SlideShare a Scribd company logo

Lecture on nk [compatibility mode]

N
NBER

1) The document outlines the nonlinear equilibrium conditions of a simple New Keynesian model without capital. It discusses formulating the model's nonlinear equations to study optimal monetary policy and higher-order solutions. 2) It presents the key components of the model, including household and firm behavior assumptions. Households maximize utility from consumption and labor. Firms set prices according to Calvo pricing and maximize profits. 3) The document derives the nonlinear equilibrium conditions that characterize household and firm optimization, including the household's intertemporal FOC and the intermediate firm's price-setting problem. It expresses the model's equilibrium objects like marginal costs and the price index.

BusinessTechnology
1 of 45
Downloaded 54 times
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
Simple New Keynesian Model  without Capital Lawrence J. Christiano
Outline • Formulate the nonlinear equilibrium conditions of the  model. – Need actual nonlinear conditions to study Ramsey‐optimal  policy, even if we want to use linearization methods to  study Ramsey.  • Ramsey will be used to define ‘output gap’ in positive model of the  economy, in which monetary policy is governed by the Taylor rule. • Later, when discussing ‘timeless perspective’, will discuss use of  Ramsey‐optimal policy in actual, real‐time implementation of  monetary policy.   – Need nonlinear equations if we were to study higher order  perturbation solutions. • Study properties of the NK model with Taylor rule,  using Dynare.
Clarida‐Gali‐Gertler Model • Households maximize:  1 E0 ∑ N logC t − exp t  t ,  t   t−1    ,   ~iid, 1 t t t0 • Subject to: Pt C t  B t1 ≤ Wt N t  R t−1 B t  T t • Intratemporal first order condition: C t exp t N t  Wt  Pt
Household Intertemporal FONC • Condition: u c,t1 Rt 1  E t u c,t 1   t1 – or, for when we do linearize later:  1  E t C t Rt C t1 1   t1  E t explogR t  − log1   t1  − Δc t1  ≃  explogR t  − E t  t1 − E t Δc t1 , c t ≡ logC t  – take log of both sides: 0  log  r t − E t  t1 − E t Δc t1 , r t  logR t  – or c t  − log − r t − E t  t1   c t1
Final Good Firms • Buy                     at prices       and sell      for  Pt Yi,t , i ∈ 0, 1 P i,t Yt • Take all prices as given (competitive) • Profits: 1 P t Yt −  P i,t Yi,t di 0 • Production function:   0 Yi,t 1 −1 −1 Yt   di,   1, • First order condition:     P i,t − 1 1− 1 Yi,t  Yt → P t   P i,t di 1− Pt 0
Intermediate Good Firms • Each ith good produced by a single monopoly  producer.  • Demand curve: P i,t − Yi,t  Yt Pt • Technology: Yi,t  expa t N i,t , Δa t  Δa t−1   a , t • Calvo Price‐setting Friction ̃ P t with probability 1 −  P i,t  , P i,t with probability 
Marginal Cost dCost 1 − Wt /P t real marginal cost  st  dwor ker  dOutput expa t  dwor ker  −1 in efficient setting   1 −  C t exp t N t  expa t 
The Intermediate Firm’s Decisions • ith firm is required to satisfy whatever  demand shows up at its posted price. • It’s only real decision is to adjust price  whenever the opportunity arises.
Intermediate Good Firm • Present discounted value of firm profits: period tj profits sent to household  marginal value of dividends to householdu c,tj /P tj revenues total cost  Et ∑ j  tj Pi,tj Yi,tj − P tj s tj Yi,tj j0 1− • Each of the         firms that can optimize price  ̃ choose      to optimize Pt in selecting price, firm only cares about future states in which it can’t reoptimize   Et ∑ j j ̃  tj Pt Yi,tj − Ptj s tj Yi,tj . j0
Intermediate Good Firm Problem • Substitute out the demand curve:  E t ∑ j  tj Pt Yi,tj − P tj s tj Yi,tj  ̃ j0   E t ∑ j  tj Ytj P P 1− − P tj s tj P − . tj ̃t ̃t j0 ̃ • Differentiate with respect to     :  Pt  E t ∑ j  tj Ytj P 1 − P t   Ptj s tj P−−1   0, tj ̃ − ̃t j0 • or  ̃ Pt −  s E t ∑ j  tj Ytj P 1  0. tj P tj  − 1 tj j0
Intermediate Good Firm Problem • Objective:   u ′ C tj  ̃ Pt −  s E t ∑ j Ytj P1  0. P tj tj Ptj  − 1 tj j0  ̃ Pt −  s → E t ∑ j P  0. tj P tj  − 1 tj j0 • or   E t ∑ j X t,j  − p t X t,j − ̃  s tj  0, −1 j0 ̃ Pt , X  1  tj  tj−1  t1 ̄ ̄ ̄ ,j≥1 ̃ pt  , X t,j  X t1,j−1 1 , j  0 Pt t,j 1, j  0.  t1 ̄
Intermediate Good Firm Problem ̃ • Want       in: pt  E t ∑ j0  j X t,j  − p t X t,j − ̃  −1 s tj 0 • Solution:  E t ∑ j0  j X t,j  −  s −1 tj ̃ pt    Kt Et ∑ j0  j X t,j  1− Ft • But, still need expressions for  Kt , Ft .
 Kt  E t ∑ j X t,j  −  s tj −1 j0  −   s t  E t ∑ j−1 1 X t1,j−1  s tj −1  t1 ̄ −1 j1  −   s t  E t 1 ∑ j X −  s −1  t1 ̄ t1,j  − 1 t1j j0 E t by LIME  −   s t   E t E 1 ∑ j X −  s −1 t1  t1 ̄ t1,j  − 1 t1j j0 exactly K t1 !  −   s t  E t 1 E t1 ∑ j X −  s −1  t1 ̄ t1,j  − 1 t1j j0 −   s t  E t 1 Kt1 −1  t1 ̄
