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Computing the steady state of linear quadratic optimization models with rational expectations a, b Hans M. Amman *, David A. Kendrick a

Department of Economics and Tinbergen Institute, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, Netherlands b Department of Economics, University of Texas, Austin, TX 78712, USA Received 12 August 1997; accepted 22 October 1997

Abstract In this paper we present a simple algorithm for computing the steady state of the Linear–Quadratic control model with rational expectations. The method uses Sims’ approach for solving rational expectations models first, and then solves the steady state through an iterative scheme. 1998 Elsevier Science S.A. Keywords: Macroeconomics; Rational Expectations; Stochastic Optimization; Computational Economics JEL classification: C63; E61

1. Introduction In recent work, Amman et al. (1995); Amman (1996), we presented a procedure that introduces RE in a linear–quadratic (LQ) control framework based on the Blanchard and Kahn (Blanchard and Kahn, 1980) method. Due to the limitations of the Blanchard and Kahn approach and the fact that we had to rely on the diagonalization of the transition matrix, this work could only deal with a limited set of models. Recently, Sims (1996) proposed a different method for solving linear models with RE allowing for a broader range of models. This method is not based on the Jordan canonical form, but uses the more widely available QZ form that is based on generalized eigenvalues and which is numerically stable. In this paper we will follow the paper of Sims and incorporate his approach for solving the steady state of the linear–quadratic optimization model with rational expectations.

2. Problem statement and solution Following Kendrick (1981), the standard single-agent stochastic linear–quadratic optimization problem is written as: *Corresponding author. Tel.: 131 20 5254252; fax: 131 20 5254254; e-mail: [email protected] 0165-1765 / 98 / $19.00 1998 Elsevier Science S.A. All rights reserved. PII S0165-1765( 97 )00263-2

H.M. Amman, D. A. Kendrick / Economics Letters 58 (1998) 185 – 191

186

Find the steady state solution of the control vector u ` and the corresponding state vector x ` that minimizes the welfare loss function

O b L (x , u ), `

J5

t

t

t

(1)

t

t 50

with 1 1 Lt 5 ] (x t 2 x¯ )9W(x t 2 x¯ ) 1 ] (u t 2 u¯ )9R(u t 2 u¯ ) 1 (x t 2 x¯ )9F(u t 2 u¯ ), 2 2 subject to the model x t11 5 Ax t 1 Bu t 1 Cz.

(2)

The vector x t [R n is the state of the economy at time t and the vector u t [R m contains the policy instruments. The initial state of the economy x 0 is known, x¯ and u¯ are target values and b is some discount factor. W, R and F are penalty matrices of conformable size. The above model is straightforward to solve for finite time and there are a number of packages available for computing its solution. However, a serious drawback for economics is that Eq. (2) does not allow for rational expectations and the model involves a finite horizon. One way of allowing RE to enter the model is to augment Eq. (2) in the following fashion

O DEx k

x t11 5 Ax t 1 Bu t 1 Cz 1

j

t t1j

1 et ,

(2a)

j51

where the matrix Dj is a parameter matrix, Et x t 11 is the expected state for time t1j at time t, k the maximum lead in the expectations formation and et is a white noise vector, see Amman et al. (1995). In order to compute the admissible set of instruments we have to eliminate the rational expectations from the model. In a previous paper, Amman and Kendrick (1997), we used Sims’ approach to solve the rational expectations in the model. Sims (1996) proposes a method based on generalized eigenvalues, see Moler and Stewart (1973) or Coleman and van Loan (1988). In order to apply Sims’ method we first put Eq. (2a) in the form 1

G0 x˜ t 11 5 G1 x˜ t 1 G2 u t 1 G3 z 1 G4 et ,

(3)

where I 2 D1 I G0,t 5 0 : 0

3

1

2 D2 0 I ...

... ... ... ?? ?

2 Dk 21 0 0 0 I

2 Dk 0 , 0 0 0

4

Note that in contrast to Sims (1996) the variable z contains exogenous variables and not random variables. Hence, the matrix P in Sims’ paper is set to zero.

H.M. Amman, D. A. Kendrick / Economics Letters 58 (1998) 185 – 191

A 0 G1 5 0 : 0

0 I 0 ?? ? ...

3

... ... ?? ?

0 0 0 , G2 5 I 0

4

B 0 , G 5 3 : 0

C 0 , G 5 4 : 0

187

I 0 , : 0

34 3 4 34

and the augmented state vector xt Ex t11 x˜ t 5 Ex t12 . : Ex t 1k21

3 4

(4)

Taking the generalized eigenvalues of Eq. (3) allows us to decompose the system matrices G0 and G1 in the following manner

L 5 QG0 Z, V 5 QG1 Z, with Z9Z5I and Q9Q5I. The matrices L and V are upper triangular matrices and the generalized eigenvalues are ;i vi,i /li,i . If we use the transformation w t 5Z9x˜ t we can write Eq. (3) as

Lw t11 5 V1 w t 1 QG2 u t 1 QG3 z 1 QG4 et .

