The Basic New Keynesian Model. Jordi Galí. June 2008

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1 The Basic New Keynesian Model by Jordi Galí June 28

2 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

3 Figure. Estimated Dynamic Response to a Monetary Policy Shock Federal funds rate GDP GDP deflator M2 Source: Christiano, Eichenbaum and Evans (999)

4

5 Households Representative household solves where subject to Z max E C t X t= Z t U (C t ; N t ) C t (i) di P t (i) C t (i) di + Q t B t B t + W t N t T t for t = ; ; 2; ::: plus solvency constraint.

6 Optimality conditions. Optimal allocation of expenditures Pt (i) C t (i) = C t implying where Z P t 2. Other optimality conditions P t P t (i) C t (i) di = P t C t Z P t (i) U n;t U c;t = W t P t Q t = E t Uc;t+ U c;t di P t P t+

7 Speci cation of utility: U(C t ; N t ) = C t N +' t + ' implied log-linear optimality conditions (aggregate variables) w t p t = c t + ' n t c t = E t fc t+ g (i t E t f t+ g ) where i t log Q t is the nominal interest rate and log is the discount rate. Ad-hoc money demand m t p t = y t i t

8 Firms Continuum of rms, indexed by i 2 [; ] Each rm produces a di erentiated good Identical technology Y t (i) = A t N t (i) Probability of resetting price in any given period: across rms (Calvo (983)). 2 [; ] : index of price stickiness Implied average price duration, independent

9 Aggregate Price Dynamics Dividing by P t : P t = (P t ) + ( ) (Pt ) t = + ( ) P t P t Log-linearization around zero in ation steady state t = ( ) (p t p t ) () or, equivalently p t = p t + ( ) p t

10 Optimal Price Setting subject to max P t X k E t Qt;t+k k= for k = ; ; 2; :::where P t Y t+kjt t+k (Y t+kjt ) Y t+kjt = (P t =P t+k ) C t+k Q t;t+k k Optimality condition: X k E t Qt;t+k Y t+kjt k= Ct+k C t where t+kjt t+k (Y t+kjt) and M Pt P t+k P t M t+kjt =

11 Equivalently, X P k E t Q t;t+k Y t t+kjt M MC t+kjt t ;t+k P t k= = where MC t+kjt t+kjt =P t+k and t ;t+k P t+k =P t Perfect Foresight, Zero In ation Steady State: P t P t = ; t ;t+k = ; Y t+kjt = Y ; Q t;t+k = k ; MC = M

12 Log-linearization around zero in ation steady state: X p t p t = ( ) () k E t f cmc t+kjt + p t+k p t g where cmc t+kjt mc t+kjt mc. k= Equivalently, where log. p t = + ( Flexible prices ( = ): X ) () k E t fmc t+kjt + p t+k g k= p t = + mc t + p t =) mc t = (symmetric equilibrium)

13 Particular Case: = (constant returns) =) MC t+kjt = MC t+k Rewriting the optimal price setting rule in recursive form: p t = E t fp t+g + ( ) cmc t + ( )p t (2) Combining () and (2): t = E t f t+ g + cmc t where ( )( )

14 Generalization to 2 (; ) (decreasing returns) De ne mc t (w t p t ) mpn t (w t p t ) (a t y t ) log( ) Using mc t+kjt = (w t+k p t+k ) (a t+k y t+kjt ) log( ), mc t+kjt = mc t+k + (y t+kjt y t+k ) = mc t+k (p t p t+k ) (3) Implied in ation dynamics where t = E t f t+ g + cmc t (4) ( )( ) +

15 Equilibrium Goods markets clearing for all i 2 [; ] and all t. Letting Y t R Y t(i) di, Y t (i) = C t (i) Y t = C t for all t. Combined with the consumer s Euler equation: y t = E t fy t+ g (i t E t f t+ g ) (5)

16 Labor market clearing N t = = = Taking logs, Z Z Yt A t N t (i) di Yt (i) A t Z di Pt (i) P t di ( ) n t = y t a t + d t where d t ( ) log R (P t(i)=p t ) di (second order). Up to a rst order approximation: y t = a t + ( ) n t

17 Marginal Cost and Output Under exible prices mc t = (w t p t ) mpn t = ( y t + ' n t ) (y t n t ) log( ) = + ' + + ' y t a t log( ) (6) mc = + ' + y n t + ' a t log( ) (7) where y where y t ( log( ))( ) =) y n t = y + ya a t +'+( ) > and ya =) cmc t = + ' + y n t ey t is the output gap +' +'+( ). (y t y n t ) (8)

18 New Keynesian Phillips Curve where + '+. Dynamic IS equation where r n t ey t = E t fey t+ g t = E t f t+ g + ey t (9) is the natural rate of interest, given by r n t (i t E t f t+ g r n t ) () + E t fy n t+g = + ya E t fa t+ g Missing block: description of monetary policy (determination of i t ).

