Online Appendix to Solution Methods for Models with Rare Disasters

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1 Online Appendix o Soluion Mehods for Models wih Rare Disasers Jesús Fernández-Villaverde and Oren Levinal In his Online Appendix, we presen he Euler condiions of he model, we develop he pricing Calvo block, we inroduce he saionary represenaion of he model, we define he variables ha we include in our simulaon, and we develop a simple example of how o implemen Taylor projecion in comparison wih perurbaion and projecion. Euler condiions Define he household s maximizaion problem as follows: { max U ψ c,k,x,l + βe V + } ) ψ s.. c + x w l r k F T = 0 [ ]) k x δ) k µ S x = 0 x k + = k exp d + θ + ). The value funcion V depends on he household s acual sock of capial k and on pas invesmen x, as well as on aggregae variables and shocks ha he household akes as given. Thus, le us use V k, and V x, o denoe he derivaives of V wih respec o k and x assuming differeniabiliy). These derivaives are obained by he envelope heorem: ψ) V ψ V k, = λ r + Q δ) 27) [ ] ) 2 ψ) V ψ V x, = Q µ S x x, 28) x x where λ and Q are he Lagrange mulipliers associaed wih he budge consrain and he evoluion law of capial hey ener he Lagrangian in negaive sign). We exclude he hird consrain from he Lagrangian and subsiue i direcly in he value funcion or he oher consrains, whenever necessary.

2 Differeniaing he Lagrangian wih respec o c, k, x, and l condiions: yields he firs-order ψ) U ψ U c, = λ 29) ) ψ) βe V + E V γ +V k,+ exp d + θ + ) ) = Q 30) λ = Q µ [ S [ x x + ψ) βe V + ]) ) γ ψ [ ] ] S x x + x x E V γ +V x,+ ) 3) ψ) U ψ U l, = λ w. 32) Subsiuing he envelope condiions 27)-28) and defining: yields equaions 6)-8) in he main ex. q = Q λ 2 The Calvo block The inermediae good producer ha is allowed o adjus prices maximizes he discouned value of is profis. Fernández-Villaverde and Rubio-Ramírez 2006, pp. 2-3) derive he firs-order condiions of his problem for expeced uiliy preferences, which yield he recursion: χ ḡ = λ mc y + βθ p E Π + χ ḡ 2 = λ Π y + βθ p E Π + ) ɛ ḡ+ ) ɛ Π + ) ḡ

3 To adjus hese condiions o Epsein-Zin preferences, divide by λ o have: ḡ λ = mc y + βθ p E λ + λ ḡ 2 λ = Π y + βθ p E λ + λ χ Π + χ Π + ) ɛ ḡ+ 33) λ + ) ɛ ) Π ḡ ) λ + Π + Noe ha β λ + λ is he sochasic discoun facor in expeced uiliy preferences. In Epsein-Zin preferences he sochasic discoun facor is given insead by 2.). Subsiuing and defining g = ḡ λ, g 2 = ḡ2 λ yields )-6). The oher condiions in he Calvo block follow direcly from Fernández-Villaverde and Rubio-Ramírez 2006, pp. 2-3). 3 The saionary represenaion of he model To saionarize he model we define: c = c z, λ = λ z ψ, r = r µ, q = q µ, x = x z, w = w z, k = ẑ = k z µ, k = k z µ, ỹ = y z, Ũ = U z, Ũ l, = U l, z, Ṽ = V z,  = A A, ˆµ = µ µ, z z. Oher re-scaled endogenous variables will be inroduced below when we lis he model condiions. Las, he derended uiliy variables are normalized by heir seady-sae value o avoid scaling problems. We define he following exogenous sae variables o make hem linear in he shocks d + = µ d + ɛ d,+ µ d) 35) log θ + = ρ θ ) log θ + ρ θ log θ + σ θ ɛ θ,+ 36) z A,+ = σ A ɛ A,+ 37) log ˆµ + = Λ µ + σ µ ɛ µ,+ 38) m + = σ m ɛ m,+ 39) ξ + = ρ ξ ξ + σ ξ ɛ ξ,+ 40) The disaser sae variable, d, is deermined by he disaser shock ɛ d,+, which akes he values or 0. The mean of his shock is µ d. Since he mean is nonzero, he shock is demeaned in 35). The sae variable log θ is he log disaser size. The sae variable z A, is inroduced 3

