This appendix presents the derivations and proofs omitted from the main text.

Transcription

1 Onlne Appendx A Appendx: Omtted Dervaton and Proof Th appendx preent the dervaton and proof omtted from the man text A Omtted dervaton n Secton Mot of the analy provded n the man text Here, we formally tate the nvetor problem and derve the optmalty condton Recall that the market portfolo the clam to all output at date Let r k z = log z Q denote the log return on th portfolo f the productvty realzed to be z Snce the payoff dtrbuton log normal, the return dtrbuton alo log normal, r k z N g log Q σ, σ A Recall that there are two type of nvetor, = o and = p, that dffer only n ther belef about the expected growth rate, denoted by g, and ther wealth hare, denoted by α Type nvetor olve the followng problem, max log c + e log U c,a,ω k where U = E [ c z γ] / γ t c + a = α y + Q and c z = a ω k exp r k z + ω k exp r f A Here, c z denote total fnancal wealth, whch equal conumpton nce the economy end at date Note that nvetor have Epten-Zn preference wth EIS coeff cent equal to one and the RRA coeff cent equal to γ > The cae wth γ = equvalent to log utlty a n the dynamc model In vew of the Epten-Zn functonal form, nvetor problem naturally plt nto two tep Condtonal on avng, a, he olve a portfolo optmzaton problem, that, U = R CE a, where [ R CE, = max E R p z γ] / γ ω k and R p z = ω k exp r k z + ω k exp r f Here, we ued the obervaton that the portfolo problem lnearly homogeneou The varable, R p z, denote the realzed portfolo return per dollar, and R CE, denote the optmal certanty-equvalent portfolo return perceved by type nvetor ntertemporal problem, A3 In turn, thee nvetor chooe aet holdng, a, that olve the max a log α y + Q a + e log R CE, a The frt order condton for th problem mple Eq 4 n the man text That, regardle of her certanty-equvalent portfolo return, the nvetor conume and ave a contant fracton of her lfetme wealth 5

2 It reman to characterze the optmal portfolo weght, ω k, a well a the certanty-equvalent return, R CE, Even though the return on the market portfolo log-normally dtrbuted ee Eq A, the portfolo return, R p z, n general not log-normally dtrbuted nce t the um of a log-normal varable and a contant Followng Campbell and Vcera, we aume nvetor olve an approxmate veron of the portfolo problem A3 n whch the log portfolo return alo normally dtrbuted Moreover, the mean and the varance of th dtrbuton are uch that the followng dentfe hold, π p, ω k π k, and σ p = ω k σ, A4 where π p, = log E [R p ] r f and σ p, = var log R p, and π k, = log E [ exp r k] r f = E [ r k] r f + σ Here, the frt lne ay that the rk premum on the nvetor portfolo meaured n log dfference of expected gro return depend lnearly on the nvetor portfolo weght and the rk premum on the market portfolo The thrd lne ay that the tandard devaton of the log portfolo return depend lnearly on the nvetor portfolo weght and the tandard devaton of the log return on the market portfolo Thee dentte hold exactly n contnuou tme In the two perod model, they hold approxmately when the perod tme-length mall Takng the log of the objectve functon n problem A3, and ung the log-normalty aumpton, the problem can be equvalently rewrtten a, log R CE, r f = max ω k π p, γ σ p,, where π p, and σ p, are defned n Eq A4 It follow that, up to an approxmaton that become exact n equlbrum, the nvetor problem turn nto tandard mean-varance optmzaton Takng the frt order condton, we obtan Eq 7 n the man text Subttutng E [ r k] = g log Q σ th expreon, the optmalty condton for type nvetor can alo be wrtten a, ω k, σ g log Q r f γ σ [cf Eq A] nto Combnng th wth the aet market clearng condton 6, we further obtan Eq 8 n the man text A Omtted dervaton n Secton 3 A Portfolo problem and t recurve formulaton The nvetor portfolo problem at ome tme t and tate can be wrtten a, V ] E c t,, ω, ω k t, t, t t, a = [ max t da = a a = a [ t ] e t log c ĩ t, d t r f + ω k r k r f + ω k Q Q Q + ω ω c dt + ω k a σ dz t abent tranton, f there a tranton to tate p A5 Note that there a unque log-normal dtrbuton for R p that enure thee dentte Specfcally, log R p N r f + π p σp, σ p 5

