Huijberts, H.

Withagen, C.A.A.M.

English

Research Papers in Economics

Economics  Letters  40  (1992)  7-11 
North-Holland 
Local  asymptotic  stability  of optimal 
steady  states  L 
Henri  Huijberts  and  Cees  Withagen  * 
Eindhoven  University  of  Technology,  Eindhoven,  Netherlands 
Received  7  May  1992 
Accepted  15  July  1992 
This  note  provides  a  sufficient  condition  for  steady  states  arising  from  economic  optimal  control  models  to  be  locally 
asymptotically  stable.  In  particular,  we  show  that  if  some  submatrix  of  the  matrix  of  eigenvectors  of  the  corresponding 
Jacobian  is  invertible,  the  stable  manifold  is  locally  y-parametrizable. 
1. Introduction 
The  use  of  optimal  control  theory  is  quite  common  in  economics.  There  is  a  large  variety  of 
fields  where  this  technique  is  employed:  growth  economics,  environmental  economics  and  many 
others.  It  often  occurs  that  there  is a  steady  state  solution  (e.g.  modified  golden  rule)  and  much 
effort  is put  into  its analysis,  justified  by the  fact  that  the  steady  state  may be  stable  in some  sense. 
The  stability  issue  itself  has  also  been  subject  to  intensive  research  [see  e.g.  Brock  and  Malliaris 
(1989)  for  an  overview].  The  emphasis  is on  global  asymptotic  stability  (GAS)  rather  than  on  local 
asymptotic  stability  (LAS);  as early  as 1976, Brock  and  Scheinkman  (1976, p.  166) have  put  forward 
that  the  literature  on  LAS  is ‘fairly  complete’.  In this  area  the  standard  references  are  Kurz  (19681, 
Levhari  and  Leviatan  (1972) and  Samuelson  (1972).  Roughly  speaking  one  proceeds  as follows.  The 
necessary  conditions  for  optimality  can  be  cast  into  a  system  of  differential  equations,  which  is 
linearized  around  the  steady  state,  if  any,  and  which  is  subsequently  analysed  in  terms  of  the 
eigenvalues  of  the  Jacobian.  The  occurrence  of  as many  eigenvalues  with  negative  real  parts  as the 
number  of  state  variables  is then  generally  deemed  sufficient  for  LAS,  the  basic  idea  being  that 
under  that  condition  there  exists  a  stable  manifold  containing  the  steady  state  [see  Burmeister 
(1968,  1980) for  applications].  This  reasoning  implicitly  ’  assumes  that  the  stable  manifold  is locally 
y-parametrizable  ‘non-degenerate’  in  the  sense  that  for  all  values  of  the  state  variables  (y)  in  a 
neighbourhood  of  the  steady  state  there  exist  values  of the  co-state  variables  so that  they  lie  all on 
the  stable  manifold.  However,  Scheinkman  (1976) provides  an example  where  this  assumption  does 
not  hold  true  (in a discrete-time  growth  model).  There  are  several  ways to  deal  with  the  degeneracy 
issue.  One  is to  make  particular  assumptions  guaranteeing  that  degeneracy  does  not  occur,  as  is 
done  by  Scheinkrnan  (1976)  in the  sequel  of  his  article;  in the  case  of  one  state  variable  it is often 
Correspondence  to:  Cees  Withagen,  Eindhoven  University  of  Technology,  Department  of  Mathematics  and  Computing 
Science,  P.O.  Box  513,  5600  MB  Eindhoven,  The  Netherlands. 
*  The  authors  are  indebted  to  Henk  Nijmijer  for  valuable  suggestions. 
’  Kurz  (1968)  makes  this  assumption  explicitly. 
0165-1765/92/$05.00  0  1992  -  Elsevier  Science  Publishers  B.V.  All  rights  reserved 8  H.  Huijberts,  C.  Withagen  /  Local  asymptotic  stability 
easy  to  depict  the  manifold  in a phase  diagram;  finally,  one  could  try  to  construct  the  manifold  for 
the  particular  model  at  hand  and  to  study  its properties.  The  latter  approach  has  proven  useful  in 
Van  Marrewijk  et  al. (1992)  but  the  analysis  turns  out  to  be  rather  cumbersome. 
The  above  observations  make  clear  that  LAS  is not  yet  completely  settled  and  raise  the  need  for 
a sufficient  condition  for  LAS  which,  preferably,  is easy  to  check.  It  is the  aim of  the  present  note 
to  provide  such  a condition. 
2. A  sufficient  condition  for  LAS 
In  this  section  we  shall  study  a system  of  differential  equations  of  the  following  type: 
i(t)  =Ax(t)  +f(t,  x(t)).  (2-l) 
Here  t  denotes  time.  x  is a 2n-vector.  A  is a given  real  2n * 2n  matrix  with  II characteristic  roots 
