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
t0
• Subject to:
Pt C t B t1 ≤ 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,t1 Rt
1 E t u c,t
1 t1
– or, for when we do linearize later:
1 E t C t Rt
C t1 1 t1
E t explogR t − log1 t1 − Δc t1
≃ explogR t − E t t1 − E t Δc t1 , c t ≡ logC t
– take log of both sides:
0 log r t − E t t1 − E t Δc t1 , r t logR t
– or
c t − log − r t − E t t1 c t1
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
7. Marginal Cost
dCost
1 − Wt /P t
real marginal cost st dwor ker
dOutput expa t
dwor ker
−1 in efficient setting
1 − C t exp t N t
expa t
9. Intermediate Good Firm
• Present discounted value of firm profits:
period tj profits sent to household
marginal value of dividends to householdu c,tj /P tj revenues total cost
Et ∑ j tj Pi,tj Yi,tj − P tj s tj Yi,tj
j0
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 ̃
tj Pt Yi,tj − Ptj s tj Yi,tj .
j0
10. Intermediate Good Firm Problem
• Substitute out the demand curve:
E t ∑ j tj Pt Yi,tj − P tj s tj Yi,tj
̃
j0
E t ∑ j tj Ytj P P 1− − P tj s tj P − .
tj
̃t ̃t
j0
̃
• Differentiate with respect to :
Pt
E t ∑ j tj Ytj P 1 − P t Ptj s tj P−−1 0,
tj
̃ − ̃t
j0
• or
̃
Pt − s
E t ∑ j
tj Ytj P 1 0.
tj
P tj − 1 tj
j0
11. Intermediate Good Firm Problem
• Objective:
u ′ C tj ̃
Pt − s
E t ∑ j Ytj P1 0.
P tj tj
Ptj − 1 tj
j0
̃
Pt − s
→ E t ∑ j
P 0.
tj
P tj − 1 tj
j0
• or
E t ∑ j X t,j − p t X t,j −
̃ s tj 0,
−1
j0
̃
Pt , X
1
tj tj−1 t1
̄ ̄ ̄ ,j≥1
̃
pt , X t,j X t1,j−1 1 , j 0
Pt t,j
1, j 0. t1
̄
12. Intermediate Good Firm Problem
̃
• Want in:
pt
E t ∑ j0 j X t,j − p t X t,j −
̃
−1
s tj 0
• Solution:
E t ∑ j0 j X t,j −
s
−1 tj
̃
pt Kt
Et ∑ j0
j X t,j 1− Ft
• But, still need expressions for Kt , Ft .
13.
Kt E t ∑ j X t,j − s tj
−1
j0
−
s t E t ∑ j−1 1 X t1,j−1 s tj
−1 t1
̄ −1
j1
−
s t E t 1 ∑ j X − s
−1 t1
̄ t1,j − 1 t1j
j0
E t by LIME
−
s t E t E 1 ∑ j X − s
−1 t1
t1
̄ t1,j
− 1 t1j
j0
exactly K t1 !
−
s t E t 1 E t1 ∑ j X − s
−1 t1
̄ t1,j
− 1 t1j
j0
−
s t E t 1 Kt1
−1 t1
̄
14. • From previous slide:
−
Kt s t E t 1 Kt1 .
−1 t1
̄
• Substituting out for marginal cost:
dCost/dlabor
s t 1 − Wt /P t
−1 −1 dOutput/dlabor
expa t
Wt
Pt
by household optimization
1 − exp t N t C t
.
−1 expa t
15. In Sum
• solution:
E t ∑ j0 j X t,j −
s
−1 tj
̃
pt Kt ,
E t ∑ j0 j X t,j 1− Ft
• Where:
exp t N t C t −
Kt 1 − t E t 1 Kt1 .
−1 expa t t1
̄
1−
F t ≡ E t ∑ X t,j
j 1−
1 E t 1 F t1
t1
̄
j0
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)
̄ t1 t1
−1 At
F t 1 E t −1 F t1 (2)
̄ t1
• 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 t1 t1
̄
• 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!
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
t0
1t 1 − Et Rt
Ct C t1 t1
̄
−1
1 − 1 − t
̄ −1
̄
2t 1 − ∗t
p∗
t 1− p t−1
3t 1 E t −1 F t1 − F t
̄ t1
C t exp t N
4t 1 − t
E t K t1 − K t
̄ t1
−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
t0
defines R 1 − Et e at Rt
1t ∗
pt Nt p t1 e N t1 t1
∗ a t1 ̄
−1
1 − 1 − t
̄ −1
̄
2t 1 − ∗t
p∗
t 1− p t−1
defines F
̄ −1
3t 1 E t t1 F t1 − F t
defines tax 4t 1 − exp t N 1 p ∗ E t K t1 − K t
̄ t1
−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
t0
−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,t1 2t , t
̄ t1 ̄ 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
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.
33. Solving for Natural Rate of Interest
• Intertemporal euler equation in natural
equilibrium:
∗ y∗
yt t1
at − 1
1
t −r ∗ − rr E t
t a t1 − 1
1
t1
• 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
36. NK IS Curve
• Euler equation in two equilibria:
Taylor rule equilibrium: y t −r t − E t t1 − rr E t y t1
Natural equilibrium: y ∗ −r ∗ − rr E t y ∗
t t t1
• Subtract: Output gap
x t −r t − E t t1 − r ∗ E t x t1
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 t1 F t1 − Ft 0 Ft 1−
C t exp t N t
1 − −1 eat
̄
E t t1 K t1 − K t 0
• Log‐linearly expand about steady state:
1−1−
t
̄ 1 x t t1
̄
• See http://faculty.wcas.northwestern.edu/~lchrist/course/solving_handout.pdf
40. Equations of Actual Equilibrium
Closed by Adding Policy Rule
E t t1 x t − t 0 (Phillips curve)
− r t − E t t1 − r ∗ E t x t1 − 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 t1 −1 0 0 t
1
1 0 0 x t1 0 −1 −
1 1
xt
0 0 0 0 r t1 1 − 1 − x −1 0 rt
0 0 0 0 r∗
t1 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 t1 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 t1 1 z t 2 z t−1 0 s t1 1 s t 0
42. Solving the Model
E t 0 z t1 1 z t 2 z t−1 0 s t1 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 BP 1 0 A 1 B 0