yes M R ĝ τ if policy changes J no M R ĝ τ Gĝ τ + Gĝ τ if policy does no change IA43 For > τ, we have µ M, γ σ + T σ σ IA44 σ M, σ + T σ σ, IA45 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 IA46 µ M σ π, σ M,, IA47 where σ π, and σ M, are given in equaions IA39 and IA46, respecively Proof of Proposiion 7: For < τ, he dynamics of he reurndm i /M i sem from an applicaion of Io s Lemma o IA40, which also yields he volailiy IA46 The expeced reurn is given by µ M Covdπ /π, dm i /Mi σ π,σ M,, which yields IA47 For > τ, he price is given in closed form as in equaion IA0, wih in place of τ in he formula An applicaion of Io s Lemma immediaely yields he diffusion erm IA45 We obain IA44 by using he closed-form soluion for he SDF diffusion in IA38 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 Rĝ τ, is 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+ IA48 IA49 where we use ω in IA and Gĝ τ M i,no τ+ /M i,yes τ + This is he same expression as in IA43, as can be easily verified by subsiuing for Rĝ τ from IA6 QED Proposiion 7 5

26 COROLLARY 3 The marke value of each firm increases a he announcemen of no policy change ie, J no M > 0 if and only if ĝ τ > g, where g is given in equaion IA8 Proof of Corollary 3: The proof follows immediaely from expression IA49, which is posiive if and only if Gĝ τ >, where Gĝ τ is given in IA8 QED Corollary 3 PROPOSITION 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 IA50 Proof of Proposiion 8: The proof follows immediaely from E τ J M p τ J yes M and subsiuing J yes M QED Proposiion 8 Rĝ τ in IA6 and J no M in IA49 COROLLARY 4 We have E τ J M < 0 if and only if where g is given in equaion IA8 and + p τj no M g < ĝ τ < g, IA5 g γ T τ σ g σ τ IA5 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 6

27 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 τ σ 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 use he fac ha Fĝ τ Gĝ τ e +γbgτt τ+ γ γ T τ σg 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 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, 7

28 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 ie, 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 σ τ 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 ie, 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 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 8 IA53

29 For < τ, he correlaion is given by Corollary A3 in he paper ρ σ M, σ + σ M, IA54 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 ρ E dm i M i E dm i M i dm j M j 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 IV 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 IA55 PROPOSITION 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 IA55 is given by V ĝ, B, B γ Φĝ,, IA56 where Φĝ, saisfies he parial differenial equaion 0 Φĝ, + { γµ + ĝ } γ γσ Φĝ, + Φĝ, σ ĝ σ + γ Φĝ, σ IA57 ĝ 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, 9

30 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 A policy change occurs a ime τ if e c B γ τ γ e γµt τ+σ g T τ γ γ σ T τ IA58 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+ } condiion E dv 0, ha is 0 V + V E db + V E dĝ + B ĝ V B I follows ha V saisfies he maringale 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 }, IA59 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 parenheses 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 B Φĝ, ; V B γ Φĝ, ĝ ĝ ĝ Φĝ, V ; γb γ ĝ B 30 γ γ B γ Φĝ, Φĝ, ĝ

31 In addiion, we have for < τ, 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 IA57 The final condiion a τ i+ follows from equaion IA59 QED Proposiion A3 STATEMENT FROM THE TEXT: 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 g T τ γ γ σ T τ IA60 Proof: See proof of Proposiion A3 QED Saemen from he ex PROPOSITION 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 B 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 γ γ σ σ + γ Ωĝ, σ ĝ wih he following boundary condiions for τ τ i+ : F ĝ, + F ĝ, ĝ } σ Ωĝ, F ĝ τ, τ p τ e γµt τ+ γ T τ bσ g σ +γγ+ T τ + p τ F ĝ τ, τ σ σ γ F ĝ, σ ĝ Ωĝ τ, τ p τ e γµt τ+ γ T τ σ g γγ σ 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, σc, IA6 3

32 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 IA6 τ+ ec+ γµt γ σgt 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 he policy 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 The final condiion a T is W B T, ĝ T, T B γ T so ha W W B We conjecure 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 3 γb γ F ĝ, ; B F ĝ, W ; ĝ τ e γµt τ+ γ T τ σg+γγ+ σ T τ γb γ B ĝ F ĝ, ĝ