3 Here, E [ ] denote the expectaton operator that correpond to the nvetor belef for tate tranton probablte The HJB equaton correpondng to th problem gven by, V a = max log c + V a r f + ω k r k r f ω c ω k, ω, c a + + λ V ω k a a V a σ + V a + ω k Q Q Q + ω V p a t A6 In vew of the log utlty, the oluton ha the functonal form n 4, whch we reproduce here, V log a a /Q = + v The frt term n the value functon capture the effect of holdng a greater captal tock or greater wealth, whch cale the nvetor conumpton proportonally at all tme and tate The econd term, v, the normalzed value functon when the nvetor hold one unt of the captal tock or wealth, a = Q Th functonal form alo mple, V a = and V a a = a The frt order condton for c then mple Eq 7 n the man text The frt order condton for ω k mple, V a a r k r f + λ V a a a Q Q Q = V a ωk a σ After ubttutng for V a, V a, V a Fnally, the frt order condton for ω mple, and rearrangng term, th alo mple Eq 8 n the man text p λ = V a a V a a = /a /a, whch Eq 9 n the man text Th complete the characterzaton of the optmalty condton A Decrpton of the New Keynean producton frm The upply de of our model feature nomnal rgdte mlar to the tandard New Keynean ettng There a contnuum of meaure one of producton frm denoted by ν Thee frm rent captal from the nvetment frm, k ν, and produce dfferentated good, y ν, ubject to the technology, y ν = Aη ν k ν A7 Here, η ν [, ] denote the frm choce of captal utlzaton We aume utlzaton free up to η ν = and nfntely cotly afterward ee our extended workng paper veron, n whch we relax th aumpton and allow for exce utlzaton at the cot of exce deprecaton The producton frm ell ther output to a compettve ector that produce the fnal output accordng to the CES technology, 5

4 ε/ε, y = y ν ε ε dν for ome ε > Thu, the demand for the frm good gven by, y ν = p ν ε y, where p ν = P ν /P A8 Here, p ν denote the frm relatve prce, whch depend on t nomnal prce, P ν, a well a the deal nomnal prce ndex, P = P ν ε dν / ε We alo aume there are ubde degned to correct the neff cence that tem from the frm monopoly power and markup In partcular, the government taxe the frm proft lump um, and redtrbute thee proft to the frm n the form of a lnear ubdy to captal Formally, we let Π ν denote the equlbrum pre-tax proft of frm ν that wll be characterzed below We aume each frm ubject to the lump-um tax determned by the average proft of all frm, T = ν Π ν dν We alo let R τ denote the after-ubdy cot of rentng captal, where R denote the equlbrum rental rate pad to nvetment frm, and τ denote a lnear ubdy pad by the government We aume the magntude of the ubdy determned by the government break-even condton, τ ν k ν dν = T A9 A Wthout prce rgdte, the frm chooe p ν, k ν, η ν [, ], y ν, to maxmze t pretax proft, Π ν p ν y ν R τ k ν, A ubject to the upply contrant n A7 and the demand contrant n A8 The optmalty condton mply, p ν = ε R τ ε A and η ν = That, the frm charge a markup over t margnal cot, and utlze t captal at full capacty In a ymmetrc-prce equlbrum, we further have, p ν = Ung Eq A7 A, th further mple, y ν = y = Ak and R = ε A + τ = A ε A That, output equal to potental output, and captal earn t margnal contrbuton to potental output n vew of the lnear ubde We focu on the alternatve ettng n whch the frm have a preet nomnal prce that equal to one another, P ν = P In partcular, the relatve prce of a frm fxed and equal to one, p ν = The frm chooe the remanng varable, k ν, η ν [, ], y ν, to maxmze t pre-tax proft, Π ν We conjecture a ymmetrc equlbrum n whch all frm chooe the ame allocaton, k, η, y, output determned by aggregate demand, y = η Ak = c d + k ι, for η [, ], A3 I and the rental rate of captal gven by, R = Aη A4 53