• From previous slide: − Kt   s t  E t 1 Kt1 . −1  t1 ̄ • Substituting out for marginal cost: dCost/dlabor  s t   1 −  Wt /P t −1 −1 dOutput/dlabor expa t  Wt  Pt by household optimization   1 −  exp t N t C t  . −1 expa t 
In Sum • solution:  E t ∑ j0  j X t,j  −  s −1 tj ̃ pt    Kt , E t ∑ j0  j X t,j  1− Ft • Where:  exp t N t C t − Kt  1 −  t    E t 1 Kt1 . −1 expa t   t1 ̄  1− F t ≡ E t ∑ X t,j  j 1−  1  E t 1 F t1  t1 ̄ j0
To Characterize Equilibrium • Have equations characterizing optimization by  firms and households. • Still need: P i,t , 0 ≤ i ≤ 1 – Expression for all the prices. Prices,                          ,  will all be different because of the price setting  frictions. – Relationship between aggregate employment and  aggregate output not simple because of price  distortions: Y t ≠ e a t N t , in general
Going for Prices • Aggregate price relationship Calvo insight: This is just a simple 1 function of last period’s 0 P 1− di 1 1− Pt  i,t aggregate price because non-optimizers chosen at random. 1 firms that reoptimize price P 1− di  firms that don’t reoptimize price P 1− di 1−  i,t i,t all reoptimizers choose same price 1   ̃ 1−   1 − P t P i,t 1− di 1− firms that don’t reoptimize price • In principle, to solve the model need all the  prices, P t , P i,t , 0 ≤ i ≤ 1 – Fortunately, that won’t be necessary. 
̃ Expression for     in terms of aggregate  pt inflation  • Conclude that this relationship holds between  prices: 1 P t  1 − ̃ 1− P t  1− P t−1 1− . – Only two variables here! • Divide by      : Pt 1 1− 1− 1 ̃ 1 − p t  1 1− t ̄ • Rearrange: 1 −1 1−  t ̄ 1− ̃ pt  1−
Relation Between Aggregate  Output and Aggregate Inputs • Technically, there is no ‘aggregate production  function’ in this model – If you know how many people are working, N, and  the state of technology, a, you don’t have enough  information to know what Y is. – Price frictions imply that resources will not be  efficiently allocated among different inputs. • Implies Y low for given a and N. How low? • Tak Yun (JME) gave a simple answer.
Tak Yun Algebra labor market clearing Y∗  0 Yi,t di 1  0 At N i,t di 1   At N t t demand curve −   Yt  1 P i,t di 0 Pt 0 P i,t  −di 1  Yt P  t Calvo insight  Y t P  P ∗  − t t −1 • Where: P∗ t ≡ 0 1 P − di i,t  ̃t  1 − P −  P ∗  −  t−1 −1 
Relationship Between Agg Inputs  and Agg Output • Rewriting previous equation:  P∗ Yt  t Y∗ t Pt  p ∗ e at N t , t • ‘efficiency distortion’: ≤1 p∗ t :  1 P i,t  P j,t , all i, j
Collecting Equilibrium Conditions • Price setting:  K t  1 −   exp t N t C t  E t   K (1) ̄ t1 t1 −1 At F t  1  E t  −1 F t1 (2) ̄ t1 • Intermediate good firm optimality and  restriction across prices: ̃ p t by restriction across prices ̃ p t by firm optimality  −1 1 Kt 1−  t ̄ 1−  (3) Ft 1−
Equilibrium Conditions • Law of motion of (Tak Yun) distortion:  −1 −1 1−  t ̄ −1   ̄t p∗  1 −   ∗ (4) t 1− p t−1 • Household Intertemporal Condition: 1  E t 1 R t (5) Ct C t1  t1 ̄ • Aggregate inputs and output: C t  p ∗ e a t N t (6) t • 6 equations, 8 unknowns: , C t , p ∗ , N t ,  t , K t , F t , R t t ̄ • System under determined!
Underdetermined System • Not surprising: we added a variable, the  nominal rate of interest. • Also, we’re counting subsidy as among the  unknowns. • Have two extra policy variables. • One way to pin them down: compute optimal  policy.
Ramsey‐Optimal Policy • 6 equations in 8 unknowns….. – Many configurations of the 8 unknowns that  satisfy the 6 equations. – Look for the best configurations (Ramsey optimal) • Value of tax subsidy and of R represent optimal policy • Finding the Ramsey optimal setting of the 6  variables involves solving a simple Lagrangian  optimization problem.
Ramsey Problem  N 1 max E 0 ∑  t  logC t − exp t  t ,p ∗ ,C t ,N t ,Rt , t ,F t ,K t t ̄ 1 t0   1t 1 − Et  Rt Ct C t1  t1 ̄  −1 1 − 1 −  t  ̄ −1   ̄   2t 1 −   ∗t p∗ t 1− p t−1   3t 1  E t  −1 F t1 − F t  ̄ t1 C t exp t N    4t 1 −   t  E t   K t1 − K t ̄ t1 −1 e at 1 1−  −1 ̄t 1−   5t F t − Kt 1−   6t C t − p ∗ e a t N t  t
Solving the Ramsey Problem  (surprisingly easy in this case) • First, substitute out consumption everywhere  N 1 max E 0 ∑  t  logN t  logp ∗ − exp t  t ,p ∗ ,N t ,Rt , t ,F t ,K t t ̄ t 1 t0 defines R 1 − Et e at  Rt   1t ∗ pt Nt p t1 e N t1  t1 ∗ a t1 ̄  −1 1 − 1 −  t  ̄ −1   ̄   2t 1 −   ∗t p∗ t 1− p t−1 defines F ̄ −1   3t 1  E t  t1 F t1 − F t  defines tax   4t 1 −   exp t N 1 p ∗  E t   K t1 − K t ̄ t1 −1 t t 1 1−  −1 ̄t 1−   5t F t − Kt  defines K 1−
Solving the Ramsey Problem, cnt’d  • Simplified problem:  N 1 max E 0 ∑   logN t t logp ∗ − exp t  t  t ,p ∗ ,N t ̄ t t 1 t0  −1 1 − 1 −  t  ̄ −1   ̄   2t 1 −   ∗t  p∗ t 1− p t−1 • First order conditions with respect to p ∗ ,  t , N t t ̄ 1 p ∗  −1 −1 p ∗   2,t1     2t ,  t  ̄ t1 ̄ t−1 , N t  exp −  t t 1−   p ∗  −1 t−1 1 • Substituting the solution for inflation into law  of motion for price distortion:  1 p∗ t  1 −   p ∗  −1 t−1 −1 .