(5)

Given the triangular structure of L and V we can partition (5) as follows

F

L11 0

GF G F

L12 L22

w 1,t11 V11 w 2,t11 5 0

V12 V22

GF G F G

F G F G

w 1,t Q1 Q1 Q1 w 2,t 1 Q 2 G2 u t 1 Q 2 G3 z 1 Q 2 G4 et ,

(6)

where the unstable eigenvalues are in lower right corner, i.e. in the matrices L22 and V22 . By forward propagation and taking expectations, it is possible to derive w 2,t as a function of future instruments and exogenous variables, Sims (Sims, 1996, page 5)

OM `

gt 5 w 2,t 5 2

j21

V 21 22 Q 2 (G2 u t 1j21 1 G3 z),

j 51

with M 5 V 21 22 L22 . In the steady state, however, Eq. (7) can be rewritten as

OM `

g 5 w2 5 2

j21

21 V 22 Q 2 (G2 u ` 1 G3 z),

j 51

where u ` is the optimal steady state solution of the control vector, which can be reduced to

(7)

H.M. Amman, D. A. Kendrick / Economics Letters 58 (1998) 185 – 191

188

21 g 5 w 2 5 2 (I 2 M)21 V 22 Q 2 (G2 u ` 1 G3 z).

(8)

Reinserting Eq. (8) into Eq. (6) and taking expectations gives us

L˜ w t11 5 V˜ w t 1 G˜2 u t 1 G˜3 z 1 G˜4 et 1 g˜ ,

(9)

with

F

G

L L˜ 5 11 0 G˜3 5

F

L12 ˜ 5 V11 , V I 0

G

F G

V12 Q , G˜2 5 1 G2 , 0 0

FQ0 G G , G˜ 5FQ0 G G , g˜ 5Fg0G, 1

1

3

4

4

Knowing that x˜ t 5Zw t we can write Eq. (9) as ˜ ˜ t 1 Bu ˜ t 1 Cz ˜ ˜ 1 e˜t , x˜ t11 5 Ax

(10)

21 ˜ Z9, B˜ 5 ZL˜ 21 G˜2 , C˜ 5 [ZL˜ 21 G˜3 ZL˜ 21 ], A˜ 5 ZL˜ V

(11)

with

FG

z 21 z˜ 5 g˜ , e˜t 5 ZL˜ G˜4 et ,

(12)

and the inverse

F

L 21 L˜ 21 5 11 0

G

2 L 21 11 L12 . I

(13)

We have to make the assumption here that L11 is nonsingular. However, the diagonal elements will generally be nonzero, so it is very likely that the matrix is nonsingular. With Eq. (10) we have removed the RE from the control model. The steady state of the LQ framework can be obtained by solving the algebraic matrix Riccati equation and tracking equation. The algebraic matrix Riccati equation has the form for the control model in Eqs. (1) and (2), see Amman et al. (1996) ˆ 5 b Aˆ 9XAˆ 2 ( b Aˆ 9XBˆ 1 Fˆ )(R9 1 b Bˆ 9XBˆ )21 ( b Bˆ 9XAˆ 1 Fˆ 9), X5W

(14)

and the tracking equation 21 ˆ ˆ 1 p) 2 Ru¯ ) 1 b Aˆ 9(XCz ˆ ˆ 1 p) 2 Wx, ˆ¯ p 5 2 ( b Aˆ 9XBˆ 1 Fˆ )( b Bˆ 9XBˆ 1 R9) ( b Bˆ 9(XCz

(15)

ˆ and Fˆ are the where X is the Riccati matrix and p the Riccati vector both for the steady state. W penalty matrices adjusted to conformable size of the augmented system in Eq. (10). In Amman and Neudecker (1997) a simple method is described for solving the Riccati matrix using a newton or quasi-newton solution method. Once we have derived the solution for X and p, it is easy to compute the steady state of the system. The optimal control is computed by the feedback equation

H.M. Amman, D. A. Kendrick / Economics Letters 58 (1998) 185 – 191

189

u ` 5 Gx ` 1 g,

(16)

G 5 2 ( b Bˆ 9XBˆ 1 R9)21 (Fˆ 9 1 b Bˆ 9XAˆ ),

(17)

21 ˆ ˆ 1 p) 2 Ru¯ ). g 5 2 ( b Bˆ 9XBˆ 1 R9) ( b Bˆ 9(XCz

(18)

with

In absence of random shocks we can compute the steady state from ˆ ` 1 Bu ˆ ` 1 Cz. ˆˆ x ` 5 Ax

(19)

With the help of the above equations, and knowing that g depends on u ` we can set up a simple iterative scheme to compute the steady state solution of the control vector. The algorithm is Step Step Step Step

0. 1. 2. 3.

