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# Lecture on nk [compatibility mode]

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• 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 treal 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≥1̃pt  , 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−1defines 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 ProblemEventually, price distortionseliminated, 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 0.92p-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
• 34. Next, Put Turn to the NK Model  with Taylor Rule
• 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, jyt  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 ruler 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 0.250.08 0.25 0.2 0.2 natural real rate0.06 0.15 0.15 actual nominal rate0.04 0.1 0.10.02 0.05 0.05 0 2 4 6 0 2 4 6 0 2 4 6 natural real rate preference shock output0.25 1 0.2 -0.10.15 -0.2 0.5 0.1 -0.3 natural output actual output0.05 -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.