DSGE Models and Optimal Monetary Policy

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DSGE Models and Optimal Monetary Policy

• Andrew P. Blake

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A framework of analysis

• Typified by Woodfords 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

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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

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Linear stochastic model

• We consider a model in state space form
• u is a vector of control instruments, s a vector
  of endogenous variables, e is a shock vector
• The model coefficients are in A, B and C

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Quadratic objective function

• Assume the following objective function
• Q and R are positive (semi-) definite symmetric
  matrices of weights
• 0 lt ? 1 is the discount factor
• We take the initial time to be 0

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How do we solve for the optimal policy?

• We have two options
• Dynamic programming
• Pontryagins minimum principle
• Both are equivalent with non-anticipatory
  behaviour
• Very different with rational expectations
• We will require both to analyse optimal policy

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Dynamic programming

• Approach due to Bellman (1957)
• Formulated the value function
• Recognised that it must have the structure

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Optimal policy rule

• First order condition (FOC) for u
• Use to solve for policy rule

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The Riccati equation

• Leaves us with an unknown in S
• Collect terms from the value function
• Drop z

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Riccati equation (cont.)

• If we substitute in for F we can obtain
• Complicated matrix quadratic in S
• Solved backwards by iteration, perhaps by

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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

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Lagrange multipliers

• Due to Pontryagin (1957)
• Formulated a system using constraints as
• ? is a vector of Lagrange multipliers
• The constrained objective function is

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FOCs

• Differentiate with respect to the three sets of
  variables

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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)

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Solutions are equivalent

• Assume that the solution to the saddlepath
  problem is
• Substitute into the FOCs to give

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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

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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.

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Bellmans dedication

• At the beginning of Bellmans book Dynamic
  Programming he dedicates it thus
• To Betty-Jo
• Whose decision processes defy analysis

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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 ?
• Why?

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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

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Pre-determined co-states (cont.)

• At time 0 this is minimised by
• We can rearrange the reaction function to
• Where
  etc

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Pre-determined co-states (cont.)

• Alternatively the value function can be written
  in terms of the x and the zs as
• The loss is

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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 ?x at time 0 to
  minimise cost
• We choose it to be 0
• This minimises the cost-to-go in period 0

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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

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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

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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

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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

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A rule for future expectations

• Assume that
• If we substitute this into the model we get

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A rule for future expectations

• The pre-determined model is
• Using the reaction function for x we get

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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

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Dynamic programming (cont.)

• FOC for u for leadership
• where
• This policy must be time consistent
• Only uses intra-period leadership

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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)

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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

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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

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Whats 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)

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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
• Lets do it!