having  negative  real  part  and  n  characteristic  roots  having  positive  real  part.  f  has  the  following 
properties. 
F.1.  f  is continuous  in  (t,  x)  for  small  I x  I  and  t 2  0  and  continuously  differentiable  in  x  for  t 
sufficiently  large. 
F.2.  f(t,  0) =  0. 
The  relation  of  this  differential  equation  with  optimal  control  problems  can  be  sketched  as 
follows.  Consider  an optimal  control  problem  (OCP)  over  an infinite  horizon  with  n state  variables 
(y)  and  m  instruments.  Assume  that  the  OCP  has  a  solution  with  continuous  state  variables  and 
piece-wise  continuous  instruments.  For  the  situation  at  hand  Toman  (1985)  and  Cesari  (1983) 
provide  existence  theorems.  It will also be  assumed  that  the  maximization  of the  Hamiltonian  with 
respect  to  the  instruments  yields  the  instruments  as continuous  functions  of the  state  variables  and 
the  co-state  variables  (4).  See  Gale  and  Nikaido  (1965) on  the  issue  of global  univalence.  Then  the 
set  of  necessary  conditions  reduces  to  a  system  of  2n  differential  equations  in  z := (y,  4):  i  = 
g(t,  z).  Assume  next  that  g(t,  z *> =  0 for  some  z *. By defining  x := z -  z * one  obtains  i  = k(t,  x> 
for  some  k.  Finally,  k  is linearized  around  0 and  (2.1)  results.  It  is worth  stressing  that  the  steps 
described  here  are  by  no  means  trivial.  But  since  we  are  interested  here  in  LAS  only,  we  depart 
from  (2.1) with  its properties  mentioned,  without  going  into  the  basic  assumptions  underlying  these 
properties. 
The  following  lemma  plays  a crucial  role  in the  LAS  literature.  It is a paraphrase  of Theorem  4.1 
in Coddington  and  Levinson  (1955). 
Lemma.  Under  the  assumptions  made  with respect  to A  and  f  and  for  any  large  t *  there  exists in 
x-space  a  real  n-dimensional  manifold  S  containing  the  origin  such  that any solution  4  of  (2.1)  with 
+(t *)  on  the  manifold  S satisfies  4(t)  -+ 0  as t +  03. 
Remark.  For  a  more  modern  geometric  treatment,  see  e.g.  Abraham  and  Marsden  (1978),  Palis 
and  De  Melo  (1982),  Shub  (1987). 
Without  loss  of  generality  we  may  partition  x  such  that  the  first  n  elements  refer  to  the  state 
variables  (y)  and  the  final  n  elements  refer  to  the  co-state  variables  (4).  A  is  partitioned 
accordingly. H.  Huijberts,  C.  withagen  /  Local  asymptotic  stability 
We  assume  that  A  can  be  written  as 
with  P  positive  definite,  Q  symmetric  and  p  a nonnegative  constant.  It will moreover  be  assumed 
that  A  has  n  eigenvalues  with  negative  real  part  and  n eigenvalues  with  positive  real  part  and  that 
all eigenvalues  are  different.  These  assumptions  are  valid  in a large  variety  of economic  models,  for 
example  those  which  satisfy  some  concavity/convexity  conditions  and  where  there  is a nonnegative 
rate  of  discount  or  rate  of  time  preference  p. 
Let  T  be  the  matrix  having  the  eigenvectors  of  A  as columns.  Since  all eigenvalues  differ,  T  is 
invertible.  T  can  be  written  as 
T= 
where  T := CT:,  TTjT  is  the  matrix  of  eigenvectors  corresponding  to  the  negative  eigenvalues. 
According  to  Theorems  4.1 and  4.2 in Coddington  and  Levinson  (1955)  there  exists  an  n-vector  of 
functions  h := col(h,,  h,,  . . . , h,)  where  hi,.  . . , h,  are  continuously  differentiable  on  an open  set  U 
in  R”, containing  the  origin,  such  that  the  stable  manifold  is given  by 
where  77  E U and  (ah/aT)(O)  =  0. 
We call the  stable  manifold  locally  y-parametrizable  if there  exists  a neighborhood  Vc  R” of the 
origin  such  that  for  every  y E I/  there  is a  q E R”  with  the  property  that  (y,  q)  is on  the  stable 
manifold.  By  the  Implicit  Function  Theorem  it  follows  that  a  sufficient  condition  for  the  stable 
manifold  to  be  locally  y-parametrizable  is that  T,  is invertible.  Our  theorem  can  now  be  stated  as 
follows. 
Theorem.  Under  the  assumptions  stated  above  T,  is  invertible  and  hence  the  stable  manifold  is 
nondegenera  te. 
Proof.  The  proof  follows  the  same  lines  as the  proof  of Theorem  7.2 in Francis  (1988). Let  A _ be 