33 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 + + 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 E dĝ + Q E db ĝ Q B ĝ B j E db j dĝ + + 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 Bτ j e σ T +σ Z j T Zj B γ B j τb γ τ B j τb 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+ : We conjecure so ha Q Q B Q B j 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 B j B γ Ωĝ, ; Q Q B j,b, ĝ, B j B γ Ωĝ,, 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 33

34 Subsiuing in he PDE, we obain 0 BB j γ Ωĝ, where we also use + 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 B j γb γ Ωĝ, σ BB j, 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 QED Proposiion A4 V Exension: Invesmen Adjusmen PROPOSITION 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, IA6 where for given c, α τ he hreshold g c, α τ saisfies he equaion where e c E τ ατ e εyes τ,t + α τ γ E τ ατ e εg τ,t + α τ γ, 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 IA63 IA64 IA65 and he equilibrium value of α τ is described below We prove Proposiion 9 ogeher wih he following saemen from he ex of he paper: 34

35 STATEMENT FROM THE TEXT: If firm i decides a ime τ o remain invesed in is risky echnology, is marke value righ afer ime τ is given by Mτ+ i M i,yes τ+ B i τ+ M i,no τ+ B i τ+ h E 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 E τ+ h e εno τ,t α τe εno τ,t + α τ γi E τ+ h ατe εno τ,t + α τ γi if policy does no change IA66 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 IA, excep ha he weighs ω τ are given by ω τ p τ p τ + p τ H τ, IA67 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 ie, he probabiliy ha condiion IA6 holds The condiion for equilibrium wih 0 < α τ < is M i yes ω τ+ M i no τ + ω Bτ+ i τ τ+ IA68 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 B τ α τ e µ+g σ where g g new if a policy change occurs a τ, and g g old oherwise T τ+σz T Z τ + α τ, IA69 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 IA9 If a firm chooses he riskless echnology insead, hen BT i Bi τ I follows ha aggregae 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 invesors expeced uiliy See Lemma A8 below 35

36 capial a ime T is B T 0 B i T di ατ 0 σ µ+gt τ e T τ+σz T Z τ σ T τ Bτ i σ τ eµ+gt T τ+σz T Z τ σ T τ+σ ZT i Zi τ di + Bτ i di α τ ατ 0 B i τ eσ Z i T Zi τ di + α τ B i τ di, 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 Bτ i eσ ZT i Zi τ ατ di ατ Bτ i α eσ ZT i Zi τ di α τ E B iτ eσ 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 Similarly, B i τdi α τ B α τ α τdi i α τ E Bτ i τ α τ Subsiuing, we obain B T E Bτ i 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 QED Lemma A6 LEMMA A7: The governmen changes is policy if and only if ĝ τ < g c, α τ, IA70 where he hreshold g c, α τ is in he claim of Proposiion 9 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 IA7 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 IA69 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 τt 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 36

37 Using his disribuion, condiion IA7 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, IA7 where wih ε i τ, T N µ ε g i, T τ σ ε σ i, T τ µ ε g i, T τ,σ ε σ i, T τ IA73 µ + g i σ T τ IA74 T τ σ i + σ T τ IA75 for i yes, no and g no ĝ τ, σ no σ τ, g yes 0, and σ yes σ g For given c, α τ, wih α τ > 0, condiion IA7 deermines a cuoff rule, as he lef-hand side 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 IA73 bu wih g i g c, α τ and σi σ no We hen have ha IA7 is saisfied if and only if ĝ τ < g c, α τ QED 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 + α τ if no policy change, where ε i τ, T are given in IA73 IA76 The value of firm i ha chooses he risky echnology is given by M i τ+ E τ+ B i T π T/π τ+ The numeraor is equal o M i τ+ π τ+ E τ+ πt BT i E τ+ B γ τ+ σ µ+gt τ α τ e T τ+σ γ Z b T bz τ + ατ Bτ+e i σ µ+gt τ T τ+σ Z b T bz τ σ T τ+σ ZT i Zi τ 37