5 To verfy that the conjectured allocaton an equlbrum, frt conder the cae n whch aggregate demand below potental output, o that y < Ak and η < In th cae, frm can reduce ther captal nput, k ν, and ncreae ther factor utlzaton, η ν, to obtan the ame level of producton Snce factor utlzaton free up to η ν =, after tax cot of captal mut be zero, R τ = Snce t margnal cot zero, and t relatve prce one, t optmal for each frm to produce accordng to the aggregate demand, whch verfe Eq A3 Ung Eq A9 and A, we further obtan, τ = Aη Combnng th wth the requrement that R τ = verfe Eq A4 Next conder the cae n whch aggregate demand equal to potental output, o that y = Ak and R τ η = In th cae, a mlar analy mple there a range of equlbra wth A and R = A Here, the frt equaton enure t optmal for the frm to meet the aggregate demand The econd equaton follow from the ubdy and the tax cheme In partcular, the frctonle benchmark allocaton A, that feature R τ A = ε ε aggregate demand equal to potental output and R = A, alo an equlbrum wth nomnal rgdte a long a the A3 Omtted dervaton and proof n Secton 4 Proof of Propoton Mot of the proof provded n the man text It reman to how that Aumpton -3 enure there ext a unque oluton, q < q and r f, to Eq 3 and 3 To th end, we defne the functon, f q, λ = + ψq δ + λ exp q Q σ The equlbrum prce the oluton to, f q, λ = gven λ Note that f q, λ a concave functon of q wth lm q f q, λ = lm q f q, λ = It dervatve, Thu, for fxed λ, t maxmzed at, Moreover, the maxmum value gven by f q, λ q = ψ λ exp q q q max λ = q + log ψ/λ f q max λ, λ = δ + ψ q + log ψ/λ + λ exp log ψ/λ σ q max = δ + ψq + ψ log ψ/λ + λ ψ σ Next note that, by Aumpton, the maxmum value trctly negatve when λ = ψ, that, f q max ψ, ψ < Note alo that dfqmax λ,λ dλ = ψ λ, whch mple that the maxmum value trctly ncreang n the range λ ψ Snce lm λ f q max λ, λ =, there ext λ mn > ψ that enure f λ mn, λ mn = By Aumpton, the actual level of optmm atfe λ λ mn, whch mple that f q max λ, λ By Aumpton, we alo have that f q, λ < It follow that, for any λ λ mn, there ext a unque prce level, q [q max, q, that olve the equaton, f q, λ = λ expq Q Our analy alo mple that the equlbrum prce atfe, fq,λ q = ψ, wth equalty only f λ = λ mn Th facltate the comparatve tatc reult n Secton 4 54

6 Next conder Eq 3, whch can be rewrtten a r f = + ψq δ + λ Q σ Q Snce q < q, th expreon decreang n λ When λ =, t trctly potve by Aumpton A λ, t approache Thu, for any q < q, there ext λ max q uch that r f f and only f λ [, λ max q ] Note alo that for any fxed λ >, r f ncreang n q Th mple that the upper bound for the tranton probablty, λ max q, ncreang n q, completng the proof Proof of Corollary Fx ome t > and let t denote the random varable that equal to f there no tate tranton over [, t], and f there at leat one tate tranton The law of total varance mple, [ k Q / t k Q V ar = E t / t V ar t k Q k Q ] + V ar t [ ] k Q / t E t k Q A5 Here, E t [ ] and V ar t [ ] denote, repectvely, the expectaton and the varance operator over the random varable, t We next calculate each component of varance For the frt component, we have, [ ] k Q E t V ar t = e λ t σ k Q t + e λ t O t Here, the frt term capture the varance condtonal on there beng no tranton, t = The varance n th cae come from the Brownan moton for k The econd term capture the average varance condtonal on there beng a tranton, t = Here, the lat term atfe, lm t O t = Dvdng by t and evaluatng the lmt, we obtan, For the econd component, we have, [ ] k Q V ar t E t k Q [ ] k Q / t lm V ar t E t t k Q [ ] Q = V ar t E t + O t, Q Q = σ A6 Q Q Q Q = λ t + λ t + O t, where Q = λ t Q + λ tq Here, O t O t denote term that atfy, lm t t = The frt lne ue the obervaton that for mall t the tate tranton change the return only through ther mpact on the prce level The econd lne calculate the varance of prce change up to term that are frt order n t Dvdng the lat lne by t and evaluatng the lmt, we obtan, [ ] lm V k Q / t t ar t E t k Q Q Q Q = λ A7 Q Combnng Eq A5, A6 and A7, the uncondtonal varance gven by, σ + λ Q Q Q, 55

7 completng the proof A4 Omtted dervaton and proof n Secton 5 We derve the equlbrum condton that we tate and ue n Secton 5 Frt note that, ung Eq 9, the optmalty condton 8 can be wrtten a, ω k, σ = r k r f Q + p Q A8 σ Q Combnng th wth the market clearng condton, we obtan, ω k,o = ω k,p = A9 Next note that by defnton, we have a o = α Q k and a p = α Q k for each {, } After pluggng thee nto Eq 9, ung k = k optmt and pemt, we obtan, nce captal doe not jump, and aggregatng over Q p = λ, A Q where λ denote the wealth-weghted average belef defned n 33 Combnng Eq A8, A9, and A, we obtan the rk balance condton 34 n the man text We next characterze nvetor equlbrum poton Combnng Eq A5 wth Eq A9 and A, nvetor wealth after tranton atfe, From Eq 9, we have p λ once more, we obtan, a a = Q Q + ω, λ A = /a Subttutng th nto the prevou expreon and ung Eq A /a Combnng th wth Eq 35, we obtan Eq 35 n the man text ω, = λ λ for each {o, p} A Fnally, we characterze the evoluton of optmt wealth hare After ubttutng a o = α Q k and ung Eq A a well a k = k, Eq A mple α α = λo λ A3 Thu, t reman to characterze the evoluton of wealth condtonal on no tranton To th end, we combne Eq A5 wth Eq A9, 4, 7 to obtan, da o a o = g + µ Q ω, dt + σ dz t After ubttutng a o = α Q k, and ung the obervaton that dq Q = µ Q dt and dk k = g dt+σ dz t, 56

8 Fgure : The phae dagram that decrbe the equlbrum wth heterogeneou belef we further obtan, dα α = ω,o dt = λ o λ dt Combnng Eq A3 and A4 mple Eq 36 n the man text A4 Proof of Propoton We analyze the oluton to the ytem n 38 ung the phae dagram over the range α [, ] and q [q p, qo ] Frt note that the ytem ha two teady tate gven by, α t, =, q t, = q p, and α t, =, q t, = q o Next note that the ytem atfe the Lpchtz condton over the relevant range Thu, the vector flow that decrbe the law of moton do not cro Next conder the locu, q = By comparng Eq 37 and 34, th locu exactly the ame a the prce that would obtan f nvetor hared the ame wealth-weghted average belef, denoted by q = q h α Ung our analy n Secton 4, we alo fnd that q h α trctly ncreang n α Moreover, q < q h α mple q t, < wherea q > q h α mple q t, > Fnally, note that α t, < for each α, Combnng thee obervaton, the phae dagram ha the hape n Fgure Th n turn mple that the ytem addle path table Gven any α t, [,, there ext a unque oluton, q t,, whch enure that lm t q t, = q p We defne the prce functon the addle path a q α Note that the prce functon atfe q α < q h α for each α,, nce the addle path cannot cro the locu, q t, = Note alo that q = q o, nce the addle path croe the other teady-tate, α t, =, q t, = q o Fnally, recall that q < q h α mple q t, < Combnng th wth α t, <, we further obtan dqα dα > for each α, Next note that, after ubttutng q t, = q α α t,, Eq 38 mple the dfferental equaton 39 n α-doman Thu, the above analy how there ext a oluton to the dfferental equaton wth q = q p and q = q o Moreover, the oluton trctly ncreang n α, and t atfe q α < q h α for each α, Note alo that th oluton unque nce the addle path unque Next conder Eq 4 whch characterze the nteret rate functon, r f α Note that drf α > nce dq α dα > recall that α = αλ o /λ α Note alo that r f α > rf >, where the latter nequalty follow nce Aumpton -3 hold for the pemtc belef Thu, the nteret rate n tate alway potve, whch verfe our conjecture and complete the proof dα 57

9 A5 Omtted dervaton n Secton 6 on equlbrum value Th ubecton derve the HJB equaton that decrbe the normalzed value functon n equlbrum It then characterze th equaton further for varou cae analyzed n Secton 6 Characterzng the normalzed value functon n equlbrum Conder the recurve veron of the portfolo problem n A6 Recall that the value functon ha the functonal form n Eq 4 Our goal to characterze the value functon per unt of captal, v correpondng to a = Q To facltate the analy, we defne, ξ = v log Q A5 Note that ξ the value functon per unt wealth correpondng to a =, and that the value functon alo atfe V a loga = + ξ We frt characterze ξ We then combne th wth Eq A5 to characterze our man object of nteret, v Conder the HJB equaton A6 We ubttute the optmal conumpton rule from Eq, 7, the contngent allocaton rule from Eq 9, and a = to characterze the value per unt wealth to obtan, ξ = log + + ξ + λ r f + ω k, log r k r f λ p + ξ ξ ω k, σ ω, A6 A we decrbe n Secton 5, the market clearng condton mply the optmal nvetment n captal and contngent ecurte atfe, ω k = and ω, = λ λ, and the prce of the contngent ecurty gven by, p /Q = λ /Q Here, λ denote the weghted average belef defned n 33 Ung thee condton, the HJB equaton become, ξ = log + + ξ + λ r k σ λ λ + λ log log Q Q λ λ + ξ ξ After ubttutng the return to captal from 4, the HJB equaton can be further mplfed a, ξ = log + + ξ ψ log Q δ + µ Q σ + λ λ λ λ λ + λ log log Q Q + ξ ξ Here, the term nde the ummaton on the econd lne, λ λ + λ λ log λ A7, zero when there are no dagreement, and t trctly potve when there are dagreement Th llutrate that peculaton ncreae the expected value for optmt a well a pemt We fnally ubttute v = ξ + log Q cf A5 nto the HJB equaton to obtan the dfferental 58

