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Two models of dynamic input demand : estimates with Canadian manufacturing data Rushton, Michael 1990

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TWO MODELS OF DYNAMIC INPUT DEMAND: ESTIMATES WITH CANADIAN MANUFACTURING DATA By MICHAEL JOHNSTONE RUSHTON B.A., The U n i v e r s i t y o f B r i t i s h Columbia, 1 9 8 0 M.A.,  The U n i v e r s i t y o f Western O n t a r i o , 1 9 8 1  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department  We accept t h i s  o f Economics  t h e s i s as conforming  to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA July 1990 ©  M i c h a e l Johnstone Rushton, 1 9 9 0  In  presenting  degree  at the  this  thesis  in  University of  partial  fulfilment  of  of  department  this thesis for or  by  his  or  scholarly purposes may be her  representatives.  permission.  of  bcovNcrw^ics  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  o<-Ao^>er 3>  ,  for  an advanced  Library shall make  it  agree that permission for extensive  It  publication of this thesis for financial gain shall not  Department  requirements  British Columbia, I agree that the  freely available for reference and study. I further copying  the  is  granted  by the  understood  that  head of copying  my or  be allowed without my written  ABSTRACT Over t h e p a s t decade t h e r e has been a number o f i n n o v a t i o n s in of  t h e e s t i m a t i o n o f i n p u t demand e q u a t i o n s . In p a r t i c u l a r , incorporating  empirical  the hypothesis  models  of the f i r m  of  have  rational been  ways  expectations  developed  and  into  improved  upon. This critique  research  agenda  of econometric  econometric  results  government  policies.  critique  evaluation,  affect  regarding Lucas  at empirical  which  of business  Lucas  effects  that  o f how  would  give  of  various  part  of h i s  investment,  which had  of t a x p o l i c i e s designed t o  investment.  i n p u t demand a r e e s t i m a t e d w i t h Canadian t h e models  rational differ.  incorporates to  expectations, Neither  of  explains well  sector,  and  but  these  e s t i m a t e d w i t h Canadian model  account  directed  T h i s t h e s i s has a d u a l purpose. F i r s t ,  of  the  suggested  take  the p o s s i b l e  specifically  models  by  current behaviour  been used p r e v i o u s l y i n t h e assessment affect  inspired  d i d not e x p l i c i t l y  of the future  misleading  perhaps  policy  models which  expectations  was  some  the  models,  manufacturing data.  degree  the  specifications to  our  of  the  results  compare  with  of  technology  knowledge,  o f t h e Canadian  Each  hypothesis  has  d a t a . We a r e i n t e r e s t e d i n whether  the behaviour  i n how  two d i s t i n c t models o f  been  either  manufacturing  the  (few)  U.S.  a p p l i c a t i o n s o f t h i s type o f model. The  second purpose  simulations  to  assess  i s t o use t h e r e s u l t s the e f f e c t  ii  of  changes  o f t h e s e models i n t o the  after-tax  rental  rate  of  manufacturing. elsewhere) policies  While  that on  innovation  capital there  attempt  expectations.  most in  thesis  is  modelling  improve  the  of  able  of  to  employment in  effects  were  estimating  expectations  reliability  and  studies  studies  p a r t i c u l a r change w h i l e remaining the  been  c a l c u l a t e the  techniques This  investment  have  to  investment, of  on  very  c a p i t a l and  estimates  is  the  of  of  a  immune t o the Lucas c r i t i q u e . I f correct,  this  estimates,  but  could also  r e s u l t s can be summarized as f o l l o w s . The  different  the  rational  effects  i n d i c a t i o n of the e m p i r i c a l importance of the Lucas The  tax  to  with  the  (and  various  prior  models  examine  Canada  of  done  in  price  l a b o u r , even though they  only  give  some  critique. two  elasticities are s i m i l a r  not  models g i v e  of  demand  for  i n many r e s p e c t s  and are e s t i m a t e d w i t h a common data s e t . I t i s a l s o the case t h a t t h e i r estimates  of the e f f e c t s  to  rate  the  rental  are  of temporary and  different.  Adjusting  permanent the  changes  reduced  form  parameters of the i n p u t demand e q u a t i o n s  t o account f o r changes i n  tax  to  policy  suggesting an  regimes  alters  the  results  t h a t the e x p l i c i t m o d e l l i n g  empirically relevant  sense.  opposite  d i r e c t i o n s f o r the  suggests  that  more  research  two  a  significant  of e x p e c t a t i o n s  However,  these  is  required  between e x p e c t a t i o n s of f u t u r e p o l i c y and  iii  into  matters i n  effects  models c o n s i d e r e d  here.  the  investment  degree,  are  in  A l l this  relationship behaviour.  TABLE OF CONTENTS Page i i v  ABSTRACT LIST OF TABLES ACKNOWLEDGEMENT S  '  vi  CHAPTER 1 - INTRODUCTION  1  1.1 The N e o c l a s s i c a l Model o f Investment  with  D i s t r i b u t e d Lags 1.2 The N e o c l a s s i c a l Model o f Investment w i t h Adjustment Costs 1.3 R a t i o n a l E x p e c t a t i o n s and t h e Lucas C r i t i q u e 1.4 M o d e l l i n g Investment with R a t i o n a l E x p e c t a t i o n s 1.5 Tax P o l i c y and Investment CHAPTER 2 - THE DATA  2 6 9 14 20 30  2.1 Wages and R e n t a l Rates 2.2 C a p i t a l and Labour Inputs 2.3 Output  30 34 35  CHAPTER 3 - THE LINEAR QUADRATIC MODEL  37  3.1 3.2 3.3 3.4 3.5  A Time S e r i e s A n a l y s i s o f t h e Data The Model Estimates Simulations Conclusions  CHAPTER 4 - EPSTEIN AND DENNY'S MODEL 4.1 4.2 4.3 4.4 4.5 4.6  The Model Estimates The Model w i t h N o n - S t a t i c E x p e c t a t i o n s Simulations with Non-Static Expectations Comparing t h e Performance o f t h e Models Conclusions  37 47 53 64 68 ....75 77 81 88 96 99 103  CHAPTER 5 - CONCLUSION  108  REFERENCES  112  iv  LIST OF TABLES Table  Page  2.1 Data  36  3.1 E s t i m a t e o f u n r e s t r i c t e d l i n e a r - q u a d r a t i c model where i n p u t demands depend on c u r r e n t i n p u t p r i c e s  70  3.2 E s t i m a t e o f u n r e s t r i c t e d l i n e a r - q u a d r a t i c model where i n p u t demands depend on l a g g e d i n p u t p r i c e s  71  3.3 E s t i m a t e o f r e s t r i c t e d l i n e a r - q u a d r a t i c model where i n p u t demands depend on c u r r e n t i n p u t p r i c e s  72  3.4 E s t i m a t e of r e s t r i c t e d l i n e a r - q u a d r a t i c model where i n p u t demands depend on l a g g e d i n p u t p r i c e s  73  3.5 F o r e c a s t various  74  v a l u e s o f the c a p i t a l conditions  s t o c k under  4.1 R e s t r i c t e d e s t i m a t e s of f l e x i b l e f u n c t i o n a l form model with s t a t i c e x p e c t a t i o n s  105  4.2 R e s t r i c t e d e s t i m a t e s o f f l e x i b l e f u n c t i o n a l form model with autoregressive expectations  106  4.3 F o r e c a s t v a l u e s of the c a p i t a l s t o c k under v a r i o u s c o n d i t i o n s , u s i n g the f l e x i b l e f u n c t i o n a l form model  107  v  ACKNOWLEDGEMENTS I  am  greatly  Professors  John  Cragg,  Rudin  for their  thank  Professors  Michael  economics provided  Ardo  departments  to  errors  my  supervisory  Helliwell  Chris  Wilson that  for their remain.  while  and  William  I  members  o f Tasmania I finished  also  Schworm,  suggestions.  The  of  Jeremy  and p a t i e n c e .  Nicol,  at the u n i v e r s i t i e s  and encouragement  committee  (supervisor),  suggestions,  Hansson,  and Tom  f o r the  support  John  criticisms,  Tretheway,  responsible  indebted  I am  of the  and Regina w r i t i n g the  thesis. Finally, believed  my thanks and my  i n me  love  no matter how many  them I was " j u s t about  done".  vi  go t o my different  family, times  who  always  I would  tell  1 CHAPTER 1 Introduction Over t h e p a s t decade t h e r e has been a number o f i n n o v a t i o n s in of  t h e e s t i m a t i o n o f i n p u t demand e q u a t i o n s . In p a r t i c u l a r , incorporating  empirical  the hypothesis  models  of the f i r m  of  have  rational been  ways  expectations  developed  and  into  improved  upon. T h i s r e s e a r c h agenda was perhaps critique  o f econometric  econometric  models which  expectations  results  government  policies.  affect  regarding Lucas  at e m p i r i c a l  evaluation,  the  take  account  that  o f how  possible  of  various  part  of h i s  effects  directed  of business  investment,  give  which  had  of tax p o l i c i e s designed t o  investment.  T h i s t h e s i s has a d u a l purpose. F i r s t , i n p u t demand a r e e s t i m a t e d w i t h Canadian of  suggested  would  specifically  models  (1976)  current behaviour  been used p r e v i o u s l y i n t h e assessment affect  which  d i d not e x p l i c i t l y  of the future  misleading  critique  policy  i n s p i r e d by t h e Lucas  t h e models  rational differ.  incorporates to  expectations, Neither  of  these  e s t i m a t e d w i t h Canadian model e x p l a i n s w e l l sector,  and  i n how  but  some  the  models,  two d i s t i n c t models o f  manufacturing  degree  the  specifications to  our  data.  hypothesis of  knowledge,  the  results  o f t h e Canadian compare  with  of  technology has  d a t a . We a r e i n t e r e s t e d i n whether  the behaviour  Each  been  either  manufacturing  the  (few)  U.S.  a p p l i c a t i o n s o f t h i s t y p e o f model. The  second purpose  i s t o use t h e r e s u l t s  o f these models t o  2  assess  the  investment several  effect  of  the  rate  of  capital  and employment i n m a n u f a c t u r i n g . While  studies  in  Canada  c a l c u l a t e the e f f e c t s studies  rental  were  done  (and  elsewhere)  of v a r i o u s t a x p o l i c i e s  prior  to  the  innovation  in  Canada  t h e r e have been  that  attempt  on investment, of  to most  techniques  in  e s t i m a t i n g models w i t h r a t i o n a l e x p e c t a t i o n s . T h i s t h e s i s w i l l a b l e t o examine the e f f e c t s of temporary the  rental  r a t e w h i l e remaining  o n l y improve  immune t o the Lucas  the r e l i a b i l i t y  give  the  remainder  a brief  of t h i s  history  critique.  i s correct,  this  o f the e s t i m a t e s , but  some i n d i c a t i o n of the e m p i r i c a l importance In  chapter,  of the Lucas  sections  1.1  critique.  policy  previous  1.1  analysis  we  have  adopted  investment, thesis.  from  the method  those  used  in  studies.  The  essence  a firm w i l l  value  and shows how  differs  The N e o c l a s s i c a l Model o f Investment  inputs,  1.4  c o n t a i n s a s h o r t survey of r e s e a r c h on the q u e s t i o n o f  the e f f e c t s o f t a x p o l i c y on investment, of  give  through  o f m o d e l l i n g i n p u t demands and  If  could  also  and s k e t c h out the o r i g i n s of the models e s t i m a t e d i n t h i s S e c t i o n 1.5  be  and permanent changes i n  t h e m o d e l l i n g of t h e e x p e c t a t i o n s p r o c e s s not  on  choose  with  o f the  assumptions purchases  of  the  theories  of  investment  i t s p a t h o f c a p i t a l accumulation, objective  income (see  neoclassical  w i t h D i s t r i b u t e d Lags  stream  Jorgenson  a l l inputs,  o f maximizing generated (1967,  including  by  the the  p.136))  capital  and of o t h e r  present firm. are  goods,  i s that  discounted  Key that in  secondary the  firm  competitive  3 markets, that  that  the f i r m  the c o n t r i b u t i o n  sells  output  of c a p i t a l  i n a c o m p e t i t i v e market, and  t o output  i s through  a flow o f  s e r v i c e s p r o p o r t i o n a l t o the stock of c a p i t a l . It use  can be presumed t h a t  itself,  t h e f i r m purchases  capital  goods t o  o r t h a t t h e f i r m l e a s e s c a p i t a l goods from a s e p a r a t e  owner. In e i t h e r case, t h e c o s t o f u s i n g a u n i t of c a p i t a l f o r one time p e r i o d , c a l l e d t h e u s e r c o s t o f c a p i t a l o r t h e r e n t a l r a t e o f capital, capital firm  i s t h e same,  stock f o r the f i r m .  suggests  decreasing obtain  and i s a key determinant  that,  i f the  with the l e v e l  any c a p i t a l  Standard  microeconomic t h e o r y o f t h e  marginal  of c a p i t a l  s t o c k i t wishes  desire a l e v e l of c a p i t a l  of the optimal  product  capital  is  i n p u t , and i f t h e f i r m  may  a t market  of  prices,  firms  will  i n p u t such t h a t t h e m a r g i n a l product o f  c a p i t a l i s equal t o the user cost. In t h e absence o f c o r p o r a t e t a x e s o r s u b s i d i e s , t h e u s e r c o s t of  one d o l l a r ' s  interest capital  rate gains  worth  plus  of c a p i t a l  the d e p r e c i a t i o n rate  resulting  from  a change  goods over t h e y e a r  (see Jorgenson  p.253)) .  taxation,  Corporate  depreciation  f o r one year  and i n t e r e s t ,  capital  i n the p r i c e  to the  minus of  together  this  with  a more  any  capital  (1967, p.143) o r Boadway  yields  f o r the user cost of c a p i t a l ;  on  i s equal  (1980,  t a x exemptions f o r  complicated  expression  i s described later  i n Chapter  2. It  has l o n g been a c c e p t e d  that  while the n e o c l a s s i c a l  of t h e f i r m can g i v e us a t h e o r y o f t h e demand f o r c a p i t a l ,  model this  i s not t h e same t h i n g as a t h e o r y o f investment. C o n s i d e r an o f t e n  c i t e d e x t r a c t from Haavelmo What we s h o u l d r e j e c t is  a  demand  (1960,  p.216):  i s the naive reasoning that  schedule  f o r investment  which  there  could  be  d e r i v e d from a c l a s s i c a l scheme o f p r o d u c e r s ' b e h a v i o r i n maximizing  profit.  