4 o capure Gaussian produciviy innovaions o log Â. The sae variable log ˆµ denoes he growh of invesmen echnology. Finally, m and ξ are he moneary shock and he ime preference shock, respecively. The following variables depend only on he exogenous variables: log  = Λ A + z A, α) d θ log ẑ = α log  + α α log ˆµ. The model condiions are given by he following equaions: Ṽ Ṽ ss ) ψ = Ũ Ũ ss Ũ = c l ) ν e ξ U c, = l ) ν e ξ Ũ l, = ν c l ) ν e ξ ) ψ Ũss Ṽ ss ) ψ + βe Ṽ+ Ṽ ss ) ẑ + ψ 4) 42) 43) 44) ψ) Ũ ) ψ Ũl, = λ w 45) ψ) Ũ ) ψ Uc, = λ 46) M + = β λ + λ ẑ + ) ψ Ṽ+ /Ṽ ss ) ψ γ ẑ+ ) ψ γ E Ṽ+ /Ṽ ss ) ẑ+ ) ) ψ γ ) E M + exp d + θ + ) [ r + + q + δ)] = q 48) ˆµ + [ ] [ ] ] [ ] ) ) 2 x = q [ S ẑ S x x ẑ ẑ + E M + q + S x+ x+ ẑ + ẑ + 49) x x x x x ỹ = c + x 50) [ ]) k δ) k x S ẑ x = 0 5) x k = k exp d θ ) 52) ẑ ˆµ )) q e = E M + ẑ + div+ + q + e 53) 47) div = ỹ w l x 54) 4

5 q f = E M + 55) χ ) ɛ g = mc ỹ d + θ p E M + g Π + χ ) ɛ g 2 = Π ỹ d Π + θ p E M + Π + +ẑ + 56) Π + ) g 2 +ẑ + 57) ɛ g = ɛ ) g 2 58) χ ) ɛ = θ p + θ p ) Π ) ɛ 59) Π ) α ) α mc = w α r α 60) α α k l = ỹ = v p = θ p R R = α α w 6) r ) α l α φ 62)  ẑ k exp d θ ) χ Π R R v p ) ɛ v p + θ p) Π ) ɛ 63) ) γr Π Π ) γπ ỹ ) γy ) R ỹ ẑ e m 64) exp Λ y ) = E M + R Π +. 65) We define he sae of he economy by he endogenous variables log k log x, log Π, log v, p log ỹ and log R, and he exogenous variables d, log θ, z A,, log ˆµ m and ξ. In he flexible price versions of he model, we use he following pricing condiions insead of 56)-60): ) α r = α∵ k exp d θ ) l α 66) ) w = α)  α k ẑ exp d θ ) l α 67) ỹ =  ẑ k exp d θ ) ) α l α φ. 68) 5

6 4 Simulaion variables The benchmark version of he model approximaes he endogenous conrol variables: ) ) log E Ṽ+ ẑ Ṽ SS +, log l l, log q e, log q f, and log k. The firs variable is an auxiliary variable inroduced ino he sysem. The oher model variables can be expressed as funcions of he approximaed variables and he given sae variables. We apply a change of variables o ensure ha variables are bound wihin heir naural domain. For insance, if x > 0, we approximae log x. Similarly, labor l mus be beween 0 and so we approximae log l l insead. The second version wih capial adjusmen coss approximaes, in addiion, he variables log q and log x back +, which are boh deermined in period. The noaion back denoes he pas value of he variable, e.g. x back variable e.g. pas invesmen) is an endogenous sae variable. x. This is required when he pas value of a conrol The hird version wih Calvo pricing approximaes, in addiion, he variables: log w, log x, log g, log Π + log Π, log Π back +, and log v p,back +, all deermined in period. We approximae log Π + log Π insead of approximaing separaely log Π and log Π. I can be shown ha his ransformaion ensures ha Π is always posiive, while keeping he number of approximaed variables as small as possible. The fourh version wih a Taylor rule ha depends on oupu growh approximaes, in addiion, log ỹ back +, which is deermined in period. The fifh version wih a smoohed Taylor rule approximaes, in addiion, he variable log R back +, which is deermined in period. The oher versions add only exogenous variables, so he number of approximaed variables does no change. 5 Perurbaion vs. Taylor projecion: a simple example As we menion in he main ex, in sandard perurbaion, we find a soluion for he variables of ineres by perurbing a volailiy of he shocks around zero. In comparison, in 6

