CALIBRATING THE (RBC + SOLOW) MODEL JANUARY 31, 2013

Transcription

1 CALIBRATING THE (RBC + SOLOW) MODEL JANUARY 3, 203

2 Inroducion STEADY STATE Deerminisic seady sae he naural poin of approximaion Shu down all shocks and se exogenous variables a heir means The idea: le economy run for many (infinie) periods Time evenually doesn maer any more Drop all ime indices c zf ( k, n) uc( c, n) = βuc( c, n) zfk( k, n) + δ c + δk = zf( k, n) ln z = ( ρ )ln z + ρ ln z z = z (a parameer of he model) z un( c, n) = u ( c, n) z n Given funcional forms and parameer values, solve for The seady sae of he model Taylor expansion around his poin ( cnk,, ) January 3, 203 2

3 Inroducion CALIBRATION PHILOSOPHY An economic model is a measuring device If model makes believable predicions along some imporan dimensions (i.e., maches some key daa ) hen maybe is predicions are believable along he novel dimensions of he model Geing some parial derivaives of he model in known direcions correc may build credibiliy ha is parial derivaives in novel direcions are a leas no grossly incorrec Make model mach some daa of ineres ofen long-run (i.e., imeaveraged daa) growh facs Preferably well-acceped sylized facs Solow growh model in he background Naural candidae: Kaldor growh facs Calibraion vs. Esimaion January 3, 203 3

4 Inroducion CALIBRATION OF BASELINE RBC MODEL Mus ake a sand on hree (relaed) poins Which daa do we wan model o mach? (even consrucing daa is challenging ) Funcional forms (uiliy, producion) Parameer values Choose funcional forms consisen wih Kaldor-plus facs (K) Capial income share and labor income share of GDP are saionary (K2) All real quaniy variables grow a same rae in he long run (K3) Real ineres rae is saionary (K4) Hours per worker are saionary (K5) (K2) requires rend produciviy o be labor-augmening (Phelps 966) Ofen sar wih RBC model ha absracs from long-run growh Bu rue calibraion begins wih model feauring only long-run growh Pus resricions on insananeous uiliy and producion forms Use (K)-(K5) o obain hese resricions Richer models: more calibraion arges and/or reaing daa differenly Monopoly markups (e.g., Dixi-Sigliz and sicky price models) Probabiliy of finding a job (e.g., labor maching models) Durable consumpion vs. non-durable consumpion January 3, 203 4

5 Calibraion of RBC Model wih Growh RBC MODEL WITH GROWTH Absen shocks, TFP grows a deerminisic rae γ Planner problem/perfec compeiion max E b u C, n ) 0 = 0 ( Red indicaes variables or parameers ha will be modified when derending he model subjec o C + K ( δ ) K = + zf( K, n X) X given sochasic process for evoluion of z and (K -, z 0, X 0 ) Trend produciviy is laboraugmening (Harrod-neural) (Makes use of fac (K5)) Flow resource consrain = γx, γ Evoluion of deerminisic componen of produciviy Suppose z = always, so only deerminisic growh Deerminisic dynamics of (C, K +, n, X + ) governed by () (2) un( C, n) = XF (, 2 K nx ) u ( C, n ) c C + + uc( C, n) = F ( K+, n+ X+ ) + δ bu ( C, n ) Labor supply funcion (aka consumpion-labor opimaliy) Capial supply funcion (aka consumpion-savings opimaliy) (3) C + K+ ( δ ) K = F( K, nx) X = γ X (4) Normalize X 0 = January 3, 203 5

6 Calibraion of RBC Model wih Growh RESTRICTIONS ON FUNCTIONAL FORMS Deerminisic dynamics of (C, K +, n, X ) governed by () (2) un( C, n) = XF (, 2 K nx ) u ( C, n ) c C + + uc( C, n) = F ( K+, n+ X+ ) + δ bu ( C, n ) C + K+ ( δ ) K = F( K, nx) (K) Capial income share and labor income share of GDP are saionary And viewing economic profis as zero (3) (4) X = γ X FKnX = K nx (, ) α α ( ) ( α 0.4) January 3, 203 6

7 Calibraion of RBC Model wih Growh RESTRICTIONS ON FUNCTIONAL FORMS Deerminisic dynamics of (C, K +, n, X ) governed by () (2) un( C, n ) K / X = ( α ) X uc ( C, n) n α uc( C, n) K / X + + = α + δ buc( C+, n+ ) n+ (K2) All real quaniy variables grow a same rae in he long run Y C K X Y C K X And scale (3) by X o make saionary α (3) (4) = = = = γ, Y K C y = y, k = k, c = c, X X X u ( C, n ) k n = ( α ) X uc( C, n ) n α uc( C, n ) k + = α + δ buc( C+, n+ ) n+ α α C + K+ ( δ ) K = K ( nx ) X = γ X c k k α α + γ k+ ( δ ) = n X = γ X α Noe long run growh rae affecs capial accumulaion even in saionary represenaion! January 3, 203 7

8 Calibraion of RBC Model wih Growh RESTRICTIONS ON FUNCTIONAL FORMS Deerminisic dynamics of (C, n, X ) governed by () (2) un( C, n ) k = ( α ) X uc( C, n) n α uc( C, n ) k = α + δ buc( C+, n+ ) n+ (K4) Hours per worker are saionary α (3) (4) c + γ k ( δ ) k = k α n X = γ X α n = n along deerminisic pah. BUT n is endogenous January 3, 203 8

