LINEARIZING AND APPROXIMATING THE RBC MODEL SEPTEMBER 7, 200 For f( x, y, z ), mulivariable Taylor liear expasio aroud ( x, yz, ) f ( x, y, z) f( x, y, z) + f ( x, y, z)( x x) + f ( x, y, z)( y y) + f ( x, y, z)( z z) Four equaios describe he dyamic soluio o RBC model Cosumpio-leisure efficiecy codiio u( c, ) = zm ( k, ) uc( c, ) Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai x y z c + k+ ( δ ) k = zm ( k, ) Law of moio for TFP z l z = ( ρ )l z + ρ l z + ε ( δ ) u ( c, ) = βe u ( c, ) + z m ( k, c c + + + k + + + z z + Sepember 7, 200 2 The Basics
The Basics 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, ) = zm ( k, ) u ( c, ) c uc( c, ) = βuc( c, ) mk( k, ) + δ c + δ k = zf( k, ) l z = ( ρ )l z + ρ l z z = z (a parameer of he model) z z Give fucios ad parameer values (ex ), solve for (c,, k) The seady-sae of he model Taylor expasio aroud his poi Sepember 7, 200 3 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 Duke 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 The Basics Sepember 7, 200 4 2
SGU Deails Defie co-sae vecor ad sae vecor y c x k z Order model s dyamic equaios i a vecor Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP Sepember 7, 200 5 f( y, y, x, x ) + + Need four marices of derivaives. Differeiae f ( y, y, x, x ) wih respec o (elemes of) y + + + SGU Deails Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP Firs derivaives wih respec o: c + + = f y + Sepember 7, 200 6 3
SGU Deails Need four marices of derivaives 2. Differeiae f ( y, y, x, x ) wih respec o (elemes of) y + + Firs derivaives wih respec o: Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP c = f y Sepember 7, 200 7 Need four marices of derivaives 3. Differeiae f ( y, y, x, x ) wih respec o (elemes of) x + + + SGU Deails Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP Firs derivaives wih respec o: k + z + = f x + Sepember 7, 200 8 4
SGU Deails Need four marices of derivaives 4. Differeiae f ( y, y, x, x ) wih respec o (elemes of) x + + Firs derivaives wih respec o: Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP k z = f x Sepember 7, 200 9 SGU Deails The model s dyamic expecaioal equaios f ( y, y, x, x ) + + Cosumpio-leisure efficiecy codiio 2 f ( y, y, x, x) E[ f( y, y, x, x) ] E + + Cosumpio-ivesme efficiecy codiio + + = f 3 ( y, y, x, x ) + + Aggregae resource cosrai 4 f ( y+, y, x+, x) Law of moio for TFP Cojecure equilibrium decisio rules Subsiue decisio rules io dyamic equaios y = g( x, ) x = h( x, ) + ηε + + Perurbaio parameer : govers size of shocks Marix of sadard deviaios of sae variables Sepember 7, 200 0 5
SGU Deails The model s dyamic expecaioal equaios [ ( +,, +, )] = E[ f( gx ( +, ), gx (, ), hx (, ) + ηε+, x] = E [ f( g( h( x, ), ), g( x, ), h( x, ) + ηε, x ] E f y y x x + F( x, ) F ( x, ) F ( x, ) x Sepember 7, 200 The model s dyamic expecaioal equaios E[ f( y+, y, x+, x) ] = E[ f( gx ( +, ), gx (, ), hx (, ) + ηε+, x] = E f( g( h( x, ), ), g( x, ), h( x, ) + ηε+, x F( x, ) Usig chai rule ad suppressig argumes [ ] Fx ( x, ) f f = y g x hx + y g x + fx h x + f + + x Holds, i paricular, a he deermiisic seady sae ( x,0) F ( x,0) = f g h + f g + f h + f x y+ x x y x x+ x x Seig shus dow shocks Sepember 7, 200 2 SGU Deails Each erm is evaluaed a he seady sae jus as Taylor heorem requires 6
SGU Deails A quadraic equaio i he elemes of g x ad h x F ( x,0) = f g h + f g + f h + f 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 = g( x, ) x = h( x, ) + ηε + + Sepember 7, 200 3 SGU Deails A quadraic equaio i he elemes of g x ad h x F ( x,0) = f g h + f g + f h + f 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 = g( x, ) x = h( x, ) + ηε + + Firs-order approximaio is y = g( x, ) g( x,0) + g ( x,0)( x x) + g ( x,0) x + x x = h( x, ) h( x,0) + h ( x,0)( x x) + h ( x,0) SGU Theorem : g ad h DONE!!! Now coduc impulse resposes, ru simulaios, abulae momes, wrie paper Sepember 7, 200 4 7
Priciples CERTAINTY EQUIVALENCE Displayed by a model if decisio rules do o deped o he sadard deviaio (i.e., he size ) of he exogeous shocks For sochasic problems wih quadraic objecive fucio ad liear cosrais, he decisio rules are ideical o hose of he osochasic problem (Ljugqvis ad Sarge (2004, p. 3)) Sepember 7, 200 5 CERTAINTY EQUIVALENCE Displayed by a model if decisio rules do o deped o he sadard deviaio (i.e., he size ) of he exogeous shocks Priciples For sochasic problems wih quadraic objecive fucio ad liear cosrais, he decisio rules are ideical o hose of he osochasic problem (Ljugqvis ad Sarge (2004, p. 3)) Here, we have y = g( x, ) g( x,0) + gx( x,0)( x x) + g ( x,0) x = h( x, ) h( x,0) + h ( x,0)( x x) + h ( x,0) + x SGU Theorem : g ad h Firs-order approximaed decisio rules do o deped o he size of he shocks, which is govered by No quie he same hig as CE, bu we ll loosely refer o i as CE Sepember 7, 200 6 8
SGU Deails SECOND-ORDER APPROXIMATION Use compued g x ad h x o cosruc liear sysem of equaios eeded o solve for coefficies of secod-order approximaio (SGU p. 762-763) SGU Theorem : g x ad h x ; bu ceraiy equivalece does o hold because g 0, h 0 Sepember 7, 200 7 LINEARIZING THE RBC MODEL Assume uc (, ) = lc ψ l ad mk (, ) = k cosumpio-leisure efficiecy codiio is (Parial) Example c k Le f ( y+, y, x+, x) = ( ) zk (ad recall y ) x z = ( ) zk Sepember 7, 200 8 9
(Parial) Example LINEARIZING THE RBC MODEL Assume uc (, ) = lc ψ l ad mk (, ) = k cosumpio-leisure efficiecy codiio is c k Le f ( y+, y, x+, x) = ( ) zk (ad recall y ) x z = ( ) zk 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, 200 9 LINEARIZING THE RBC MODEL Assume uc (, ) = lc ψ l ad mk (, ) = k cosumpio-leisure efficiecy codiio is (Parial) Example c k Le f ( y+, y, x+, x) = ( ) zk (ad recall y ) x z = ( ) zk 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 + ( ) z k 2 Sepember 7, 200 20 0
(Parial) Example LINEARIZING THE RBC MODEL Assume uc (, ) = lc ψ l ad mk (, ) = k cosumpio-leisure efficiecy codiio is c k Le f ( y+, y, x+, x) = ( ) zk (ad recall y ) x z = ( ) zk 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 0 0 Sepember 7, 200 2 LINEARIZING THE RBC MODEL Assume uc (, ) = lc ψ l ad mk (, ) = k cosumpio-leisure efficiecy codiio is (Parial) Example c k Le f ( y+, y, x+, x) = ( ) zk (ad recall y ) x z = ( ) zk 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 k k ( ) z ( ) + Sepember 7, 200 22
(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 + ( ) zk 2 0 0 k ( ) z ( ) k f x + How o compue derivaives? By had (feasible for small models) Schmi-Grohe ad Uribe Malab aalyical rouies Your ow Maple or Mahemaica programs Dyare package Sepember 7, 200 23 2