10 equaton, v = log + log Q + ψ log Q δ σ λ λ + λ λ log λ + v + λ v v Here, we have canceled term by ung the obervaton that ξ have thu obtaned Eq 43 n the man text = v log Q = v µq We Solvng for the value functon n the common belef benchmark Next conder the benchmark wth common belef In th cae, the prce level tatonary, q = q for each ee Secton 4 Then, the HJB equaton 43 mple the value functon are alo tatonary, v = v, wth value that atfy, v = log + q + ψq δ σ + λ v v Conder the ame equaton for Multplyng that equaton wth λ and the above equaton wth + λ, and addng up, we obtan a cloed form oluton, v = log + q + ψq δ σ, A8 where q = β q + β q and σ = β σ + β σ, and β = + λ + λ + λ Here, the weght β and β can be thought of a capturng the dcounted expected tme the economy pend n each tate note that the economy tart n tate and the nvetor dcount the future at rate The value n a tate the um of the utlty from the dcounted average of current conumpton and the preent value of the rk-adjuted growth rate All ele equal, the value decreang n the weghted average rk, σ, but t ncreang n the weghted-average prce level, q Note alo that the weght the dcounted expected tme atfy the followng property, β = + λ + λ + λ > β = λ + λ + λ Here, β rep β the dcounted tme the nvetor pend n tate when he tart n tate rep n the other tate Thu, β > β mple that the economy pend more dcounted tme n the tate t tart wth Combnng th obervaton wth q < q = q and σ > σ, Eq A8 mple v < v Intutvely, nvetor have a lower expected value when they are n the hgh-rk tate nce they expect aet prce to be lower and the rk to be hgher Next note that {v } defned a the oluton to the ame equaton ytem wth q = q for each The gap value, w = v v, can be calculated by ubtractng the correpondng equaton for v an v Wth ome algebra, we obtan, w = q q + ψ A9 That, the gap value proportonal to the weghted-average prce gap relatve to the frt bet Note alo that we have q q = and q q < Snce β,, th mple w < for each {, } Snce β > β, we further obtan w < w < 59

11 Solvng the value functon { wth belef } dagreement Wth belef dagreement, the value functon and t component, v, v,, w, can be wrtten a functon of optmt wealth hare, {, v α, v, α, w α }, that olve approprate ordnary dfferental equaton, Recall that the prce level n each tate can be wrtten a a functon of optmt wealth hare, q = q α where we alo have, q α = q Pluggng n thee prce functon, and ung the evoluton of α from Eq 36, the HJB equaton 43 can be wrtten a, v α = log + q α + v α λo λ p α α + λ ψq α δ σ λ λ α + λ log v α λo λ α λ λ p +α λ v α For each {o, p}, the value functon, v α, are found by olvng th ytem of ODE For =, {,} the boundary condton are that the value, {v o }, are the ame a the value n the common belef benchmark characterzed n Secton 4 when all nvetor have the optmtc belef For = p, the boundary condton are that the value, {v p }, are the ame a the value n the common belef benchmark when all nvetor have the pemtc belef Lkewe, the frt-bet value functon, v, α, are found by olvng the analogou ytem after {,} replacng q α wth q and changng the boundary condton approprately Fnally, after ubttutng the prce functon nto Eq 45, the gap-value functon, w α, are found by olvng the followng, ytem wth approprate boundary condton, w α = + ψ q α q w α λ o λ p α α α + λ w α λo λ α w α Fgure 7 n the man text plot the oluton to thee dfferental equaton for a partcular parameterzaton A6 Omtted dervaton n Secton 6 on macroprudental polcy Recall that macroprudental polcy nduce optmt to chooe allocaton a f they have more pemtc belef,,, that atfy, λ o and λ o We next how that th allocaton can be mplemented wth portfolo retrcton on optmt We then how that the planner Pareto problem reduce to olvng problem 46 n the man text Fnally, we derve the equlbrum value functon that reult form macroprudental polcy and preent the proof of Propoton 3 and 4 Implementng the polcy wth rk lmt Conder the equlbrum that would obtan f optmt had the planner-nduced belef, Ung our analy n Secton 5, optmt equlbrum portfolo are gven by, ω k,o,pl = and ω,o,pl = λ pl for each t, A3 We frt how that the planner can mplement the polcy by requrng optmt to hold exactly thee portfolo weght We wll then relax thee portfolo contrant nto nequalty retrcton ee Eq A3 Formally, an optmt olve the HJB problem A6 wth the addtonal contrant A3 In vew of log utlty, we conjecture that the value functon ha the ame functonal form 4 wth potentally dfferent normalzed value, ξ o, v o, that reflect the contrant Ung th functonal form, the optmalty condton for conumpton reman unchanged, c = a o [cf Eq 7] Pluggng th equaton and the portfolo holdng n A3 nto the objectve functon n A6 verfe that the value functon ha the conjectured 6