the  diagonal  matrix  with  the  stable  eigenvalues  of  A  as  its  diagonal  elements.  Decompose  A 
according  to 
A  =H+pB 
where 
(2.2) 10  H.  Huijberts,  C.  Withagen  /  Local  asymptotic  stability 
Introduce  the  symplectic  matrix 
Then  it  is easy  to  check  that  .ZZZ  is symmetric  and  JB  + BJ  = J.  We  have 
AT=  TA_. 
Premultiplying  (2.3) by  TTJ  we  obtain 
(2.3) 
TT~~T=  TT~TA_e  TT~HT=  TT~TA_-pTT~~T.  (2.4) 
Now  the  left-hand  side  of  (2.4)  is  symmetric,  hence  so  is the  right-hand  side  of  (2.4).  Using  the 
skew-symmetry  of  J  this  yields 
=  7;T~7;(1_e(p~-h_)TT~T=  TT~FA_.  (2.5) 
Since  p r  0, the  set  of  eigenvalues  of  pZ -A_  is in C+. Moreover,  the  set  of  eigenvalues  of  A_  is 
in  C-.  Hence  the  sets  of  eigenvalues  of  (pZ -  A_)  and  A_  respectively  are  disjoint.  Then  (2.5) 
implies  [cf. Gantmacher  (1960)] that  FTJT  =  0,  or 
T;Tl  =  TTT,.  (2.6) 
Now  assume  that  Tl  is not  ivertible,  i.e.,  there  is an  x #  0 such  that  T,x  = 0.  Premultiply  (2.31 by 
(IO)  to  obtain 
ST,  + PT,  =  T,A_.  (2.7) 
Premultiply  (2.7) by  x’TF  and  postmultiply  by  x: 
xTT,TSTlx  +xTT;PT2x  =xTT,TT,A_x  exTT,TpT,x  =xTTTT,h_x  =  0.  (2.8) 
Since  P >  0,  (2.8)  implies  that 
T,x  =  0. 
However,  this  implies  that 
x = 0, 
(2.9) 
which  contradicts  the  fact  that  (pz) has  full  column  rank.  Hence  T,  is invertible. H.  Huijberts,  C.  Withagen  /  Local  asymptotic  stability  11 
3.  Conclusion 
This  paper  offers  a sufficient  condition  for  the  local  asymptotic  stability  of optimal  steady  states. 
The  condition  is  easy  to  check  in  terms  of  the  properties  of  the  submatrices  constituting  the 
Jacobian  of the  linearized  system  of differential  equations,  but,  indeed,  many  economic  problems  in 
e.g.  growth  theory  produce  a Jacobian  with  the  desired  properties,  so that  tedious  calculations  to 
establish  that  the  stable  manifold  is locally  y-parametrizable  can  be  left  undone. 
References 
Abraham,  R.  and  J.E.  Marsden,  1978,  Foundations  of  Mechanics,  Second  edition  (Benjamin/Cummings,  Reading,  MA). 
Brock  W.  and  A.  Malliaris,  1989,  Differential  equations,  stability  and  chaos  in  dynamic  economics  (North-Holland, 
Amsterdam). 
Brock,  W.  and  J.  Scheinkman,  1976,  Global  asymptotic  stability  of  optimal  control  systems  with  applications  to  the  theory  of 
economic  growth,  Journal  of  Economic  Theory  12,  164-190. 
Burmeister,  E.,  1968,  The  role  of  the  Jacobian  determinant  in  the  two-sector  model,  International  Economic  Review  9, 
195-203. 
Burmeister,  E.,  1980,  Capital  theory  and  dynamics  (Cambridge  University  Press,  Cambridge). 
Cesari,  L.,  1983,  Optimization-theory  and  applications  (Springer,  Berlin). 
Coddington,  E.  and  N.  Levinson,  1955,  Theory  of  ordinary  differential  equations  (McGraw-Hill,  New  York). 
Francis,  B.,  1987,  A  course  in  H,  control  theory  (Springer,  Berlin). 
Gale,  D.  and  Nikaido,  H,  1965,  The  Jacobian  matrix  and  global  univalence  of  mappings,  Mathematische  Annalen,  81-93. 
Gantmacher,  F.,  1960,  Matrix  theory  I  (Chelsea  Publishing  Company,  New  York). 
Kurz,  M.,  1968,  The  general  instability  of  a  class  of  competitive  growth  processes,  Review  of  Economic  Studies  35,  155-174. 
Levhari,  D.  and  N.  Leviatan,  1972,  On  stability  in  the  saddle-point  sense,  Journal  of  Economic  Theory  4,  88-93. 
Palis,  J.  and  W.  de  Melo,  1982;  Geometric  theory  of  dynamical  systems  (Springer,  New  York). 
Samuelson,  P.,  1972,  The  general  saddlepoint  property  of  optimal-control  motions,  Journal  of  Economic  Theory  5,  102-120. 
Scheinkman,  J.,  1976,  On  optimal  steady  states  of  n-sector  growth  models  when  utility  is  discounted,  Journal  of  Economic 
Theory  12,  11-30. 
Shub,  M.,  1987,  Global  stability  of  dynamical  systems  (Springer,  Berlin). 
Toman,  M.,  1985,  Optimal  control  with  an  infinite  horizon,  Journal  of  Economic  Dynamics  and  Control  9,  291-316. 
Van  Marrewijk,  Ch.,  C.  de  Vries  and  C.  Withagen,  1992,  Optimal  localized  production  experience  and  schooling, 
International  Economic  Review,  forthcoming. 

Local asymptotic stability of optimal steady states.

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics