Graduate Macroeconomics 2 Problem set 4. - Solutions

Transcription

1 Graduae Macroeconomics Problem se. - Soluions In his problem, we calibrae he Roemberg and Woodford (995) model of imperfec compeiion. Since he model and is equilibrium condiions are discussed a lengh in he lecure noes, here we focus on he log-linearizaion and implemenaion in Dnare. The.mod file corresponding o quesion i is called ps qi.mod. The echnolog dnamics is an AR() process which describes he deviaions from sead sae echnolog level: z = ρ z z + ε. According o he model, he number of firms (denoe wih N raher han I, no o confuse wih invesmen) follows a sluggish dnamics: log N = κ log(n np ) + ( κ) log N. () Here κ is he speed of adjusmen, and N np consisen wih no profis. To log linearize his equaion, we need wo new variables, he log deviaion of he number of firms from is BGP value (n ), and he log-deviaion of he no profi number of firms from is BGP value (n np ). The no-profi number of firms in and ouside of he sead sae is given b: N np = µ φµ F (K, Z H ) In he RW paper for simplici and for comparabili (wih anoher paper) he assume ha N np = N Z, where N is a parameer, and Z is he echnolog level. (Noe ha his is probabl a good approximaion if µ is consan. When he markups var exogenousl, hen N np will have a more complex pah ouside of he BGP, which will depend on k, z, h and ˆµ (he log deviaion of he markup from is sead sae value).) Wih he RW simplificaion () becomes: log N = κ log(n Z ) + ( κ) log N.

2 This holds in he sead sae as well: log N = κ log(n Z ) + ( κ) log N. We can subrac o wrie log deviaions as n = κz + ( κ)n. Tha is, he number of firms is exogenous, and we don reall need N for he impulse response funcions. Noe ha his implies ha unless here are echnolog shocks (i.e. z ), he number of firms will no deviae from is BGP value. In preparaion for quesion, assume ha here are governmen purchases financed from lump sum axes. On he BGP i is rue ha: G =.Y, bu he log deviaions of governmen spending from is BGP pah follow an AR() dnamics: g = ρ g g + ε g. Noe ha since now we have governmen spending, which is denoed b G, we will denoe he deerminisic growh rae of he econom wih γ insead of G. Le s collec all he log-linearized equaions: oupu: capial accumulaion: = µs k k + µs h [z + h ] (µ )i γk + = ( δ)k + Y K C Y Y K MPK: R r = ηs h (R + δ)[z + h k ] MPL: w = ( ηs k )z + ηs k [k h ] c G Y Y K g marginal uili: λ = c labor suppl: h = ε h [w + λ ] Euler equaion: λ = E [λ + + r + ] number of firms: n = κz + ( κ)n We have jus rewrien he equaions wih he noaion s k for capial share in value added, s h for labor share in value added, γ for he growh rae, s I for he sead-sae invesmen rae, η for he elasici of subsiuion beween capial

3 .5 x k x n.5 6 x k.5 x n x h x c x u x h 6 x c x u x r x w.5 z x r 6 x w.5 z Figure : Response o a producivi shock: influence of ρ z (lef:., righ:.9) and labor (assumed o be one in RW), and ε h for he Frisch elasici of labor suppl. We also made use of he uniar iner-emporal elasici of subsiuion, so ha Λ = /C and λ = c. µ η s I s h s k γ δ ɛ hw R Y...58 s h..5.6 C Y G Y K ρ z κ ρ g γ ( δ) s I Table : Parameers values Quesion Using he calibraion for he parameers given in he above Table, we plo he response of he econom o a producivi shock wih differen values for ρ and κ in Figure and Figure. In Figure we see ha we need persisen shocks o ge persisen responses (as alwas). Also observe in Figure ha labor flucuaions are much smaller if he number of firms does no adjus quickl: people will enjo increased profis and will wan o consume more leisure and work less. Tha is, in he presence of + The felici funcion is U (C, H ) = log C χ H ɛh + ɛ. The parameer χ disappears from he h linearized equaions. ɛ h is he Frisch elasici of labor suppl defined as he elasici of hours wih respec o wage keeping consan he marginal uili of wealh (see labor suppl Euler equaion).

4 6 x k.5 x n.5 x k.5 n x h 6 x c x u 6 x h x c x u x r 6 x w.5 z x r x w.5 z Figure : Response o a producivi shock: influence of κ (lef:., righ: ) profis, he wealh effec dominaes he subsiuion effec. (To check his, run he code wih no markups (µ = ), which exhibis a bigger labor response o echnolog shocks.) Since labor hours should be highl pro-cclical, we ma prefer specificaions wih rapid enr of firms (high κ), and hence low wealh effecs. A relaed observaion is ha wages are ver pro-cclical (ou have o pa people a lo o couner he wealh effec). This is so even wih he fairl elasic labor suppl (ε hw = ). However, given he Solon, Barsk and Parker (99) paper, we are no oo much worried abou his. Quesion Wha happens if ou go direcl o a Cobb-Douglas producion funcion? The hree equaions involving he producion funcion are he oupu, MPK and MPL. The firs one sas ha oupu is a weighed average of facor usages, wih he weighs corresponding o facor shares. This equaion would be exac under Cobb- Douglas, since here he facor shares are consan. = αk + ( α)[z + h ] However, here are differen was o calculae he marginal producs, MPK and MPL. To see his, le us consider a simple case wihou markups, Also see p. 7 in RW (995).

5 Express marginal producs as a funcion of relaive inpu usages: ( ) α R = F K (K, Z H ) + δ = αk α (Z H ) α Z H + δ = α + δ, K ( ) α W = F H (K, Z H ) = ( α)z K α (Z H ) α K = ( α)z. Z H Log-linearizing hese ields ( Z R H ) α r = α ( α)[z + h k ] = (R + δ)( α)[z + h k ], K w = z + α [k (z + h )], where we used he expression for he renal rae in he BGP in simplifing he expression for he log-deviaion of he renal rae. These are equivalen o he original MPL and MPK equaions. Now wrie marginal producs as a funcion of average producs: The log-linear versions are R = αk α (Z H ) α + δ = α Y + δ K W = ( α)z K α (Z H ) α = ( α) Y H R r = α Y K [ k ] = α Y K ( α)[z + h k ] w = h = z + α [k (z + h )] so ou would be emped o calibrae he model o he sead sae capial oupu raio, which is usuall inferred from he saving and depreciaion raes (Y /K = (γ +δ)/s I ). However, he original MPK equaion offers an alernaive calibraion o he real ineres rae, which ma be more observable. We use he MPK equaion in he calibraion, which remains exacl he same when considering markups: R = F K(K, Z H ) µ ( ) R = α Z H α K + δ + δ = α µ Kα (Z H ) α + δ = α µ ( ) α Z H + δ K 5

6 Log-linearized: ) α R r = α ( Z H µ K }{{ ( α)[z + h k ] = (R + δ)( α)[z + h k ], } =R ( δ) Quesion Here we assume ha markups follow an exogenous AR() process ˆµ = ρ µ ˆµ + ν µ. The markup now is variable, denoe is level b µ µ ˆµ. The onl wo equaions ha change are he firm s FOCs, as hese are he onl wo where he variable µ shows up. (Noe : µ also shows up in he producion funcion, bu onl insead of a sead sae value (φi /Y ), so here nohing changes. Noe : if we would no adop RW simplifing assumpion, ha N np in he low-linearizaion of N np = N Z, hen ˆµ would show up as well.) The wo firs order condiions are now: R = F (K, Z H ) µ + δ µ W = Z F (K, Z H ) The log-linearized equaions are as follows: ˆµ + w = R r = F F ( F K F F F FF F Z H µ F F F F (z + h k ) F ) z + F K F FF F F µ ˆµ (k h ) where F = F (K, Z H ), and in he calibraion we can use from he sead sae reurn on capial ha F /µ = R ( δ). According o Figure markup shocks produce roughl similar paerns o hose of a producion shock. I even improves a bi he response of emplomen: now emplomen reacs wice as much as wages and slighl more han oupu. For markup shocks, here are wo wealh effecs ha go in he opposie direcion. The firs wealh effec is negaive, and is driven b he decrease in he real wage. The second wealh effec is posiive, and i sems from he increase in profis. These wo wealh effecs offse each oher (he almos cancel each oher ou), herefore he iner-emporal subsiuion effec becomes more prominen. Since 6

7 x k 5 x h x k 5 x h x c 8 x u 5 x r x c 6 x u 5 x r 6 8 x w.5 m x w.5 m Figure : Response o a markup shock: influence of ρ m (lef:., righ:.9) x k 5 x h x k 5 x h x c 6 x u 5 x r x c 6 x u 5 x r x w.5 m x w.5 m Figure : Response o a markup shock: influence of κ (lef:., righ: ) 7

8 wages decrease, he subsiuion effec leads o a sharp reducion in emplomen. (Conras his o producivi shocks: boh wealh effecs go in he same direcion, making he emplomen response weaker.) Hence, we could consider couner-cclical markup shocks as a source of flucuaions. While exogenous markup flucuaions are implausible. Firs, given ha he markup onl depends on demand elasiciies, shocks on markup in he model would resul from shifs in preferences. Those shifs could explain flucuaions in some secors bu no in he whole econom. Second, his implies an increase in profis and a decrease in oupu. This means ha recessions are periods in which firms s profis increase, a resul which is hard o reconcile wih daa. Endogenous markup flucuaions could amplif he ccle (cf. lecure ). Noe ha he persisence of he markup shocks, as alwas, plas an imporan role. The speed of adjusmen is no imporan (has no effec), as here are no echnolog shocks (see discussion before Quesion ). Quesion Nex include governmen spending wih a sead sae raio o GDP of % and a persisence of.9. For his we need o add he log deviaions of governmen spending from is BGP pah (as given in he quesion and on page of hese soluions). We also have o modif he capial accumulaion equaion, as unil now he econom s resource consrain was Y = C + I, and now we have Y = C + I + G. The modified capial accumulaion is he following (as on page of hese soluions): γk + = ( δ)k + Y K C Y Y K c G Y Y K g Inuiivel, an increase in governmen spending should cause consumpion o fall and labor suppl o increase because of is negaive wealh effec. Governmen spending affecs he iner-emporal budge consrain of he agen b lowering he presen value of he afer ax income, leading o he negaive wealh effec. Le s r he following calibraion and see if our inuiion is correc. Calibraion: ρ g =.9 (high persisence), κ =. (sluggish adjusmen o new condiions ) and µ =. (imperfec compeiion). A persisen shock o governmen spending has he following effec in he Compare o figures 9. and 9. in RW (995). 8

9 model: consumpion, capial and real wages decrease while oupu, real ineres rae and labor suppl increase. The labor suppl increases and hen graduall reurns o normal; in conras o wha occurs wih echnolog shocks (as in Quesion and ), i never falls below is normal value. Because echnolog is unchanged and he capial sock moves ver lile, he movemens in oupu are small and rack he changes in emplomen fairl closel. Consumpion decreases a he ime of he shock and hen graduall reurns o normal. The increase in emplomen and he fall in he capial sock cause he wage o fall and he ineres rae o rise. The anicipaed wage movemens afer he shock are small and posiive. Hence he increase in labor suppl sems from he iner-emporal subsiuion effec due o he increase in he ineres rae, and from he wealh effec due o he governmen use of more oupu. Furhermore, because he rise in governmen purchases is no permanen, agens also respond b decreasing heir capial: his is a direc implicaion of he capial accumulaion equaion. Similar o a echnolog shock, he persisence of movemens in governmen purchases has imporan effecs on how he econom responds o a shock. If ou r an alernae calibraion wih a emporar shock (for example: ρ g = ) o governmen spending (i.e. he movemens in purchases are much shorer-lived) he direcion of he response in consumpion, ineres rae, labor suppl ec remain he same bu heir magniude is reduced. In his case, much of he response akes he form of reducions in capial holdings, while he magniude of shifs in oher variables is decreased. Noice in Figure 5 ha here is ver lile effec of ransior governmen purchases (he amplificaion problem). The amplificaion problem is case of governmen spending shocks is parl due o he weak share of governmen spending in he GDP chosen in he calibraion. Aggregae demand, in log deviaions, is given b: = C Y c + I Y i + G Y g Given ha G /Y =., he response of o g is necessaril weak. The negaive consumpion response (crowding ou) impl ha he effecs of governmen spending shocks on GDP become even smaller. 9

10 x k x.5 x h x k 6 x 6 x h x c 8 x u x r x c x u.5 x r x w.5 g x w.5 g Figure 5: Response o a governmen shock: influence of ρ g (lef:., righ:.9)