Four equations describe the dynamic solution to RBC model. Consumption-leisure efficiency condition. Consumption-investment efficiency condition

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1 LINEAR APPROXIMATION OF THE BASELINE RBC MODEL FEBRUARY, 202 Iroducio For f(, y, z ), mulivariable Taylor liear epasio aroud (, yz, ) f (, y, z) f(, y, z) + f (, y, z)( ) + f (, y, z)( y y) + f (, 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 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 + February, 202 2

2 Iroducio STEADY STATE Deermiisic seady sae he aural local poi of approimaio Shu dow all shocks ad se eogeous 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 = zm( k, ) l z = ( ρ )l z + ρ l z z = z (a parameer of he model) z z Give fucioal forms ad parameer values, solve for (c,, k) The seady sae of he model Taylor epasio aroud his poi February, LINEARIZATION ALGORITHMS Schmi-Grohe ad Uribe (2004 JEDC) A perurbaio algorihm A class of mehods used o fid a approimae soluio o a problem ha cao be solved eacly, by sarig from he eac soluio of a relaed problem Applicable if he problem ca be formulaed by addig a small erm o he descripio of he eacly-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 Iroducio February,

3 Iroducio Defie co-sae vecor ad sae vecor y = k = z February, Defie co-sae vecor ad sae vecor SGU Deails y = 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 February, f( y, y,, ) + + 3

4 SGU Deails Need four marices of derivaives. Differeiae f ( y, y,, ) wih respec o (elemes of) y Firs derivaives wih respec o: c + + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP = f y + February, Need four marices of derivaives 2. Differeiae f ( y, y,, ) wih respec o (elemes of) y + + SGU Deails Firs derivaives wih respec o: Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP c = f y February,

5 SGU Deails Need four marices of derivaives 3. Differeiae f ( y, y,, ) wih respec o (elemes of) Firs derivaives wih respec o: k + z + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP = f + February, Need four marices of derivaives 4. Differeiae f ( y, y,, ) wih respec o (elemes of) + + SGU Deails Firs derivaives wih respec o: Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP k z = f February,

6 SGU Deails The model s dyamic epecaioal equaios f ( y, y,, ) + + Cosumpio-leisure efficiecy codiio 2 f ( y, y,, ) E[ f( y, y,, ) ] E + + Cosumpio-ivesme efficiecy codiio + + = f 3 ( y, y,, ) + + Aggregae resource cosrai 4 f ( y+, y, +, ) Law of moio for TFP February, 202 SGU Deails The model s dyamic epecaioal equaios f ( y, y,, ) + + Cosumpio-leisure efficiecy codiio 2 f ( y, y,, ) E[ f( y, y,, ) ] E + + Cosumpio-ivesme efficiecy codiio + + = f 3 ( y, y,, ) + + Aggregae resource cosrai 4 f ( y+, y, +, ) Law of moio for TFP Cojecure equilibrium decisio rules Noe: g(.) ad h(.) are ime ivaria fucios! Perurbaio parameer : y = g(, ) govers size of shocks = h(, ) + ηε Subsiue decisio rules io dyamic equaios + + Mari of sadard deviaios of sae variables February,

7 SGU Deails The model s dyamic epecaioal equaios [ ( +,, +, )] = E[ f( g ( +, ), g (, ), h (, ) + ηε+, ] = E [ f( g( h(, ) + ηε, ), g(, ), h(, ) + ηε, ] E f y y + + F(, ) F (, ) F (, ) February, The model s dyamic epecaioal equaios E[ f( y+, y, +, ) ] = E[ f( g ( +, ), g (, ), h (, ) + ηε+, ] = E f( g( h(, ) + ηε+, ), g(, ), h(, ) + ηε+, F(, ) [ ] Usig chai rule ad suppressig argumes F f f (, ) = fy g h + fy g + h SGU Deails February,

8 SGU Deails The model s dyamic epecaioal equaios E[ f( y+, y, +, ) ] = E[ f( g ( +, ), g (, ), h (, ) + ηε+, ] = E f( g( h(, ) + ηε+, ), g(, ), h(, ) + ηε+, F(, ) Usig chai rule ad suppressig argumes [ ] F (, ) f f = y g h + y g + f h + f + + February, The model s dyamic epecaioal equaios E[ f( y+, y, +, ) ] = E[ f( g ( +, ), g (, ), h (, ) + ηε+, ] = E f( g( h(, ) + ηε+, ), g(, ), h(, ) + ηε+, F(, ) Usig chai rule ad suppressig argumes [ ] F (, ) f f = y g h + y g + f h + f + + Holds, i paricular, a he deermiisic seady sae (,0) F (,0) = f g h + f g + f h + f y+ y + Seig shus dow shocks February, SGU Deails Each erm is evaluaed a he seady sae jus as Taylor heorem requires 8