The  demand  f o r investment  s i m p l y be d e r i v e d from t h e demand f o r capital.  cannot  Demand f o r  a f i n i t e a d d i t i o n t o t h e s t o c k o f c a p i t a l can l e a d t o any rate  of  investment,  depending  on  from  almost  the additional  zero  to  infinity,  hypothesis  we  introduce  r e g a r d i n g t h e speed o f r e a c t i o n o f c a p i t a l - u s e r s . I t h i n k t h a t t h e sooner t h i s  n a i v e , and unfounded, t h e o r y o f t h e  demand-for-investment we s h a l l  have a chance  constructing capricious investment. This (1975,  p.  schedule  more  i s abandoned,  o f making  powerful  theories  short-run variations (Emphasis  some  real  t h e sooner progress i n  to deal  i n the rate  with the of private  i n original).  view  has not been u n i v e r s a l l y  304)  p o i n t s out t h a t  accepted  however.  i f we have determined  f o r c a p i t a l a t time t and a t time t + 1 then we have a l s o the r a t e o f n e t investment seems t o be s a y i n g , investment  will  over t h a t time  suggests  Bliss,  not n e c e s s a r i l y  t h e demand determined  i n t e r v a l . What Haavelmo  i s that  be e q u a l  Bliss  t h e demand  t o t h e change  for  net  in  the  l e v e l s o f c a p i t a l s t o c k t h a t equates m a r g i n a l p r o d u c t t o u s e r c o s t from  one  period  to  (described  below)  firms w i l l  capital.  In any case,  t h e next,  because  n o t always  Haavelmo,  f o r various choose  when d i s c u s s i n g  those  reasons  l e v e l s of  t h e 'demand f o r  5  investment'  i s referring  t o what f i r m s  d i s c u s s i n g t h e 'demand f o r c a p i t a l ' of c a p i t a l  f i r m s would choose  actually  he i s r e f e r r i n g t o t h e l e v e l s  i n a w o r l d o f no time  d e l i v e r y o f c a p i t a l goods and no adjustment the l e v e l s o f c a p i t a l Jorgenson's  lags  costs with respect t o  t o t h e problem  be t h e l e v e l  posed  of the c a p i t a l  by Haavelmo i s as  s t o c k a t time t t h a t  s a t i s f i e s t h e c o n d i t i o n o f m a r g i n a l p r o d u c t of c a p i t a l user  cost.  initiated  Assume t h a t such t h a t  on t h e  input.  solution  follows. Let K  do, whereas when  each  period  new investment  t h e b a c k l o g o f uncompleted  equals the  projects are  projects  i s equal  * t o t h e d i f f e r e n c e between Let  s t o c k of c a p i t a l  K .  P/(L) be a power s e r i e s i n t h e l a g o p e r a t o r L , which d e s c r i b e s  the  rate  at  completed, gross plus  and t h e a c t u a l  which  investment  projects  in  progress  and which i s g i v e n . I f 5 i s t h e d e p r e c i a t i o n  investment replacement  i n any p e r i o d investment,  e q u a l s investment  then  gross  become  r a t e , and  i n new p r o j e c t s  investment  a t time  tis  g i v e n by  I  fc  w(L)[K*  =  (Jorgenson  (1963,  - K*_ ] 1  + 5K  (1.1)  f c  pp.249-51)).  The demand  for capital  i s thus  t r a n s f o r m e d i n t o a 'demand f o r investment' by the a p p l i c a t i o n o f a rational  distributed  l a g process  (see Jorgenson  (1966)  for  an  a n a l y s i s o f the p r o p e r t i e s o f such a p r o c e s s ) . Jorgenson parameters  (1963)  and H a l l  o f investment  and Jorgenson  functions l i k e  (1.1),  (1967)  estimate the  and from these a r e  6 able t o i n f e r in  user  t h e response  costs  critical  of H a l l  specification something choosing  and output  of  more  o f investment prices.  Eisner  and Jorgenson's technology,  general  w i t h r e s p e c t t o changes and N a d i r i  reliance  when  i s called  on  t h e Cobb-Douglas  t h e evidence for,  (1968) a r e  suggests  and o f t h e i r  an a p p r o p r i a t e l a g s t r u c t u r e ,  since  that  method o f  i t turns  out t h a t  r e s u l t s a r e h i g h l y dependent on t h e r e s t r i c t i o n s p l a c e d on w(L). Gaudet,  May,  and M c F e t r i d g e  model w i t h a d i s t r i b u t e d  apply  (1976)  l a g t o Canadian  manufacturing  t o 1 9 7 3 ) , u s i n g a CES p r o d u c t i o n f u n c t i o n Eisner  and N a d i r i ) . They  c o u l d be u s e f u l that  the user  suggest  i n the a n a l y s i s cost  that  the n e o c l a s s i c a l  (which i s recommended by  the r e s u l t s  of tax p o l i c i e s ,  of c a p i t a l  has a  d a t a (1952  o f t h e model  since they  statistically  find  significant  e f f e c t on investment. How t h e m o d e l l i n g o f investment  has changed from  of Jorgenson over t h e p a s t two decades three  t h e method  i s t h e s u b j e c t o f t h e next  sections.  1.2 The N e o c l a s s i c a l Model o f Investment The  concept  neoclassical  o f adjustment  costs  t h e o r y o f investment  w i t h Adjustment  Costs  was  incorporated into the  because,  i n t h e words o f Lucas  (1967b, p.78) : ...many s t u d e n t s  o f investment  b e h a v i o r have r e c o g n i z e d  the i n c o n g r u i t y o f d e v e l o p i n g a r i g o r o u s economic t h e o r y of t h e d e t e r m i n a t i o n o f [K^] and then combining an ad hoc t h e o r y o f adjustment.  t h i s with  7 When we speak function also  o f adjustment  output  on  the  particular,  c o s t s we mean t h a t  depends not o n l y upon t h e l e v e l s rate  at  the f a s t e r  which  these  the l e v e l s  levels  (1970),  (1968), Lucas  and Treadway  costs  as convex.  level  of  absolute  capital  (1967a,  (1969,  That  changing.  a r e changing, t h e  Mortensen  (1973), Schramm  1970, 1971) a l l s p e c i f y  i n any  t o the f i r m  period  adjustment  o f changing i t s  i s increasing  v a l u e o f t h e change and i s i n c r e a s i n g  r a t e . A l l o f t h e s t u d i e s mentioned  a t an  with the increasing  above base adjustment  c o s t s on  the r a t e o f n e t investment, w i t h t h e e x c e p t i o n o f Gould, who g r o s s investment, and Lucas  In  paribus.  1967b),  i s , the cost  input  of i n p u t s , b u t  are  of inputs  lower w i l l be t h e r a t e o f output, ceteris Gould  i n the production  uses  (1967a), who uses g r o s s investment as  a proportion of c a p i t a l stock. S i n c e i t i s assumed i n t h e s e s t u d i e s t h a t c a p i t a l markets a r e c o m p e t i t i v e , t h e r a t i o n a l e g i v e n f o r adjustment there  i s net investment  capital  and  investment. devoted  labour, This  output  than  i t would  i s because  to i n s t a l l i n g ,  adjustment  costs  model - i . e . i t remains present  some  be  how  incorporated  t h e case t h a t  d i s c o u n t e d v a l u e o f cash  investment  has t h e same  the stocks of  and  capital  o f net must  t o work with,  be  t h e new  (1978, Chapter 3 ) ) .  flow,  and s e l l outputs i n c o m p e t i t i v e markets net  given  i n t h e absence  labour  and l e a r n i n g  l e v e l s o f i n p u t s (see N i c k e l l With  i s less,  c o s t s i s t h a t when  form  into  the  neoclassical  f i r m s seek t o maximize t h e and t h a t  they buy i n p u t s  - t h e r e s u l t i n g demand f o r  as was  o b t a i n e d by  Jorgenson.  8 That i s , n e t investment i s p r o p o r t i o n a l t o t h e d i f f e r e n c e between the  'target  reduced  stock'  form  of c a p i t a l  i s called  between t h i s  result  model  a  gives  equilibrium  adjustment  i s that  function  accumulation,  disequilibrium model  K .  This  The d i f f e r e n c e  t h e adjustment  that  (1975,  towards  cost  i s the r e s u l t  whereas  (see B l i s s  t h e movement  capital  accelerator.  and Jorgenson's  n e t investment  cost  and a c t u a l  fc  the f l e x i b l e  capital  essentially  K  K  Jorgenson's p.  *  305)).  of is  In t h e  i s not immediate  because i t would not be p r o f i t maximizing t o do s o . In Jorgenson's distributed there  l a g model movement towards K  a r e time l a g s  cannot  be a v o i d e d  involved no matter  *  i s not immediate because  i n completing what  price  capital  the f i r m  projects  that  i s w i l l i n g to  1 pay. In  e i t h e r the d i s t r i b u t e d  information existing its  about  future  prices  l a g o r t h e adjustment i s valuable  cost  t o the f i r m .  problem o f maximizing t h e p r e s e n t then t h e c h o i c e  discounted  value  of future  o f n e t investment i n t h e c u r r e n t  w i l l determine t h e c o n s t r a i n t a t t h e b e g i n n i n g o f t h e next prices  See lags  I f the  l e v e l o f c a p i t a l i n p u t a t any time c o n s t r a i n s t h e f i r m i n  cash flow,  If  model  Almon  a r e changing  i n some way over  (1965) f o r an e m p i r i c a l  between c a p i t a l  appropriations  time,  period period.  t h e way t h e y a r e  study o f such u n a v o i d a b l e time and e x p e n d i t u r e s .  The  notion  of u n a v o i d a b l e d e l a y s between i n i t i a t i n g a c a p i t a l p r o j e c t and t h e capital  being  a v a i l a b l e f o r production  services  " t i m e - t o - b u i l d " model o f Kydland and P r e s c o t t  reappears  (1982).  i n the  9 changing current  will net  affect  future  investment.  target  The  capital  therefore of  investment which f o l l o w e d the i n t r o d u c t i o n of adjustment c o s t s  was  1.3  Rational Expectations  expectations.  as c u r r e n t l y a p p l i e d t o economic models  (1961). He d e s c r i b e s the i d e a as f o l l o w s :  ...expectations  of  firms  (or,  more  subjective probability distribution be  empirical  and t h e Lucas C r i t i q u e  Rational expectations dates from Muth  to  and  models  the i n t r o d u c t i o n of r a t i o n a l  innovation  stocks  distributed,  f o r the  generally,  of outcomes) t e n d  same i n f o r m a t i o n  p r e d i c t i o n of the t h e o r y  (or the  the  set,  about  to the  "objective" probability  d i s t r i b u t i o n s of outcomes). In p r a c t i c e , the (1) the  information  s e t p o s t u l a t e d by m o d e l l e r s  s t r u c t u r e of the model i t s e l f ,  and  (2) a l l p a s t  includes values  of  assumption of r a t i o n a l e x p e c t a t i o n s  is  relevant variables. An  i m p l i c a t i o n of the  t h a t f o r e c a s t e r r o r s of f i r m s s h o u l d be any  v a r i a b l e s i n the  s i n c e any that  information  random and u n c o r r e l a t e d t o  s e t or t o past  values  of e r r o r s ,  such c o r r e l a t i o n would be u s e f u l i n f o r m a t i o n t o the f i r m  should  be  incorporated  into  d i s t r i b u t i o n of outcomes. T h i s i s why  the  subjective  probability  r a t i o n a l expectations  really  amounts t o a c o n s i s t e n c y c o n d i t i o n ; except f o r a random e r r o r term the  expectations  of  firms  should  be  c o n s i s t e n t with  the  outcomes  of the model which i n c o r p o r a t e s the f i r m s . We  have  mentioned  earlier  that  corporate  tax  policy  influences  the user cost  adjustment  costs  to  help  current  firms w i l l  determine period.  relationship  of c a p i t a l ,  i n t h e presence o f  want t o make f o r e c a s t s  the optimal  This  and t h a t  level  means t h a t  between t h e r a t e  of user  o f net investment  under  rational  costs  for  the  expectations the  o f net investment  and t h e c u r r e n t  user cost of c a p i t a l w i l l vary according t o expectations of future user  costs.  Yet H a l l  and Jorgenson  (1967)  treat  the  between investment and u s e r c o s t s as though i t were The of  method o f H a l l  tax policy  on  and Jorgenson  investment  was  stable.  f o r estimating  as  follows.  relation  the e f f e c t s  First,  derive  r e l a t i o n between t h e u s e r c o s t and t h e o p t i m a l c a p i t a l s t o c k  a  from  * a  static  relates  perspective,  K^. Then e s t i m a t e  the d i s t r i b u t e d  * investment t o l a g g e d v a l u e s o f K . Then ask how K would  have been d i f f e r e n t had u s e r c o s t s been d i f f e r e n t , some  hypothetical  distributed ask  alternative  lag function  which  tax  policy.  say because o f  Then,  was estimated  using  given  the  actual  data,  how t h e path o f investment would have been d i f f e r e n t . The  is  l a g that  *  c r i t i q u e o f econometric p o l i c y e v a l u a t i o n by Lucas  based  on  econometric  model,  example, w i l l analysts  the  idea  not be i n v a r i a n t  should  distinguish  of  Marschak).  Structural  regime,  reduced  say t h e d i s t r i b u t e d  parameters  analysis,  that  models  (a  parameters  l a g function  t o changes i n p o l i c y .  between warning  parameters,  structural made  as  of  policy  and reduced  early  an  w(L) f o r Thus  as  f o r t h e purposes  a r e those which a r e i n v a r i a n t where by p o l i c y  form  (1976)  form  1953  of  by  policy  t o changes i n t h e p o l i c y  regime we mean t h e p r o c e s s  which  guides  year  t o year  investment reduced in  changes t o t h e t a x s t r u c t u r e .  where e x p e c t a t i o n s o f f u t u r e  form o f t h e investment  the p o l i c y  regime,  Since  rental  equation w i l l  the r e s u l t s  of H a l l  i n a model o f  rates  matter t h e  change w i t h  changes  and Jorgenson  on t h e  e f f e c t s o f t a x p o l i c y on investment a r e not r e l i a b l e . The  r e s e a r c h agenda  devise  means  expectations (in  models  of  estimating  and which of  suggested by t h e Lucas  the  could  firm  models  which  identify  these  included  the s t r u c t u r a l usually  parameters).  