China s Model of Managing the Financial System

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1 Chna odel of anagng he Fnancal Syem arku K Brunnermeer chael Sockn We Xong Inerne Appendx Th nerne appendx preen proof of he propoon n he man paper Proof of Propoon A We dere he perfec nformaon equlbrum wh radng by he goernmen We fr conjecure ha when + and N are oberable o he goernmen and neor he ock prce ake he lnear form P = p + + p N N + p g G Gen ha ddend are D = + σ D ε D he ock prce mu reac o a deermnc un hf n + by he preen alue of ddend derng from ha hock R f ρ follow ha p = R f ρ The nnoaon o + and N are he only ource of rk and from he perpece of all economc agen he condonal expecaon and arance of R + are E [R + F = p N ρn R f N R f p g G V ar [R + F = σ D + σ R f + p ρ Nσ N + p gσ G Snce all neor are dencal when and N are oberable follow ha n he CARA- Normal enronmen all neor hae an dencal mean-arance demand for he rky ae: X S = γ E [R + F V ar [R + F = γ In he goernmen nerenon rule ϑ N deermned by U G = up γ σ σ D + ϑ p N ρn R f N R f p g G σ D + + p N σ N + p gσ G X G = ϑ N N + ϑ N σ N G R f ρ σ + p Nσ N + p gσ G

2 Fnally by mpong marke-clearng we arre a N = p N ρ N r N γ σ D + + p N σ N + ϑ N N ϑ N σ NG = R f p g γ σ D + + p N σ N + p gσ G whch by machng coeff cen reeal ha γ σ D + p N ρn R f Th confrm he conjecured equlbrum + ϑ N = + p N σ N + p gσ G p N ρn R f ϑ N σ R f N = p g ϑ N Rearrangng h equaon for p N and ubung for p g we arre a he quadrac equaon for p N ρn R f ϑn + σ R f G p N ϑ N + Rf ρ N γσ N ϑ N p N + σ D σ + = σ N R f ρ σ N from whch follow ha p N ha wo roo p N ϑ N = ρ N R f γσ N ϑ N ± R f ρn γσ N ϑ N 4 + ρn R f R f + ρn R f R f G ϑ N ϑ N σ σ D G σ N + IA R f ρ σ σ N ϑ N ϑ N σ G Recognzng ha wo negae oluon for P N ex f he expreon under he quare roo nonnegae follow ha he marke break down occur wheneer R f < ρ N + ϑ N γ ρn R + f ϑn σ R f G σ D ϑ σn + σ N R f ρ σ N Gen ha V ar P F = σ D + goernmen opmzaon problem conequenly reduce o U G ρn R f ϑn = up γ + γ σ + ϑ N R f ϑ N + p N σ N + p gσ G ubung for p g he σ G p Nσ N

3 and from he wo marke-clearng condon rercon on he coeff cen p N and p g ha ϑ N deermned by ϑ N = p N ρn R f γ σ D ρn R f ϑ N R f ϑ N σ G p N σ N To eablh ha he lnear equlbrum he unque ymmerc equlbrum we expre each neor opmzaon problem a U = max E [e γr W +X + +σ D ε D + +P + RP X For an arbary prce funcon P he FOC for he neor holdng of he rky ae X E [ + + σ D ε D + + P + R f P e γx + +σ D ε D + +P + RP F = Subung h wh he marke-clearng condon we arre a E X = ϑ N N + ϑ N σ N G [ + + σ D ε D + + P + R f P e γx + +σ D ε D + +P + RP F = Snce P + canno be a funcon of ε D + a P + forward-lookng for he new generaon of neor a me + he aboe can be rewren a P = R f + γ R f σ D ϑ N N + ϑ N σ N G IA + [ R E e γ ϑ N N +ϑ N σ N G P + P f + E [e γ ϑ N N +ϑ N σ N G P + F F Th defne a funconal equaon whoe fxed pon he prce funconal P To ee ha he lnear equlbrum we dered aboe ole h funconal equaon we rewre equaon IA a P = R f + γ R f σ D ϑ N N + ϑ N σ N G + R f u log E [ e up + F u= γ ϑn N +ϑ N σ N G and conjecure ha P = R f ρ + + p N N + p g G from whch follow ha p N afe equaon IA Th erfe ha he lnear prce equlbrum afe h more general equlbrum condon 3

