dsge models and optimal monetary policy andrew p. blake
TRANSCRIPT
![Page 1: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/1.jpg)
DSGE Models and Optimal Monetary Policy
Andrew P. Blake
![Page 2: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/2.jpg)
A framework of analysis
• Typified by Woodford’s Interest and Prices– Sometimes called DSGE models– Also known as NNS models
• Strongly micro-founded models
• Prominent role for monetary policy
• Optimising agents and policymakers
![Page 3: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/3.jpg)
What do we assume?
• Model is stochastic, linear, time invariant• Objective function can be approximated
very well by a quadratic• That the solutions are certainty equivalent
– Not always clear that they are
• Agents (when they form them) have rational expectations or fixed coefficient extrapolative expectations
![Page 4: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/4.jpg)
Linear stochastic model
• We consider a model in state space form:
• u is a vector of control instruments, s a vector of endogenous variables, ε is a shock vector
• The model coefficients are in A, B and C
11 tttt CBuAss
![Page 5: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/5.jpg)
Quadratic objective function
• Assume the following objective function:
• Q and R are positive (semi-) definite symmetric matrices of weights
• 0 < ρ ≤ 1 is the discount factor
• We take the initial time to be 0
0
0 2
1min
ttttt
tu RuuQssV
t
![Page 6: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/6.jpg)
How do we solve for the optimal policy?
• We have two options:– Dynamic programming– Pontryagin’s minimum principle
• Both are equivalent with non-anticipatory behaviour
• Very different with rational expectations• We will require both to analyse optimal
policy
![Page 7: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/7.jpg)
Dynamic programming
• Approach due to Bellman (1957)
• Formulated the value function:
• Recognised that it must have the structure:
112
1min
2
1 ttttttuttt SssRuuQssSssV
t
)()(min ttttttttut BuAsSBuAsRuuQssVt
![Page 8: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/8.jpg)
Optimal policy rule
• First order condition (FOC) for u:
• Use to solve for policy rule:
0)(0
tttt
t BuAsSBRuu
V
t
tt
Fs
SAsBSBBRu
1)(
![Page 9: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/9.jpg)
The Riccati equation
• Leaves us with an unknown in S
• Collect terms from the value function:
• Drop z:tt
tttttt
sBFASBFAs
RFsFsQssSss
)()(
)()( BFASBFARFFQS
![Page 10: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/10.jpg)
Riccati equation (cont.)
• If we substitute in for F we can obtain:
• Complicated matrix quadratic in S
• Solved ‘backwards’ by iteration, perhaps by:
SABSBBRSBASAAQS 12 )(
ASBBSBRBSAASAQS jjjjj 11
112
1 )(
![Page 11: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/11.jpg)
Properties of the solution
• ‘Principle of optimality’• The optimal policy depends on the unknown S• S must satisfy the Riccati equation• Once you solve for S you can define the policy rule
and evaluate the welfare loss• S does not depend on s or u only on the model and
the objective function• The initial values do not affect the optimal control
![Page 12: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/12.jpg)
Lagrange multipliers
• Due to Pontryagin (1957)
• Formulated a system using constraints as:
• λ is a vector of Lagrange multipliers:
• The constrained objective function is:
)( 11 kkkkkkkkk
k sBuAsRuuQssH
tk
kt HV
![Page 13: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/13.jpg)
FOCs
• Differentiate with respect to the three sets of variables:
00
00
00
1
1
1
tttt
t
tttt
t
ttt
t
sBuAsH
AQss
H
BRuu
H
![Page 14: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/14.jpg)
Hamiltonian system
• Use the FOCs to yield the Hamiltonian system:
• This system is saddlepath stable• Need to eliminate the co-states to determine the
solution• NB: Now in the form of a (singular) rational
expectations model (discussed later)
t
t
t
t s
IQ
As
A
BRBI
0
0 1
11
![Page 15: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/15.jpg)
Solutions are equivalent
• Assume that the solution to the saddlepath problem is
• Substitute into the FOCs to give:
00 1
tttt
t SsSsAQss
H
tt Ss
00 1
ttt
t SsBRuu
H
![Page 16: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/16.jpg)
Equivalence (cont.)
• We can combine these with the model and eliminate s to give:
• Same solution for S that we had before• Pontryagin and Bellman give the same answer• Norman (1974, IER) showed them to be
stochastically equivalent• Kalman (1961) developed certainty equivalence
SABSBBRSBASAAQS 12 )(
![Page 17: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/17.jpg)
What happens with RE?
• Modify the model to:
• Now we have z as predetermined variables and x as jump variables
• Model has a saddlepath structure on its own• Solved using Blanchard-Kahn etc.
tt
tet
t uB
B
x
z
AA
AA
x
z
2
1
2221
1211
1
1
![Page 18: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/18.jpg)
Bellman’s dedication
• At the beginning of Bellman’s book Dynamic Programming he dedicates it thus:
To Betty-Jo
Whose decision processes defy analysis
![Page 19: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/19.jpg)
Control with RE
• How do rational expectations affect the optimal policy?– Somewhat unbelievably - no change– Best policy characterised by the same algebra
• However, we need to be careful about the jump variables, and Betty-Jo
• We now obtain pre-determined values for the co-states λ
• Why?
