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

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, 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.