Solution to Ramsey Problem Eventually, price distortions eliminated, regardless of shocks 1 p ∗  1 −   p t−1  −1 t ∗ −1 When price distortions p∗ gone, so is inflation.  t  t−1 ̄ p∗t N t  exp − t Efficient (‘first best’) 1 allocations in real economy 1−  −1  C t  p ∗ e at N t . t Consumption corresponds to efficient allocations in real economy, eventually when price distortions gone
Eventually, Optimal (Ramsey) Equilibrium and  Efficient Allocations in Real Economy Coincide Convergence of price distortion 0.98 0.96 0.94   0. 75,   10 0.92 p-star 0.9 1 0.88 p∗ t  1 −   p ∗  −1 t−1 −1 0.86 0.84 0.82 0.8 2 4 6 8 10 12 14 16 18 20
• The Ramsey allocations are eventually the  best allocations in the economy without price  frictions (i.e., ‘first best allocations’) • Refer to the Ramsey allocations as the ‘natural  allocations’…. – Natural consumption, natural rate of interest, etc.
Equations of the NK Model Under the  Optimal Policy (‘Natural Equilibrium’) • Output and employment is (eventually) y∗  at − 1 t , n∗  − 1 t t 1 t 1 • Intertemporal Euler equation after taking logs  and ignoring variance adjustment term: y ∗  −r ∗ − E t  ∗ − rr  E t y ∗ , rr  − log t t t1 t1 • Inflation in Ramsey equilibrium is (eventually)  zero.
Solving for Natural Rate of Interest • Intertemporal euler equation in natural  equilibrium: ∗ y∗ yt t1 at − 1 1  t  −r ∗ − rr  E t t a t1 − 1 1  t1 • Back out the natural rate: r ∗  rr  Δa t  t 1 1 1 −  t • Shocks:  t   t−1    , Δa t  Δa t−1   t t
Next, Put Turn to the NK Model  with Taylor Rule
Taylor Rule • Taylor rule: designed, so that in steady state,  1 ̄ inflation is zero (             ) • Employment subsidy extinguishes monopoly  power in steady state:  1 −    1 −1
NK IS Curve • Euler equation in two equilibria: Taylor rule equilibrium: y t  −r t − E t  t1 − rr  E t y t1 Natural equilibrium: y ∗  −r ∗ − rr  E t y ∗ t t t1 • Subtract: Output gap x t  −r t − E t  t1 − r ∗   E t x t1 t
Output in NK Equilibrium • Agg output relation:  0 if P i,t  P j,t for all i, j yt  logp ∗ t  nt  at , logp ∗ t  . ≤0 otherwise • To first order approximation,  ̂t ̂ t−1 p ∗ ≈ p ∗  0   t , → p ∗ ≈ 1 ̄ t
Price Setting Equations • Log‐linearly expand the price setting equations  about steady state. 1 1− −1 ̄t − Kt  0 1− 1 ̄ −1 E t  t1 F t1 − Ft  0 Ft 1−   C t exp t N t 1 −  −1 eat ̄  E t  t1 K t1 − K t  0 • Log‐linearly expand about steady state:  1−1−  t  ̄  1  x t   t1 ̄ • See http://faculty.wcas.northwestern.edu/~lchrist/course/solving_handout.pdf
Taylor Rule • Policy rule r t  r t−1  1 − rr     t   x x t   u t , , x t ≡ y t − y ∗ t
Equations of Actual Equilibrium Closed by Adding Policy Rule  E t  t1  x t −  t  0 (Phillips curve) − r t − E t  t1 − r ∗   E t x t1 − x t  0 (IS equation) t r t−1  1 −    t  1 −  x x t − r t  0 (policy rule) r ∗ − Δa t − 1 1 −  t  0 (definition of natural rate) t 1
Solving the Model Δa t  0 Δa t−1 t st    t 0   t−1  t s t  Ps t−1   t  0 0 0  t1 −1  0 0 t 1  1 0 0 x t1 0 −1 − 1 1  xt  0 0 0 0 r t1 1 −   1 −  x −1 0 rt 0 0 0 0 r∗ t1 0 0 0 1 rr ∗ t 0 0 0 0  t−1 0 0 0 0 0 0 0 0 0 x t−1 0 0 0 0 0   s t1  st 0 0  0 r t−1 0 0 0 0 0 0 0 0 0 r∗ t−1 0 0 0 − −  1 −  1 E t  0 z t1   1 z t   2 z t−1   0 s t1   1 s t   0
Solving the Model E t  0 z t1   1 z t   2 z t−1   0 s t1   1 s t   0 s t − Ps t−1 −  t  0. • Solution: z t  Az t−1  Bs t • As before:  0 A2   1 A   2 I  0, F   0   0 BP   1   0 A   1 B  0
 x  0,    1. 5,   0. 99,   1,   0. 2,   0. 75,   0,   0. 2,   0. 5. Dynamic Response to a Technology Shock inflation output gap nominal rate 0.2 0.15 0.03 0.15 natural nominal rate 0.1 actual nominal rate 0.02 0.1 0.01 0.05 0.05 0 2 4 6 0 2 4 6 0 2 4 6 natural real rate log, technology output 0.2 0.15 1 1.2 natural output 1.15 0.1 actual output 0.5 1.1 0.05 1.05 0 1 0 2 4 6 0 2 4 6 0 2 4 6 employment 0.2 0.15 0.1 natural employment 0.05 actual employment 0 0 5 10
Dynamic Response to a Preference Shock inflation output gap nominal rate 0.1 0.25 0.08 0.25 0.2 0.2 natural real rate 0.06 0.15 0.15 actual nominal rate 0.04 0.1 0.1 0.02 0.05 0.05 0 2 4 6 0 2 4 6 0 2 4 6 natural real rate preference shock output 0.25 1 0.2 -0.1 0.15 -0.2 0.5 0.1 -0.3 natural output actual output 0.05 -0.4 0 -0.5 0 2 4 6 0 2 4 6 0 2 4 6 employment -0.1 -0.2 natural employment actual employment -0.3 -0.4 -0.5 0 2 4 6
Conclusion of NK Model Analysis • We studied examples in which the Taylor rule  moves the interest rate in the right direction  in response to shocks. • However, the move is not strong enough. Will  consider modifications of the Taylor rule using  Dynare.