rational parameters  taken  to  the  parameters  e s t i m a t e d t h e m o d e l l e r c o u l d determine how reduced form  parameters  would  policy  the  be  and adjustment  for different  With  was t o  technology  change  cost  are  critique  structural  regimes,  and c o u l d  more a c c u r a t e l y a s s e s s t h e impacts o f v a r i o u s p o l i c i e s . Although much o f t h i s r e s e a r c h was f o c u s s e d on t h e e s t i m a t i o n 2 of  complete  macroeconomic  models ,  we  direct  our a t t e n t i o n  to  t e c h n i q u e s o f e s t i m a t i n g models o f i n p u t demand i n t h e presence o f adjustment A  firm  returns levels  c o s t s , c a l l e d dynamic models o f i n p u t demand.  from  which  seeks  production subject  of inputs,  adjustment  i n p u t s , and market p r i c e s time  i n a way  See Chow (1983, example.  t o maximize  beyond  t o the constraints  costs  t o changing  11), T a y l o r  i s faced  these  with  stream  of  f o r i n p u t s and output which  i t s control,  Chapter  the discounted  existing  levels  of  e v o l v e over  a problem  (1979), o r W a l l i s  of  of  (1980) f o r  12  optimal the the  control.  firm's  3  The f i r s t  maximization  transversality  of  problem  include  by  f o r the s o l u t i o n t o  the Euler  The t e c h n i q u e  estimation"  t h e model  d i r e c t l y . Kennan  conditions  conditions.  "limited-information parameters  order  referred  involves  estimating  e q u a t i o n s and t o as  estimating  the Euler  the  equations  d e s c r i b e s how t h e E u l e r e q u a t i o n s might be  (1979) 4  estimated  efficiently.  Applications  of t h i s  method  t o dynamic  models o f i n p u t demand a r e d e s c r i b e d i n t h e f o l l o w i n g s e c t i o n . Under the  model  the technique i s solved  estimated. Sargent rule  There  (1980,  lagged  values  as " f u l l - i n f o r m a t i o n e s t i m a t i o n "  f o r a l l the f i r s t - o r d e r  a r e two methods  1981,  f o r input  known  1982)  demands of input  solve  of achieving  i n terms  variables  of current  period  and f u t u r e  then use r e s u l t s o f p r e d i c t i o n t h e o r y prices  this.  f o r the firm's  i n the current levels  conditions  (past  which can be e s t i m a t e d  v a r i a b l e s - lagged  input  prices.  future  They  expected  observations  of  p r i c e s and knowledge o f  the model which determines t h e s e v a r i a b l e s ) . T h i s demand e q u a t i o n  decision  as a f u n c t i o n o f  t o express  useful i n forecasting relevant  Hansen and  optimal  expected  information  and then  since  gives  an i n p u t  the right-hand  l e v e l s and t h e c u r r e n t  information  side set -  are a l l observed v a r i a b l e s . Technology parameters o f t h e f i r m and  A  standard  incorporates  reference  i s Chow  i n t o the optimal  i n r a t i o n a l expectations  (1975)  .  For a  treatment  c o n t r o l problem recent  see Sargent  (1987,  Chapter  developments 1).  4  A l s o see Hansen  (1982)  and Hansen and S i n g l e t o n  which  (1982) .  13 parameters which a r e used i n t h e p r o j e c t i o n o f c u r r e n t into  expected  future  Simultaneously, estimated.  of  will  rule  and  tests test  of r a t i o n a l  p. x v i i )  "hallmark"  refer  although yield  (1980a,  expectations.  expectations  1981)  the estimation different  parameterization  takes  from  of standard  constraint  "transition  a  remains  models;  Lucas  and  Sargent  restrictions the firm's  cross-parameter  and Sargent.  optimal  as a  decision  which  methods  Chow uses t h e where t h e  represents  the  problem.  dynamic  When t h e  i t remains t r u e t h a t t h e r e  testing  have  and does not  prices i s incorporated  i n i t s optimization  the model o f t h e f i r m and r a t i o n a l  approach,  c o n t r o l theory,  and output  restrictions  Full-information  different  full-information  equation",  f o r the f i r m  somewhat  parameters of t h e system a r e e s t i m a t e d , are  cross-equation  r e l a t e d t o t h e model used i n  Hansen  model o f t h e movement o f i n p u t the  the  this  prices.  results  into  of  in  o f t h e model o f t h e f i r m and  f o r i n p u t demand i s e x p l i c i t l y  Chow  directly.  restrictions  t o the cross-equation  of r a t i o n a l  forecasting  estimated  cross-equation  amount t o a j o i n t  the hypothesis  (1981,  be  estimation,  restrictions  are  t h e model used t o f o r e c a s t f u t u r e p r i c e s i s a l s o  There  simultaneous  prices  information  the j o i n t  hypothesis  of  expectations. so f a r o n l y  been  applied to  problems where t h e f i r m ' s o b j e c t i v e f u n c t i o n can be d e s c r i b e d i n linear  or quadratic  functions,  at l e a s t  cross-equation  terms.  This  i n t h e reduced  restrictions  leads  to linear  form parameters,  are non-linear  input  demand  although t h e  and extremely  complex  14  even  f o r models with  functions  also  only  allow  two  the  "certainty-equivalence";  inputs.  modeller the  the  invoke to  the  the  w o r l d i s the  objective  principle  firm's  same as  of  optimal  i t would  be  firm perfect foresight.  Comparisons of the of  to  solution  c o n t r o l problem i n the u n c e r t a i n had  Linear-quadratic  input  demand West  (1986).  numerically, standard  are  two  made by  compares and  errors  methods as  that  and  the  slightly,  and  (1986)  limited-  finds only  West  applied  to  dynamic models  Prucha  and  full-information  Nadiri methods  f u l l - i n f o r m a t i o n method  and  that  i t s parameter  lowers  estimates  t e n d t o be more b i a s e d than l i m i t e d - i n f o r m a t i o n e s t i m a t e s when the model  is  techniques  misspecified. by  "considerable  Prucha gains  Monte and  Carlo  Nadiri,  in statistical  comparisons on  the  efficiency"  of  other  the  two  hand,  find  from  using  of  input  (p.209)  f u l l - i n f o r m a t i o n methods.  1.4  Modelling  Investment w i t h R a t i o n a l  Full-information demand  is  usually  linear-quadratic scalar  output  and  Meese  E p s t e i n and (1980),  carried  objective  of  out  function  a  dynamic  with for  the the  i s a column-vector of  form of the p r o d u c t i o n  See  estimation  Expectations  function, excluding  Yatchew  Sargent  (1985),  (1978),  assumption firm.*'  inputs  If  y  then the  of is  a a  usual  adjustment c o s t s i s  Hansen and  and West  model  (1986)  Sargent  (1980,  f o r examples.  1981),  15  '  t -  y  a  '  X  t  +  t '  X  A  x  t  /  2  ( 1  where a  i s a v e c t o r and  A  matrix.  Convex adjustment  is a  symmetric  c o s t s are  also  and  negative  specified  -  2 )  definite  as q u a d r a t i c ,  and c o u l d be r e p r e s e n t e d by  ( x  t  "  t - l  X  )  ,  B  (  X  t  " Vl  ) / 2  ( 1  -  3 )  where B i s a symmetric m a t r i x . The  theoretical  results  of  such  a model  of  the  firm  r a t i o n a l e x p e c t a t i o n s are d e s c r i b e d by Lucas and P r e s c o t t who and  are p r i m a r i l y concerned by  Sargent  (1979,  with the e q u i l i b r i u m p r i c e of  Chapter  14,  and  1981)  p a r t i c u l a r the o p t i m a l d e c i s i o n r u l e f o r the The  first  e m p i r i c a l use  who  are  adjustment  (1978) ,  who  firm.  Sargent  b e i n g exogenous with r e s p e c t t o l a b o u r demand justified begins  on  the  basis  the p a p e r ) . Kennan  model u s i n g Sargent's where  of  labour  causality  (1988),  on  tests  capital, in  of t h i s model i s Sargent  costs present.  (1971),  considers  c o n s i d e r s the demand f o r l a b o u r , both s t r a i g h t - t i m e and when t h e r e  under  over-time,  takes  wages  ( t h i s assumption with  which  as is  Sargent  the o t h e r hand, e s t i m a t e s  a  f i r m t o g e t h e r with endogenous l a b o u r supply,  suppliers  also  have  linear-quadratic  objective  6 functions.  Kennan's Nickell  model  is  drawn  from  (1986) f o r a complete  Sargent survey  (1979,  Chapter  16).  of dynamic models of  See  labour  Meese demand  (1980) uses f u l l - i n f o r m a t i o n techniques  for  capital  q u a r t e r l y data  and  from 1947  with  estimates labour model  of  user  expectations two  respect  demand  for  costs  to  and  model...is capable  a  equation  and  of  latter.  the to  rejects  the  a  of  penalty  model,  an  he  and  wages  The  as  simultaneously  each f o r demand f o r autoregressive  hypothesis  of  restrictions  constrained  task.  rational  are  Few  version  software  of  routines  former highly  f u n c t i o n , . where  u n r e s t r i c t e d four  the are  complexity...  appending a " c o n c e n t r a t e d  comparing  the  penalty  likelihood function  of the model. A l i k e l i h o o d the  restricted  equation  restrictions  at  version  vector any  to  ratio what  autoregression,  significance  level  2%.  E p s t e i n and Yatchew find  (1985) take the t h e o r e t i c a l model used by  a reparameterization  somewhat s i m p l i f i e s  demand.  the  restrictions  theoretical  g r e a t e r than  that  costs  bivariate  The  (using  (pp.149-50):  i s c a r r i e d out by with  and  a  of e s t i m a t i n g a model of such  amounts  Meese,  and  wages.  difficult  weights the v a r i o u s test  of u s e r  model: one  capital,  the  to a l i n e a r - q u a d r a t i c  levels,  the  estimation  function"  according  input  n o n - l i n e a r . Meese remarks  Estimation  manufacturing  imposes r e s t r i c t i o n s between parameters of the  equations  The  U.S.  the m o d e l l i n g  a four equation  and  by  t o 1974)  model. A f t e r j u s t i f y i n g exogenous  labour  to estimate  the  of the  estimation  estimating  of the  equations  restricted  model.  It  is  this  simplified  amendment, w i t h full  Canadian d a t a  d i s c u s s i o n of  until  version  the  that  in this  Epstein  and  is  estimated,  thesis  without  i n Chapter  Yatchew  method  3,  so  a  i s deferred  later. We  now  turn  estimating  our  dynamic  attention  models  of  to  alternative  input  demand  methods  with  of  rational  expectations. A model of Tobin's investment  known  as  (1969) has g e n e r a t e d a method of m o d e l l i n g  "q-theory".  Tobin's  q  i s the  nominal  market v a l u e of a f i r m t o the nominal  capital  s t o c k e v a l u a t e d at replacement  than of  one  the  should lead to p o s i t i v e new  investment, then  be  capital  will  assuming  positively  be  there  are  related  to  the  v a l u e of the f i r m ' s  investment, than  convex the  of  c o s t . A v a l u e of q g r e a t e r  net  greater  ratio  s i n c e the  i t s c o s t . The  adjustment  current  rate  costs,  value  of  value of  should  q.  It  is  assumed t h a t the v a l u e of the f i r m ' s e q u i t y c a p t u r e s the market's e x p e c t a t i o n s about the f u t u r e v a l u e of c a p i t a l . Hayashi  (1982) makes  model investment q"  we  two  s h o u l d use  important  observations.  "marginal  q"  as the e x p l a n a t o r y v a r i a b l e ,  where the  change i n the market v a l u e of the  firm  c a p i t a l d i v i d e d by the p r i c e of a u n i t is  the t o t a l market v a l u e of the  of  the  capital  stock  at  r a t h e r than  former  of c a p i t a l ,  values.  i s the  marginal unit  and the the t o t a l  Second,  model u s i n g m a r g i n a l q and the n e o c l a s s i c a l model w i t h  a  to  "average  f o r an a d d i t i o n a l  f i r m d i v i d e d by  replacement  First,  of  latter value  q-theory adjustment  18 c o s t s and  r a t i o n a l expectations  7 theories .  are e q u i v a l e n t  Q-theory models of investment have been e s t i m a t e d by Abel  (1980), Summers  (1981), and  McKibbin and  Siegloff  Hayashi, (1988).  A  problem w i t h e m p i r i c a l a p p l i c a t i o n of q-theory i s t h a t : ...(with)  the  investor  use  of  perception  opportunities...the But  the  must  relevant  the  valuation  to  volatility  of  (Bosworth  be  taken  commenting  stock  i t "begs the q u e s t i o n ,  [ f a c t o r s determine v a l u e s  (1980, p.  since  r e f e r r e d to Euler  estimated  sacrificed that  we  equations directly.  not  linear-quadratic.  as  of  the  the  i s that  if  we  fluctuations in e x p l a i n what  restrict  some  i s used,  ourselves  allows  the  dynamic i n p u t demand  "limited-information"  firm's  Although  technique  This  77)  i n ] the s t o c k market".  earlier  when t h i s  need  stock  Summers  i t does not  A second a l t e r n a t i v e method t o m o d e l l i n g  are  be  130).  market t o e x p l a i n  where the  in  on  r e l y on v a r i a t i o n s i n the  that  toto.  may  "noise"  Abel  is  in  investment the  infer  investment  A f u r t h e r problem, mentioned by  investment,  to  physical  information  fluctuations.  (1981), p.  market  of  information  overshadowed by market  stock  to  optimal  control  useful there  problem  information is  the  technologies  possible  method,  use  of  for  this  is  advantage that  are  production  f u n c t i o n s t h a t more c l o s e l y f i t the f a c t s .  Hayashi  credits  Lucas  a l t h o u g h t h e y d i d not put  and  Prescott  (1971)  i t i n t h e s e terms.  insight,  Pindyck techniques  to  quadratic is  and  Rotemberg  a model with  adjustment  estimated  annual  (1983a) and  1949-76  for  techniques,  but  with  quarterly  U.S.  Bischoff  (1986)  1959 are  aware,  While  that  data  their  results  user  costs, since this  The  (1986a)  function,  for  1955-80. Kokkelenberg  and  production  t o Canadian  data,  a  variable  1962-83).  the  Carmichael,  short-run data  from  Mohnen,  Their tests  fail  to  the model. I t i s i n t e r e s t i n g  statistic "impact  of c a p i t a l  will  we and  c o s t f u n c t i o n t o Quebec  also  elasticity"  ( f o r a shock t o user  be  with  -  reject to  note  respect to  estimated  t o be  in  0.0 98  and  this the  c o s t s t h a t i s permanent  approach t o e s t i m a t i n g dynamic  form  cost  to  manufacturing  r a t i o n a l e x p e c t a t i o n s i s from E p s t e i n and Denny  variable  uses  for  r e c o g n i z e d as permanent) t o be -0.271.  