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!