7 Taylor projecion as we would do in a projecion), we ake accoun of he rue volailiy of he shocks. An example should clarify his poin. Imagine ha we are dealing wih he sochasic neoclassical growh model wih fixed labor supply, full depreciaion, and no persisence of he produciviy shock hese wo assumpions allows us o derive simple analyic expressions). The social planner problem of his model can be wrien as: max E 0 =0 β log c s.. c + k + = e z k α, z N 0, σ), where E 0 is he condiional expecaion operaor, β is he discoun facor, c is consumpion, k is capial, and z is he produciviy shock wih volailiy σ. To ease he presenaion, we will swich now o he recursive noaion, where we drop he ime subindex and where for an arbirary variable x, we have ha x = x +. Thus, consumpion can be wrien in erms of he policy funcion c = c k, z) and from he resource consrain of he economy k = e z k α c k, z). ge: If we subsiue c k, z) and k = e z k α c k, z) in he Euler equaion of he model, we c k, z) + αβe e z e z k α c k, z)) α c e z k α c k, z), z ) = 0. From his Euler equaion, we can find he deerminisic seady sae of he model: k ss = αβ) α c ss = k α ss k ss. In a firs-order perurbaion, we posulae an approximaion for he policy funcion of he form: c k, z) = θ 0 + θ k k ss ) + θ 2 z, where we are already aking advanage of he cerainy equivalence propery of firs-order approximaions o drop he erm on σ. 7

8 If we plug his policy funcion ino he equilibrium condiions before, we ge: θ 0 + θ k k ss ) + θ 2 z + αβe e z e z k α θ 0 θ k k ss ) θ 2 z) α = 0. θ 0 + θ e z k α θ 0 θ k k ss ) θ 2 z k ss ) + θ 2 z To find θ 0, we firs evaluae he previous expression a he deerminisic seady-sae value of he sae variables k = k ss and z = 0): e z kss α θ 0 ) α + αβe = 0 θ 0 θ 0 + θ kss α θ 0 k ss ) + θ 2 z and hen ake σ 0 o ge: kss α θ 0 ) α + αβ θ 0 θ 0 + θ kss α θ 0 k ss ) = 0. This equaion has a zero a θ 0 = c ss = kss α k ss. This resul is naural: he leading consan erm of a perurbaion around he deerminisic seady sae of he policy funcion of an endogenous variable is jus he seady-sae value of such a variable in pracice, his resul is jus assumed wihou solving for i explicily). To find θ and θ 2, we ake derivaives of he Euler equaion wih respec o capial and produciviy, evaluae hem a he deerminisic seady sae, and ake σ 0, and solve for he unknown coefficiens. The algebra is sraighforward, bu edious. Noe, however, ha he procedure is recursive: we solve firs for θ 0, and when his coefficien is known, for θ and θ 2. In a Taylor projecion, up o firs order, we also posulae: c = θ 0 + θ k k ss ) + θ 2 z. In his Taylor projecion, we will ake our approximaion around k ss, 0) o make he comparison wih perurbaion easier, bu oher approximaion poins are possible. As before, we subsiue in he equilibrium condiion: θ 0 + θ k k ss ) + θ 2 z +αβe e z e z k α θ 0 θ k k ss ) θ 2 z) α = 0, 69) θ 0 + θ e z k α θ 0 θ k k ss ) θ 2 z k ss ) + θ 2 z 8