final  functional  (1948-71  same  approximation  f o r the e l a s t i c i t y  elasticity  and immediately  data  model  the  q u a r t e r l y U.S.  (annual  here  long-run  polynomial  imposed by  They f i n d  Shapiro  from  function,  t h e r e are no s t u d i e s of t h i s type, of which  apply  restrictions  thesis.  data  cost  e x p e c t a t i o n s . The  manufacturing  (1989) a p p l y a t r a n s l o g  manufacturing the  a  restricted  Cobb-Douglas  c o s t f u n c t i o n on  t o 1977.  Vigeant  U.S.  manufacturing use  apply l i m i t e d - i n f o r m a t i o n  rational  (1983b)).  a  b)  a translog  c o s t s , and  with  variable  (1983a,  is  chosen  minimization  for  the  problem,  value where  i n p u t demands (1983). A  f u n c t i o n of adjustment  with  flexible a  firm's  costs  are  p r e s e n t . A l i m i t e d s p e c i f i c a t i o n of e x p e c t a t i o n s i s allowed; i t i s supposed t h a t r e a l  input p r i c e s  follow f i r s t - o r d e r autoregressive  processes.  While t h i s may  expectations, useful is  i n that  not be c o n s i s t e n t with  there  may  be  other  f o r f o r e c a s t i n g input p r i c e s ,  usually specified  anyway.  A  more  Neither dynamic  information  i n practice i n rational of  available  expectations  this  model  models  i s found  in  thesis.  the l i n e a r - q u a d r a t i c r a t i o n a l  input  rational"  i t c l o s e l y approximates what  complete d i s c u s s i o n  Chapter 4 of t h i s  "fully  demand,  nor t h e model  expectations  of E p s t e i n  model of  and Denny,  have 9  been e s t i m a t e d , One  of  the  estimate,  t o our knowledge, with  two  and  principal  compare,  Canadian aggregate  contributions  these  two  of  methods  demands, u s i n g Canadian manufacturing  this of  thesis  data. i s to  estimating  input  data.  1.5 Tax P o l i c y And Investment Since investment, and  the  introduction  which p r o v i d e d  therefore  corporate these  t a x p o l i c i e s on  investment.  neoclassical  between user to  theories  costs  investment,  models t o c o n s i d e r  H a l l and Jorgenson  and Berndt  a link  tax policy,  have been u s i n g  Morrison  of  the e f f e c t s  of  capital,  researchers of v a r i o u s  (1967) i s f r e q u e n t l y c i t e d as t h e  (1981) e s t i m a t e  of  seminal  a s t a t i c expectations version  of a model s i m i l a r t o E p s t e i n and Denny's. 9 Bernstein Epstein  (1986)  and Denny's  estimates  a  model with  static  expectations  the pooled  f i r m s engaged i n r e s e a r c h and development.  data  version  of  of some Canadian  article and  in this  investment  neoclassical  field. since  models  g i v e n by C h i r i n k o  The number o f s t u d i e s  Hall  and Jorgenson  of investment  on U.S.  i s immense;  tax p o l i c y a survey o f  and t a x p o l i c y i n the U.S. i s  (1986, 1987) . C h i r i n k o and E i s n e r  (1983) compare  the e m p i r i c a l p r e d i c t i o n s of a number o f U.S. macroeconomic models r e g a r d i n g t a x p o l i c y and investment. Here  we  investment,  will  confine  which w i l l  the d i s c u s s i o n  later  t o Canadian s t u d i e s  be compared  t o the r e s u l t s  the various  studies  of  obtained  in this thesis.  done  Bird  (1980)  in  Canada  quasi-empirical. investment  classifies by The  i n Canada  Review Committee  three two  types: major  survey,  surveys  are H e l l i w e l l  (1975). H e l l i w e l l  have  been  econometric,  and  on  (1966)  that  tax  policy  and t h e Tax  considers  Measures  the b e h a v i o u r  l a r g e f i r m s , 35 of which a r e i n manufacturing, t h e o t h e r s resources  and  of 70  deal i n  o r s e r v i c e s . These f i r m s are taken from those which were  i n t e r v i e w e d by the Royal Commission on Banking and which were a l s o sent  questionnaires  initiatives  by t h e Royal Commission  are examined.  One  is a  1961  on T a x a t i o n .  change  to  Two  tax  depreciation  allowances, which a l l o w e d  d e p r e c i a t i o n at double t h e normal r a t e s  in  was  the  year  depreciation  an in  asset  following  purchased,  years.  This  with  normal  provision  lasted  January 1, 1964. Summarizing the r e s u l t s of the surveys, (1968, p. 128)  says the measure  was  not "...thought  have had a n o t i c e a b l e i n f l u e n c e on t h e i r investment The o t h e r  i n i t i a t i v e i s a 1963 p r o p o s a l  rates  of  until  Helliwell  by f i r m s  to  expenditures".  t o a l l o w 50% s t r a i g h t - l i n e  depreciation the two  on machinery  and  y e a r s commencing June 14,  25% Canadian  processing  was  to expire  i n June 1965,  end  of  Helliwell  1966.  responses  1963  for capital by  firms  purchased  in  which are e i t h e r  owned and c o n t r o l l e d or are i n v o l v e d i n m a n u f a c t u r i n g  and  impact  equipment  by  i n designated  which  incentive  of  slow  a l t h o u g h i t was  (1966,  managers  of t h i s  areas  pp. in  was  The  change  measure  l a t e r extended  170-72) p r o v i d e s  general  to  growth.  suggest  the  a  number  that  timing  of  t o the  the  of  main  investment  p r o j e c t s . F o r example, one company spokesman s a i d : Although we  wouldn't undertake  accelerated  depreciation,  equipment is The in  the  e a r l y t o a l l o w us  Tax  t o take  will  order  whatever  Measures Review Committee was  effects  on  firms  i n the 1972  accelerated  depreciation  t h e i r survey, t h e y ...83  per  our  advantage  cent o f the  operations.  would  investment  s e c t o r . With  tax  incentives  These i n c e n t i v e s  allowances  respondents  have  and 1,288  lower firms  consisted  corporate  investment  impact  on  Tax Measures Review Committee  10  (1975, p.  9)  the  their  expenditures  t a x measures were a n t i c i p a t e d by  respondents...  tax  responding to  anticipated that  some p o s i t i v e  ...Increased  of the  cent o f the  the  especially interested  found:  measures  result  of  f e d e r a l budget.  r a t e s i n the m a n u f a c t u r i n g  tax  probably  o f the  obtainable.  introduced of  we  a p r o j e c t because  as 47  a  per  The  positive  responses  o b t a i n e d are  r e s u l t s of H e l l i w e l l , and as we with e m p i r i c a l  s t u d i e s . May  surprising  g i v e n the  survey  s h a l l see l a t e r are not c o n s i s t e n t  (1979,  p. 73)  c l a i m s "The  f i n d i n g s of  the Committee were g r e e t e d w i t h a good d e a l of s k e p t i c i s m by the p r o f e s s i o n a l and p o l i t i c a l communities".  both  I t i s w e l l known t h a t  economists are o f t e n s k e p t i c a l r e g a r d i n g any r e s u l t s o f s u r v e y s . By has  quasi-empirical  i n mind  formal  way  Hyndman  studies  research that than  o f investment  "...uses  numbers,  i n the econometric  (1974), Harman and Johnson  Bird but  studies".  (1980,  p.  i n a much  Three  examples  (1978), and Johnson  42) less are  and S c a r t h  (1979) . Hyndman's concern changes Tax  i s the  f o r manufacturing  Measures  Review  effects  firms  Committee  that study  o f the  1972  were a l s o  corporate tax  the  f o c u s of  discussed e a r l i e r .  Hyndman  does not a c t u a l l y c a l c u l a t e the e f f e c t s on investment, but assesses  the  production,  effects  of  tax  changes  l e a v i n g the reader t o i n f e r  might be. He says the 1972 and  the  equipment  on  the  i n manufacturing  by  at most  20%,  which  rather  costs  what the o v e r a l l  changes lowered u s e r c o s t s o f  the  of  effects  machinery he  claims  i n c r e a s e s the p r i c e o f f i n a l output r e l a t i v e t o c o s t s by about  3%.  Harman and Johnson e s t i m a t e t h e e l a s t i c i t y of investment w i t h r e s p e c t t o u s e r c o s t s o f c a p i t a l u s i n g a model f i r s t Coen  (1971), i n which investment depends on new  past  investment, and the u s e r c o s t  suggested by  o r d e r s , cash flow,  r e l a t i v e t o wages. The  are used t o c a l c u l a t e t h e impacts of v a r i o u s investment from the  1963  federal  budget  t o the  1972  budget.  results  incentives  S i n c e the o n l y  figures the  r e p o r t e d are the p r e s e n t v a l u e o f i n d u c e d investment  budget  initiatives  i t is  difficult  to  infer  the  from  relevant  elasticities. Johnson emphasis  and  on the  Scarth, ratio  "tax e x p e n d i t u r e " by model  of  tax c r e d i t  The  of  model  and  t h e government. H a l l is  used  and  investment  r a t h e r parameter  Harman  Johnson,  of i n d u c e d investment  investment  investment  like  a  to  t o the  and  calculate  itself  i s not  v a l u e s are imposed,  level  much  of  Jorgenson's the  l o w e r i n g o f the  place  (1967)  effects  of  corporate tax  actually  of  an  rate.  estimated,  so the impact  the  but  investment  i n c e n t i v e s on investment i s g i v e n a p r i o r i . Turning behaviour,  to  the  econometric  M c F e t r i d g e and May Wilson's capital  capital  to  and  aforementioned  elasticity -0.67.  of  Capital proportions  is  are  investment  lag  linking  user costs range  variables  are  to  the  estimated  putty-clay fixed  "backlogged" p r o j e c t s  of  output. A  models  are  research  Canada  appropriations.  investment  Similar  model  distributed  capital  funds,  on  of  on  investment  Wilson  (1967),  (1976), and B r a i t h w a i t e (1983).  s t o c k depends on  corporate the  a  category  studies  econometric  and  1 1  projects  final  in for  the  At of  assumes  completed any  time  capital,  of  reduced  interest  rate  U.S.  particular  completed  the  forms The  optimal  data  long  found  sense  that but  of  involving  i s estimated  projects,  (Wilson (1967, p. 3 6 ) ) .  investment  availability  estimated.  with  putty-clay  run as  somewhat  factor not  for  lower  elasticities:  (1971) found that  Jorgenson  -0.23. W i l s o n  t h e peak  investment  (1963) o b t a i n e d -0.38 and B i s c h o f f  also  found,  response  as d i d t h e U.S.  t o an i n t e r e s t  rate  studies, shock was  one year f o l l o w i n g t h e shock (the model i s q u a r t e r l y ) . McFetridge Jorgenson's, relative The  and May e s t i m a t e a model o f investment  with  investment  input p r i c e s  impact  elasticity  cost of c a p i t a l ,  being  and depending of c a p i t a l  log-linear  in  similar to output  and  on an ad hoc l a g s t r u c t u r e .  s t o c k with  respect t o the user  a^ i n t h e i r n o t a t i o n , i s -0.08, and t h e l o n g run 12  elasticity,  Tf^, i s -0.43.  accelerated  d e p r e c i a t i o n changes o f 1972. T h e i r e s t i m a t e  e x t r a g r o s s investment about  one-quarter  The r e s u l t s  a r e used  t o analyse the of the  i n d u c e d by t h e change i n t a x p o l i c y i s o n l y  the  estimate 13  of  the  Tax  Measures  Review  Committee r e f e r r e d t o e a r l i e r . Braithwaite  embeds  h i s model  o f investment  i n t h e Economic  C o u n c i l o f Canada's macroeconomic model CANDIDE 2.0. Investment i s modelled  as depending  relative  t o the user  stock;  i.e.  on d i s t r i b u t e d  l a g s o f t h e v a l u e o f output  cost of c a p i t a l ,  no p r o d u c t i o n  function  and l e v e l s is explicitly  of the c a p i t a l d e s c r i b e d . The  In Chapters 3 and 4 o f t h i s t h e s i s t h e e l a s t i c i t y o f c a p i t a l w i t h r e s p e c t t o u s e r c o s t i s e s t i m a t e d and can be compared d i r e c t l y t o M c F e t r i d g e and May's r e s u l t s . 13 May larger study.  (1979) remarks t h a t t h e Tax Measures Review Committee c l a i m e d effects  of tax p o l i c y  on  investment  than  any  empirical  26 estimates On  of h i s investment  e q u a t i o n are d i f f i c u l t  to  interpret.  the o t h e r hand, a s i m u l a t i o n o f B r a i t h w a i t e ' s which w i l l  interest  to  us  i s an  i n c r e a s e i n the  investment  tax  be  of  credit.  He  c o n s i d e r s a permanent i n c r e a s e , i n 1 9 8 0 , by a f a c t o r of 1.8 i n the investment  tax  credit.  investment  tax c r e d i t  increasing  the  machinery  and  base  Since  in  i n Canada rate  to  equipment  was  1980  the  7%,  the  1 2 . 6 % . The  base  of  experiment  effect  i n manufacturing,  rate  on  the  involves  investment  relative  to  the  case, e x p r e s s e d i n terms of m i l l i o n s of 1971 d o l l a r s i s :  in  base  14  year  1980  1981  1982  1983  1984  1985  change  +30  +82  +126  +134  +125  +68  In  1 9 8 3 , where the e f f e c t  of  the base case g r o s s Each  different  of  the  Bird  described i n  t o e s t i m a t i n g the  ( 1 9 8 0 , p.  46)  remarks  the e f f e c t s of i n c e n t i v e s on investment above  must  Babel".  feel  Yet  with  as  though the  peaks,  he  has  exception  this  section  effects  Feldstein  1 4  Braithwaite  i s 3.2%  the  into  Tax  quite  tax p o l i c y to  from the s t u d i e s  wandered of  of  take  "anyone t r y i n g  Committee t h e r e does seem t o be some consensus capital i s fairly  the e f f e c t  investment.  studies  approaches  investment.  on investment  the  on  discern reviewed  Tower  Measures  of  Review  t h a t the demand f o r  i n e l a s t i c w i t h r e s p e c t t o the u s e r c o s t .  (1982) e l o q u e n t l y s t a t e s the case f o r c o n s i d e r i n g a  ( 1 9 8 3 , p.  67)  wide range of t h e o r e t i c a l models of investment when attempting  to  ask  be  any  question,  invariant  for  looking  t o model s e l e c t i o n .  