More Related Content

PDF
Optimalpolicyhandout
NBER
 
PDF
Chapter 5 heterogeneous
NBER
 
PDF
Chapter 1 nonlinear
NBER
 
PDF
Chapter 4 likelihood
NBER
 
PDF
Label propagation - Semisupervised Learning with Applications to NLP
David Przybilla
 
PDF
Paris2012 session2
Cdiscount
 
PDF
State Space Model
Cdiscount
 
PDF
Ada boost brown boost performance with noisy data
Shadhin Rahman
 
Optimalpolicyhandout
NBER
 
Chapter 5 heterogeneous
NBER
 
Chapter 1 nonlinear
NBER
 
Chapter 4 likelihood
NBER
 
Label propagation - Semisupervised Learning with Applications to NLP
David Przybilla
 
Paris2012 session2
Cdiscount
 
State Space Model
Cdiscount
 
Ada boost brown boost performance with noisy data
Shadhin Rahman
 

What's hot (18)

PDF
100 things I know
r-uribe
 
PDF
Parameter Estimation in Stochastic Differential Equations by Continuous Optim...
SSA KPI
 
PDF
Datamining 4th Adaboost
sesejun
 
PDF
Portfolios and Risk Premia for the Long Run
guasoni
 
DOC
Neural network Algos formulas
Zarnigar Altaf
 
PPT
Fdtd
physics Imposible
 
PDF
Paris2012 session1
Cdiscount
 
PDF
UT Austin - Portugal Lectures on Portfolio Choice
guasoni
 
PDF
Optimal control of coupled PDE networks with automated code generation
Delta Pi Systems
 
PDF
Paris2012 session3b
Cdiscount
 
PDF
Olivier Cappé's talk at BigMC March 2011
BigMC
 
KEY
Unit Circle - Trigonometry
Simon Borgert
 
PDF
Numerical solution of spatiotemporal models from ecology
Kyrre Wahl Kongsgård
 
PDF
WCSB 2012
Colin Gillespie
 
KEY
Trig identities
Simon Borgert
 
PDF
Biomathematics
Francis Mponda
 
PPT
Final Present Pap1on relibility
ketan gajjar
 
PDF
Boris Blagov. Financial Crises and Time-Varying Risk Premia in a Small Open E...
Eesti Pank
 
100 things I know
r-uribe
 
Parameter Estimation in Stochastic Differential Equations by Continuous Optim...
SSA KPI
 
Datamining 4th Adaboost
sesejun
 
Portfolios and Risk Premia for the Long Run
guasoni
 
Neural network Algos formulas
Zarnigar Altaf
 
Fdtd
physics Imposible
 
Paris2012 session1
Cdiscount
 
UT Austin - Portugal Lectures on Portfolio Choice
guasoni
 
Optimal control of coupled PDE networks with automated code generation
Delta Pi Systems
 
Paris2012 session3b
Cdiscount
 
Olivier Cappé's talk at BigMC March 2011
BigMC
 
Unit Circle - Trigonometry
Simon Borgert
 
Numerical solution of spatiotemporal models from ecology
Kyrre Wahl Kongsgård
 
WCSB 2012
Colin Gillespie
 
Trig identities
Simon Borgert
 
Biomathematics
Francis Mponda
 
Final Present Pap1on relibility
ketan gajjar
 
Boris Blagov. Financial Crises and Time-Varying Risk Premia in a Small Open E...
Eesti Pank
 
Ad

Viewers also liked (10)

PDF
Chapter 2 pertubation
NBER
 
PDF
Lecture on solving1
NBER
 
PDF
Chapter 3 projection
NBER
 
PDF
Csvfrictions
NBER
 
PDF
Dynare exercise
NBER
 
PDF
Applications: Prediction
NBER
 
PDF
Econometrics of High-Dimensional Sparse Models
NBER
 
PDF
Big Data Analysis
NBER
 
PDF
High-Dimensional Methods: Examples for Inference on Structural Effects
NBER
 
PDF
Nuts and bolts
NBER
 
Chapter 2 pertubation
NBER
 
Lecture on solving1
NBER
 
Chapter 3 projection
NBER
 
Csvfrictions
NBER
 
Dynare exercise
NBER
 
Applications: Prediction
NBER
 
Econometrics of High-Dimensional Sparse Models
NBER
 
Big Data Analysis
NBER
 
High-Dimensional Methods: Examples for Inference on Structural Effects
NBER
 
Nuts and bolts
NBER
 
Ad

Similar to Lecture on nk [compatibility mode] (20)

PDF
A NON LINEAR MODEL OF THE NEW ECONOMIC GEOGRAPHY
Vítor João Pereira Domingues Martinho
 
PDF
Slides
bushman325
 
PDF
Exchange confirm
NBER
 
PDF
Research Inventy : International Journal of Engineering and Science
researchinventy
 
PDF
THE IMPORTANCE OF INCREASING RETURNS TO SCALE IN THE PROCESS OF AGGLOMERATION...
Vítor João Pereira Domingues Martinho
 
PDF
Monte-Carlo method for Two-Stage SLP
SSA KPI
 
PDF
Macroeconometrics of Investment and the User Cost of Capital Presentation Sample
Thethach Chuaprapaisilp
 
PDF
Kt2418201822
IJERA Editor
 
PDF
On problem-of-parameters-identification-of-dynamic-object
Cemal Ardil
 
PDF
Reinforcement Learning
Jungyeol
 
PDF
IJCER (www.ijceronline.com) International Journal of computational Engineerin...
ijceronline
 
PDF
Real time information reconstruction -The Prediction Market
raghavr186
 
PDF
20 sameer sharma 186-206
Alexander Decker
 
PDF
Common and private ownership of exhaustible resources: theoretical implicat...
alexandersurkov
 
PDF
Markov Tutorial CDC Shanghai 2009
Sean Meyn
 
DOC
225.doc
butest
 
PPTX
Systems Of Differential Equations
JDagenais
 
PDF
Interest rate derivatives
Springer
 
PDF
Elett02
Elyas Khan
 
PDF
Welcome to International Journal of Engineering Research and Development (IJERD)
IJERD Editor
 
A NON LINEAR MODEL OF THE NEW ECONOMIC GEOGRAPHY
Vítor João Pereira Domingues Martinho
 
Slides
bushman325
 
Exchange confirm
NBER
 
Research Inventy : International Journal of Engineering and Science
researchinventy
 
THE IMPORTANCE OF INCREASING RETURNS TO SCALE IN THE PROCESS OF AGGLOMERATION...
Vítor João Pereira Domingues Martinho
 
Monte-Carlo method for Two-Stage SLP
SSA KPI
 
Macroeconometrics of Investment and the User Cost of Capital Presentation Sample
Thethach Chuaprapaisilp
 
Kt2418201822
IJERA Editor
 
On problem-of-parameters-identification-of-dynamic-object
Cemal Ardil
 
Reinforcement Learning
Jungyeol
 
IJCER (www.ijceronline.com) International Journal of computational Engineerin...
ijceronline
 
Real time information reconstruction -The Prediction Market
raghavr186
 
20 sameer sharma 186-206
Alexander Decker
 
Common and private ownership of exhaustible resources: theoretical implicat...
alexandersurkov
 