4 Now defne he operaor T : B R B R T f = R f γ R f σ D ϑ N N + ϑ N σ N G+ R f E [ f e γ ϑ N N+ϑ N σ N Gf E [e γ ϑ N N+ϑ N σ N Gf N G for f B R where B R he pace of connuou funcon ψ bounded n he ψ norm f ψ = up x fx ψx where ψ x = + x b ha polynomal growh for ome b Snce + G and N are arko procee { + N G } are uff cen ac for he condonng n he aboe condonal expecaon We now eablh ha T a conracon map by erfyng ha T afe Blackwell Suff cency condon Snce P defne an ae prce mu be he cae ha f P + weakly ncreae { + G + N + } whch a FOSD hf n he drbuon of P + hen weakly preferred by any aere agen whoe uly ncreang n wealh Conequenly neor would demand more o earn he hgher reurn whch would bd up he prce oday Thu T f T g for f g and T afe monooncy Furhermore T f + c = R f γ R f σ D ϑ N N + ϑ N σ N G + R f E [f + c e γ ϑ N N+ϑ N σ N Gf +c E [e γ ϑ N N+ϑ N σ N Gf +c N G = R γ f R f σ D ϑ N N + ϑ N σ N G + [ R E f e γ ϑ N N+ϑ N σ N Gf f E [e γ ϑ N N+ϑ N σ N Gf N G = T f + R f c N G N G + R c f N G T f + R f cν and T afe dcounng nce R f Weghed Conracon appng Theorem of Boyd 99 > Therefore T a rc conracon map by he Snce a conracon map ha a mo one fxed pon and an equlbrum wh lnear P ex mu be he unque equlbrum n he economy a lea whn he cla of funcon connuou and bounded n he ψ norm Proof of Propoon Noe from he arance of he exce ae payoff ha V ar [R + F = σ D + σ R f + p ρ Nσ N 4

5 and hu he exce olally dren by he p N σ N erm Conder now he expreon for he le negae roo of p N from Propoon n he abence of goernmen nerenon: p N = ρ N R f + Rf ρ N σ D σ + γσ N γσ N σ N R f ρ σ N Gen h expreon follow ha p Nσ N = ρn R f + ρ N R f Rf ρ N σ N γ γ γσ N σ D + σ R f ρ 4 σ D σ N Dfferenang wh repec o σ N we fnd wh ome manpulaon ha p N σ N σ N = Rf ρ N γσ N + Rf ρ N γσ N Rf ρ N γσ N R f ρn γσ N 4 σ D σ N σ 4 D σ N σ + R f ρ σ + R f ρ σ N + R f ρ σ σ N σ N Thu from he econd par of he aboe expreon uff cen for p N σ N σ N σ 4 N + 4 σ R D + σ R f σ f ρ N N > ρ γ > ha By Decare Rule of Sgn he aboe ha only one poe roo for σ N uch ha p N σ N > σ N f σ N > σ N 4 σ = D + + R f ρ N γ σ D + Proded ha σ N > σ N hen p N σ N σ N R f ρ N σ γσ 4 D + N σ N when ε = or > and olaly hghe cloe o marke breakdown when = ε for ε arbrarly mall arke breakdown occur R f ρ σ σ N σ N = R f ρ N αγ σ D + 5

6 Furhermore a ε and σ N R f ρ N αγ σ D + R f σ ρ hen p Nσ N σ D + σ R f ρ Conequenly he maxmum condonal exce payoff arance before breakdown occur V ar [R + F σ D + Proof of Propoon A To arre a he belef of neor we fr characerze he marke belef baed on only he publc nformaon e F To dere he marke belef we proceed n eeral ep Fr we aume he marke poeror belef of + N jonly Gauan + N N ˆ + ˆN Σ where [ ˆ + ˆN Σ = = E [[ + N [ Σ Σ N F Σ N Σ NN Sandard reul for he Kalman Fler eablh ha he law of moon of he condonal expecaon of he marke poeror belef ˆ + ˆN [ ˆ + ˆN [ ρ = ρ N [ ˆ ˆN + k [ D ˆ η H p ρ ˆ p N ρ N ˆN where [[ [ k + D ˆ = Co N η H p ρ ˆ F p N ρ N ˆN [[ D ˆ V ar η H p ρ ˆ F p N ρ N ˆN he Kalman Gan and he condonal arance Σ eole deermncally accordng o [ [ [ Σ ρ = Σ ρ σ + ρ N ρ N σ N [[ [ k D ˆ Co η H p ρ ˆ F p N ρ N ˆN N 6

