LINEAR APPROXIMATION OF THE BASELINE RBC MODEL SEPTEMBER 7, 203
Iroducio LINEARIZATION OF THE RBC MODEL For f( xyz,, ) = 0, mulivariable Taylor liear expasio aroud f( xyz,, ) f( xyz,, ) + f( xyz,, )( x x) + f( xyz,, )( y y) + f( xyz,, )( z z) x y z Four equaios describe he dyamic soluio o RBC model Cosumpio-leisure efficiecy codiio u( c, ) = zm ( k, ) uc( c, ) Cosumpio-ivesme efficiecy codiio u ( c, ) = βe u ( c, ) δ + z m ( k, Aggregae resource cosrai c + k ( δ ) k = zm ( k, ) Law of moio for TFP l z = ( ) l z + l z + + ( ) c c + + + k + + ρ ρ ε z + z z + ( xyz,, ) Sepember 7, 203 2
Iroducio STEADY STATE Deermiisic seady sae he aural local poi of approximaio Shu dow all shocks ad se exogeous variables a heir meas The Idea: Le ecoomy ru for may (ifiie) periods Time eveually does maer ay more Drop all ime idices u( c, ) = u( c, ) c zm ( k, ) uc( c, ) = βuc( c, ) mk( k, ) + δ c + δ k = zm( k, ) Give fucioal forms ad parameer values, solve for (c,, k) The seady sae of he model Taylor expasio aroud his poi Sepember 7, 203 3 l z = ( ρ )l z + ρ l z z = z (a parameer of he model) z z
Iroducio LINEARIZATION ALGORITHMS Schmi-Grohe ad Uribe (2004 JEDC) A perurbaio algorihm A class of mehods used o fid a approximae soluio o a problem ha cao be solved exacly, by sarig from he exac soluio of a relaed problem Applicable if he problem ca be formulaed by addig a small erm o he descripio of he exacly-solvable problem Malab code available hrough Columbia Dep. of Ecoomics web sie Uhlig (999, chaper i Compuaioal Mehods for he Sudy of Dyamic Ecoomies) Uses a geeralized eige-decomposiio Typically implemeed wih Schur decomposiio (Sims algorihm) Malab code available a hp://www2.wiwi.hu-berli.de/isiue/wpol/hml/oolki.hm Sepember 7, 203 4
SGU Deails LINEARIZATION OF THE RBC MODEL Defie co-sae vecor ad sae vecor y c = x k = z Order model s dyamic equaios i a vecor f( y, y, x, x ) = 0 + + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP Sepember 7, 203 5
SGU Deails LINEARIZATION OF THE RBC MODEL Need four marices of derivaives f( y, y, x, x ). Differeiae wih respec o (elemes of) y + + + Firs derivaives wih respec o: c + + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai = f y + Law of moio for TFP Sepember 7, 203 6
SGU Deails LINEARIZATION OF THE RBC MODEL Need four marices of derivaives f( y, y, x, x ) 2. Differeiae wih respec o (elemes of) y + + Firs derivaives wih respec o: c Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio = f y Aggregae resource cosrai Law of moio for TFP Sepember 7, 203 7
SGU Deails LINEARIZATION OF THE RBC MODEL Need four marices of derivaives f( y, y, x, x ) 3. Differeiae wih respec o (elemes of) x + + + Firs derivaives wih respec o: k + z + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai = f x + Law of moio for TFP Sepember 7, 203 8
SGU Deails LINEARIZATION OF THE RBC MODEL Need four marices of derivaives f( y, y, x, x ) 4. Differeiae wih respec o (elemes of) x + + Firs derivaives wih respec o: k z Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio = f x Aggregae resource cosrai Law of moio for TFP Sepember 7, 203 9
SGU Deails LINEARIZATION OF THE RBC MODEL The model s dyamic expecaioal equaios f ( y, y, x, x ) + + 2 f ( y+, y, x+, x) E[ f( y+, y, x+, x) ] = E f 3 ( y, y, x, x ) + + 4 f ( y+, y, x+, x) Cojecure equilibrium decisio rules Noe: g(.) ad h(.) are ime ivaria fucios! y x Subsiue decisio rules io dyamic equaios = gx (, σ ) = hx (, σ ) + ησε + + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP Perurbaio parameer : govers size of shocks Marix of sadard deviaios of sae variables Sepember 7, 203 0
SGU Deails LINEARIZATION OF THE RBC MODEL The model s dyamic expecaioal equaios [ ( +,, +, )] = 0 = E[ f( gx ( +, σ), gx (, σ), hx (, σ) + ησε+, x] = E[ f( ghx ( (, σ) + ησε, σ), gx (, σ), hx (, σ) + ησε, x] E f y y x x + + F( x, σ ) F ( x, σ ) = 0 Fσ ( x, σ ) = 0 x Sepember 7, 203
SGU Deails LINEARIZATION OF THE RBC MODEL The model s dyamic expecaioal equaios [ ( +,, +, )] = 0 = E[ f( gx ( +, σ), gx (, σ), hx (, σ) + ησε+, x] = E[ f( ghx ( (, σ) + ησε, σ), gx (, σ), hx (, σ) + ησε, x] E f y y x x + + F( x, σ ) Usig chai rule ad suppressig argumes Fx ( x σ ) = fy gx h + f g + f h + f + + = 0, x y x x x x Sepember 7, 203 2
SGU Deails LINEARIZATION OF THE RBC MODEL The model s dyamic expecaioal equaios F ( x, ) = f + f + + x [ ( +,, +, )] = 0 = E[ f( gx ( +, σ), gx (, σ), hx (, σ) + ησε+, x] = E[ f( ghx ( (, σ) + ησε, σ), gx (, σ), hx (, σ) + ησε, x] E f y y x x + + F( x, σ ) σ g x hx y g f x h f + + x y x x = 0 Usig chai rule ad suppressig argumes Holds, i paricular, a he deermiisic seady sae F ( x,0) = f g h + f g + f h + f = 0 x y x x y x x x x + + ( x,0) Seig σ = 0 shus dow shocks Each erm is evaluaed a he seady sae jus as Taylor heorem requires Sepember 7, 203 3
SGU Deails LINEARIZATION OF THE RBC MODEL A quadraic equaio i he elemes of g x ad h x evaluaed a he seady sae F ( x,0) = f ( x,0) g ( x,0) h ( x,0) + f ( x, 0) g ( x,0) + f ( x,0) h ( x, 0) + f ( x,0) = 0 x y x x y x x x x + + Solve umerically for he elemes of g x ad h x (use fsolve i Malab) Recall cojecured equilibrium decisio rules y x = gx (, σ ) = hx (, σ ) + ησε + + Firs-order approximaio is = 0 y= gx (, σ) gx (,0) + g( x,0)( x x) + g( x,0) σ x x = hx (, σ) hx (,0) + h( x,0)( x x) + h( x,0) σ + x SGU Theorem : g σ = 0 ad h σ = 0 DONE!!! = 0 Now coduc impulse resposes, abulae busiess cycle momes, wrie paper σ σ Sepember 7, 203 4
Macro Fudameals CERTAINTY EQUIVALENCE Displayed by a model if decisio rules do o deped o he sadard deviaio of exogeous uceraiy e.g., PRECAUTIONARY SAVINGS! For sochasic problems wih quadraic objecive fucio ad liear cosrais, he decisio rules are ideical o hose of he osochasic problem Here, we have = 0 y = gx (, σ) gx (,0) + gx( x,0)( x x) + gσ ( x,0) x = hx (, σ) hx (,0) + h( x,0)( x x) + h( x,0) + x SGU Theorem : g σ = 0 ad h σ = 0 = 0 Firs-order approximaed decisio rules do o deped o he size of he shocks, which is govered by σ No he same hig as exac CE, bu refer o i as CE σ σ σ Sepember 7, 203 5