19 Equilibrium under a Simple Interest Rate Rule i t = + t + y ey t + v t () where v t is exogenous (possibly stochastic) with zero mean. Equilibrium Dynamics: combining (9), (), and () eyt Et fey = A t+ g T + B t E t f t+ g T (br t n v t ) (2) where A T ; B + ( + y ) T and + y +

20 Uniqueness () A T has both eigenvalues within the unit circle Given and y, (Bullard and Mitra (22)): is necessary and su cient. ( ) + ( ) y >

21 E ects of a Monetary Policy Shock Set br n t = (no real shocks). Let v t follow an AR() process v t = v v t + " v t Calibration: v = :5, = :5, y = :5=4, = :99, = ' =, = 2=3, = 4. Dynamic e ects of an exogenous increase in the nominal rate (Figure ): Exercise: analytical solution

22 Figure 3.: Effects of a Monetary Policy Shock (Interest Rate Rule))

23 E ects of a Technology Shock Set v t = (no monetary shocks). Technology process: a t = a a t + " a t : Implied natural rate: br n t = ya ( a ) a t Dynamic e ects of a technology shock ( a = :9) (Figure 2) Exercise: AR() process for a t

24 Figure 3.2: Effects of a Technology Shock (Interest Rate Rule)

25 Equilibrium under an Exogenous Money Growth Process Money market clearing m t = m m t + " m t (3) b lt = by t bi t (4) = ey t + by n t bi t (5) where l t m t p t denotes (log) real money balances. Substituting (4) into (): ( + ) ey t = E t fey t+ g + b l t + E t f t+ g + br t n by t n (6) Furthermore, we have b lt = b l t + t m t (7)

26 Equilibrium dynamics 2 ey t A M; 4 t b lt where A M; = A M; 4 3 E t fey t+ g E t f t+ g b lt 5 ; A M; B M brn t by n t m t (8) 5 ; B M Uniqueness () A M A M; A M; has two eigenvalues inside and one outside the unit circle.

27 E ects of a Monetary Policy Shock Set br t n = yt n = (no real shocks). Money growth process m t = m m t + " m t (9) where m 2 [; ) Figure 3 (based on m = :5) E ects of a Technology Shock Set m t = (no monetary shocks). Technology process: a t = a a t + " a t : Figure 4 (based on a = :9). Empirical Evidence

28 Figure 3.3: Effects of a Monetary Policy Shock (Money Growth Rule)

29 Figure 3.4: Effects of a Technology Shock (Money Growth Rule)

30 Figure 3.5: Estimated Effects of a Permanent Technology Shock Source: Galí (999)

31 Technical Appendix Optimal Allocation of Consumption Expenditures Maximization of C t for any given expenditure level R P t(i) C t (i) di Z t can be formalized by means of the Lagrangean Z Z L = C t (i) di P t (i) C t (i) di Z t The associated rst order conditions are: for all i 2 [; ]. Thus, for any two goods (i; j) we have: C t (i) Ct = Pt (i) C t (i) = C t (j) Pt (i) P t (j) which can be plugged into the expression for consumption expenditures to yield Pt (i) Z t C t (i) = P t P t for all i 2 [; ]. The latter condition can then be substituted into the de nition of C t, yielding Z P t (i) C t (i) di = P t C t Combining the two previous equations we obtain the demand schedule: Pt (i) C t (i) = P t C t Log-Linearized Euler Equation

32 We can rewrite the Euler equation as = E t fexp(i t c t+ t+ )g (2) In a perfect foresight steady state with constant in ation and constant growth we must have: with the steady state real rate being given by i = + + r i = + A rst order Taylor expansion of exp(i t c t+ t+ ) around that steady state yields: exp(i t c t+ t+ ) ' + (i t i) (c t+ ) ( t+ ) = + i t c t+ t+ which can be used in (2) to obtain, after some rearrangement of terms, the log-linearized Euler equation Aggregate Price Level Dynamics c t = E t fc t+ g (i t E t f t+ g ) Let S(t) [; ] denote the set of rms which do not re-optimize their posted price in period t. The aggregate price level evolves according to P t = Z P t (i) di + ( ) (Pt ) S(t) = (P t ) + ( ) (P t )

33 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, with total mass reduced to. Equivalently, dividing both sides by P t : t = + ( ) where t Pt P t. Notice that in a steady state with zero in ation Pt = P t : Log-linearization around a zero in ation ( = ) steady state implies: P t (2) P t t = ( ) (p t p t ) (22) Price Dispersion From the de nition of the price index: = = Z Z " Pt (i) di P t ' + ( ) thus implying the second order approximation expf( )(p t (i) p t )g di Z p t ' E i fp t (i)g + (p t (i) p t ) di + ( ) 2 Z ( )2 2 Z (p t (i) p t ) 2 di (p t (i) p t ) 2 di

34 where E i fp t (i)g R p t(i) di is the cross-sectional mean of (log) prices. In addition, Z Pt (i) P t Z di = exp (p t(i) p t ) di Z ' (p t (i) p t ) di + 2 Z (p t (i) p t ) 2 di 2 ' + Z ( ) (p t (i) p t ) 2 di + 2 Z (p t (i) p t ) 2 di 2 2 = + Z (p t (i) p t ) 2 di 2 ' + 2 var ifp t (i)g > where, and where the last equality follows from the observation that, up to second order, + Z (p t (i) p t ) 2 di ' Z (p t (i) E i fp t (i)g) 2 di var i fp t (i)g Finally, using the de nition of d t we obtain d t ( ) log Z Pt (i) P t di ' 2 var ifp t (i)g

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