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DSGE Models and Optimal Monetary Policy
Andrew P. Blake
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
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
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
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
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
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
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)(
The Riccati equation
• Leaves us with an unknown in S
• Collect terms from the value function:
• Drop z:tt
tttttt
sBFASBFAs
RFsFsQssSss
)()(
)()( BFASBFARFFQS
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 )(
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
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
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
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
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
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 )(
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
Bellman’s dedication
• At the beginning of Bellman’s book Dynamic Programming he dedicates it thus:
To Betty-Jo
Whose decision processes defy analysis
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?
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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)
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!