12 functonal form For later reference, we alo obtan that the optmt unt-wealth value functon atfe [cf Eq A5], ξ o = log + ω k,o,pl,o,pl + ω p r f + ω k,o,pl σ + ξ o r k r f + λ o log ω,o,pl a o a o + ξ o ξo A3 Here, ao a = + ω k,o,pl Q Q o Q n vew of the budget contrant of problem A6 Hence, the value functon ha a mlar characterzaton a before [cf Eq A6] wth the dfference that optmt portfolo holdng reflect the contrant Snce pemt are uncontraned, ther optmalty condton are unchanged It follow that the equlbrum take the form n Secton 5 wth the dfference that nvetor belef are replaced by ther a-f belef, λ,pl Th verfe that the planner can mplement the polcy ung the portfolo retrcton n A3 We next how that thee retrcton can be relaxed to the followng nequalty contrant, ω k,o,pl for each, A3 ω,o,pl t, ω,o t, λo,pl λ pl t, and ω,o,pl t, ω,o t, λo,pl λ pl t, In partcular, we wll etablh that all nequalty contrant bnd, whch mple that optmt optmally chooe the portfolo weght n Eq A3 Thu, our earler analy contnue to apply when optmt are ubject to the more relaxed retrcton n A3 The reult follow from the aumpton that the planner-nduced belef are more pemtc than optmt actual belef, λ o and λ o To ee th formally, note that the optmalty condton for captal gven by the followng generalzaton of Eq 8, ω k,o,pl σ σ r k r f + λ o wth complementary lackne Note alo that, a o a o Q Q Q and ω k,o,pl, A33 λ o a o a o Q Q Q = λ o λ pl Q Q Q λ pl Q Q Q for each Here, the equalty follow from Eq A36 and the nequalty follow by conderng eparately the two cae, {, } For =, the nequalty hold nce Q Q > and the belef atfy, λ o For =, the nequalty hold nce Q Q < and the belef atfy, λ o Note alo that n equlbrum the return to captal atfe the rk balance condton [cf Eq 34], σ = r k r f + λ pl Q σ Q Combnng thee expreon mple, σ σ r k r f + λ o a o Q Q a o Q, whch n turn mple the optmalty condton A33 atfed wth ω k,o,pl = A mlar analy how that optmt alo chooe the corner allocaton n contngent ecurte, ω,o,pl t, = ω,o t, and ω,o,pl t, = ω,o t,, verfyng that the portfolo contrant A3 can be relaxed to the nequalty contrant n A3 6

