Equilibrium Conditions for the Simple New Keynesian Model

Transcription

1 Equilibrium Conditions for the Simple New Keynesian Model Lawrence J. Christiano August 4, 04

2 Baseline NK model with no capital and with a competitive labor market. private sector equilibrium conditions Details: Use the equilibrium conditions of this model: as a base to introduce financial frictions to illustrate the application of solution methods.

3 Households Problem: max E 0 Â b t t=0! log C t exp (t t ) N+j t, t t = lt + t + # t t j s.t. P t C t + B t+ W t N t + R t B t + Profits net of taxes t First order conditions: = be t C t C t+ exp (t t ) C t N j t = W t P t. R t p t+ (5)

4 Goods Production Ahomogeneousfinalgoodisproducedusingthefollowing (Dixit-Stiglitz) production function: # Z Y t = 0 % Y # # # # i,t dj. Each intermediate good, Y i,t, is produced as follows: Y i,t = =exp(a t ) z} { A t N i,t, a t = ra t + # a t Before discussing the firms that operate these production functions, we briefly investigate the socially e cient ( First Best ) allocation of labor across i, for given N t : N t = Z 0 N it di

5 E cient Sectoral Allocation of Labor With Dixit-Stiglitz final good production function, there is a socially optimal allocation of resources to all the intermediate activities, Y i,t It is optimal to run them all at the same rate, i.e., Y i,t = Y j,t for all i, j [0, ]. For given N t, it is optimal to set N i,t = N j,t for all i, j [0, ] In this case, final output is given by Y t = e a t N t. Best way to see this is to suppose that labor is not allocated equally to all activities. But, this can happen in a million di erent ways when there is a continuum of inputs! Explore one simple deviation from N i,t = N j,t for all i, j [0, ].

6 SupposeLaborNot AllocatedEqually Example: N it N t i 0, N t i,,0. Notethatthisisaparticulardistributionof laboracrossactivities: 0 Nit di N t N t N t

7 LaborNot AllocatedEqually,cnt d Y t Yi,t di 0 0 Y i,t e at 0 di N i,t Yi,t di di Ni,t di e at N t di 0 Nt e at N t di 0 e at N t e at N t f di di

8 f Efficient Resource Allocation Means Equal Labor Across All Sectors f

9 Final Goods Production Final good firms: maximize profits: Z P t Y t P i,t Y i,t dj, 0 subject to: # Z Y t = 0 % # Y # # # i,t dj. Foncs: * + # Pt Y i,t = Y t! P i,t "cross price restrictions" * Z z } { P t = 0 P ( #) i,t di + #

10 Intermediate Goods Production Demand curve for i th monopolist: Production function: * + # Pt Y i,t = Y t. P i,t Y i,t = exp (a t ) N i,t, a t = ra t + # a t Calvo Price-Setting Friction:, P i,t = P t with probability q with probability q Real marginal cost: P i,t. s t = dcost dworker doutput dworker = minimize monopoly distortion by setting = # # z } { ( n) W t P t exp (a t )

11 Optimal Price Setting by Intermediate Goods Producers Let p t P t, p t P t Optimal price setting: where Note: p t = K t F t, P t P t. K t = # # s t + bqe t p t+ # K t+() F t = + bqe t p t+ # t+.() K t = # # s t + bqe t p t+ # # # s t+ + (bq) E t p t+ # # # s t+ +...

12 Goods and Price Equilibrium Conditions Cross-price restrictions imply, given the Calvo price-stickiness: h i P t = ( q) P ( #) t + qp ( #) # t. Dividing latter by P t and solving: " # q p # # t p t =! K t = q F t " # q p t # # q (3) Relationship between aggregate output and aggregate inputs: C t = pt A t N t, (6) where p t = 4( q) q p (# ) t q! # # + q p# t p t 3 5 (4)

13 Tak Yun Distortion: a Closer Look Tak Yun showed (JME): pt = p 5 p t, pt 6 4( q) q p (# )! # # t + q p# t q p t 3 5 Distortion, p t, increasing function of lagged distortion, p t. Current shocks a ect current distortion via p t only. Derivatives: p 5 pt, p t 6 = (p t ) #q p # t 4 p t pt! 3 q p(# ) # t 5 q p! 5 pt, pt 6 p = t pt q p t. #

14 Linear Expansion of Tak Yun Distortion in Undistorted Steady State Linearizing about p t = p, p t = p : dp t = p ( p, p ) d p t + p ( p, p ) dp t, where dx t x t x, for x t = p t, p t, p t. In an undistorted steady state (i.e., p t = p t = p t = ) : so that p (, ) = 0, p (, ) = q. dpt = 0 d p t + qdpt! pt = q + qpt Often, people that linearize NK model ignore p t. Reflects that they linearize the model around a price-undistorted steady state.

15 Current Period Tak Yun Distortion as a Function of Current Inflation Graph conditioned on two alternative values for p* t and θ = 0.75, ε = Tak Yun distortion, p * t lagged Tak Yun distortion = (i.e., no distortion) lagged Tak Yun distortion = π t

16 Ignoring Tak Yun Distortion, a Mistake? Tak Yun Distortion and Quarterly US CPI Inflation (Gross) Tak Yun distortion inflation

17 Linearizing around E cient Steady State In steady state (assuming p =, n = # # ) p =, K = F = bq, s = #, Da = t = 0, N = # Linearizing the Tack Yun distortion, (4): p t =, t large enough Denote the output gap in ratio form by X t : X t A t exp C t 7 t t +j 8 = p t N t exp * + tt, + j where the denominator is the socially e cient ( First Best ) level of consumption.

18 Output Gap and First Best Consumption, Employment Explained above that with socially e cient sectoral allocation of labor, Y t = exp (a t ) N t. First best level of employment and consumption is solution to 9 Nt best = arg max,log [exp (a t ) N] exp (t t ) N+j N + j so, N best t * = exp t t + j + *, C best t = exp a t t + t + j Then, using expression on previous slide for X t (and, x t ˆX t ): x t = ˆN t + dt t +j

19 NK IS Curve, Baseline Model The intertemporal Euler equation, (5), after substituting for C t in terms of X t : where X t A t exp then, use to obtain: 7 t t +j 8 = be t 7 X t+ A t+ exp = E t X t X t+ Rt+ R t p t+, t t+ +j Rt+ * b exp a t+ a t t + t+ t t + j dz t u t = ẑ t + û t, * \ + ut = û t ẑ t ; ˆX t = E t ˆX t+ 5 ˆR t b p t+ ˆR t+ 6= z t 8 R t p t+

20 NK IS Curve, Baseline Model We now want to establish (when the steady state is e cient) the following result: 5 E t ˆR t b p t+ ˆR t+ 6 = r t E t p t+ rt, where Note: r t log R t, r t E t log R t+, p t+ log p t+. Z t = exp (z t ), where z t log Z t Ẑ t dz t Z = d exp (z t) = Zdz t Z Z = dz t. The result follows easily from the following facts in e cient steady state: log R = log R, log p = 0.

21 NK IS Curve, Baseline Model Substituting ; ˆX t = E t ˆX t+ 5 ˆR t b p t+ ˆR t+6=, xt ˆX t, we obtain NK IS curve: x t = E t x t+ E t [r t p t+ rt ] Also, # rt = log (b) + E t a t+ a t t % t+ t t. + j

22 Linearized Marginal Cost in Baseline Model Marginal cost (using da t = a t, dt t = t t because a = t = 0): s t = ( n) w t, w t = exp (t t ) N j t A C t t! b w t = t t + a t + ( + j) ˆN t Then, ŝ t = b w t a t = (j + ) h tt j+ + ˆN t i = (j + ) x t

23 Linearized Phillips Curve in Baseline Model Log-linearize equilibrium conditions, ()-(3), around steady state: Substitute (3) into () ˆF t + ˆK t = ( bq) ŝ t + bq 5 6 #b p t+ + ˆK t+ () ˆF t = bq (# ) b p t+ + bq ˆF t+ () ˆK t = ˆF t + q q b p t (3) q q ˆp t = ( bq) ŝ t + bq * #b p t+ + ˆF t+ + q + q ˆp t+ Simplify the latter using (), to obtain the NK Phillips curve: p t = ( q)( bq) q ŝ t + bp t+

24 The Linearized Private Sector Equilibrium Conditions x t = x t+ [r t p t+ rt ] p t = ( q)( bq) q ŝ t + bp t+ ŝ t = (j + ) x t h i rt = log (b) + E t a t+ a t t t+ t t +j

25 Nonlinear Private Sector Equilibrium Conditions K t = # # s t + bqe t p t+ # K t+() F t = + bqe t p t+ # t+.() K t F t = p t = " q p t # q 4( q) # # R t = be t (5) C t C t+ p t+ C t = pt A t N t. (6) (3) q p (# )! # # t + q p# t q p t 3 5 (4)