demand w i l l  be  adjustment models  and  estimated.  costs  we  when  obtain  for  In t h i s  Each  of  changing  results t h e s i s two  the  models  input  estimates  under  i s c l e a r that  possible  ways  of  investment, but other  studies  the  the  i t i s hoped t h a t  described  models of  For  above. We  each  of  a  of  results  will  be  not  exhaust a l l  the  r e n t a l rate  will  complement  generate e l a s t i c i t i e s  very d i f f e r e n t  from those  found by  case can a l s o be made t h a t the models used here  designed  of to  an  improvement  improve  the  way  over input  previous  studies;  in  the data  the  demand models  are  t o be used i n e s t i m a t i o n  costs  others. represent techniques estimated,  which have o n l y v e r y r e c e n t l y been developed, are put t o In Chapter 2 ,  on  i n t e r e s t e d i n whether  models of i n p u t demand w i t h adjustment  something  the  priori  r a t i o n a l expectations  The  face  expectations.  effects the  input  firms  alternative  models used here w i l l  calculating  seem t o  assumes  levels.  s p e c i f i c a t i o n s of s t a t i c and n o n - s t a t i c It  which  use.  i s described  detail. The  linear-quadratic  estimated  in  properties  of the  compatible applied. Yatchew  Chapter  with The  3.  data the  model Some  is  described  analysis  of  in the  detail  time-series  i s undertaken t o ensure t h a t the theoretical  simplified  estimation  (1985) i s used. The  the model i s d e s c r i b e d ,  and  model  to  which  procedure  method of d o i n g  of  policy  and  they  data  are  will  be  Epstein  and  analysis  with  some s i m u l a t i o n s are c a r r i e d out  which  28 c o n s i d e r the e f f e c t s o f changes i n u s e r c o s t s on In  Chapter  4  the  model  of  Epstein  investment.  and  Denny  (1983)  is  d e s c r i b e d and e s t i m a t e d . S i m u l a t i o n s s i m i l a r t o those i n Chapter 3 are of  done f o r the purposes  o f comparison,  the dependence of p o l i c y  simulation  and p r o v i d e an results  on  indication  the  particular  useful  f o r three  model of dynamic i n p u t demand chosen. The  results  reasons.  First,  usefulness  of  these  they  of  such  three  provide  chapters  us  models  with  and  are  more  whether  i n f o r m a t i o n on there  are  advantages  i n u s i n g one type o f s p e c i f i c a t i o n r a t h e r than  Chapter  contains  4  (1981) "P t e s t " , model  where  which,  the  the  rental  application  of  Davidson  f o r each o f the two  other  hypothesis.Second, of  an  model  is  taken  and  the  distinct another.  MacKinnon's  models, e v a l u a t e s the as  the  alternative  t h e y w i l l p r o v i d e some e s t i m a t e s of the e f f e c t s  rate  of  capital  on  investment,  which  can  then  be  compared t o p r e v i o u s s t u d i e s . T h i r d , we w i l l o b t a i n some e m p i r i c a l e v i d e n c e on the dependence o f the reduced form parameters demand e q u a t i o n s on e x p e c t a t i o n s . A l a n B l i n d e r has The  Lucas  critique  may  be  correct,  but  of input  remarked: ^ 1  I have  seen  no  p e r s u a s i v e e v i d e n c e i n any sphere t o i n d i c a t e t h a t i t i s empirically made.  The  regimes cause  In Klamer  i m p o r t a n t . The big  cause  question  empirical  is  whether  case  i s yet t o  changes  in  be  policy  l a r g e changes i n c o e f f i c i e n t s . Maybe they  just very t i n y  changes.  ( 1 9 8 3 , p. 1 6 6 ) .  We  will  changes  provide in  coefficients Chapter comparisons  evidence  the  'tax  on  the  policy  q u e s t i o n of the  regime'  affect  degree t o  the  reduced  which form  of i n p u t demand e q u a t i o n s . 5 with  concludes previous  the  thesis.  studies,  The  will  p o s s i b l e f u t u r e r e s e a r c h w i l l be d e s c r i b e d .  results, be  and  the  summarized,  and  30 CHAPTER 2 The In t h i s  Data  c h a p t e r a l l d a t a t h a t w i l l be used i n e s t i m a t i n g the  i n p u t demand models i n Chapters 3 and 4 are d e s c r i b e d .  2.1  Wages and R e n t a l Rates Table  rental  2.1  rate  CANDIDE  lists  the  for capital  2.0  Database,  data  used  i s from  and  in this  the  thesis.  Economic  The  nominal  C o u n c i l of  i s described i n d e t a i l  Canada  by B r a i t h w a i t e  (1983). The  following  capital  is  competitive  taken firm  derivation from  of  Boadway  which  uses  the  implicit  (1980).  capital,  a  unit  of  capital  q.  According to  investment  i n the  absence  of  any  Imagine  k,  C a l l the m a r g i n a l p r o d u c t of c a p i t a l MPK  rental  to  rate  a  produce  of  perfectly output,  y.  and the purchase p r i c e of  the  neo-classical  adjustment  lags  or  theory  of  adjustment  c o s t s i n c a p i t a l s t o c k , the f i r m w i l l purchase u n i t s of c a p i t a l t o the p o i n t where the p r i c e of a u n i t  up  of c a p i t a l i s e q u a l t o i t s  n e t - o f - t a x p r e s e n t d i s c o u n t e d v a l u e o f m a r g i n a l revenue  product:  00  q  = S P MPK (l-U )e"  fc  s  s  ( R + 6 ) ( S _ t )  s  ds +  q u Z (l-ITC )+q ITC t  t  t  t  t  t  (2.1) where p rate,  s  R  capital,  i s the p r i c e is  the  o f output  interest  rate,  at time 6  is  s, u  the  i s the p r o f i t s  depreciation  Z i s the p r e s e n t d i s c o u n t e d v a l u e o f d e d u c t i o n s  rate  tax on  allowed  f o r d e p r e c i a t i o n and i n t e r e s t c o s t s , and ITC i s the investment  tax  credit case  rate. in  that  Canada,  depreciation the  Note  and  investment  i t i s presumed here,  that  the  interest  tax  amount  allowances  credit.  of  as  i s actually  capital  eligible  i s reduced by  Differentiating  the for  the amount o f  equation  (2.1)  with  respect to t gives:  q =(R+5)q (1-u Z (1-ITC J ) - I T C - ( 1 - u J p MPK +q u Z (1-ITC )+o_ITC t t t t t t t t t t t t t t t  where q  i s the time d e r i v a t i v e of q .  fc  MPK  S o l v i n g f o r MPK^  fc  =  t  ( (R + 5) q t  gives:  - q ) (1 - u Z ) (1 - ITC ) / ( (1 - u. ) p^.) t t t t t t (2.2)  The  right  hand s i d e o f t h i s  r a t e of c a p i t a l , f o r one  capital  s t o c k i s the l e v e l  time p e r i o d .  e q u a l s the r e a l i m p l i c i t the  implicit  rental  which i s the n e t - o f - t a x c o s t o f u s i n g a u n i t  capital  In  e q u a t i o n i s the r e a l  series  At each p o i n t  i n time  the  desired  of  capital  where the m a r g i n a l product  r e n t a l r a t e , h e r e a f t e r denoted r .  for  the  nominal  rental  rate  in  Canadian  m a n u f a c t u r i n g g i v e n i n T a b l e 2.1,  the investment goods p r i c e  q^  The  i s set  equal  to  1  in  1971.  p r o v i d e s d a t a f o r the nominal will  be  p^_, a l s o  deflated listed  by  use i n the model of Chapter  in  Economic  4.  sector  output  t o generate the r e a l  this  index Canada series  price  series  index r^_ f o r  3, and the s e r i e s w i l l be d e f l a t e d  sector material  the model of Chapter  C o u n c i l of  r e n t a l r a t e on c a p i t a l ;  the m a n u f a c t u r i n g  i n T a b l e 2.1,  the m a n u f a c t u r i n g  of  input p r i c e  Since p  index m^  by  when used  i s normalized to equal  1 in  32 1971,  r  output  i s then dollars  t h e a f t e r - t a x annual  of using  cost  one 1971 d o l l a r ' s  i n 1971 worth  manufacturing  o f machinery and  equipment. Braithewaite detail,  so o n l y  (1983,  a brief  pp  9-15)  describes  description i s given  here.  rate  approximate  outset  that  rental  r a t e , d u e t o t h e many c o m p l e x i t i e s  due  the rental  The equity being  can only  tax, while taking f u l l  interest cost  rate  R  of capital  obtained Ideally,  corporate  the true  o f t h e t a x system, and behaviour  account of others.  i s a weighted  average  and t h e a f t e r - t a x bond  by  dividing  we  tax rate  unavailable marginal  total  would  taxes  paid  of the expected  rate,  the weights  tax rates  might  be  "effective" by  tax  net taxable  t o use a measure t h e average recent  on  1  rate,  income.  of the  rate.  research  calculated  rate  marginal  This  data  how  effective  s e e Boadway  (1987)  was  or  (1987).  a body o f r e s e a r c h  on t h e e f f e c t s o f changes i n t h e  t a x s y s t e m on t h e f i n a n c i n g d e c i s i o n s  beyond t h e scope o f t h i s  such e f f e c t s i n our data.  capital.  the effects of the real rental  than  t o u s ; f o r some  There e x i s t s  research.  i s the  want  rather  Boadway, B r u c e , a n d M i n t z  corporate  u  i n t r y i n g t o evaluate  investment,  effective  tax rate  great  We n o t e a t t h e  determined by t h e h i s t o r i c a l e q u i t y share o f t o t a l The  is  data  in  t o o u r i g n o r i n g some o f t h e p o s s i b l e e f f e c t s o n f i r m  of t h e corporate  on  the data  of firms, but i t  t h e s i s t o attempt t o i n c o r p o r a t e any  See A u e r b a c h  (1983) f o r a s u r v e y  of this  33 Equation the  (2.2)  assumption  Canada  d e s c r i b i n g the r e n t a l r a t e of c a p i t a l  that  there  loss-offsetting  estimates  of  how  this  is  is  full  loss-offsetting.  imperfect.  Mintz  imperfection  (1988)  might  contains  In  fact  provides  affect  in  some  effective  tax  r a t e s i n Canada. The  investment t a x c r e d i t  government budget of detail  by  Timbrell  1975.  (ITC)  The  (1975).  was  terms of the Boadway  and  d e s c r i b e the changes t o the ITC up t o Briefly, structures  the  and  capital  will  measure  of  ITC  is  a  tax  machinery and refer  the  simply  value  are  Kitchen  credit  on  described (1984,  gross  (note  machinery  capital  ITC  that  and  stock  in this  used  for  was  the end it  of 1978  the base r a t e was  is  (2.1)  the  reduced by e x a c t l y the  before  the  ITC  real  per  cent.  r e n t a l r a t e on  rate; i f r  i s introduced,  five  At and  1984.  effective ITC  ITC  credit.  i n c r e a s e d t o seven per cent,  remained at t h i s l e v e l through In e q u a t i o n  the  The  depreciation  When i n t r o d u c e d ,  r a t e of  in  thesis  equipment).  investment t a x  base  p.146)  investment  allowances i s reduced by the amount of the the  in  1984.  equipment to  of  i n t r o d u c e d i n the f e d e r a l  i t is  i s the -  ITC)  capital  real r e n t a l rate after  the  ITC  is  i n t r o d u c e d . T h i s i s the r e s u l t of assuming t h a t the purchase p r i c e of c a p i t a l goods, q,  i s determined i n a c o m p e t i t i v e i n t e r n a t i o n a l  market, of which the Canadian m a n u f a c t u r i n g s e c t o r i s but  a  part.  leased,  then we  I f we  imagine  that  capital  i s a c t u a l l y rented  or  small  c o u l d d e s c r i b e the model as assuming t h a t the l e s s e e bears  the burden of the  corporate  tax  and  r e c e i v e s the  b e n e f i t s of  the  investment other  tax credit.  studies  of taxation  general equilibrium pp.383-4).  T h i s has been  Note  and investment;  analysis  that  t h e working  i n Hamilton  we  ignore  assumption  in  see, f o r example, t h e and Whalley  the  treatment  (1989, e s p . by  foreign  governments o f c o r p o r a t e income earned i n Canada by m u l t i - n a t i o n a l firms.  This s i m p l i f i e s  series,  but  Hartman  this  the c o n s t r u c t i o n  simplification  o f the r e a l  i s perhaps  rental  rate  justifiable;  see  (1985) f o r an e x p l a n a t i o n o f why t h e f o r e i g n t a x r a t e s can  be i r r e l e v a n t t o t h e investment d e c i s i o n o f a m u l t i - n a t i o n a l The nominal wage d a t a a r e a l s o from t h e CANDIDE 2.0 and  a r e average  this  series  p^_,  so t h a t  output  i s deflated w^_  i n manufacturing.  by t h e m a n u f a c t u r i n g  i s average  hourly earnings  Database,  In Chapter  output  price  3  index  i n 1971 m a n u f a c t u r i n g  dollars.  The m  hourly earnings  firm.  output p r i c e index p^ and t h e m a t e r i a l i n p u t p r i c e  are constructed  from  Statistics  Canada's  Input-Output  index data  (annual c a t a l o g u e s 15-201E and 15-202E), p r i c e s b e i n g i m p l i e d and revealed  by  dividing  the  current  dollar  statistics  with  the  c o n s t a n t d o l l a r s t a t i s t i c s . P r i c e s f o r each s e c t o r a r e weighted i n the  index by t h e s e c t o r ' s  share o f t o t a l m a n u f a c t u r i n g  output i n  that year.  2.2 C a p i t a l and Labour Capital Table  and l a b o u r  2.1. C a p i t a l ,  manufacturing  Inputs inputs  f o r manufacturing  k^_, i s machinery  are l i s t e d i n  and equipment  i n Canadian  as g i v e n by t h e Economic C o u n c i l o f Canada, and i s  35 measured i n terms (1983,  pp.  o f m i l l i o n s o f 1971  15-16) d e s c r i b e s  measured i n terms  services,  such and  w t  worth.  of Canada's d a t a b a s e .  that  l '  r  t^t'  annual  t  both measured i n terms  a n n u a  l  on  o f m i l l i o n s o f 1971  real  Statistics Economy, capital  output  Canada's  various and  the  simply  total  from  i n the  output  on  from are  capital  labour  services,  dollars  (note t h a t  manufacturing  are the  index).  of  1961  reveals  non-manufactured  o t h e r s e c t o r s produce  goods  of  the  manufactured  sector",  manufacturing  goods p r o d u c e d  is  the k  and  fc  output  sector,  (e.g.  some goods  services)  not  of  the  produces  that  some  (e.g. t h e a g r i c u l t u r e goods).  Here  output p r i c e index i s derived and  are  does  with the d e f i n i t i o n of  sector,  1  i s taken  and  and  from  Canadian  Inspection  some m a n u f a c t u r e d  Note a l s o t h a t t h e i m p l i c i t  of  Since  goods.  