The Basic New Keynesian Model

University:
Boston University
Course:
CAS EC 203 | Empirical Economics 1
Academic year:

2023
Views:

211

Pages:

27
Author:

5CelestialChronicle

The Basic New Keynesian Model Motivation and Outline Evidence on Money, Output, and Prices: Short Run E¤ects of Monetary Policy Shocks (i) persistent e¤ects on real variables (ii) slow adjustment of aggregate price level (iii) liquidity e¤ect Micro Evidence on Price-setting Behavior: signi…cant price and wage rigidities Failure of Classical Monetary Models A Baseline Model with Nominal Rigidities monopolistic competition sticky prices (staggered price setting) competitive labor markets, closed economy, no capital accumulation Households Representative household solves max E0 1 X t U (Ct; Nt) t=0 where Ct Z 1 Ct(i) 1 1 1 di 0 subject to Z 1 Pt(i)Ct(i) di + QtBt 0 for t = 0; 1; 2; ::: plus solvency constraint. Bt 1 + W t Nt Tt Optimality conditions 1. Optimal allocation of expenditures Pt(i) Pt Ct(i) = implying Z Ct 1 Pt(i)Ct(i) di = PtCt 0 where Pt Z 1 1 1 Pt(i)1 di 0 2. Other optimality conditions Un;t Wt = Uc;t Pt Uc;t+1 Pt Q t = Et Uc;t Pt+1 Speci…cation of utility: Ct1 U (Ct; Nt) = 1 Nt1+' 1+' implied log-linear optimality conditions (aggregate variables) pt = ct + 'nt 1 ct = Etfct+1g (it Etf t+1g wt where it log Qt is the nominal interest rate and discount rate. Ad-hoc money demand mt pt = yt it ) log is the Firms Continuum of …rms, indexed by i 2 [0; 1] Each …rm produces a di¤erentiated good Identical technology Yt(i) = AtNt(i)1 Probability of resetting price in any given period: 1 across …rms (Calvo (1983)). 2 [0; 1] : index of price stickiness Implied average price duration 1 1 , independent Aggregate Price Dynamics Pt = Dividing by Pt 1 (Pt 1) 1 + (1 )(Pt ) : 1 t = + (1 Pt ) Pt 1 1 1 1 1 Log-linearization around zero in‡ation steady state t = (1 )(pt pt 1 ) + (1 )pt or, equivalently pt = pt 1 (1) Optimal Price Setting max Pt subject to 1 X k t+k (Yt+kjt ) Et Qt;t+k Pt Yt+kjt k=0 Yt+kjt = (Pt =Pt+k ) Ct+k for k = 0; 1; 2; :::where Qt;t+k Optimality condition: 1 X where t+kjt k k Ct+k Ct Et Qt;t+k Yt+kjt Pt k=0 0 t+k (Yt+kjt ) and M 1 Pt Pt+k M t+kjt =0 Equivalently, 1 X k Et Qt;t+k Yt+kjt k=0 where M Ct+kjt t+kjt =Pt+k and Pt Pt 1 MM Ct+kjt t 1;t+k Pt+k =Pt =0 t 1;t+k 1 Perfect Foresight, Zero In‡ation Steady State: Pt =1 ; Pt 1 t 1;t+k =1 ; Yt+kjt = Y ; Qt;t+k = k ; MC = 1 M Log-linearization around zero in‡ation steady state: 1 X pt pt 1 = (1 ) ( )k Etfmc c t+kjt + pt+k k=0 where mc c t+kjt mct+kjt pt 1 g mc. Equivalently, pt = + (1 ) 1 X k=0 where log 1. ( )k Etfmct+kjt + pt+k g Flexible prices ( = 0): pt = =) mct = + mct + pt (symmetric equilibrium) Particular Case: = 0 (constant returns) =) M Ct+kjt = M Ct+k Rewriting the optimal price setting rule in recursive form: pt = )mc c t + (1 Etfpt+1g + (1 Combining (1) and (2): t where = Et f (1 t+1 g )(1 + mc ct ) )pt (2) Generalization to De…ne 2 (0; 1) (decreasing returns) mct Using mct+kjt = (wt+k (wt