7 I raghforward o compue ha [[ [ + D ˆ Co N η H p ρ ˆ F p N ρ N ˆN = ρ Σ p ρ Σ + σ + p N ρ ρ N Σ N ρ N Σ N p ρ ρ N Σ N + p N ρ N ΣNN + σ N and ha [[ Ω D ˆ = V ar η p ρ ˆ F p N ρ N ˆN Σ + σ D p ρ Σ + p N ρ N Σ N = p p ρ Σ + p N ρ N Σ N ρ Σ + σ + p p N ρ ρ N Σ N +p N ρ N ΣNN + σ N We conder he deermnc eady-ae of he Kalman Fler and conequenly drop all me ubcrp from condonal arance We hall erfy exence a he end of he proof For η H F F I can expre η H a η H = p + + p N N = p ˆ + + p N ˆN from whch follow ha p + ˆ + + pn N ˆN = A a conequence mu be ha he marke belef abou and N are ex-po correlaed afer oberng he ock prce nnoaon proce η by akng arance and coarance wh + ˆ + and N ˆN Σ N = p p N Σ Σ NN = p p N Σ θn = uch ha we hae he hree dene p p N Σ θθ Conequenly a n He and Wang 995 we need o only compue Σ Updang he marke belef o he prae belef of economc agen can be done n a manner mlar o ha n He and Wang 995 Snce he marke belef ac a a normal pror for neor who obere he normally drbued prae gnal hey updae her belef by Baye Law n accordance wh a lnear updang rule The poeror of neor 7

8 N ˆ + Σ [ + where ˆ + = E [ + F and Σ = E ˆ + F are gen by and ˆ + = ˆ + + Σ Σ + τ ˆ + Σ = Σ + τ Th characerze he belef of neor gen he marke belef Snce he goernmen doe no rade n h benchmark neor hae no ncene o learn abou he goernmen behaor and herefore he nformaon acquon decon ral Gen ha neor each acqure a prae gnal andard reul for CARA uly wh normally drbued prce and payoff eablh ha he opmal radng polcy of neor X gen by where and X = E [ D + + P + R f P F γv ar [D + + P + F + p ρ R f ˆ + ˆ + + pn ρn R f ˆN [ [ + pˆ p ˆ k ˆ + p ρ ˆ + ˆ + + pn ρ N ˆN ˆN = γϕ Ω ϕ [ Ω = Ω ϕ = [ p ρ ρ N + k [ pˆ p Σ θθ Σ θθ + τ [ p ρ ρ N he condonal arance of D + and P + wh repec o F I can rewre he aboe a [ + p ρ R f [ + pˆ p k ˆ + ˆ p X ρ ρ N + + pn ρn R f ˆN = γϕ Ω ϕ by recognzng ha Σ N = p p N Σ Subung for ˆ + and recognzng from aboe ha and herefore ha ˆN = N + p + ˆ + p N ˆN = ˆN p ˆ p + ˆ + = N + p + ˆ p + ˆ N p N p + ˆ + N 8

9 we arre a [ ϕ X = p ρ ρ N Σ Σ +τ ˆ + + ρn R f p N N + p + ˆ + γϕ Ω ϕ Aggregang oer he demand of neor and mpong marke-clearng we arre a he wo equaon for p and p N ϕ [ p ρ ρ N Σ + ρ Σ + τ N R f p = ρn R f p N = γϕ Ω ϕ Th complee our characerzaon of he lnear equlbrum Proof of Propoon A3 To arre a he belef of neor and he goernmen we fr characerze he marke belef baed on he publc nformaon e F To dere he marke belef we proceed n eeral ep Fr we aume he marke poeror belef of + N G + jonly Gauan + N G + N ˆ + ˆN Ĝ + Σ where ˆ + ˆN Ĝ + G = E Σ = + N G + G Σ Σ N F Σ N Σ G Σ NN Σ NG Σ G Σ NG Σ G G Sandard reul for he Kalman Fler hen eablh ha he law of moon of he condonal expecaon of he marke poeror belef ˆ + ˆN ˆ + ˆN Ĝ + G = ρ ρ N ˆ ˆN Ĝ G + K D ˆ η p ρ ˆ p N ρ N ˆN G G 9

10 where K = Co V ar + N G + G D ˆ η p ρ ˆ p N ρ N ˆN F G G D ˆ F η p ρ ˆ p N ρ N ˆN G G he Kalman Gan and ha he condonal arance Σ o Σ = ρ ρ N K Co Σ ρ ρ N D ˆ η p ρ ˆ p N ρ N ˆN G G eole deermncally accordng + σ σ N σ G + N G + F I raghforward o compue ha + Co N D ˆ G + η p ρ ˆ p N ρ N ˆN F G G G ρ Σ p ρ Σ + σ + p N ρ ρ N Σ N ρ Σ G = ρ N Σ N p ρ ρ N Σ N + p N ρ N ΣNN + σ N ρ N Σ NG p G σ G Σ G p ρ Σ G + p N ρ N Σ NG Σ G G and ha Ω = V ar = D ˆ η p ρ ˆ p N ρ N ˆN G G Σ F + σ D p ρ Σ + p N ρ N Σ N p ρ Σ + σ p ρ Σ + p N ρ N Σ N +p p N ρ ρ N Σ N +p N ρ N ΣNN + σ N +p G σ G Σ G p ρ Σ G + p N ρ N Σ NG Σ G p ρ Σ G + p N ρ N Σ NG Σ G G

11 Snce η F F I can expre η a η = p + p N N + p G G + = p ˆ + p N ˆN + p G Ĝ + from whch follow ha p ˆ + pn N ˆN + p G G + Ĝ + = A a conequence mu be ha he marke belef abou and N are ex-po correlaed afer oberng he ock prce nnoaon proce η by akng arance and coarance wh + ˆ + and N ˆN : Σ θn = p Σ p G Σ θg p N p N Σ NN = p Σ N p G Σ NG p N p N Σ NG = p Σ G p G Σ G G p N p N Th complee our characerzaon of he marke belef Proof of Propoon A4 uch ha we hae he hree dene Updang he marke belef o each neor prae belef can be done n a manner mlar o ha n He and Wang 995 Noe ha he marke belef ac a he pror for neor who obere he normally drbued prae gnal The poeror of neor N ˆ + ˆN Ĝ + Σ where ˆ + ˆN Ĝ + = E [ N G + F and Σ = + ˆ + + ˆ E N ˆN + N ˆN F are gen by G + Ĝ + G + Ĝ + where Γ = Co = + N G + ˆ + ˆN Ĝ + Σ Σ G Σ N Σ NG Σ G Σ G G = ˆ + ˆN Ĝ + [ ˆ + g Ĝ + + Γ [ ˆ + g Ĝ + [[ F V ar ˆ + g F Ĝ + [ Σ + a τ Σ G Σ G Σ G G + [ a τ g

12 and Σ = Σ Γ Σ Σ G Σ N Σ NG Σ G Σ G G Snce G publcly reealed common knowledge and peculaor need no updae her belef abou wh her prae nformaon Th characerze he belef of neor gen he marke belef Proof of Corollary Afer he yem ha run for a uff cenly long me nal condon wll dmnh and he condonal arance of he Kalman Fler for he marke belef Σ wll ele down o deermnc coarance-aonary eady-ae To ee h le u conjecure ha Σ Σ In h propoed eady-ae Γ Γ where Γ gen by Σ Σ θg [ Γ = Σ N Σ NG Σ + a τ Σ G Σ G Σ G G Σ G Σ G G + [ a τ g Conequenly nce Γ ndeed conan o Σ Furhermore he eady-ae Kalman Gan K gen by K = ρ Σ p ρ Σ + σ + pn ρ ρ N Σ N ρ Σ G ρ N Σ N p ρ ρ N Σ N + p N ρ N Σ NN + σn ρn Σ NG p G σ G Σ G p ρ Σ θg + p N ρ N Σ NG Σ G G Ω where Ω = Σ + σ D p ρ Σ + p N ρ ρ N Σ N Σ G ρ Σ + σ p ρ Σ + p N ρ ρ N Σ N p +p p N ρ ρ N Σ N ρ N Σ NN + σn +p N +p G σ G p ρ Σ G + p N ρ N Σ NG Σ G p ρ Σ G + p N ρ N Σ NG Σ G G Conequenly nce we hae conruced a eady-ae for he Kalman Fler for he marke belef uch a eady-ae ex

