Optimal policy computation with Dynare MONFISPOL workshop, Stresa - PowerPoint PPT Presentation

Optimal policy computation with Dynare MONFISPOL workshop, Stresa Michel Juillard 1 March 12, 2010 1 Bank of France and CEPREMAP

Introduction Dynare currently implements two manners to compute optimal policy in DSGE models ◮ optimal rule under commitment (Ramsey policy) ◮ optimal simple rules

A New Keynesian model Thanks to Eleni Iliopulos! Utility function: ln C t − φ N 1 + γ t U t = 1 + γ 1 U c , t = C t U N , t − φ N γ = t

Recursive equilibrium � � R t 1 1 = β E t C t C t + 1 π t + 1 � π t + 1 � π t ( π t − 1 ) = β E t ( π t + 1 − 1 ) C t C t + 1 � φ C t N γ � + ε A t N t − ε − 1 t C t A t ω ε A t N t = C t + ω 2 ( π t − 1 ) 2 ln A t = ρ ln A t − 1 + ǫ t

Ramsey problem � ∞ ln C t − φ N 1 + γ � β t t max E 0 1 + γ t = 0 � 1 � �� R t 1 − β E t − µ 1 , t C t C t + 1 π t + 1 � π t � π t + 1 � ( π t − 1 ) − β E t − µ 2 , t ( π t + 1 − 1 ) C t C t + 1 � φ C t N γ �� − ε A t N t − ε − 1 t C t A t ω ε � 2 ( π t − 1 ) 2 � A t N t − C t − ω − µ 3 , t � − µ 4 , t ( ln A t − ρ ln A t − 1 − ǫ t ) µ i , t = λ i , t β t where λ i , t is a Lagrange multiplier.

F .O.C. C t R t − 1 � � + 1 µ 1 , t − 1 [ 2 π t − 1 ] µ 2 , t − µ 2 , t − 1 = µ 3 , t ω ( C t π 2 t � � R t − 1 π t ( π t − 1 ) + A t N t − µ 1 , t 1 1 + µ 1 , t − 1 − µ 2 , t ( ε − 1 ) C t C 2 C 2 C 2 t π t ω t t 1 + µ 2 , t − 1 [ π t ( π t − 1 )] = µ 3 , t C 2 t A t t + φ ε φ N γ ω N γ C t ω ( ε − 1 ) = µ 3 , t A t t µ 2 , t ( γ + 1 ) − µ 2 , t N t µ 2 , t ( ε − 1 ) + µ 3 , t N t + µ 4 , t − ρβµ 4 , t + 1 = 0 C t A t A t ω � � 1 µ 1 , t β E t = 0 C t + 1 π t + 1

Dynare code var pai, c, n, r, a; varexo u; parameters beta, rho, epsilon, omega, phi, gamma; beta=0.99; gamma=3; omega=17; epsilon=8; phi=1; rho=0.95;

Dynare code (continued) model; a = rho*a(-1)+u; 1/c = beta*r/(c(+1)*pai(+1)); pai*(pai-1)/c = beta*pai(+1)*(pai(+1)-1)/c(+1) +epsilon*phi*n^(gamma+1)/omega -exp(a)*n*(epsilon-1)/(omega*c); exp(a)*n = c+(omega/2)*(pai-1)^2; end;

Dynare code (continued) initval; pai=1; r=1/beta; c=0.96717; n=0.96717; a=0; end; shocks; var u; stderr 0.008; end; planner_objective(ln(c) -phi*((n^(1+gamma))/(1+gamma))); ramsey_policy(planner_discount=0.99,order=1);

Ramsey policy: General nonlinear case ∞ � β τ − t U ( y τ ) E t max { y τ } ∞ τ = t τ = 0 s.t. E τ f ( y τ + 1 , y τ , y τ − 1 , ε τ ) = 0 y t ∈ R n : endogenous variables ε t ∈ R p : stochastic shocks and f : R 3 n + p → R m There are n − m free policy instruments.