pt ) (wt pt ) pt+k ) mpnt 1 (at 1 1 1 yt ) (at+k mct+kjt = mct+k + = mct+k log(1 yt+kjt) (yt+kjt 1 (pt 1 log(1 ) ), yt+k ) pt+k ) (3) Implied in‡ation dynamics t where (1 = Et f )(1 t+1 g ) + mc ct 1 1 + (4) Equilibrium Goods markets clearing Yt(i) = Ct(i) for all i 2 [0; 1] and all t. R1 1 1 Letting Yt di Y (i) 0 t 1 , Yt = Ct for all t. Combined with the consumer’s Euler equation: yt = Etfyt+1g 1 (it Etf t+1 g ) (5) Labor market clearing Nt = Z 1 Nt(i) di 0 = Z 1 0 = Yt At Yt(i) At 1 Z 1 1 1 di 1 0 Pt(i) Pt 1 di Taking logs, (1 where dt (1 ) log R1 0 )nt = yt (Pt(i)=Pt) 1 at + dt di (second order). Up to a …rst order approximation: yt = at + (1 ) nt Marginal Cost and Output mct = (wt pt) mpnt = ( yt + 'nt) (yt '+ yt = + 1 Under ‡exible prices '+ mc = + ytn 1 =) ytn = where y where yt ( ytn log(1 ))(1 ) +'+ (1 ) > 0 and =) mc ct = yet is the output gap nt) log(1 ) 1+' at log(1 1 1+' at 1 y log(1 ) (6) ) (7) + ya '+ + 1 ya at 1+' +'+ (1 (yt ). ytn) (8) New Keynesian Phillips Curve t where + '+ . 1 = Etf t+1 g + yet (9) Dynamic IS equation 1 yet = Etfe yt+1g (it Etf t+1 g rtn) (10) where rtn is the natural rate of interest, given by rtn = Missing block: + + n Etf yt+1 g ya Et f at+1 g description of monetary policy (determination of it). Equilibrium under a Simple Interest Rate Rule it = + t + et yy + vt (11) where vt is exogenous (possibly stochastic) with zero mean. Equilibrium Dynamics: yet t where AT and 1 + y+ combining (9), (10), and (11) = AT Etfe yt+1g + BT (b rtn Etf t+1g 1 + ( + y) ; BT vt ) (12) 1 Uniqueness () AT has both eigenvalues within the unit circle Given 0 and y 0, (Bullard and Mitra (2002)): ( is necessary and su¢ cient. 1) + (1 ) y >0 E¤ects of a Monetary Policy Shock Set rbtn = 0 (no real shocks). Let vt follow an AR(1) process vt = v vt 1 + "vt Calibration: v = 0:5, = 1:5, y = 0:5=4, = 0:99, = ' = 1, = 2=3, = 4. Dynamic e¤ects of an exogenous increase in the nominal rate (Figure 1): Exercise: analytical solution E¤ects of a Technology Shock Set vt = 0 (no monetary shocks). Technology process: at = a at 1 + "at: Implied natural rate: rbtn = ya (1 Dynamic e¤ects of a technology shock ( Exercise: AR(1) process for at a a )at = 0:9) (Figure 2) Equilibrium under an Exogenous Money Growth Process mt = m mt 1 + "m t (13) Money market clearing where lt mt b bit lt = ybt = yet + ybtn (14) (15) bit pt denotes (log) real money balances. Substituting (14) into (10): (1 + ) yet = Furthermore, we have Etfe yt+1g + b lt + Etf b lt 1 =b lt + t t+1 g mt + rbtn ybtn (16) (17) Equilibrium dynamics 3 2 3 2 n 3 2 Etfe yt+1g yet rbt AM;0 4 t 5 = AM;1 4 Etf t+1g 5 + BM 4 ybtn 5 b b mt lt 1 lt 1 where 2 3 2 3 1+ 0 0 1 1 0 5 ; AM;1 4 0 0 5 ; BM AM;0 4 0 1 1 0 0 1 (18) 2 3 1 0 40 0 0 5 0 0 1 1 Uniqueness () AM AM;0 AM;1 has two eigenvalues inside and one outside the unit circle. E¤ects of a Monetary Policy Shock Set rbtn = ytn = 0 (no real shocks). Money growth process mt = where m 2 [0; 1) Figure 3 (based on m m mt 1 + "m t = 0:5) E¤ects of a Technology Shock Set mt = 0 (no monetary shocks). Technology process: at = Figure 4 (based on Empirical Evidence a = 0:9). a at 1 + "at Technical Appendix Optimal Allocation of Consumption Expenditures R1 Maximization of Ct for any given expenditure level 0 Pt (i) Ct (i) di L= Z 1 1 1 Ct (i) 1 di 0 Zt can be formalized by means of the Lagrangean Z 1 Pt (i) Ct (i)di Zt 0 The associated …rst order conditions are: Ct (i) 1 1 Ct = Pt (i) for all i 2 [0; 1]. Thus, for any two goods (i; j) we have: Ct (i) = Ct (j) Pt (i) Pt (j) which can be plugged into the expression for consumption expenditures to yield Ct (i) = Pt (i) Pt Zt Pt for all i 2 [0; 1]. The latter condition can then be substituted into the de…nition of Ct , yielding Z 1 Pt (i) Ct (i) di = Pt Ct 0 Combining the two previous equations we obtain the demand schedule: Ct (i) = Log-Linearized Euler Equation Pt (i) Pt Ct We can rewrite the Euler equation as 1 = Et fexp(it In a perfect foresight steady state with constant in‡ation ct+1 and constant growth i= + (19) )g t+1 we must have: + with the steady state real rate being given by r i = A …rst order Taylor expansion of exp(it exp(it ct+1 ct+1 t+1 + ) around that steady state yields: ) ' 1 + (it = 1 + it t+1 i) ( ct+1 ct+1 ) ( t+1 ) t+1 which can be used in (19) to obtain, after some rearrangement of terms, the log-linearized Euler equation 1 ct = Et fct+1 g (it Et f t+1 g ) Aggregate Price Level Dynamics Let S(t) [0; 1] denote the set of …rms which do not re-optimize their posted price in period t. The aggregate price level evolves according to Pt = Z 1 1 Pt 1 (i) di + (1 1 )(Pt ) S(t) = (Pt 1 )1 + (1 )(Pt )1 1 1 1 where the second equality follows from the fact that the distribution of prices among …rms not adjusting in period t corresponds to the distribution of e¤ective prices in period t 1, with total mass reduced to . Equivalently, dividing both sides by Pt 1 : 1 t = + (1 Pt Pt 1 ) 1 (20) where t PPt t 1 . Notice that in a steady state with zero in‡ation Pt = Pt 1 : Log-linearization around a zero in‡ation ( = 1) steady state implies: t = (1 )(pt (21) pt 1 ) Price Dispersion From the de…nition of the price index: 1 = Z 1 0 = Z Pt (i) Pt 1 " di 1 expf(1 0 ' 1 + (1 ) )(pt (i) Z pt )gdi 1 (pt (i) pt )di + )2 (1 2 0 Z 1 (pt (i) 0 thus implying the second order approximation pt ' Ei fpt (i)g + (1 ) 2 Z 0 1 (pt (i) pt )2 di pt )2 di where Ei fpt (i)g In addition, R1 pt (i) di is the cross-sectional mean of (log) prices. 0 Z 0 1 Pt (i) Pt 1 di = Z 1 exp 1 Z 1 0 ' 1 1 0 1 (1 ' 1+ 2 1 1 = 1+ 2 1 1 ' 1+ 2 1 where 1 1 + ) (pt (i) 2 1 pt )di + 2 (pt (i) Z pt ) di 1 1 1 pt )2 di + 2 (pt (i) 0 Z 1 1 (pt (i) Z 1 (pt (i) 0 2 1 Z pt )2 di 1 (pt (i) 0 pt )2 di 0 1 vari fpt (i)g > 1 , and where the last equality follows from the observation that, up to second order, Z 1 Z 1 2 (pt (i) pt ) di ' (pt (i) Ei fpt (i)g)2 di 0 0 vari fpt (i)g Finally, using the de…nition of dt we obtain dt (1 ) log Z 0 1 Pt (i) Pt 1 di ' 1 2 vari fpt (i)g pt )2 di