9 SGU Deails A quadraic equaio i he elemes of g ad h evaluaed a he seady sae F (,0) = f (,0) g (,0) h (,0) + f (, 0) g (,0) + f (,0) h (, 0) + f (,0) y+ y + Solve umerically for he elemes of g ad h (use fsolve i Malab) Recall cojecured equilibrium decisio rules y = g(, ) = h(, ) + ηε + + February, A quadraic equaio i he elemes of g ad h evaluaed a he seady sae y+ y + Solve umerically for he elemes of g ad h (use fsolve i Malab) Recall cojecured equilibrium decisio rules Firs-order approimaio is y = g(, ) g(,0) + g (,0)( ) + g (,0) = h(, ) h(,0) + h (,0)( ) + h (,0) + SGU Deails F (,0) = f (,0) g (,0) h (,0) + f (, 0) g (,0) + f (,0) h (, 0) + f (,0) y = g(, ) = h(, ) + ηε + + SGU Theorem : g ad h DONE!!! Now coduc impulse resposes, abulae busiess cycle momes, wrie paper February,

10 Macro Fudameals CERTAINTY EQUIVALENCE Displayed by a model if decisio rules do o deped o he sadard deviaio of eogeous 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 y = g(, ) g(,0) + g(,0)( ) + g (,0) = h(, ) h(,0) + h (,0)( ) + h (,0) + SGU Theorem : g ad h Firs-order approimaed decisio rules do o deped o he size of he shocks, which is govered by No he same hig as eac CE, bu refer o i as CE February, LINEARIZING THE RBC MODEL Assume uc (, ) = lc ψ l ad mk (, ) = k α cosumpio-leisure efficiecy codiio is (Parial) Eample k Le f ( y+, y, +, ) = ( α) zk (ad recall y = ) = z ( α) zk February,

11 (Parial) Eample LINEARIZING THE RBC MODEL Assume uc (, ) = lc ψ l ad mk (, ) = k α cosumpio-leisure efficiecy codiio is k Le f ( y+, y, +, ) = ( α) zk (ad recall y = ) = z ( α) zk Compue firs row of mari f y+ c + + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP February, LINEARIZING THE RBC MODEL Assume uc (, ) = lc ψ l ad mk (, ) = k α cosumpio-leisure efficiecy codiio is (Parial) Eample k Le f ( y+, y, +, ) = ( α) zk (ad recall y = ) = z ( α) zk Compue firs row of mari f y+ c + + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP 0 0 February,

12 (Parial) Eample LINEARIZING THE RBC MODEL Assume uc (, ) = lc ψ l ad mk (, ) = k α cosumpio-leisure efficiecy codiio is k Le f ( y+, y, +, ) = ( α) zk (ad recall y = ) = z ( α) zk Compue firs row of mari f y Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP c ψ + α ( α ) z k 2 α α February, LINEARIZING THE RBC MODEL Assume uc (, ) = lc ψ l ad mk (, ) = k α cosumpio-leisure efficiecy codiio is (Parial) Eample k Le f ( y+, y, +, ) = ( α) zk (ad recall y = ) = z ( α) zk Compue firs row of mari f + k + z + Cosumpio-leisure efficiecy codiio Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP 0 0 February,

13 (Parial) Eample LINEARIZING THE RBC MODEL Assume uc (, ) = lc ψ l ad mk (, ) = k α cosumpio-leisure efficiecy codiio is k Le f ( y+, y, +, ) = ( α) zk (ad recall y = ) = z ( α) zk Compue firs row of mari f k z Cosumpio-leisure efficiecy codiio α k k α ( α ) z ( α ) α Cosumpio-ivesme efficiecy codiio Aggregae resource cosrai Law of moio for TFP α α + February, LINEARIZING THE RBC MODEL I deermiisic seady sae, he firs rows of f y+, f y, f +, f are f y+ 0 0 (Parial) Eample f y f + ψ + α( α) zk α k α( α) z ( ) k α f α α α + How o compue derivaives f y+, f y, f +, f? By had (feasible for small models) Schmi-Grohe ad Uribe Malab aalyical rouies Your ow Maple or Mahemaica programs Dyare package February,

14 CALIBRATION? Solvig for he seady sae? Choosig parameer values? Ne: calibraio of he baselie represeaive-age (RBC + growh) model February,