Markov Tutorial CDC Shanghai 2009
Sean Meyn
 
225.doc
butest
 
Systems Of Differential Equations
JDagenais
 
Interest rate derivatives
Springer
 
Elett02
Elyas Khan
 
Welcome to International Journal of Engineering Research and Development (IJERD)
IJERD Editor
 

More from NBER (20)

PDF
FISCAL STIMULUS IN ECONOMIC UNIONS: WHAT ROLE FOR STATES
NBER
 
PDF
Business in the United States Who Owns it and How Much Tax They Pay
NBER
 
PDF
Redistribution through Minimum Wage Regulation: An Analysis of Program Linkag...
NBER
 
PDF
The Distributional E ffects of U.S. Clean Energy Tax Credits
NBER
 
PDF
An Experimental Evaluation of Strategies to Increase Property Tax Compliance:...
NBER
 
PDF
Nbe rtopicsandrecomvlecture1
NBER
 
PDF
Nbe rcausalpredictionv111 lecture2
NBER
 
PDF
Nber slides11 lecture2
NBER
 
PPTX
Recommenders, Topics, and Text
NBER
 
PPTX
Machine Learning and Causal Inference
NBER
 
PDF
Introduction to Supervised ML Concepts and Algorithms
NBER
 
PDF
Jackson nber-slides2014 lecture3
NBER
 
PDF
Jackson nber-slides2014 lecture1
NBER
 
PDF
Acemoglu lecture2
NBER
 
PDF
Acemoglu lecture4
NBER
 
PPTX
The NBER Working Paper Series at 20,000 - Joshua Gans
NBER
 
PPTX
The NBER Working Paper Series at 20,000 - Claudia Goldin
NBER
 
PPT
The NBER Working Paper Series at 20,000 - James Poterba
NBER
 
PPTX
The NBER Working Paper Series at 20,000 - Scott Stern
NBER
 
PDF
The NBER Working Paper Series at 20,000 - Glenn Ellison
NBER
 
FISCAL STIMULUS IN ECONOMIC UNIONS: WHAT ROLE FOR STATES
NBER
 
Business in the United States Who Owns it and How Much Tax They Pay
NBER
 
Redistribution through Minimum Wage Regulation: An Analysis of Program Linkag...
NBER
 
The Distributional E ffects of U.S. Clean Energy Tax Credits
NBER
 
An Experimental Evaluation of Strategies to Increase Property Tax Compliance:...
NBER
 
Nbe rtopicsandrecomvlecture1
NBER
 
Nbe rcausalpredictionv111 lecture2
NBER
 
Nber slides11 lecture2
NBER
 
Recommenders, Topics, and Text
NBER
 
Machine Learning and Causal Inference
NBER
 
Introduction to Supervised ML Concepts and Algorithms
NBER
 
Jackson nber-slides2014 lecture3
NBER
 
Jackson nber-slides2014 lecture1
NBER
 
Acemoglu lecture2
NBER
 
Acemoglu lecture4
NBER
 
The NBER Working Paper Series at 20,000 - Joshua Gans
NBER
 
The NBER Working Paper Series at 20,000 - Claudia Goldin
NBER
 
The NBER Working Paper Series at 20,000 - James Poterba
NBER
 
The NBER Working Paper Series at 20,000 - Scott Stern
NBER
 
The NBER Working Paper Series at 20,000 - Glenn Ellison
NBER
 

Recently uploaded (20)

PDF
COST SHEET- Tender and Quotation unit 2.pdf
MANJU N
 
PPTX
Dragon_Fruit_Cultivation_in Nepal ppt.pptx
UmeshTimilsina1
 
PPTX
job Avenue by vinith.pptxvnbvnvnvbnvbnbmnbmbh
kaniece
 
PPTX
ICG2025_ICG 6th steering committee 30-8-24.pptx
AtiarRahaman5
 
PDF
Reconciliation AND MEMORANDUM RECONCILATION
MANJU N
 
PDF
Laughter Yoga Basic Learning Workshop Manual
yeechensee
 
PPT
340036916-American-Literature-Literary-Period-Overview.ppt
carinaherdiles611
 
PPTX
The Marketing Journey - Tracey Phillips - Marketing Matters 7-2025.pptx
shannonzipoy2
 
PPTX
AI-assistance in Knowledge Collection and Curation supporting Safe and Sustai...
Barry Hardy
 
PDF
Ôn tập tiếng anh trong kinh doanh nâng cao
thaoonguyenn2311
 
PDF
Elevate Cleaning Efficiency Using Tallfly Hair Remover Roller Factory Expertise
farsookmon97766765
 
PPTX
HR Introduction Slide (1).pptx on hr intro
suhanidwivedi227
 
PDF
Nidhal Samdaie CV - International Business Consultant
Nidhal Samdaie
 
PDF
pdfcoffee.com-opt-b1plus-sb-answers.pdfvi
thaoonguyenn2311
 
PDF
IFRS Notes in your pocket for study all the time
jjj81507
 
PDF
How to Get Business Funding for Small Business Fast
Jake Thornhill
 
PDF
SIMNET Inc – 2023’s Most Trusted IT Services & Solution Provider
sidnik500
 
PDF
Power and position in leadershipDOC-20250808-WA0011..pdf
meghavinikumar2006
 
DOCX
unit 2 cost accounting- Tender and Quotation & Reconciliation Statement
MANJU N
 
PDF
A Brief Introduction About Julia Allison
Julia Allison
 
COST SHEET- Tender and Quotation unit 2.pdf
MANJU N
 
Dragon_Fruit_Cultivation_in Nepal ppt.pptx
UmeshTimilsina1
 
job Avenue by vinith.pptxvnbvnvnvbnvbnbmnbmbh
kaniece
 
ICG2025_ICG 6th steering committee 30-8-24.pptx
AtiarRahaman5
 
Reconciliation AND MEMORANDUM RECONCILATION
MANJU N
 
Laughter Yoga Basic Learning Workshop Manual
yeechensee
 
340036916-American-Literature-Literary-Period-Overview.ppt
carinaherdiles611
 
The Marketing Journey - Tracey Phillips - Marketing Matters 7-2025.pptx
shannonzipoy2
 
AI-assistance in Knowledge Collection and Curation supporting Safe and Sustai...
Barry Hardy
 
Ôn tập tiếng anh trong kinh doanh nâng cao
thaoonguyenn2311
 
Elevate Cleaning Efficiency Using Tallfly Hair Remover Roller Factory Expertise
farsookmon97766765
 