9 and evaluae i a he deerminisic seady-sae value of he sae variables k = k ss and z = 0): Bu now we do no le σ 0. e z kss α θ 0 ) α + αβe = 0. θ 0 θ 0 + θ kss α θ 0 k ss ) + θ 2 z Noe, in paricular, ha his means we sill have an expecaion operaor E and a z. Furhermore, i also means ha we mus simulaneously solve for θ 0, θ, and θ 2, and no recursively as in perurbaion. To do so, we ake derivaives of equaion 69 wih respec o k and z and evaluae hem a he deerminisic seady sae. This operaion gives us hree equaions 69 and he wo derivaives) on hree unknowns θ 0, θ, and θ 2 ) ha can be solved wih a sandard Newon algorihm. In general, he presence of he expecaion operaor will imply ha he θ 0 from firs-order perurbaion and he θ 0 from Taylor projecion will be differen. To see his, we can plug θ 0 = c ss afer equaion 69 and verify ha: unless σ = 0. c ss + αβk α ss E e z c ss + θ 2 z 0 To furher illusrae his poin, we will implemen a simple calibraion of he model wih α = 0.3 and β = Produciviy, insead of being a normal disribuion as before, is now a wo-poin process: z = [ log0.4), 0. ] 0.9 log0.4) P rob = [0., 0.9]. This calibraion assumes 0% probabiliy of a 60% fall in TFP and 90% probabiliy of a 0.7% increase he mean of z is sill zero). The soluions for he θ s are repored in Table 27. Noe he difference beween he Taylor projecion θ s and he perurbaion θ s. 9

10 Table 27: Soluion for θ s Parameer Taylor projecion Perurbaion θ θ θ Taylor projecion vs. projecion: a simple example We can coninue our previous example wih he sochasic neoclassical growh model wih full depreciaion, bu now comparing Taylor projecion wih a sandard projecion. The firs seps of a Taylor projecion and a sandard projecion are he same. In boh cases we posulae a policy funcion: c = θ 0 + θ k k ss ) + θ 2 z. For his example and o make he comparison wih perurbaion easier, we cener he policy funcion around k ss, even if oher approximaion poins are possible. As we did in previous cases, we subsiue in he equilibrium condiion o ge a residual funcion: R k, z, θ 0, θ, θ 2 ) = θ 0 + θ k k ss ) + θ 2 z e z e z k α θ 0 θ k k ss ) θ 2 z) α +αβe θ 0 + θ e z k α θ 0 θ k k ss ) θ 2 z k ss ) + θ 2 z, 70) bu we do no impose ha his residual funcion is zero. Insead, we express i as an explici funcion of k, z, θ 0, θ, and θ 2. In Taylor projecion, we find he values of θ 0, θ, and θ 2 ha solve: R k ss, 0, θ 0, θ, θ 2 ) = 0 7) R k, z, θ 0, θ, θ 2 ) k = 0, 72) kss,0 R k, z, θ 0, θ, θ 2 ) z = 0, 73) kss,0 0

11 In comparison, projecion selecs hree poins k, z ), k 2, z 2 ), and k 3, z 3 ) one of hese poins can be k ss, 0); here are differen choices of how o underake his selecion) and finds he values of θ 0, θ, and θ 2 ha solve: R k, z, θ 0, θ, θ 2 ) = 0 74) R k 2, z 2, θ 0, θ, θ 2 ) = 0 75) R k 3, z 3, θ 0, θ, θ 2 ) = 0 76) In boh cases, we have hree equaions 7-73 for Taylor projecion, for projecion) in hree unknowns θ 0, θ, θ 2 ) ha come from he residual funcion R k, z, θ 0, θ, θ 2 ), bu in he former case we deal wih he level and wo parial derivaives of he funcion a one poin and in he laer we deal wih he level of he funcion a hree differen poins.