y ,  that the manufacturing sector  manufactured  produce  1984.  manufacturing  the d e f i n i t i o n of output c o n s i s t e n t  output  to  "manufacturing the  sector,  Statistics  manufacturing  tables  sectors  the  Input-Output  represent  forestry  of  issues  labour  Input-Output some  taken  is  the data  expenditures  expenditures  1^,  Output The  from  i s also  Note t h a t  d e f l a t o r i s a manufacturing sector output p r i c e  2.3  Braithewaite  i t i s c a l c u l a t e d . Labour,  o f m i l l i o n s o f manhours, and  t h e Economic C o u n c i l constructed  how  dollars'  not  from  b y t h e e n t i r e economy.  the  we  and make  inputs. from  output  the of  36  TABLE 2.1 Data N  Year  P  y  w  1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984  0. 8315 0. 8393 0. 8486 0. 8588 0. 8714 0. 8926 0. 9104 0. 9254 0. 9524 0. 9792 1. 0000 1. 0417 1. 1449 1. 3677 1. 5331 1. 6209 1. 7471 1. 9204 2 .1896 2 .4646 2 .7481 2 .9396 3. 0411 3. 1385  29649 .6 32324 .9 34734 .9 37951 .9 41292 .6 44149 .2 45235 .4 48126 .1 51338 .9 50617 .6 53479 .1 57571 .9 62669 .9 65105 .2 61241 .8 65056 .3 66613 .1 70069 .5 73236 .8 71942 .4 72967 .5 65851 .7 69292 .3 76700 .9  2 .18 2 .24 2 .30 2 .36 2 .63 2 .78 2 .99 3 .23 3 .41 3 .65 3 .91 4 .18 4 .57 5 .27 6 .13 6 .92 7 .62 8 .10 8 .81 9 .55 11 .01 12 .31 13 .21 13 .49  p i s a price  1  r  2789 2903 3010 3201 3184 3377 3367 3328 3445 3370 3346 3466 3630 3702 3513 3599 3564 3702 3877 3920 3911 3517 3484 3621  N  k  .113 .121 .121 .114 .119 .122 .132 .138 .144 .149 .143 .158 .135 .148 .163 .164 .190 .211 .236 .274 .344 .356 .362 .424  m 0 .7978 0 .8130 0 .8254 0 .8400 0 .8596 0 .8866 0 . 9067 0 .9240 0 . 9574 0 .9834 1 .0000 1 .0478 1 .1694 1 .3894 1 .5391 1 .6136 1 .7349 1 . 9069 2 .1880 2 .4243 2 .6130 2 .7331 2 .7886 2 .9028  9049. 3 9300. 3 9531. 0 10207 .5 11055. 8 12183. 2 12954. 7 13327. 5 13907. 3 14742. 6 15336. 4 15811. 3 16696. 2 17813. 4 18715. 3 19404. 4 20004. 6 20376. 6 21013. 3 21982. 7 23286. 0 23805. 9 23698. 8 23603. 4  index o f goods and s e r v i c e s produced by t h e Canadian  m a n u f a c t u r i n g s e c t o r , y i s t h e q u a n t i t y o f such goods and s e r v i c e s N measured i n m i l l i o n s o f 1971 d o l l a r s , w i s average hourly earnings  i n manufacturing  manhours the  measured  of labour i n manufacturing  implicit  rental  rate  of  in  current  measured i n m i l l i o n s ,  machinery  cost  of r e n t i n g  one  k i s t h e s t o c k o f machinery  and  worth  worth,  and m  i s a price  measured i n m i l l i o n s index  Canadian m a n u f a c t u r i n g  sector.  The  and y  source  Statistics source  f o r p, m, of  the  Canadian  f o r a l l other  data  Economic C o u n c i l o f Canada.  is in  1971 d o l l a r ' s  i n manufacturing  r  equipment  measured i n t h e c u r r e n t d o l l a r  equipment  1 is N  and  manufacturing  of c a p i t a l ,  dollars,  of m a t e r i a l  i s Statistics Economy  i n p u t s used  Canada,  (various  i s t h e C AND IDE  o f 1971  2.0  dollar's by t h e  Input-Output  issues), databank  and t h e of the  37 CHAPTER 3 The L i n e a r Q u a d r a t i c In t h i s  c h a p t e r a model o f i n p u t  estimated  with  described  i n Chapter  necessary  t o examine  data,  since  t h e data  from  2.  Model demand i s s p e c i f i e d and i s  t h e Canadian  Before  t h e model  some o f t h e time  t h e model  requires  that  manufacturing  sector  i s described,  i tis  series  properties  t h e data  satisfy  of the certain  conditions.  3.1 A Time S e r i e s A n a l y s i s o f t h e Data We now examine some o f t h e time s e r i e s p r o p e r t i e s o f t h e wage and  rental  series are  rate  are s t a t i o n a r y ,  exogenous  motivation  with  for this  demand i s l a t e r firms  data.  We  ask i n t u r n  ( i ) whether t h e two  time  and ( i i ) whether t h e wage and r e n t a l  rate  respect  t o the l e v e l s of input  examination  i s that  e s t i m a t e d we w i l l  i s t o assume t h a t  i n p u t p r i c e s can r e p r e s e n t  of future  a linear  t h e way f i r m s made  We d e f l a t e t h e nominal  rental rate  m a n u f a c t u r i n g output p r i c e index g i v e n Assume f o r now rental  rate  process,  (the assumption  and wage  each  follow  of input  input  time  p r i c e s . One  s e r i e s model o f  expectations.  and nominal  wage by t h e  i n T a b l e 2.1.  i s justified a  how  first-order  below) t h a t t h e autoregressive  which we w r i t e as  + e  ' t  when t h e model  need some way t o r e p r e s e n t  might have formed e x p e c t a t i o n s  p o s s i b l e method  demands. The  It  = v +6 w + e 2 22 t-1 2t  (3.1)  (3.2)  38  where  i  I r 2 are  =  Hansen  and  Sargent  necessary c o n d i t i o n rule  in  the  R  is  the  it  will  be  stationary  expectations  chapter i s  <  rate.  In  whether  and  e s t i m a t e s of 9^  =  (3.1)  .9211  Fuller  (.0473)  (1976)  | 2• e  and  <  1  2  w  8  922  ^°  under  hypotheses  9^=  1  the  null  cannot  be  used  to  "significantly different" appropriate test  D i c k e y and  Fuller  where  n 0 t  h  following  a  and  decision  model  to  be  5 (1+R) '  <  more  stringent  22  "  a  |&1  I n  o  can  we  t  n  e  be  words,  r  considered  from for  v  that  =  l f  the  S O  .8271  are  in  estimates  of  distributions t h e  6  whether  =  errors  standard  e  6  obtain  standard  922  establish  e s t :  "-  and  8^  °-  '-  m a t e s  are  2  1.  8  <  1 and  (1979). Nelson and  examine whether U.S. The  (3.2)  demonstrates  and  trend.  5  that  input  section  and  and  of  to  fact  quadratic  this  r  standard errors  An  firm's  (1+R)'  < 1 and  the  processes.  parentheses.  above  the  linear  {9^1  tested:  determined  From OLS (.1222)  convergence of  discount  r e s t r i c t i o n s are  errors.  (1981, p.136) e s t a b l i s h  for  rational  estimated i n t h i s where  random  aggregate  equations  922  Plosser  output  are  <  1  escr:  (1982) use  is  stationary  estimated  by  Dea  ^  this  test  around  ordinary  t  - t-1>  =  ( W  t  " t-1>  =  E  W  "lO  a  20  +  +  a  a  ii t-l r  21 t-l W  +  +  a  a  !2  22  ( r  ( W  t-l  t-l  " W  - t-2 W  +  }  +  U  U  l t  2t  ( 3  a  least  squares:  ( r  by  ( 3  '  -  4 )  3 )  Enough l a g g e d v a l u e s of the dependent v a r i a b l e s o f ( 3 . 3 ) and (3.4) are i n c l u d e d on t h e r i g h t hand s i d e s of these e q u a t i o n s error  terms are w h i t e - n o i s e ;  u n t i l the  one l a g t u r n e d out t o be  sufficient  f o r t h e s e s e r i e s . OLS e s t i m a t e s of ( 3 . 3 ) and (3.4) a r e :  r  - r  _ = .024(.017) - .191(.131)r  f c  + .044(.238)<r  1  - r  w^ - w^ = .376 (.183) - .090 (.049)w^ , + . 381 (. 197) (w^ t t-1 t-1 t-1  )  - w  „)  X.—2.  where s t a n d a r d e r r o r s a r e i n p a r e n t h e s e s . The &2\  a  r  e  null  n e  test  of s t a t i o n a r i t y  9 tive  of  non-stationarity  i s (n - p ) a ^ (1 - o ^ )  respectively,  whether e s t i m a t e s  of  and  and s i g n i f i c a n t l y d i f f e r e n t from zero; i f so, t h e  a  hypothesis  statistic  asks  is  where  1  rejected. i =  n i s the number o f o b s e r v a t i o n s  1,2  The  test  f o r r and  w  (22) and p i s t h e  number o f r i g h t hand s i d e v a r i a b l e s i n the r e g r e s s i o n ( 3 ) . So f o r r  this  Fuller  test  statistic  equals  (1976, p. 371) p r o v i d e s t h e d i s t r i b u t i o n  ( i n h i s n o t a t i o n the t e s t From  his  Table  non-stationarity approximately  West points have  -3.788 and f o r w i t equals  (1988)  8.5.1  rejected  we  find  statistic  i s d i s t r i b u t e d as n(p that  i s not r e j e c t e d :  the  t h e 0.1  null  - 1)).  hypothesis  of  significance  level i s  random-walk  behavior"  -10.2.^  i n h i s d i s c u s s i o n of  out t h a t with here,  statistic  of t h i s  -2.751.  i t will  "near  o n l y a s m a l l number o f o b s e r v a t i o n s , as  we  be  be  as a p o s s i b i l i t y  unlikely even  that  n o n - s t a t i o n a r i t y can  i f the true value  of the  lagged  The  only  stationarity  study  of  real  given  by  Sigurdson  wages  in  Canada  wage  Sargent all  and  I  Stewart a  am  Meese  aware  that  i s the  (1990) ;  random  to reject  f o r the  (1978),  which  wages i n Canada  follow  (1980) are unable hourly  of  suggest  Altonji  and  the h y p o t h e s i s t h a t  entire  U.S.  (1980),  economy  and  the  preliminary results  they  walk.  examines  real  Ashenfelter  the r e a l  follows a  E p s t e i n and  that  average  random  Yatchew  walk.  (1985)  e m p i r i c a l models of dynamic f a c t o r demand e s t i m a t e d with  manufacturing explicit  sector  test  of  data;  none  stationarity  of  for  these real  studies  input  U.S.  contains  prices,  are  an  although  s t a t i o n a r i t y i s imposed by d e t r e n d i n g the d a t a . We  now  c o n s i d e r the p r o p e r t i e s o f the  rental  rate  series i n  more d e t a i l . The  r  r e s u l t of OLS  = .0222 +  estimation i s  .8271r _ t  1  (.0159) (.1222)  with  standard  .1609,  suggesting  Applying  and  no  Gujarati's  p o s i t i v e and 1984)  errors  i n parentheses. serial (1978,  runs.  The  correlation  so  provides  this  p.  5%  negative s e r i a l test  246)  "runs  the  further  18  for runs,  evidence  statistic error  test",  values  are 7 runs and  0.8.  of  h  ( f o r the annual  critical  some  term parameter i s as low as  Durbin  correlation  11 n e g a t i v e r e s i d u a l s 11  The  we  is  terms. find  12  sample 1962  to  positive  and  respectively,  for  no  serial  correlation. The  software SHAZAM (White  1978)  7 different  heteroskedasticity test  examination  of  the  relationship  p r o v i d e s the r e s e a r c h e r w i t h statistics,  between  estimated  some t r a n s f o r m a t i o n of them) and the independent predicted  values  of  the  regression  a l l involving  (or  some  residuals  an (or  v a r i a b l e s , o r the transformation  of  2 them). A l l 7 t e s t  are x  statistics  w i t h 1 degree of freedom.  The  maximum s t a t i s t i c of the s e t of seven f o r the r e n t a l r a t e e q u a t i o n is  1.740, so one We  Harvey  now  i s l e d t o assume h o m o s k e d a s t i c i t y .  consider  (1981,  pp.  151-4)  cumulative  sum  residuals.  Harvey's  equation  forward  and  (2.10),  residuals  tests  tend  p. to  recursive  of  structural  discusses  how  breaks one  cumulative  sum  t-test  recursive  156) be  on  helps  the  same  estimation  of  is  of  whether  series.  analyse  (see  the  for  his  recursive  t-statistic and  the  recursive  residuals  The  -0.6084  might  squares  identify sign.  i n the  for  our  our  backward  r e c u r s i v e e s t i m a t i o n i s +0.4285, each w i t h 20 degrees  of freedom.  The  present  plots  of  the  cumulative  sum  of  squares  also  no  evidence of m i s s p e c i f i c a t i o n . Finally Goldfeld  and  observations numerator the  we  and  consider  the  sets  tests  for  tests  have  Quandt the 19  F-statistic  Chow  i n the  s e q u e n t i a l Chow t e s t s  structural 2  denominator.  break.  degrees  of  The  critical  10%  with  freedom  and 23  in  the  value  for  i s exceeded at one p o i n t i n the 2 sample, s p e c i f i c a l l y between 1972 and 1973. Examining the p l o t of  S i n c e we  i s 2.61.  of  This  are t e s t i n g f o r s t r u c t u r a l break without  asking a p r i o r i  42 the r e s i d u a l s o f t h e AR(1) r e g r e s s i o n we f i n d t h e l a r g e s t r e s i d u a l in  terms  of absolute  value  occurs  i n 1973, t h e f i r s t  allowances  were  i n effect  full  year  accelerated  depreciation  (see Boadway  and K i t c h e n  (1984 pp. 128-9) f o r d e t a i l s ) . The G o l d f e l d and Quandt  (1973) t e s t f o r s t r u c t u r a l break, which i s based on t h e r a t i o o f the  sum o f squared  before the  a break  residuals  from  regressions  and a f t e r , does not y i e l d  Chow t e s t , an F - s t a t i s t i c )  exceeds t h e 10% c r i t i c a l  using  a test  a t any p o s s i b l e  t h e sample  statistic break  (like  point  that  value.  Of t h e Chow t e s t s , t h e G o l d f e l d and Quandt t e s t s , and v a r i o u s tests  on t h e p a t t e r n  gives  any e v i d e n c e o f s t r u c t u r a l break. C a s u a l o b s e r v a t i o n  data any  of recursive  and o f t h e r e s i d u a l s  residuals,  only  o f t h e AR(1) e s t i m a t i o n  our  entire  linear-quadratic the  of the  does not f i n d  obvious break i n t h e time s e r i e s . There a r e two i m p l i c a t i o n s o f t h i s . F i r s t ,  use  t h e Chow t e s t  1961  to  1984  data  r a t i o n a l expectations  we w i l l be a b l e t o  s e t when  estimating  the  model, which r e q u i r e s  that  input p r i c e s follow s t a t i o n a r y processes  (we examine t h e wage  l a t e r ) . We need not e s t i m a t e f o r d i f f e r e n t "regimes". Second, interpreting  i t suggests  that  any  in  change  introduction  of accelerated  in  o f t h e investment  the rate  perhaps corporate  depreciation  one  should  tax  rules  allowances,  tax credit,  or a  be -  wary  say t h e  o r a change  change  p r o f i t s t a x r a t e - as a change i n t h e " p o l i c y regime". Sims  where i t might occur, i t i s not s u r p r i s i n g t h a t t e s t i s s i g n i f i c a n t a t t h e 10% l e v e l .  of  i n the (1982,  a t l e a s t one Chow  43 p.108) w r i t e s : ...it can  i s a mistake t o t h i n k  only  choice  be d e s c r i b e d ,  A  decisions  o r even o f t e n  among permanent  authorities.  that  about p o l i c y  be d e s c r i b e d ,  as  r u l e s of behavior f o r the p o l i c y  policy  action  i s better  portrayed  as  implementation o f a f i x e d o r s l o w l y changing r u l e . This  i s a possible  Canada.  way  Suppose t h a t  stabilize  t o some  about  degree t h e r e a l  the r e a l  corporate  t h e r u l e t h e government  because o f t h e d e s i g n causes  to think  rental  of the corporate  rental  rate  to  tax policy i n  i s following  rate  of c a p i t a l . I f ,  income t a x , h i g h  rise  beyond  i s to  levels  inflation which t h e  government t h i n k s  appropriate,  may be i n t r o d u c e d  t o o f f s e t the harmful e f f e c t s of i n f l a t i o n .  from t h i s angle,  s p e c i a l t a x c r e d i t s and allowances  t h e i n t r o d u c t i o n o f t h e ITC might not r e p r e s e n t  "regime change" a t a l l , but r a t h e r r u l e t h a t was a l r e a d y This effects tax  generates  i s simply  problems  f o r those  of tax p o l i c y ,  who  expectations  of a  presumed  rational,  f o r the counter-factual  wish t o examine t h e  f o r example  c r e d i t , i n p a r t i c u l a r . I f one i s u s i n g are  a manifestation  a  i n place.  o f one aspect  expectations  Seen  t h e investment  a model where  how  can  one  firms' specify  p o l i c y o f no investment t a x  c r e d i t ? Indeed, how does one s p e c i f y t h e c o u n t e r - f a c t u a l p o l i c y i n the  model?  system  Should  remain  one assume  unchanged,  that  then  one  t h e other i s , as  parts a  of the t a x  counter-factual,  c o n s i d e r i n g what would have been a change i n regime. This models  problem  a r e used  i s examined i n simulations  s e r i e s of the r e n t a l r a t e .  further  below,  f o r some  where  estimated  counter-factual  time  We now examine t h e s t a t i o n a r i t y p r o p e r t i e s  o f t h e r e a l wage  rate. The  w  r e s u l t o f OLS e s t i m a t i o n i s  = .3613 + .9211w  t  1  (.1750) (.0473)  with  standard  1.8012,  errors  suggesting  i n parentheses. there  might  be  The D u r b i n serial  h statistic is  correlation  i n the  r e s i d u a l s . There a r e 13 p o s i t i v e , and 10 n e g a t i v e , r e s i d u a l s , 9 runs. The 5% c r i t i c a l (see  Gujarati  against of  serial  (1978,  v a l u e s f o r s e r i a l c o r r e l a t i o n a r e 7 and 18 pp.  440-1)),  correlation  as w e l l .  the r e s i d u a l s  with  yields  no c l e a r  so  there  i s some  Casual observation  evidence  evidence of a p l o t  f o r or against  serial  correlation. 2 The  values  of the 7  %  statistics  for  heteroskedasticity  g i v e n by SHAZAM range from 0.280 t o 2.437. The 10% c r i t i c a l with  one  degree  evidence o f  t-test  yields of  statistic  freedom  i s 2.70 6,  so  there  i s no  strong  heteroskedasticity.  Regarding squares  of  value  the  the recursive no c a s u a l  evidence  cumulative  o f -0.953  residuals,  the cumulative  of m i s s p e c i f i c a t i o n .  residuals  f o r t h e forward  (1981,  p.  recursive  156)  sum o f Harvey's  yields  residuals,  a  which  does not l e a d one t o suspect m i s s p e c i f i c a t i o n ,  but a s t a t i s t i c o f  -2.967  This  f o r t h e backward  recursive  some s o r t o f m i s s p e c i f i c a t i o n , clear  residuals.  but t h e p l o t  does  of r e s i d u a l s  suggest g i v e s no  i n d i c a t i o n where any s t r u c t u r a l change i n t h e s e r i e s  have taken  place.  might  The degrees sample this  Chow t e s t  statistic  o f freedom.  has an F d i s t r i b u t i o n  The h i g h e s t  i s d i v i d e d between  statistic  wage level  increased  over  the f i r s t  o f s t r u c t u r a l break.  i n Table part  the s e r i e s  i s stationary  t h e parameters  increase  above suggest mean  o f 2.62, i f one used t h e e s t i m a t e s o f t h e AR(1) r e g r e s s i o n ,  which perhaps p r o v i d e s  equation  given  i t has an e s t i m a t e d  i n t h e wage over some time  discernable  2.1 r e v e a l s t h a t t h e r e a l  o f t h e sample but seemed t o  o f f t h e r e a f t e r . The OLS e s t i m a t e s  Given a 1961 v a l u e  no  when t h e  i s l e s s than t h e 5% s i g n i f i c a n c e l e v e l o f 3.52. G o l d f e l d and  A c a s u a l look a t t h e data  of  i s obtained  (2, 19)  1964 and 1965, where i t i s 2.554, but  Quandt t e s t s s i m i l a r l y g i v e no evidence  if  with  pattern  that  o f 4.58.  we  obtained  one would p r e d i c t an  f o l l o w e d by a l e v e l l i n g o f f ,  t h e i n t u i t i o n behind  why t h e r e  i n t h e r e s i d u a l s . A time  (3.2) proved t o be i n s i g n i f i c a n t ,  with  seems t o be  trend  added t o  a t - s t a t i s t i c of  o n l y 0.548. Note t h a t will  assume  i n t h e models we e s t i m a t e firms  form  expectations  i n Chapters 3 and 4, we using  these  simple  3 autoregressive  processes,  so t h e r e s i d u a l s o f these  translate  the  into  in  expectations  models  a r e t o be d e s c r i b e d  forecast  errors  as r a t i o n a l  regressions  by  firms.  i n t h e model,  If  there  s h o u l d be no i n f o r m a t i o n embodied i n t h e r e s i d u a l s , and i t i s f o r this  reason  we have examined t h e p r o p e r t i e s  the r e n t a l r a t e and wage equations  This  i s standard  practice  of the r e s i d u a l s of  i n such depth.  i n empirical  applications  models; see a l l t h e papers r e f e r r e d t o i n Chapter 1.4.  of  these  Modelling simplified not  input  demands  in  the  manufacturing  i f i t can be assumed i n t h e model t h a t  caused  ( i n t h e Granger  (1969)  - Sims  (1972)  r e n t a l r a t e s on c a p i t a l , t h e demand f o r c a p i t a l , labour,  and t h a t  real  simplifies  decisions  the s p e c i f i c a t i o n  to  input  Sargent (1978), Meese others (and  to obtain  wages a r e  sense)  This  real  assumption  of the firms' input d e c i s i o n  i s no feedback from t h e f i r m s '  prices.  The  assumption  (1980), and E p s t e i n and Yatchew  i s used  mentioned  papers  Yatchew) f o r t e s t i n g t h e exogeneity  by  Meese  and  by  (1985) among  a t r a c t a b l e s o l u t i o n . The technique  i n t h e above  by  o r t h e demand f o r  o r t h e demand f o r l a b o u r .  r u l e s , because i t means t h a t t h e r e input  real  is  r e n t a l r a t e s a r e not caused by r e a l wages,  the demand f o r c a p i t a l , greatly  sector  used  here  Epstein  and  o f i n p u t p r i c e s i s from Geweke  (1978) . We b e g i n by e s t i m a t i n g t h e v e c t o r  s  t  where S real  "  a  +  V t - i  +  Vt-2  +  b  t  +  autoregression  e  ( 3 t  = (k , 1 , r , w. )', k i s c a p i t a l ,  rental  rate  of c a p i t a l ,  w  1 i s labour,  i s t h e wage,  a  and b  -  5 )  r i s the a r e 4x1  v e c t o r s o f parameters, A^ and A^ a r e 4x4 m a t r i c e s  o f parameters, t  is  u n c o r r e l a t e d but  time, and  i s a vector  perhaps c o r r e l a t e d a c r o s s We  consider  two  equations.  alternatives to u n r e s t r i c t e d estimation  (3.5):(i) that the lagged the r and w e q u a t i o n s ,  of errors, s e r i a l l y  of  k and 1 terms have zero c o e f f i c i e n t s i n  and ( i i ) t h a t t h e lagged  r and w terms have  zero c o e f f i c i e n t s i n t h e k and 1 e q u a t i o n s . The  evidence  i s that there  i s stronger  c a u s a l i t y from  lagged  prices  to  current  input  levels  l e v e l s t o c u r r e n t p r i c e s . The null  hypothesis  (i)  is  than  there  i s from  lagged  likelihood ratio test  18.225  and  the  Wald  input  statistic  test  for  statistic  is  degrees  of  2 23.329  (each  of  freedom). The  which  is  likelihood  ratio  statistic  i s 21.630 and the Wald t e s t s t a t i s t i c With  the  confidence  small  sample  whether  the  x  distributed  we  are  8  f o r n u l l hypothesis  (ii)  i s 27.224.  using  causality  with  is  we  cannot  say  statistically  with  any  significant.  U s i n g the l i k e l i h o o d r a t i o and Wald t e s t s t a t i s t i c s as g i v e n above leads  one  to  reject  q u e s t i o n . But ratio  statistics  and  (2.7).  i n a way  K i s the  This  40,  the  by  of  the  to  correct  likelihood (1982)  where T i s the  sample  u n r e s t r i c t e d model  problem  that  with  i s meaningless this  i f the Nelson and  small  a  of  using  f o r our  sample we  and  purposes, simply  K or  cannot  Schwert c o r r e c t i o n i s a p p r o p r i a t e .  model  is called  linear  quadratic  because  the  quadratic  s p e c i f i c a t i o n of both the adjustment c o s t s and the output to  i n p u t demands which are  assumption of the model i s t h a t input  large  Model  The  leads  in  Schwert  i n the  f o r the  variables the  Nelson and  (T - K) /T  number of parameters  amended s t a t i s t i c  say a n y t h i n g ,  The  any  t o examine a small-sample model. Since T = 22  perhaps warns us  3.2  of  suggested by  statistics  i s meant  sample t h e o r y =  exogeneity  E p s t e i n and Yatchew suggest m o d i f y i n g  by m u l t i p l y i n g the size  the  prices  rationally. as  a  linear  formed  efficiently.  More s p e c i f i c a l l y  input p r i c e s .  f i r m s form e x p e c t a t i o n s  Rational  expectations  in real  result  of  expectations using  function  are  available  of  An  future  defined  as  information  i t means i n d i v i d u a l s and  firms  in  48 the model make use o f (a) p a s t knowledge  of  the  expectations. are  of  the  o f v a r i a b l e s and (b)  economic  model  In t h i s model o f i n p u t demand r a t i o n a l  i n c o r p o r a t e d by assuming t h a t f i r m s know p a s t  input  prices,  represent for  structure  observations  and t h a t  an VAR(l)  specification  in  forming  expectations  values  of r e a l  can be  used t o  t h e "model" f i r m s use t o make f o r e c a s t s . See Chapter  a survey  o f models o f t h i s  type  1  t h a t have been a n a l y s e d and  estimated. In Epstein  this  chapter  and Yatchew  the  estimation  (1985) . The assumptions  used i n t h e i r paper a r e not d i f f e r e n t rational  expectations  parameterization  technique  models  from  and f u n c t i o n a l  from o t h e r l i n e a r  of  input  of the estimating  i s taken  demand,  equations  form  quadratic but  the  i s different.  Its  advantage i s t h a t t h e c r o s s - e q u a t i o n r e s t r i c t i o n s t h a t a r e i m p l i e d in  rational  specified identical hand,  expectations than  in  models  o r Chow  models o f i n p u t  other  parameterizations  (e.g. Hansen  (1980b,  demand  1981,  and Sargent 1983)  on  a r e more  using (1980,  simply  essentially 1981)  on one  t h e other) . F o r ease  of  r e f e r e n c e E p s t e i n and Yatchew's n o t a t i o n i s used. A f i r m produces output  y w i t h i n p u t s c a p i t a l k and l a b o u r 1.  D e f i n e t h e v e c t o r x^_ = (k^ 1^) ', and t h e p r o d u c t i o n f u n c t i o n i s  y  = a'x  t  + x ' A x / 2 + (x t  t  fc  - x _ )'B(x t  1  - x _ ) / 2 + S {t), 1  t  t  1  (3.6) where  a  matrices capture definite  is a  2x1  vector  o f parameters,  A  and B  o f parameters, and S^" (t) i s a s c a l a r changes i n t e c h n o l o g y . and B  i s diagonal  time  a r e each  2x2  t r e n d meant t o  M a t r i x A i s symmetric and n e g a t i v e and  negative  definite.  Hansen  and  Sargent terms  (1981)  a n d Chow  (1980b,  i n t h e adjustment  1981) a l l o w n o n - z e r o  cost matrix  B, b u t t h i s  of c o m p l i c a t i n g t h e s o l u t i o n o f t h e model Costs (3.6)  i n v o l v i n g B.  because  the costs  Treadway changes based  of adjusting input They  levels  (1969, p . 2 2 9 ) ) . Note i n input  levels.  on  the percentage  in  t h e model  "internal  change  i n Lucas  i n input  (3.1) a d j u s t m e n t  (1968)  lost  costs  (see  depend on n e t costs  are  they a r e based  One c o u l d  are capturing  costs"  output  adjustment (1967a)  levels.  costs  adjustment  of  adjustment  I n Gould  on g r o s s i n v e s t m e n t , w h i l e  the cost  considerably.  i n terms  that  carries  a r e c a p t u r e d by t h e term i n  are called  are expressed  off-diagonal  imagine  that  the disruption  i n v o l v e d i n c h a n g i n g t h e l e v e l o f a n y i n p u t , s o t h a t no a d j u s t m e n t costs arise The w^_,  from p u r e l y replacement investment.  real  rental  and t h e v e c t o r  time  t  =  0 a  firm  rate  of capital  of input chooses  i s r  and t h e r e a l  prices  i s w r i t t e n z^_ =  a  f o r setting  rule  wage i s  (r^_ w^_) ' . A t  x^ t o s o l v e t h e  problem  max E  0  ^ p  t  [a'x + x ^ / 2 t  + (x  f c  - x ^ ' B ^  - x^)/2  + S ( t ) - z 'x ]  (3.7)  1  fc  subject t o x of  discount.  g i v e n , where p = Each  time p e r i o d  t o t h e problem, making  (1 + R) the firm  and R i s a constant  rate  recalculates the solution  u s e o f a n y new i n f o r m a t i o n .  Input p r i c e s f o l l o w t h e process  (3.8)  50  w h e r e t h e 2x1 v e c t o r v a n d t h e 2x2 m a t r i x  0 a r e parameters and e  i s a random e r r o r t e r m . An  advantage  the problem has  of the linear  quadratic  specification  i s that  (3.7) c a n b e s o l v e d u n d e r t h e a s s u m p t i o n t h a t t h e f i r m  perfect  foresight;  i . e . c e r t a i n t y equivalence  applies.  The  s o l u t i o n has t h e form  x* - x _ t  where  x  level" and  t-1  - i >,  (3.9)  t  i s the optimal  decision  at time  t, x  i s the "target  o f x a t time t , and M i s t h e "adjustment m a t r i x "  (Epstein  Y a t c h e w (1985, p p . 