HR Introduction Slide (1).pptx on hr intro
suhanidwivedi227
 
Nidhal Samdaie CV - International Business Consultant
Nidhal Samdaie
 
pdfcoffee.com-opt-b1plus-sb-answers.pdfvi
thaoonguyenn2311
 
IFRS Notes in your pocket for study all the time
jjj81507
 
How to Get Business Funding for Small Business Fast
Jake Thornhill
 
SIMNET Inc – 2023’s Most Trusted IT Services & Solution Provider
sidnik500
 
Power and position in leadershipDOC-20250808-WA0011..pdf
meghavinikumar2006
 
unit 2 cost accounting- Tender and Quotation & Reconciliation Statement
MANJU N
 
A Brief Introduction About Julia Allison
Julia Allison
 

Lecture on nk [compatibility mode]

  • 1. Simple New Keynesian Model  without Capital Lawrence J. Christiano
  • 2. Outline • Formulate the nonlinear equilibrium conditions of the  model. – Need actual nonlinear conditions to study Ramsey‐optimal  policy, even if we want to use linearization methods to  study Ramsey.  • Ramsey will be used to define ‘output gap’ in positive model of the  economy, in which monetary policy is governed by the Taylor rule. • Later, when discussing ‘timeless perspective’, will discuss use of  Ramsey‐optimal policy in actual, real‐time implementation of  monetary policy.   – Need nonlinear equations if we were to study higher order  perturbation solutions. • Study properties of the NK model with Taylor rule,  using Dynare.
  • 3. Clarida‐Gali‐Gertler Model • Households maximize:  1 E0 ∑ N logC t − exp t  t ,  t   t−1    ,   ~iid, 1 t t t0 • Subject to: Pt C t  B t1 ≤ Wt N t  R t−1 B t  T t • Intratemporal first order condition: C t exp t N t  Wt  Pt
  • 4. Household Intertemporal FONC • Condition: u c,t1 Rt 1  E t u c,t 1   t1 – or, for when we do linearize later:  1  E t C t Rt C t1 1   t1  E t explogR t  − log1   t1  − Δc t1  ≃  explogR t  − E t  t1 − E t Δc t1 , c t ≡ logC t  – take log of both sides: 0  log  r t − E t  t1 − E t Δc t1 , r t  logR t  – or c t  − log − r t − E t  t1   c t1
  • 5. Final Good Firms • Buy                     at prices       and sell      for  Pt Yi,t , i ∈ 0, 1 P i,t Yt • Take all prices as given (competitive) • Profits: 1 P t Yt −  P i,t Yi,t di 0 • Production function:   0 Yi,t 1 −1 −1 Yt   di,   1, • First order condition:     P i,t − 1 1− 1 Yi,t  Yt → P t   P i,t di 1− Pt 0
  • 6. Intermediate Good Firms • Each ith good produced by a single monopoly  producer.  • Demand curve: P i,t − Yi,t  Yt Pt • Technology: Yi,t  expa t N i,t , Δa t  Δa t−1   a , t • Calvo Price‐setting Friction ̃ P t with probability 1 −  P i,t  , P i,t with probability 
  • 7. Marginal Cost dCost 1 − Wt /P t real marginal cost  st  dwor ker  dOutput expa t  dwor ker  −1 in efficient setting   1 −  C t exp t N t  expa t 
  • 8. The Intermediate Firm’s Decisions • ith firm is required to satisfy whatever  demand shows up at its posted price. • It’s only real decision is to adjust price  whenever the opportunity arises.
  • 9. Intermediate Good Firm • Present discounted value of firm profits: period tj profits sent to household  marginal value of dividends to householdu c,tj /P tj revenues total cost  Et ∑ j  tj Pi,tj Yi,tj − P tj s tj Yi,tj j0 1− • Each of the         firms that can optimize price  ̃ choose      to optimize Pt in selecting price, firm only cares about future states in which it can’t reoptimize   Et ∑ j j ̃  tj Pt Yi,tj − Ptj s tj Yi,tj . j0
  • 10. Intermediate Good Firm Problem • Substitute out the demand curve:  E t ∑ j  tj Pt Yi,tj − P tj s tj Yi,tj  ̃ j0   E t ∑ j  tj Ytj P P 1− − P tj s tj P − . tj ̃t ̃t j0 ̃ • Differentiate with respect to     :  Pt  E t ∑ j  tj Ytj P 1 − P t   Ptj s tj P−−1   0, tj ̃ − ̃t j0 • or  ̃ Pt −  s E t ∑ j  tj Ytj P 1  0. tj P tj  − 1 tj j0
  • 11. Intermediate Good Firm Problem • Objective:   u ′ C tj  ̃ Pt −  s E t ∑ j Ytj P1  0. P tj tj Ptj  − 1 tj j0  ̃ Pt −  s → E t ∑ j P  0. tj P tj  − 1 tj j0 • or   E t ∑ j X t,j  − p t X t,j − ̃  s tj  0, −1 j0 ̃ Pt , X  1  tj  tj−1  t1 ̄ ̄ ̄ ,j≥1 ̃ pt  , X t,j  X t1,j−1 1 , j  0 Pt t,j 1, j  0.  t1 ̄
  • 12. Intermediate Good Firm Problem ̃ • Want       in: pt  E t ∑ j0  j X t,j  − p t X t,j − ̃  −1 s tj 0 • Solution:  E t ∑ j0  j X t,j  −  s −1 tj ̃ pt    Kt Et ∑ j0  j X t,j  1− Ft • But, still need expressions for  Kt , Ft .
  • 13.  Kt  E t ∑ j X t,j  −  s tj −1 j0  −   s t  E t ∑ j−1 1 X t1,j−1  s tj −1  t1 ̄ −1 j1  −   s t  E t 1 ∑ j X −  s −1  t1 ̄ t1,j  − 1 t1j j0 E t by LIME  −   s t   E t E 1 ∑ j X −  s −1 t1  t1 ̄ t1,j  − 1 t1j j0 exactly K t1 !  −   s t  E t 1 E t1 ∑ j X −  s −1  t1 ̄ t1,j  − 1 t1j j0 −   s t  E t 1 Kt1 −1  t1 ̄
  • 14. • From previous slide: − Kt   s t  E t 1 Kt1 . −1  t1 ̄ • Substituting out for marginal cost: dCost/dlabor  s t   1 −  Wt /P t −1 −1 dOutput/dlabor expa t  Wt  Pt by household optimization   1 −  exp t N t C t  . −1 expa t 