13 Smplfyng the planner problem Recall that, to trace the Pareto{ fronter, } we allow the planner to a denote type nvetor do a one-tme wealth tranfer among the nvetor at tme Let V expected value n equlbrum when he tart wth wealth a and the planner commt to mplement the { polcy, t } Then, the planner Pareto problem can be wrtten a, max γ o V, o α, Q, k, + γ p V p, α, Q, k, A34, α, Here, γ o, γ p wth at leat one trct nequalty denote the Pareto weght, and Q, denote the endogenou equlbrum prce that obtan under the planner polcy Next recall that the nvetor value functon wth macroprudental polcy ha the ame functonal form n 4 wth potentally dfferent ξ o, v o for optmt that reflect the contrant After ubttutng a = α k Q, the functonal form mple, V = v + log α + log k Ung th expreon, the planner problem A34 can be rewrtten a, max γ o v, o + γ p v p γ o log α o, + γ p log α o,, +, α, t + γo + γ p log k, Here, the lat term that feature captal a contant that doen t affect optmzaton The econd term lnk the planner choce of wealth redtrbuton, α o,, α p,, to her Pareto weght, γo, γ p Specfcally, the frt order condton wth repect to optmt wealth hare mple γo γ p = α, α, Thu, the planner effectvely maxmze the frt term after ubttutng γ o and γ p repectvely wth the optmal choce of α, and α, Th lead to the mplfed problem 46 n the man text Characterzng the value functon wth macroprudental polcy We frt how that the normalzed value functon, v, are characterzed a the oluton to the followng dfferental equaton ytem, v v = log + q + ψq δ σ λ,pl λ pl + λ log λ,pl λ pl + λ v v A35 Th a generalzaton of Eq 43 n whch nvetor poton are calculated accordng to ther a-f belef, λ,pl, but the tranton probablte are calculated accordng to ther actual belef, λ Frt conder the pemt Snce they are uncontraned, ther value functon characterzed by olvng the earler equaton ytem A3 In th cae, equaton A35 alo hold nce t the ame a the earler equaton Next conder the optmt In th cae, the analy n Secton 5 and Appendx A4 apple wth a-f belef In partcular, we have, a o a o = α α Q Q = λo,pl λ pl Q Q A36 6

14 Pluggng th expreon a well a Eq A3 nto Eq A3, optmt unt-wealth value functon atfe, ξ o = log + + ξo + λ o r k σ λ pl + λ o log log Q + ξ o ξo Th the ame a Eq A3 wth the dfference that the a-f belef,, are ued to calculate ther poton on and the payoff from the contngent ecurte, wherea the actual belef, λ o, are ued to calculate the tranton probablte Ung the ame tep after Eq A3, we alo obtan A35 wth = o We next characterze the frt-bet and the gap value functon, v, and w, that we ue n the man text By defnton, the frt-bet value functon olve the ame dfferental equaton A35 after ubttutng q = q It follow that the gap value functon w = v v,, olve, w w = Q λ pl, + ψ q q + λ w w, whch the ame a the dfferental equaton 45 wthout macroprudental polcy The latter affect the path of prce, q, but t doe not affect how thee prce tranlate nto gap value Note alo that, a before, the value functon can be wrtten a functon of optmt wealth hare, { v α, v, α, w α } For completene, we alo characterze the dfferental equaton that thee, functon atfy n equlbrum wth macroprudental polcy Combnng Eq A35 wth the evoluton of optmt wealth hare condtonal on no tranton, α = λ p α α, the value functon, v α, are found by olvng,, v α = log + q α + λ,pl v α λ p α α + λ ψq α δ σ λ pl + λ λ log,pl λ pl v v α α λo,pl λ pl, wth approprate boundary condton Lkewe, the frt-bet value functon, v, α, are found by {,} olvng the analogou ytem after replacng q α wth q Fnally, combnng Eq 45 wth the evoluton of optmt wealth hare, the gap-value functon, w α, are found by olvng Eq 48 n the man, text Proof of Propoton 3 For th and the next proof, we fnd t ueful to work wth the tranformed tate varable, α b log, whch mple α = α + exp b A37 The varable, b, vare between, and provde a dfferent meaure of optmm, whch we refer to a bullhne Note that there a one-to-one relaton between optmt wealth hare, α,, and the bullhne, b R =, + Optmt wealth dynamc n 36 become partcularly mple when expreed n term of bullhne, 63

15 { b ḃ = λ p, f there no tate change, = b + log log λ p, f there a tate change A38 Wth a lght abue of notaton, we alo let q b and w b denote, repectvely, the prce functon and the gap value functon n term of bullhne Note alo that, nce db dα = α α, we have the dentte, q b = α α q α and w b = α α w α α A39 Ung th obervaton, the dfferental equaton for the prce functon, Eq 39, can be wrtten n term of bullhne a, q b λ p = + ψq δ + λ α Q Q σ A4 Lkewe, the dfferental equaton for the gap value functon, Eq 48 can be wrtten n term of bullhne a, w b = We next turn to the proof + ψ w q b q λ p b + λ w b w b A4 To etablh the comparatve tatc of the gap value functon, we frt decrbe t a a fxed pont of a contracton mappng Recall that, n the tme doman, the gap value functon olve the HJB equaton 45 Integratng th equaton forward, we obtan, w b, = e +λ t + ψ q b q + λ w b dt, A4 for each {, } and b, R Here, b denote bullhne condtonal on there not beng a tranton before tme t, wherea b gven a-f belef, λ,pl we further obtan, denote the bullhne f there a tranton at tme t Solvng Eq A38 b = b, t b = b, t λ p, A43 + log log λ p λ p Hence, Eq A4 decrbe the value functon a a oluton to an ntegral equaton gven the cloed form oluton for bullhne n A43 Let B R denote the et of bounded value functon over R Gven ome contnuaton value functon, w b B R, we defne the functon, T w b B R, o that T w b, = e +λ t + ψ q b q + λ w b dt, A44 for each and b, R Note that the reultng value functon bounded nce the prce functon, q b, bounded n partcular, t le between q p and q It can be checked that operator T a contracton mappng wth repect to the up norm In partcular, t ha a fxed pont, whch correpond to the gap value functon, w b We next how that the value functon ha trctly potve dervatve wth repect to bullhne a well a optmt wealth hare To th end, we frt note that the value functon dfferentable nce t olve 64