Lagrangian The Lagrangian is written ∞ � β τ − t U ( y τ ) − [ λ τ ] η [ f ( y τ + 1 , y τ , y τ − 1 , ε τ )] η L ( y t − 1 , ε t ) = E t τ = t where λ τ is a vector of m Lagrange multipliers We adopt tensor notations because later on we will have to deal with the second order derivatives of [ λ ( s t )] η [ F ( s t )] η γ . It turns out that it is the discounted value of the Lagrange multipliers that are stationary and not the multipliers themselves. It is therefore handy to rewrite the Lagrangian as β τ − t � U ( y τ ) − [ µ τ ] η [ f ( y τ + 1 , y τ , y τ − 1 , ε τ )] η � ∞ � L ( y t − 1 , ε t ) = E t τ = t whith µ t = λ τ /β τ − t .

Optimization problem reformulated The optimization problem becomes ∞ β τ − t � U ( y τ ) − [ µ τ ] η [ f ( y τ + 1 , y τ , y τ − 1 , ε τ )] η � � E t max min { y τ } ∞ { λ τ } ∞ τ = t τ = t τ = t for y t − 1 given.

First order necessary conditions Derivatives of the Lagrangian with respect to endogenous variables: � ∂ L � � = E t [ U 1 ( y t )] α − [ µ t ] η [ f 2 ( y t + 1 , y t , y t − 1 , ε t )] η ∂ y t α α � − β [ µ t + 1 ] η [ f 3 ( y t + 2 , y t + 1 , y t , ε t + 1 )] η τ = t α � ∂ L � � = E t [ U 1 ( y τ )] α − [ µ τ ] η [ f 2 ( y τ + 1 , y τ , y τ − 1 , ε τ )] η ∂ y τ α α − β [ µ τ + 1 ] η [ f 3 ( y τ + 2 , y τ + 1 , y τ , ε τ + 1 )] η α � − β − 1 [ µ τ − 1 ] η [ f 1 ( y τ , y τ − 1 , y τ − 2 , ε τ − 1 )] α τ > t

First order conditions The first order conditions of this optimization problem are � E t [ U 1 ( y τ )] α − [ µ τ ] η [ f 2 ( y τ + 1 , y τ , y τ − 1 , ε τ )] η α − β [ µ τ + 1 ] η [ f 3 ( y τ + 2 , y τ + 1 , y τ , ε τ + 1 )] η α � − β − 1 [ µ τ − 1 ] η [ f 1 ( y τ , y τ − 1 , y τ − 2 , ε τ − 1 )] η = 0 α [ f ( y τ + 1 , y τ , y τ − 1 , ε τ )] η = 0 with µ t − 1 = 0 and where U 1 () is the Jacobian of function U () with respect to y τ and f i () is the first order partial derivative of f () with respect to the i th argument.

Cautionary remark The First Order Conditions for optimality are only necessary conditions for a maximum. Levine, Pearlman and Pierse (2008) propose algorithms to check a sufficient condition. I still need to adapt it to the present framework where the dynamic conditions, f () are given as implicit functions and variables can be both predetermined and forward looking.

Nature of the solution The above system of equations is nothing but a larger system of nonlinear rational expectation equations. As such, it can be approximated either to first order or to second order. The solution takes the form � y t � g ( y t − 2 , y t − 1 , µ t − 1 , ε t − 1 , ε t ) = ˆ µ t The optimal policy is then directly obtained as part of the set of g () functions.