  • 15. In Sum • solution:  E t ∑ j0  j X t,j  −  s −1 tj ̃ pt    Kt , E t ∑ j0  j X t,j  1− Ft • Where:  exp t N t C t − Kt  1 −  t    E t 1 Kt1 . −1 expa t   t1 ̄  1− F t ≡ E t ∑ X t,j  j 1−  1  E t 1 F t1  t1 ̄ j0
  • 16. To Characterize Equilibrium • Have equations characterizing optimization by  firms and households. • Still need: P i,t , 0 ≤ i ≤ 1 – Expression for all the prices. Prices,                          ,  will all be different because of the price setting  frictions. – Relationship between aggregate employment and  aggregate output not simple because of price  distortions: Y t ≠ e a t N t , in general
  • 17. Going for Prices • Aggregate price relationship Calvo insight: This is just a simple 1 function of last period’s 0 P 1− di 1 1− Pt  i,t aggregate price because non-optimizers chosen at random. 1 firms that reoptimize price P 1− di  firms that don’t reoptimize price P 1− di 1−  i,t i,t all reoptimizers choose same price 1   ̃ 1−   1 − P t P i,t 1− di 1− firms that don’t reoptimize price • In principle, to solve the model need all the  prices, P t , P i,t , 0 ≤ i ≤ 1 – Fortunately, that won’t be necessary. 
  • 18. ̃ Expression for     in terms of aggregate  pt inflation  • Conclude that this relationship holds between  prices: 1 P t  1 − ̃ 1− P t  1− P t−1 1− . – Only two variables here! • Divide by      : Pt 1 1− 1− 1 ̃ 1 − p t  1 1− t ̄ • Rearrange: 1 −1 1−  t ̄ 1− ̃ pt  1−
  • 19. Relation Between Aggregate  Output and Aggregate Inputs • Technically, there is no ‘aggregate production  function’ in this model – If you know how many people are working, N, and  the state of technology, a, you don’t have enough  information to know what Y is. – Price frictions imply that resources will not be  efficiently allocated among different inputs. • Implies Y low for given a and N. How low? • Tak Yun (JME) gave a simple answer.
  • 20. Tak Yun Algebra labor market clearing Y∗  0 Yi,t di 1  0 At N i,t di 1   At N t t demand curve −   Yt  1 P i,t di 0 Pt 0 P i,t  −di 1  Yt P  t Calvo insight  Y t P  P ∗  − t t −1 • Where: P∗ t ≡ 0 1 P − di i,t  ̃t  1 − P −  P ∗  −  t−1 −1 
  • 21. Relationship Between Agg Inputs  and Agg Output • Rewriting previous equation:  P∗ Yt  t Y∗ t Pt  p ∗ e at N t , t • ‘efficiency distortion’: ≤1 p∗ t :  1 P i,t  P j,t , all i, j
  • 22. Collecting Equilibrium Conditions • Price setting:  K t  1 −   exp t N t C t  E t   K (1) ̄ t1 t1 −1 At F t  1  E t  −1 F t1 (2) ̄ t1 • Intermediate good firm optimality and  restriction across prices: ̃ p t by restriction across prices ̃ p t by firm optimality  −1 1 Kt 1−  t ̄ 1−  (3) Ft 1−
  • 23. Equilibrium Conditions • Law of motion of (Tak Yun) distortion:  −1 −1 1−  t ̄ −1   ̄t p∗  1 −   ∗ (4) t 1− p t−1 • Household Intertemporal Condition: 1  E t 1 R t (5) Ct C t1  t1 ̄ • Aggregate inputs and output: C t  p ∗ e a t N t (6) t • 6 equations, 8 unknowns: , C t , p ∗ , N t ,  t , K t , F t , R t t ̄ • System under determined!
  • 24. Underdetermined System • Not surprising: we added a variable, the  nominal rate of interest. • Also, we’re counting subsidy as among the  unknowns. • Have two extra policy variables. • One way to pin them down: compute optimal  policy.
  • 25. Ramsey‐Optimal Policy • 6 equations in 8 unknowns….. – Many configurations of the 8 unknowns that  satisfy the 6 equations. – Look for the best configurations (Ramsey optimal) • Value of tax subsidy and of R represent optimal policy • Finding the Ramsey optimal setting of the 6  variables involves solving a simple Lagrangian  optimization problem.
  • 26. Ramsey Problem  N 1 max E 0 ∑  t  logC t − exp t  t ,p ∗ ,C t ,N t ,Rt , t ,F t ,K t t ̄ 1 t0   1t 1 − Et  Rt Ct C t1  t1 ̄  −1 1 − 1 −  t  ̄ −1   ̄   2t 1 −   ∗t p∗ t 1− p t−1   3t 1  E t  −1 F t1 − F t  ̄ t1 C t exp t N    4t 1 −   t  E t   K t1 − K t ̄ t1 −1 e at 1 1−  −1 ̄t 1−   5t F t − Kt 1−   6t C t − p ∗ e a t N t  t
  • 27. Solving the Ramsey Problem  (surprisingly easy in this case) • First, substitute out consumption everywhere  N 1 max E 0 ∑  t  logN t  logp ∗ − exp t  t ,p ∗ ,N t ,Rt , t ,F t ,K t t ̄ t 1 t0 defines R 1 − Et e at  Rt   1t ∗ pt Nt p t1 e N t1  t1 ∗ a t1 ̄  −1 1 − 1 −  t  ̄ −1   ̄   2t 1 −   ∗t p∗ t 1− p t−1 defines F ̄ −1   3t 1  E t  t1 F t1 − F t  defines tax   4t 1 −   exp t N 1 p ∗  E t   K t1 − K t ̄ t1 −1 t t 1 1−  −1 ̄t 1−   5t F t − Kt  defines K 1−
  • 28. Solving the Ramsey Problem, cnt’d  • Simplified problem:  N 1 max E 0 ∑   logN t t logp ∗ − exp t  t  t ,p ∗ ,N t ̄ t t 1 t0  −1 1 − 1 −  t  ̄ −1   ̄   2t 1 −   ∗t  p∗ t 1− p t−1 • First order conditions with respect to p ∗ ,  t , N t t ̄ 1 p ∗  −1 −1 p ∗   2,t1     2t ,  t  ̄ t1 ̄ t−1 , N t  exp −  t t 1−   p ∗  −1 t−1 1 • Substituting the solution for inflation into law  of motion for price distortion:  1 p∗ t  1 −   p ∗  −1 t−1 −1 .