16 the dfferental equaton 48 Next, we mplctly dfferentate the ntegral equaton A4 wth repect to b,, and ue Eq A43, to obtan, w b, = e +λ t + ψ q b + λ w b dt A45 Note from Eq A4 that the dervatve of the prce functon, qb, bounded Thu, Eq A45 decrbe the dervatve of the value functon, w b,, a a fxed pont of a correpondng operator T b over bounded functon whch related to but dfferent than the earler operator, T Th operator alo a contracton mappng wth repect to the up norm Snce qb > for each b, and λ > for each, t can further be een that the fxed pont atfe, w b, > for each b and {, } Ung Eq A39, we alo obtan w α α > for each α, and {, } Next conder the comparatve tatc of the fxed pont wth repect to macroprudental polcy We mplctly dfferentate the ntegral equaton A4 wth repect to, and ue Eq A43, to obtan, w b, w b, = = Note alo that, ung Eq A43 mple, e +λ t λ e +λ t λ db t, d w b t, w b t, dt = t + evaluatng the partal dervatve at = λ, we obtan, + w b t, db t, d dt, Pluggng th nto the prevou ytem, and w b, w b, = h b, + = where h b, = e +λ t λ e +λ t λ e +λ t λ w b t, dt, w b t, w b t, dt, t + λ o dt A46 Note that the functon, h b, bounded nce the dervatve functon, w b, bounded ee A45 Hence, w Eq A46 decrbe the partal dervatve functon,, a a fxed pont of a correpondng b λ o,pl =λ o operator T λ over bounded functon whch related to but dfferent than the earler operator, T Snce h b bounded, t can be checked that the operator T λ alo a contracton mappng wth repect to the up norm In partcular, t ha a fxed pont, whch correpond to the partal dervatve functon The analy o far apple generally We next conder the pecal cae, λ o = λ p, and how that t mple the partal dervatve are trctly potve In th cae, λ = λ for each {o, p} In addton, Eq A43 mple b t, = b, Ung thee obervaton, for each b,, we have, h b, = w b, = w b, e +λt λ t + dt λ λ + > + λ + λ + λ Here, the nequalty follow from our earler reult that w b, > Snce h b > for each b, and λ >, 65

17 t can further be een that the fxed pont that olve A46 atfe w b > for each b and {, } Ung Eq A39, we alo obtan w α > for each α, and {, } Proof of Propoton 4 dervatve functon, w b A mlar analy a n the proof of Propoton 3 mple that the partal, characterzed a the fxed pont of a contracton mappng over bounded functon the analogue of Eq A46 for tate In partcular, the partal dervatve ext and t bounded Moreover, nce the correpondng contracton mappng take contnuou functon nto contnuou functon, the partal dervatve alo contnuou over b R Ung Eq A39, we further obtan that the partal dervatve, w α, contnuou over α, Next note that w lm α w α ext and equal to the value functon accordng to type belef when all nvetor are optmtc In partcular, the aet prce are gven by q = q and q = q o, and the tranton probablte are evaluated accordng to type belef Then, followng the ame tep a n our analy of value functon n Appendx A5, we obtan, w = + ψ β q o + β q o q, where β = + λ + λ + λ Here, β denote the expected dcount tme the nvetor pend n tate accordng to type belef We conder th equaton for = and take the dervatve wth repect to to obtan, w = + ψ β d dq o < Here, the nequalty follow nce reducng optmt optmm reduce the prce level n the common belef benchmark ee Secton 4 w Note that the nequalty, <, hold for each tate and each belef type Ung the contnuty w of the partal dervatve functon, α, we conclude that there ext α uch that w α < =λ o for each, and α α,, completng the proof 66