Set the iteration counter n 50 and set the instruments u n` , to an initial value. Compute g n using Eq. (8). Compute X, p, G, g, x n`11 and u n`11 as described above. Set n 5 n 11 and goto Step 0 until convergence is reached.

3. An example In this section we will present a simple example that be checked by hand to illustrate the algorithm. Consider a simple macro model with output, x t , consumption, c t , investment, i t , government expenditures, gt , and taxes tt . The problem can then be stated as: Find the steady state solution of the control vector u ` and the corresponding state vector x ` that minimizes the welfare loss function

O

1 ` J5] 0.90 t h(x t 2 1600)2 1 g 2t j, 2 t50

(20)

for the model x t11 5 c t11 1 i t 11 1 gt11 ,

(21)

c t11 5 0.8(x t 2 tt ) 1 200,

(22)

i t 11 5 0.2Et x t12 1 100 1 et ,

(23)

gt11 5 u t ,

(24)

tt 11 5 0.25x t 11 .

(25)

If we reduce the above model to one equation for output we get

H.M. Amman, D. A. Kendrick / Economics Letters 58 (1998) 185 – 191

190

x t11 5 0.6x t 1 u t 1 0.2Et x t12 1 300 1 et ,

(26)

which leads to the augmented system

F

GF G F

2 0.2 0

1 1

GF G F G F G Fe G

x t11 0.6 5 Et x t 12 0

0 1

xt 1 300 t 1 u 1 zt 1 , x t11 0 t 0 0

(27)

Apply the QZ factorization, Coleman and van Loan (1988), to compute the generalized eigenvalues of the model gives us the time invariant solution 2

F1.0822 0 0.8202 Z 5F 0.5719

G F 2 0.5719 0.6523 , Q 5F 0.8203 G 2 0.7580

G

2 0.9136 0.7546 , V5 0.1848 0

L5

0.3979 , 0.7951

(28)

G

0.7580 , 0.6523

(29)

so the eigenvalues are h0.7546 / 1.0822, 0.7952 / 0.1848j5h0.6972, 4.328j and the ordering of the system is such that the unstable root 4.328 is in the lower right corner, so no reordering is required. The other components are

F 148.3243 C˜ 5F 103.4153 0.2966 A˜ 5 0.2068

G

F

G 0.1205 . 1.3031G

0.5745 0.2944 , B˜ 5 , 0.4006 0.3447 0.7580 0.5285

(30) (31)

The adjusted penalty matrices are in this case

F

˜ 5 1 W 0

G

FG

0 0 R 5 [1] F˜ 5 . 0 0

(32)

The Riccati matrix and the tracking vector have the solution

F1.0938 0.1818

X5

G

F

G

0.1818 2 2116.7492 , p5 , 0.3520 2 1000.8060

(33)

leading to a steady state of x ` 51585.66 and u ` 517.13.

4. Summary In this paper we have presented a method for solving the steady state solution of the Linear– Quadratic control model augmented with rational expectations. Our solution method is based on Sims’s method of generalized eigenvalues. By using an iterative scheme, the reduced model can be fitted into a standard linear–quadratic framework that allows us to derive the optimal policy instruments for the model with rational expectations. 2

As G0 is invertible we could have used a Schur decomposition in this example too.

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References Amman, H.M., 1996. Numerical optimization methods for dynamic optimization problems. In: Amman, H.M., Kendrick, D.A., Rust, J. (Eds.), Handbook of Computational Economics, 13, pp. 579–618. Appeared in the series: Handbooks in Economics, Arrow, K.J., Intriligator, M.D. (Eds.), North-Holland Publishers. Amman, H.M., Kendrick, D.A., 1997. Linear–quadratic optimization for models with rational expectations. Research Memorandum, University of Amsterdam. Amman, H.M., Kendrick, D.A., Achath, S., 1995. Solving stochastic optimization models with learning and rational expectations. Economics Letters 48, 9–13. Amman, H.M., Kendrick, D.A., Neudecker, H., 1996. Numerical steady state solutions for nonlinear dynamic optimization models. Research Memorandum, University of Amsterdam. Amman, H.M., Neudecker, H., 1997. Numerical solution methods of the algebraic matrix Riccati equation. Journal of Economic Dynamics and Control 21, 363–370. Blanchard, O.J., Kahn, C.M., 1980. The solution of linear difference models under rational expectations. Econometrica 48, 1305–1311. Coleman, T.F., van Loan, C., 1988. Handbook for matrix computations. Siam, Philadelphia. Kendrick, D.A., 1981. Stochastic Control for Economic Models. McGraw-Hill, New York. Moler, C.B., Stewart, G.W., 1973. An algorithm for generalized matrix eigenvalue problems. SIAM Journal on Numerical Analysis 10, 241–256. Sims, C.A. 1996. Solving linear rational expectations models. Research Paper, Department of Economics, Yale University.