38 Because Z i T Zi τ is independen of all he oher random variables, upon aking he expecaion we obain E τ+ πt BT i B γ τ+b i τ+ E τ+ e εyes τ,t α τ e εyes τ,t + α τ γ if policy changes B γ τ+b i τ+ E τ+ e εno τ,t α τ e εno τ,t + α τ γ if policy does no change, IA77 where ε yes τ, T and ε no τ, T are defined in IA73 Equaion IA66 follows from IA77 and IA76 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 Pr ĝ τ < gc, α τ ĝ τ, α τ p τ Using he law of ieraed expecaions, we obain π τ c:bg τ<gc,α τ φ c c; σ c, σ c dc p τ π yes τ+ + p τ π no τ+ IA78 E τ πt B i T pτ E τ+ πt B i T yes + p τ E τ+ πt B i T no, IA79 where E τ+ π T B i T yes and E τ+ π T B i T no are given in IA77 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 IA67 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 π 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 Bi τ and hus a marke price Mi τ 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 > Bτ i For here o be an equilibrium wih a fracion α τ of firms choosing he risky echnology, we mus have he indifference condiion Mi τ, oherwise all firms would op for eiher he risky or he Bτ i riskless echnology From he pricing formula of socks IA66, and because B τ+ B τ capial does no jump a he announcemen, his indifference condiion implies condiion IA68 To summarize, he Nash equilibrium is as follows: Given ĝ τ, le here be α τ such ha condiion IA68 is saisfied Given ha he fracion α τ of firms choose he risky echnology, le gc, α τ be 38 τ+

39 he soluion o equaion IA63 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 Mτ i/bi τ and hus also he equilibrium condiion IA68 QED Proposiion 9 LEMMA A8: The equilibrium condiion IA68, 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 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 max E τ max p τ E τ α τ γ α τ where B γ T γ yes + p τ E τ B γ T B T yes B τ ατ e εyes,t + α τ B T no B τ ατ e εno,t + α τ γ no, and where ε yes, T and ε no, T are defined in he paper Subsiue B γ ατ T max E τ B γ p τ E τ e εyes,t + α τ γ τ ατ α τ γ γ + p τ E τ e εno,t + α τ γ Assuming an inerior soluion 0 < α τ <, he FOC wih respec o α τ is 0 p τ E τ ατ e εyes,t + α τ γ e ε yes,t or equivalenly, + 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 ατ e ε yes,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 39

40 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 B i τ QED Lemma A8 VI Exension: Differen Policy Exposures LEMMA A9 Learning: Le g N0, σ g, 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 Above, dẑ is a N vecor of Brownian moions defined by σ g dẑ σ ds E ds + β β σ The profiabiliy process for firm i in secor n under he filered probabiliy measure is hen dπ i µ + β n ĝ d + σdẑn, + σ dz i IA80 Proof of Lemma A9: The proof of he learning dynamics follows from an applicaion of he Kalman Bucy filer see, for example, 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 IA8 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 40 w n τ e βn gt τ+σz nt Z nτ,

41 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 N BT n, where B n T 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 Λ n n T τ+σz n,t Z n,τ σ e µ+βn gt τ σ T τ+σz nt Z nτ σ T τ T τ+σ Z i,t Z i,τ di Λ n B i τe σ Z i,t Z i,τ di B n τ e µ+βn gt σ T +σz nt Z n IA8 Thus, or where B T B T N BT n n For laer reference, noe ha N n B τ e µt τ σ B n τ e µ+βn gt τ σ T τ+σz nt Z nτ N T τ n w n τ Bn τ B τ w n τ e βn gt τ+σz nt Z nτ, β T τg + σ Z T Z τ N β T τĝ τ, σ τ T τ ββ + σ T τ IA83 IA84 QED Lemma A0 PROPOSITION 0 The governmen changes is policy a ime τ if and only if ĝ τ < g c,w τ, IA85 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 4

42 and φx a, b, denoes he mulivariae normal densiy N a T β, b T ββ + σ T IA86 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 IA83, 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 IA84, we can express he expecaion as N γ e c wτ n e xn φ x 0, σg, τ dx < R n n R n n N γ wτ n e xn φ x ĝ τ, σ τ, τ dx, IA87 where φx a, b, denoes he mulivariae normal disribuion as in IA86 Given w τ, he lefhand side of IA87 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 n ĝ τ < g w τ, c QED Proposiion 0 PROPOSITION 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,, IA88 γ w n e xn φ x ĝ, σ, dx and φx ĝ, σ is he mulivariae normal disribuion as in IA86 b A τ, righ before he announcemen, he sochasic discoun facor is given by π τ Bτ γ e σ γµt τ+γ n T τ p τ Ω w τ, 0, σ g, τ+ + p τ Ω w τ, ĝ τ, σ τ, τ+, IA89 For noaional convenience, we suppress he dependence of he SDF on a consan Lagrange muliplier 4