  • 29. Solution to Ramsey Problem Eventually, price distortions eliminated, regardless of shocks 1 p ∗  1 −   p t−1  −1 t ∗ −1 When price distortions p∗ gone, so is inflation.  t  t−1 ̄ p∗t N t  exp − t Efficient (‘first best’) 1 allocations in real economy 1−  −1  C t  p ∗ e at N t . t Consumption corresponds to efficient allocations in real economy, eventually when price distortions gone
  • 30. Eventually, Optimal (Ramsey) Equilibrium and  Efficient Allocations in Real Economy Coincide Convergence of price distortion 0.98 0.96 0.94   0. 75,   10 0.92 p-star 0.9 1 0.88 p∗ t  1 −   p ∗  −1 t−1 −1 0.86 0.84 0.82 0.8 2 4 6 8 10 12 14 16 18 20
  • 31. • The Ramsey allocations are eventually the  best allocations in the economy without price  frictions (i.e., ‘first best allocations’) • Refer to the Ramsey allocations as the ‘natural  allocations’…. – Natural consumption, natural rate of interest, etc.
  • 32. Equations of the NK Model Under the  Optimal Policy (‘Natural Equilibrium’) • Output and employment is (eventually) y∗  at − 1 t , n∗  − 1 t t 1 t 1 • Intertemporal Euler equation after taking logs  and ignoring variance adjustment term: y ∗  −r ∗ − E t  ∗ − rr  E t y ∗ , rr  − log t t t1 t1 • Inflation in Ramsey equilibrium is (eventually)  zero.
  • 33. Solving for Natural Rate of Interest • Intertemporal euler equation in natural  equilibrium: ∗ y∗ yt t1 at − 1 1  t  −r ∗ − rr  E t t a t1 − 1 1  t1 • Back out the natural rate: r ∗  rr  Δa t  t 1 1 1 −  t • Shocks:  t   t−1    , Δa t  Δa t−1   t t
  • 35. Taylor Rule • Taylor rule: designed, so that in steady state,  1 ̄ inflation is zero (             ) • Employment subsidy extinguishes monopoly  power in steady state:  1 −    1 −1
  • 36. NK IS Curve • Euler equation in two equilibria: Taylor rule equilibrium: y t  −r t − E t  t1 − rr  E t y t1 Natural equilibrium: y ∗  −r ∗ − rr  E t y ∗ t t t1 • Subtract: Output gap x t  −r t − E t  t1 − r ∗   E t x t1 t
  • 37. Output in NK Equilibrium • Agg output relation:  0 if P i,t  P j,t for all i, j yt  logp ∗ t  nt  at , logp ∗ t  . ≤0 otherwise • To first order approximation,  ̂t ̂ t−1 p ∗ ≈ p ∗  0   t , → p ∗ ≈ 1 ̄ t
  • 38. Price Setting Equations • Log‐linearly expand the price setting equations  about steady state. 1 1− −1 ̄t − Kt  0 1− 1 ̄ −1 E t  t1 F t1 − Ft  0 Ft 1−   C t exp t N t 1 −  −1 eat ̄  E t  t1 K t1 − K t  0 • Log‐linearly expand about steady state:  1−1−  t  ̄  1  x t   t1 ̄ • See http://faculty.wcas.northwestern.edu/~lchrist/course/solving_handout.pdf
  • 39. Taylor Rule • Policy rule r t  r t−1  1 − rr     t   x x t   u t , , x t ≡ y t − y ∗ t
  • 40. Equations of Actual Equilibrium Closed by Adding Policy Rule  E t  t1  x t −  t  0 (Phillips curve) − r t − E t  t1 − r ∗   E t x t1 − x t  0 (IS equation) t r t−1  1 −    t  1 −  x x t − r t  0 (policy rule) r ∗ − Δa t − 1 1 −  t  0 (definition of natural rate) t 1
  • 41. Solving the Model Δa t  0 Δa t−1 t st    t 0   t−1  t s t  Ps t−1   t  0 0 0  t1 −1  0 0 t 1  1 0 0 x t1 0 −1 − 1 1  xt  0 0 0 0 r t1 1 −   1 −  x −1 0 rt 0 0 0 0 r∗ t1 0 0 0 1 rr ∗ t 0 0 0 0  t−1 0 0 0 0 0 0 0 0 0 x t−1 0 0 0 0 0   s t1  st 0 0  0 r t−1 0 0 0 0 0 0 0 0 0 r∗ t−1 0 0 0 − −  1 −  1 E t  0 z t1   1 z t   2 z t−1   0 s t1   1 s t   0
  • 42. Solving the Model E t  0 z t1   1 z t   2 z t−1   0 s t1   1 s t   0 s t − Ps t−1 −  t  0. • Solution: z t  Az t−1  Bs t • As before:  0 A2   1 A   2 I  0, F   0   0 BP   1   0 A   1 B  0
  • 43.  x  0,    1. 5,   0. 99,   1,   0. 2,   0. 75,   0,   0. 2,   0. 5. Dynamic Response to a Technology Shock inflation output gap nominal rate 0.2 0.15 0.03 0.15 natural nominal rate 0.1 actual nominal rate 0.02 0.1 0.01 0.05 0.05 0 2 4 6 0 2 4 6 0 2 4 6 natural real rate log, technology output 0.2 0.15 1 1.2 natural output 1.15 0.1 actual output 0.5 1.1 0.05 1.05 0 1 0 2 4 6 0 2 4 6 0 2 4 6 employment 0.2 0.15 0.1 natural employment 0.05 actual employment 0 0 5 10
  • 44. Dynamic Response to a Preference Shock inflation output gap nominal rate 0.1 0.25 0.08 0.25 0.2 0.2 natural real rate 0.06 0.15 0.15 actual nominal rate 0.04 0.1 0.1 0.02 0.05 0.05 0 2 4 6 0 2 4 6 0 2 4 6 natural real rate preference shock output 0.25 1 0.2 -0.1 0.15 -0.2 0.5 0.1 -0.3 natural output actual output 0.05 -0.4 0 -0.5 0 2 4 6 0 2 4 6 0 2 4 6 employment -0.1 -0.2 natural employment actual employment -0.3 -0.4 -0.5 0 2 4 6
  • 45. Conclusion of NK Model Analysis • We studied examples in which the Taylor rule  moves the interest rate in the right direction  in response to shocks. • However, the move is not strong enough. Will  consider modifications of the Taylor rule using  Dynare.