The steady state problem The steady state is solution of ′ � U 1 (¯ y ) − ¯ f 2 (¯ y , ¯ y , ¯ y , 0 ) + β f 3 (¯ y , ¯ y , ¯ y , 0 ) µ � + β − 1 f 1 (¯ y , ¯ y , ¯ y , 0 ) = 0 f (¯ y , ¯ y , ¯ y , 0 ) = 0

Computing the steady state y , it is possible to use the first matrix equation For a given value ˜ above to obtain the value of ˜ µ that minimizes the sum of square residuals, e : M f 2 (˜ y , ˜ y , ˜ y , ˜ x , 0 ) + β f 3 (˜ y , ˜ y , ˜ y , ˜ x , 0 ) = + β − 1 f 1 (˜ y , ˜ y , ˜ y , 0 ) y ) M ′ � MM ′ � − 1 U 1 (˜ µ ′ ˜ = e ′ U 1 (˜ y ) − ˜ µ ′ M = y must satisfy the m equations Furthermore, ˜ f (˜ y , ˜ y , ˜ y , 0 ) = 0 It is possible to build a sytem of equations whith only n unkowns y , but we must provide n − m independent measures of the ˜ residuals e . Independent in the sense that the derivatives of y must be linearly independent. these measures with respect to ˜

A QR trick At the steady state, the following must hold exactly U 1 (˜ y ) = ¯ µ ′ M This can only be if � � M M ⋆ = U 1 (˜ y ) is of rank m The reordered QR decomposition of M ⋆ is such that M ⋆ E Q R = ( m + 1 ) × n n × n ( m + 1 ) × ( m + 1 ) ( m + 1 ) × n where E is a permutation matrix, Q an orthogonal matrix and R a triangular matrix with diagonal elements ordered in decreasing size.

A QR trick (continued) ◮ When U 1 (˜ y ) = ¯ µ ′ M doesn’t hold exactly M ⋆ is full rank ( m + 1) and the n − m last elements of R may be different from zero. µ ′ M holds exactly M ⋆ has rank m and the ◮ When U 1 (˜ y ) = ¯ n − m last elements of R are zero. ◮ The last n − m elements of the last row of R provide the n − m independent measures of the residuals e y as input ◮ In practice, we build a nonlinear function with ˜ and that returns the n − m last elements of the last row of R and f (˜ y , ˜ y , ˜ y , 0 ) . At the solution, when ˜ y = ¯ y , this function must return zeros.

When an analytical solution is available Let’s write y x the n − m policy instruments and y y , the m remaining endogenous variables, such that � y ⋆ � y ⋆ = x . y ⋆ y The analytical solution is given by y ⋆ y = ˜ f ( y ⋆ x ) . We have then f ( y ⋆ , y ⋆ , y ⋆ , 0 ) = 0 Combining this analytical solution with the n − m residuals computed above, we can reduce the steady state problem to a system of n − m nonlinear equations that is much easier to solve.

First order approximation of the FOCs � �� � γ − [ � α − β − 1 [ � E t [ U 11 ] αγ y t µ t ] η [ f 2 ] η µ t + 1 ] η [ f 3 ] η µ t − 1 ] η [ f 1 ] η α − β [ � α � � � �� �� � γ + β − 1 [ f 13 ] η �� � γ + � γ β [ f 13 ] η y t + 2 y t − 2 [ f 12 ] η γα + β [ f 23 ] η y t + 1 − [¯ µ ] η γα αγ γα � � �� � � �� � γ + � γ αγ + β − 1 [ f 12 ] η αγ + β − 1 [ f 11 ] η [ f 23 ] η y t − 1 [ f 33 ] η αγ + β [ f 22 ] η y t + αγ αγ �� ε t ] δ + β [ f 34 ] η ε t + 1 ] δ + β − 1 [ f 14 ] η + [ f 24 ] η ε t − 1 ]] δ αδ [ � αδ [ � αδ [ � = 0 � �� � γ + [ f 2 ] ηγ �� � γ + [ f 3 ] ηγ �� � γ + [ f 4 ] ηδ [ � ε t ] δ � E t [ f 1 ] ηγ y t + 1 y t y t − 1 = 0 y t = y t − ¯ y , � µ and f ij the second order where � µ t = µ t − ¯ derivatives corresponding to the i th and the j th argument of the f () function.

The first pitfall A naive approach ot linear-quadratic appoximation that would consider a linear approximation of the dynamics of the system and a second order approximation of the objective function, ignores the second order derivatives f ij that enter in the first order approximation of the dynamics of the model under optimal policy.