![Page 20: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/20.jpg)
Pre-determined co-states
• Look at the value function
• Remember the reaction function is:
• So the cost can be written as
• We can minimise the cost by choosing some co-states and letting x jump
ttt SssV 21
tt
t
xt
zt Ss
x
z
SS
SS
2221
1211
ttt sV 21
![Page 21: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/21.jpg)
Pre-determined co-states (cont.)
• At time 0 this is minimised by:
• We can rearrange the reaction function to:
• Where etc
02
1
2
1 000
0
0000
z
x
z
xzxzV
xt
t
t
zt z
NN
NN
x
2221
1211
1222221
122121111 , SNSSSSN
![Page 22: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/22.jpg)
Pre-determined co-states (cont.)
• Alternatively the value function can be written in terms of the x and the z’s as:
• The loss is:
x
x zSTTzV
0
0002
10
xt
txt
t
t
t zT
z
NN
I
x
z
2221
0
![Page 23: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/23.jpg)
Cost-to-go
• At time 0, z0 is predetermined
• x0 is not, and can be any value
• In fact is a function of z0 (and implicitly u)
• We can choose the value of λx at time 0 to minimise cost
• We choose it to be 0
• This minimises the cost-to-go in period 0
![Page 24: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/24.jpg)
Time inconsistency
• This is true at time 0
• Time passes, maybe just one period
• Time 1 ‘becomes time 0’
• Same optimality conditions apply
• We should reset the co-states to 0
• The optimal policy is time inconsistent
![Page 25: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/25.jpg)
Different to non-RE
• We established before that the non-RE solution did not depend on the initial conditions (or any z)
• Now it directly does• Can we use the same solution methods?
– DP or LM?– Yes, as long as we ‘re-assign’ the co-states
• However, we are implicitly using the LM solution as it is ‘open-loop’ – the policy depends directly on the initial conditions
![Page 26: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/26.jpg)
Where does this fit in?
• Originally established in 1980s– Clearest statement Currie and Levine (1993)– Re-discovered in recent US literature– Ljungqvist and Sargent Recursive
Macroeconomic Theory (2000, and new edition)
• Compare with Stokey and Lucas
![Page 27: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/27.jpg)
How do we deal with time inconsistency?
• Why not use the ‘principle of optimality’
• Start at the end and work back
• How do we incorporate this into the RE control problem?– Assume expectations about the future are
‘fixed’ in some way– Optimise subject to these expectations
![Page 28: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/28.jpg)
A rule for future expectations
• Assume that:
• If we substitute this into the model we get:
111 ttet zNx
tttt
ttt
tttt
uKzJ
uBBNANA
zAANANAx
)()(
)()(
2111
12122
121111
12122
![Page 29: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/29.jpg)
A rule for future expectations
• The ‘pre-determined’ model is:
• Using the reaction function for x we get:
ttt
ttttt
uBzA
uKBBzJAAz
ˆˆ
)()( 2112111
tttt uBxAzAz 112111
![Page 30: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/30.jpg)
Dynamic programming solution
• To calculate the best policy we need to make assumptions about leadership
• What is the effect on x of changes in u?
• If we assume no leadership it is zero
• Otherwise it is K, need to use:
tt
t
t
t
t
t
t
t
t
t Kx
V
u
V
u
x
x
V
u
V
0
![Page 31: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/31.jpg)
Dynamic programming (cont.)
• FOC for u for leadership:
where:
• This policy must be time consistent
• Only uses intra-period leadership
tt
tttttttttt
zF
zQJQKASBBSBRu
ˆ
))(ˆˆ()ˆˆˆ( 212211
1
ttt KQKRR 22ˆ
![Page 32: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/32.jpg)
Dynamic programming (cont.)
• This is known in the dynamic game literature as feedback Stackelberg
• Also need to solve for S– Substitute in using relations above
• Can also assume that x unaffected by u– Feedback Nash equilibrium
• Developed by Oudiz and Sachs (1985)
![Page 33: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/33.jpg)
Dynamic programming (cont.)
• Key assumption that we condition on a rule for expectations
• Could condition on a time path (LM)
• Time consistent by construction– Principle of optimality
• Many other policies have similar properties
• Stochastic properties now matter
![Page 34: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/34.jpg)
Time consistency
• Not the only time consistent solutions
• Could use Lagrange multipliers
• DP is not only time consistent it is subgame perfect
• Much stronger requirement– See Blake (2004) for discussion
![Page 35: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/35.jpg)
What’s new with DSGE models?
• Woodford and others have derived welfare loss functions that are quadratic and depend only on the variances of inflation and output
• These are approximations to the true social utility functions
• Can apply LQ control as above to these models• Parameters of the model appear in the loss
function and vice versa (e.g. discount factor)
![Page 36: DSGE Models and Optimal Monetary Policy Andrew P. Blake](https://reader030.vdocuments.us/reader030/viewer/2022032709/56649ec05503460f94bcb6ac/html5/thumbnails/36.jpg)
DGSE models in WinSolve
• Can set up micro-founded models
• Can set up micro-founded loss functions
• Can explore optimal monetary policy– Time inconsistent– Time consistent– Taylor-type approximations
• Let’s do it!