9 Calibraion of RBC Model wih Growh RESTRICTIONS ON FUNCTIONAL FORMS Deerminisic dynamics of C governed by () (2) un( C, n) k = ( α ) X uc( C, n) n α uc( C, n) k = α + δ buc( C+, n) n Implied already (RHS of (2)) is (K3) Real ineres rae is saionary Final sep funcional form for uiliy? α (3) c + γ k ( δ ) k = k α n α Observaions Opimal choice of labor ( n ) mus be independen of X (from ()) Requires offseing income and subsiuion effecs of wages (long-run produciviy) on labor supply IMRS can only depend on C + /C (from (2)), which in urn = γ Two requiremens ogeher imply [ Cv n ] σ ( ) if σ > 0, σ uc (, n) = σ ln C + v( n) if σ = King, Plosser, Rebelo (988 JME) January 3, 203 9

10 RBC Model wih Growh STEADY STATE Seady sae cnk,, solves (), (2), (3) ( ) A dynamic phenomenon! No saic! Economy is moving exacly along is long-run (i.e., deerminisic) growh pah Balanced growh pah Scale of absolue quaniy oucomes wihin model is meaningless Wha does, e.g., c =.56 mean? Relaive quaniy oucomes are inerpreable Provide calibraion arges e.g., c / y = 0.70, k / y = 25. (if annual measuremen) Time use and ineremporal price oucomes wihin model are inerpreable Provide calibraion arges n is fracion of ime spen in paid marke work Empirical: n 0.30 Reurn on capial α k α + δ n Use ss calibraion arges o se parameer values, given funcional forms January 3, 203 0

11 Transformed RBC Model RBC MODEL WITHOUT GROWTH Ofen sar insead wih [ cv n ] σ ( ) if σ > 0, σ uc (, n) = σ ln c + v( n) if σ = and = 0 c = C /X β uc (, n) For his ransformed model o deliver same seady sae (relaive quaniies, ime use, and r), require Resource consrain c k ( ) k zk α + γ δ n α + = β bγ σ Subjecive discoun facor (King and Rebelo, p. 945) Typical assumpion γ = omis growh alogeher Wha if rend growh rae flucuaes, γ? Typical represenaion canno accommodae rend shocks because γ = Trend shocks fairly common in small-open-economy models Affecs discoun facor and capial accumulaion equaion January 3, 203

12 Calibraion of RBC Model: Typical Approach BASELINE RBC MODEL Assuming γ = complee calibraion? Daa: long-run labor income share of GDP 0.60 Cobb-Douglas F(.) implies α α wn ( α) k n = = α α = 0.40 α α Fk (, n) kn Daa: long-run raio of (annual) gross invesmen o capial sock 0.07 k ( δ) k δk = = 0.07 k k δ = 0.07 (annual) or 0.08 (quarerly) Daa: long-run raio of (annual) oupu o capial sock 0.4 Seady-sae Euler equaion k α α α n αf( k, n) = β + δ = β + δ k k β = 0.95 (annual) or 0.99 (quarerly) OR Daa: avg. ne real reurn on capial 5% per year (e.g, reurn on S&P500) Seady-sae Euler equaion f ( kn k, ) = +δ β = 0.96 (annual) or 0.99 (quarerly) β January 3, 203 2

13 Calibraion of RBC Model: Typical Approach BASELINE RBC MODEL Uiliy parameers Daa: IES is around uniy(?) or lower Implies σ > (Recall: IES = /σ for ime-separable CRRA uiliy) σ = a convenional value Labor subuiliy Common form [ cv n ] σ ( ) if σ > 0, σ uc (, n) = σ ln c + v( n) if σ = ψ / + η vn ( ) = / η n + η measures Frisch elasiciy of labor supply (use C-L opimaliy condiion) Calibrae Ψ o hi n 0.3 Empirical evidence on Frisch elasiciy? January 3, 203 3

14 Calibraion of RBC Model: Typical Approach BASELINE RBC MODEL Labor supply elasiciy conroversial Micro evidence: very low η (subsanially) smaller han one Macro evidence: very high η (subsanially) larger han one Tension beween macro and micro evidence no useful way o frame he conroversy Micro sudies pick up inensive margin of labor supply Macro sudies pick up (mosly) exensive margin of labor supply And oher fricions in allocaion of workers o jobs Chey (20 Economerica): uses boh macro and micro evidence o pu bounds on labor supply elasiciy Common in DSGE models: η > January 3, 203 4

15 Calibraion of RBC Model: Typical Approach BASELINE RBC MODEL Exogenous process for TFP (deviaions from long-run rend produciviy) ln z = ( ρ )lnz + ρ ln z + ε + z z + ε : 2 + iid N(0, σz) Normalize z = Only governs absolue scale of model, which is arbirary Wha does, e.g., c =.56 mean? Consruc ime-series for z using Daa on labor, (derended) capial, and (derended) oupu AR() esimaion Quarerly frequency ρ = 0.95 and σ = z z January 3, 203 5

16 Summary USING THE RBC (OR ANY DSGE) MODEL. Dream up/consruc/wrie fully-ariculaed model Ideally o answer quesions moivaed by daa and wih hypoheses 2. Choose parameer values Perhaps exremely rigorously, if goal is o mach cerain empirical facs very precisely Perhaps adoping generally-acceped values, if goal is o illusrae some insigh 3. Solve for deerminisic seady sae (balanced growh pah) 4. Solve for dynamic decision rules (e.g., linear approximaion, second-order approximaion, global approximaion) 5. Conduc informaive baery of experimens (impulse responses, simulaions, ec.) o ry o falsify hypoheses 6. Tabulae resuls, wrie a (good!) paper, ge i published January 3, 203 6