(Parial) Example LINEARIZING THE RBC MODEL uc (, ) = l c ψ l Assume ad cosumpio-leisure efficiecy codiio is c k Le f ( y+, y, x+, x) = ( α) zk = 0 (ad recall y = ) x = z mk (, ) = k α ( α) zk = 0 Sepember 7, 203 6
(Parial) Example LINEARIZING THE RBC MODEL uc (, ) = l c ψ l Assume ad cosumpio-leisure efficiecy codiio is c k Le f ( y+, y, x+, x) = ( α) zk = 0 (ad recall y = ) x = z mk (, ) = k α ( α) zk = 0 Compue firs row of marix f y+ c + + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP Sepember 7, 203 7
(Parial) Example LINEARIZING THE RBC MODEL uc (, ) = l c ψ l Assume ad cosumpio-leisure efficiecy codiio is c k Le f ( y+, y, x+, x) = ( α) zk = 0 (ad recall y = ) x = z mk (, ) = k α ( α) zk = 0 Compue firs row of marix f y+ c + + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP 0 Sepember 7, 203 8
(Parial) Example LINEARIZING THE RBC MODEL uc (, ) = l c ψ l Assume ad cosumpio-leisure efficiecy codiio is c k Le f ( y+, y, x+, x) = ( α) zk = 0 (ad recall y = ) x = z mk (, ) = k α ( α) zk = 0 Compue firs row of marix f y+ c + + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP 0 0 Sepember 7, 203 9
(Parial) Example LINEARIZING THE RBC MODEL uc (, ) = l c ψ l Assume ad cosumpio-leisure efficiecy codiio is c k Le f ( y+, y, x+, x) = ( α) zk = 0 (ad recall y = ) x = z mk (, ) = k α ( α) zk = 0 Compue firs row of marix f y Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP c ψ + α( α) zk 2 Sepember 7, 203 20
(Parial) Example LINEARIZING THE RBC MODEL uc (, ) = l c ψ l Assume ad cosumpio-leisure efficiecy codiio is c k Le f ( y+, y, x+, x) = ( α) zk = 0 (ad recall y = ) x = z mk (, ) = k α ( α) zk = 0 Compue firs row of marix f x+ k + z + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP Sepember 7, 203 2
(Parial) Example LINEARIZING THE RBC MODEL uc (, ) = l c ψ l Assume ad cosumpio-leisure efficiecy codiio is c k Le f ( y+, y, x+, x) = ( α) zk = 0 (ad recall y = ) x = z mk (, ) = k α ( α) zk = 0 Compue firs row of marix f x k z Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP α k α( α) z ( α) α k α α+ Sepember 7, 203 22
(Parial) Example LINEARIZING THE RBC MODEL uc (, ) = l c ψ l Assume ad cosumpio-leisure efficiecy codiio is c k Le f ( y+, y, x+, x) = ( α) zk = 0 (ad recall y = ) x = z mk (, ) = k α ( α) zk = 0 Compue firs row of marix f x Cosumpio-leisure efficiecy codiio k z Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP Sepember 7, 203 23
(Parial) Example LINEARIZING THE RBC MODEL I deermiisic seady sae, he firs rows of f y+, f y, f x+, f x are f y+ 0 0 f y f x+ f x ψ + α( α) zk 2 0 0 α k α( α) z ( ) k α α α α + Sepember 7, 203 24
(Parial) Example LINEARIZING THE RBC MODEL I deermiisic seady sae, he firs rows of f y+, f y, f x+, f x are f y+ 0 0 f y f x+ ψ + α( α) zk 2 0 0 f x α k α( α) z ( ) k α α α α + How o compue derivaives f y+, f y, f x+, f x? By had (feasible for small models) Schmi-Grohe ad Uribe (or your ow!) Malab aalyical/symbolic rouies Your ow Maple or Mahemaica programs Dyare package Sepember 7, 203 25
CALIBRATION? Solvig for he seady sae? Choosig parameer values? Nex: calibraio of he baselie represeaive-age (RBC + growh) model Sepember 7, 203 26