13 Proof of Propoon A5 Smlar o he problem for he goernmen conenen o defne he ae ecor Ψ = [ˆ + ˆN Ĝ+ G wh law of moon Ψ + = ρ ρ N Ψ + K ε + and ε + F N 4 Ω gen by D + ˆ ε + + = η + p ρ ˆ + p N ρ N ˆN G + Ĝ + wh Ω gen n he proof of Corollary Gen ha exce payoff are normally drbued we can decompoe R + a R + = E [ R + F + φ ε S + [ Σ = ςψ + φ ω + a τ [ Σ G ˆ Σ G Σ G G + [ a τ g + g + φ ε S Ĝ + + [ Σ φ G G + [ a ω τ g [ Σ G ˆ Σ G Σ + a τ + g = ςψ + Ĝ + Σ + a τ Σ G G + [ a τ g Σ G + φ ε S + wh ς = [ + pˆ ρ R f p N ρn R f p g R f pĝ R f p g pˆ p φ = + K pĝ p G p g In h decompoon we hae updaed he neor belef equenally from he marke belef followng Baye Rule a E [ R + F = E [ [ Σ R + F + φ ω + a τ [ Σ θg ˆ Σ θg Σ G G + [ a τ g + g Ĝ + [ Σ φ G G + [ a ω τ g [ Σ G ˆ Σ G Σ + a τ + g = ςψ + Ĝ + Σ + a τ Σ G G + [ a τ g Σ G 3

14 where a n Propoon 4 [ ω = Co ε + = [ + G + F Σ Σ G p ρ Σ + p N ρ N Σ θn p ρ Σ G + p N ρ N Σ NG ρ Σ G Σ G G Smlarly by Baye Rule ε S + F N Ω S where [ Σ G G + [ a ω τ g Σ G Σ G Ω S = Ω Σ + a τ ω Σ + a τ Σ G G + [ a τ g Σ G Sandard reul eablh ha he neor problem equalen o he mean-arance opmzaon program up X { R f W + X E [ R + F γ X V ar [ } R + F Imporanly nce he neor hae o form condonal expecaon abou exce payoff a + hey mu form condonal expecaon abou he goernmen fuure radng E [G + F Gen ha he neor are prce-aker from he FOC we ee ha he opmal nemen of neor n he rky ae gen by X = = γ E [R + F γv ar [R + F ςψ + φ ω φ Ω φ ΣG G + [ a τ g Σ G Σ G Σ + a τ ˆ + g Ĝ + Σ +a τ Σ G G +[ a τ g Σ G φ ω ΣG G + [ a τ g Σ G Σ θg Σ + a τ Σ +a τ Σ G G +[ a τ g Σ G Th complee our characerzaon of he opmal radng polcy of he neor Proof of Propoon A6 Each neor face he opmzaon problem A gen n he man paper I hen follow ha neor wll chooe o learn abou he payoff fundamenal e a = wh probably λ: Q < λ = Q = Q > 4 ω φ

15 where Q = φ φ = φ ω [ Σ +τ Σ G G +τ g Gen ω we can expand ou h condon o arre a + pˆ p K + pĝ p g K 3 + pĝ p G K 4 Σ θg + + pˆ p K + pĝ p g K 3 + pĝ p G K 4 p ρ Σ G + p g Σ G G + p N ρ N Σ NG Q = Σ G G + τ g + pˆ p K + pĝ p g K 3 + pĝ p G K 4 Σ + + pˆ p K + pĝ p g K 3 + pĝ p G K 4 p ρ Σ + p g Σ G + p N ρ N Σ N Σ + τ [ [ Recognzng ha φ + ω = Co R + F G we can rewre he aboe more gener- + ally a Q = Co [ R + G + F Σ G G + τ Proof of Propoon A7 g Co [ R + + F Σ + τ Snce he goernmen doe no hae any addonal nformaon o ha of he marke ha he marke belef A decrbed n he man paper conenen o defne he ae ecor Ψ whch follow a VAR proce n he coarance-aonary equlbrum of he economy gen by Propoon A5 Gen he reul n Propoon A5 he goernmen polcy rule X G = ϑ ˆN ˆN + V ar [ϑ ˆN ˆN F {a } G ω φ and follow ha V ar [ P + F V ar [ X G + F = = H φ Ω φ where ϑ = [ ϑ ˆN and H = + Σ G G ϑ K Ω K ϑ + ϑ K Ω K ϑ ϑ K 5 Σ G p ρ Σ G + p N ρ N Σ NG Σ G G

16 In addon he prce olaly can be expreed a V ar [ P + F = φ Ω φ Fnally we can expre he condonal uncerany abou he deaon n he ae prce from fundamenal a F = V ar [ P + pˆ + F = V ar φ ε + pˆ + ρ ˆ + F = φ Ω φ + p ˆ ρ Σ + σ pˆ φ ρ Σ p ρ Σ + σ + pn ρ ρ N Σ N ρ Σ G I follow n he coarance-aonary equlbrum ha we can expre he goernmen objece a U G = up γ σ ϑ φ Ω φ γ F ψh Reference Boyd John H 99 Recure Uly and he Ramey Problem Journal of Economc Theory