239-40)). The m a t r i x M s o l v e s t h e e q u a t i o n  M  and  f c  = M<x  1  2  x  - (1 + R ) B  -1  AM - RM - B  -1 A ( l + R) = 0 ,  (3.10)  i s given by  x  = A  _  1  (J  t  - a),  (3.11)  where  J  t  = D Z (I + D ) " s=t  D = AB  ( S  t  +  ^E  z , t s  (3.12)  (3.13)  (1 + R) + R - M',  w h e r e R i s a 2x2 d i a g o n a l  matrix  with  every  diagonal  entry  equal  t o R, a n d I i s a 2x2 i d e n t i t y The  vector  J  matrix.  i s a weighted  average  future input p r i c e s . I f expectations  for  a l l s  however, and  =  J  t,...,oo,  then  i s clearly  =  going  of current  are static,  z. W i t h  and expected  say that E z = z t s  rational  expectations,  t o d e p e n d somehow o n c u r r e n t  prices  on t h e p a r a m e t e r s o f t h e model u s e d t o f o r e c a s t f u t u r e p r i c e s ,  n a m e l y v a n d 6. The s o l u t i o n f o r J  given  by E p s t e i n  and Yatchew  ( p . 241) i s a s f o l l o w s :  J  f c  = a + 0z ,  (3.14)  t  where a a n d £ a r e d e f i n e d b y  v = |3 D a  (3.15)  0 = (3 ( ( I + D)|3 - D)  (3.16)  X  and  1  and  In  D i s a s d e f i n e d i n (3.13).  the following  chapter  assumed t o be i n d e p e n d e n t be  a diagonal matrix.  simplifying  4  the input  price  f r o m one a n o t h e r ;  autoregressions  are  i . e . 0 i s assumed t o  W h i l e one m i g h t t h i n k t h i s c o u l d be a u s e f u l  assumption  i n this  adding  a  model, number  i n fact of  i t would  restrictions  greatly  complicate  matters,  to the  estimation.  We do n o t know how much d i f f e r e n c e i n t h e r e s u l t s o f  52 As a f i n a l P  =  BM.  The  step i n deriving  simplified  p a r a m e t e r i z a t i o n o f E p s t e i n and  refered to earlier w i l l of  the estimating equations, define  d e f i n e t h e i n p u t demand e q u a t i o n s i n t e r m s  a , B, P, a n d J ^ . E q u a t i o n  (3.10) c a n b e s o l v e d f o r A a s  A = P / ( l + R) - B + B ( I + M)  This  represents  Yatchew  1  .  (3.17)  parameter r e s t r i c t i o n s  on the s o l u t i o n t o  (3.7).  Other r e s t r i c t i o n s d e r i v e d i n Lucas (1967b) are  M has 2 r e a l e i g e n v a l u e s between -1 and 0  (3.18)  P i s symmetric and p o s i t i v e d e f i n i t e .  (3.19)  and  W r i t i n g the s o l u t i o n o f the model i n the form i n which i t i s t o be e s t i m a t e d we have  x  =  (I + B ~ P ) x _ 1  t  1  - B  _ 1  PA  1  ( J  t  - a) + u  (3.20)  fc  and  z  the  t  -  v  +  models  different  9  z  t-i  of  +  this  v  ( 3  and  the  following  chapter  are  treatment o f the e v o l u t i o n of i n p u t p r i c e s .  due  -  to  2 1 )  this  53  where A  i s as  are B, P, and  defined  in  a, a, and  (3.17) . The  parameters  Technology i s c o m p l e t e l y  a, w h i l e a and  as  the  /3 are the parameters r e l a t i n g v and  to  be  technology  parameters.  as P i s c o n f i n e d t o be  positive definite,  and  (but i s s a t i s f i e d by the data terms u^_ are meant t o errors they  i n the  should  data"  be  (Epstein  independent  how  applies  to  governing  (1976) the  and  Yatchew p.  the  residuals e  of  are p r e s e n t e d , of  change  through  (3.16) and  changes  the  the  (3.14),  relation  input  the  reduced  change when v and t h a t t h i s change  3.3  form  is  imposed  see below). E r r o r  from  and  principle  (3.21),  but  J  policy  change  of  and  v  and  8.  see  evaluation  i n the  policy  (e.g. a change i n c o r p o r a t e  changes the  between  i t i s i n t e r e s t i n g to  demands. A  parameters  changes the parameters r e l a t i n g sum,  a  confined  i s not  243) . In  econometric  r e n t a l r a t e s on c a p i t a l would  and  and  (3.19)  i s not  (3.18) as we  P,  "random e r r o r s of o p t i m i z a t i o n  critique  problem of  policy)  case,  input  imposed.  the e s t i m a t e s  Lucas'  symmetric but  i n any  P,  i n terms of a  Restriction  restriction  reflect  t h i s r e s t r i c t i o n i s not Before  S to  (3.17) i s imposed w r i t i n g (3.20) u s i n g B,  only  half-imposed,  estimated  d e s c r i b e d by B,  demands. In the e s t i m a t i o n v and S are e x p r e s s e d |3. R e s t r i c t i o n  t o be  tax  This  in  turn,  parameters a and  0,  which  prices,  which  current  input  i n p u t p r i c e s t o i n p u t demands. In  parameters  of  the  input  demand  equations  8 change. Lucas warns p o l i c y a n a l y s t s t o  realize  occurs.  Estimates Epstein  and  Yatchew  assume  that  firms  can  observe  this  period's input p r i c e s before having to decide t h i s period's input levels.  Chow  (1980b,  1981),  in his  formulation  i d e n t i c a l model, assumes f i r m s must based  only  on  observations  Since  here  we  are  not  reduced  input  and  3.2  l e v e l s must be  under  in  although not s i g n i f i c a n t l y  so.  3.4.  the  model,  (3.21),  assumption  of  the  under the  after period t  that  the  period t  estimates  of  SHAZAM's  squares.  the  the  two  s e t s of  Surprisingly, labour  i n both t a b l e s  equation  models are  estimates,  given  is  positive,  i n Tables  3.3  In the t a b l e s the f o l l o w i n g n o t a t i o n i s used:  "p P =  prices.  chosen b e f o r e p e r i o d t p r i c e s become known,  wages  Restricted  of  u n r e s t r i c t e d models are e s t i m a t e d with  on  levels  estimates  (3.20) and  a l t h o u g h magnitudes change s l i g h t l y .  and  validity  parameter s i g n s change a c r o s s the  coefficient  input  earlier)  unrestricted  the  (1978)) t h r e e - s t a g e l e a s t  No  otherwise  estimated.  show the  known and  r e s p e c t i v e l y . The  the  the  (and  t h a t f i r m s choose p e r i o d t i n p u t l e v e l s  become  (White  about  form of the 4 e q u a t i o n model  assumption prices  3.1  an  choose p e r i o d t  p e r i o d t-1  certain  both s p e c i f i c a t i o n s w i l l be Tables  of  of  1  P 1-2  The  p  2  ' l  P 3-J  •°  models  maximum-likelihood  °"  B  B =  B  2-  are  from  maximization  process  expectations  version  1962 were of  V  to  - 2-  chosen the  by  model.  Starting first A  P  the  data  P  22-  non-linear  i s annual values  estimating  selection  12  P  - 21  a  with  SHAZAM. The  /3 = - 2-  a  1984.  " ll P  a =  estimated  o p t i o n of  manufacturing  a =  of  Canadian for a  the  static  different  55 starting  values  converged  i n t h e neighbourhood  of the f i n a l  on t h e e s t i m a t e s shown.  T a b l e 3.3 g i v e s e s t i m a t e s under t h e assumption a b l e t o observe The  estimates  parameters  o f Table  are very  simple  input  bi-variate  specification interest  then  the input  3.4  choose  Comparing  gives  any y e a r ' s  these  input  results  elasticity  Table  constant  terms.  This  levels  f o r any s i m u l a t i o n s o f rate.  t h e assumption before  3.3, we  prices  find  t h e reduced  that  firms  are  observed.  i n Table  3.4 t h a t  to rental rates,  i s only approximately  3.3, but otherwise  with r e s p e c t  with  the demand f o r c a p i t a l does respond impact  inelastic  to a  under  t o Table  t h e demands  reduce  no p o s s i b i l i t y  estimates  from t h e  equations  autoregression leaves  different  levels.  demand  r e g a r d i n g changes t o t h e r e n t a l  Table must  prices;  input  3.1. In p a r t i c u l a r ,  f o r c a p i t a l and l a b o u r a r e each c o m p l e t e l y both  that firms are  current input p r i c e s before s e t t i n g  i m p l i e d reduced-form  unrestricted  to  solution a l l  although t h e  -.004, and i t does not i n forms  a r e t h e same between  the two c a s e s . The  e s t i m a t e s o f s t r u c t u r a l parameters a r e v e r y d i f f e r e n t i n  magnitude a c r o s s t h e two s p e c i f i c a t i o n s , same  sign  across  t h e two  tables.  But  although since  a l l are of the  M,  the  adjustment  m a t r i x , e q u a l s B ^P, t h e a b s o l u t e v a l u e s o f B and P a r e not g o i n g t o be w e l l i d e n t i f i e d . Since  the model  of Table  c o n c e i v a b l y be o f i n t e r e s t r a t e s on investment,  3.4  i s the only  t o one examining  the r e s t  of t h i s  one which  the e f f e c t s  chapter w i l l  could  of r e n t a l  focus on t h i s  model. The  restriction  (3.18),  that  M  have  2  real  eigenvalues  between 0 and -1, i s s a t i s f i e d ,  although  e s t i m a t i o n . S i n c e M i s g i v e n by B P,  1.9576  ,0003  -.1883  not imposed i n the  the e s t i m a t e d v a l u e of M i s  1  -.1144  i t was  M  and  the  eigenvalues  Restrictions definite  and  positive,  of M  are  -.1237 and  which are not  that  B  be  which i s the  -.17 90.  satisfied  negative  are  definite.  In  positive  particular  costs to  a d j u s t i n g the l a b o u r i n p u t . A l s o , c o n s i d e r the m a t r i x A,  from the  constant  f u n c t i o n , which  or  decreasing  i t suggests  is  negative  production  wrong s i g n ;  t h a t P be  s h o u l d be  returns  to  negative  scale.  been g i v e n above; the p o s i t i v e element an i n c r e a s i n g m a r g i n a l product Virtually fail.  a l l aspects  Adjustment  product marginal  of  product  of  are  at  various  that  Its  the  the  "labour  wrong  data  side"  sign,  points  of  and  the  completely capital constant 3654.7.  reduced  indicate  on  the  term, So  the  and  form the  is  where  result  actually  the  this  model  marginal  a  negative  "capital of  side", capital  decreasing. is a  i n e l a s t i c with r e s p e c t t o both  stock,  has  2 implies  the  adjustment c o s t s are p o s i t i v e , and the m a r g i n a l product  In  ensure  implied value  i n column 2 row  i s i n c r e a s i n g . Yet  i s found t o be p o s i t i v e and  to  of l a b o u r .  costs  labour  definite  mean  significant  just of  the  effect  of  demand i n the c a p i t a l e q u a t i o n  labour  demand t h a t  i n p u t p r i c e s and  an  AR(1)  process labour  process is  to with  estimated  input  on  is the a at  capital  i s simply a term c a p t u r i n g t h i s  AR  process. It of U.S.  this  i s i n t e r e s t i n g t o note t h a t E p s t e i n and Yatchew's same 4 e q u a t i o n  model w i t h  annual m a n u f a c t u r i n g  rational  results  expectations,  data from 1948 t o 1977, a r e v e r y  using  similar  t o t h e r e s u l t s o b t a i n e d here  (see t h e i r Table 5, p. 249).  cases e s t i m a t e d s i g n s a r e  > 0, P^ and P^ < 0 (so n e i t h e r case  s a t i s f i e s t h e r e s t r i c t i o n P p o s i t i v e d e f i n i t e ) , and  In both  < 0 and B^  > 0 (so both cases have B^ b e i n g t h e wrong s i g n ) . The  marginal  restrictions determinant  significance  of the t e s t  of the cross  i s found by t a k i n g t h e d i f f e r e n c e  equation  i n the l o g of the  o f t h e sigma m a t r i c e s , m u l t i p l y i n g t h i s by t h e number  of o b s e r v a t i o n s  (23), and comparing t h i s  test  statistic  with the  2 X  distribution  significance suggest The  with  of  3  degrees  of  the r e s t r i c t i o n s  freedom.  i s only  2.53%,  The  marginal  which  would  r e j e c t i o n o f t h e model by t h e d a t a . impact  e l a s t i c i t i e s o f i n p u t demands, as o f 1975,  are:  s h o r t run e l a s t i c i t y w i t h r e s p e c t t o : r  w  capital  -.004  +.004  labour  +.000  +.000  Long run e l a s t i c i t i e s  a r e found  by a p p l y i n g e q u a t i o n  (3.11),  which d e s c r i b e s how steady s t a t e demands change with r e s p e c t t o a change  in J . A  represented  by  a  permanent change  change  i n input  i n J , which  is a  prices  would  be  weighted  index  of  58 c u r r e n t and expected f u t u r e i n p u t p r i c e s . The change i n t h e t a r g e t v a l u e s o f i n p u t s a t time t , g i v e n by x^, with r e s p e c t t o a change in  i s g i v e n by our e s t i m a t e o f A  . From t h e e s t i m a t e s g i v e n i n  Table 3.4 t h e i m p l i e d e s t i m a t e o f A i s  -.00005997 .002329  At 3513.0,  1975  .002329 .9458  capital  respectively,  and  labour  input  and t h e r e n t a l  levels  are  rental  and  r a t e and t h e wage r a t e a r e  .10 6 and 4.00. In t h a t same y e a r t h e elements o f J the  18715.3  r a t e and t h e wage a r e .113 and 4.134  corresponding (from  equation  (3.14) and t h e e s t i m a t e s o f a and P g i v e n i n T a b l e 3.4). A p p l y i n g equation in  1975  (3.11) we o b t a i n t h e t a r g e t l e v e l s o f c a p i t a l of  24925.5  and  3613.5.  So  at  1975  and l a b o u r  the  long  run  e l a s t i c i t i e s o f i n p u t demand, where t h i s means t h e r e l a t i v e change i n t a r g e t i n p u t l e v e l s p e r r e l a t i v e change i n J , a r e :  l o n g run e l a s t i c i t y w i t h r e s p e c t t o :  As  r  w  capital  -.069  + .006  labour  + .001  + .001  l o n g as M i s a s t a b l e m a t r i x ,  between -1 and 0, and our e s t i m a t e  i . e . has 2 r e a l  eigenvalues  i s a stable matrix,  then t h e  method  of  relative we  calculating  long-run  elasticities  change i n t a r g e t v a l u e s  looked  at the  relative  by  examining  the  i s the same as we would f i n d i f  change  i n long  run  actual  values  of  inputs. Consider the f o l l o w i n g . I f  x  = (I + M)x _  fc  fc  1  + MA  -1  a - MA  -1  J ,  (3.22)  f c  then  2 x ^ + ( I + ( I + M ) + ( I + M ) +...+ t-1 s -1 -1 -1 (I+M))MA a-MA J - (I + M) MA J , t+s t+s-1  x^, =(I+M) t+s  s+1  ... - (I + M ) M A J . S  (3.23)  - 1  t  I f we change a l l J , T = t , t+1, x  t+s  ...,co, by AJ, then t h e change i n  is  Ax^ , = - ( 1 + t+s  (I+M)  + (I+M)  2  + ... + ( I + M )  s  -1 ) MA AJ , (3.24)  and as s  co, i f M i s s t a b l e , t h i s converges  Ax^ , = - ( I - (I + M))" MA~ AJ = A t+s 1  So A ^ g i v e s  us  the change  1  _ 1  to  AJ.  i n the t a r g e t  (3.25)  input  levels  and  the  change i n the a c t u a l l e v e l s i f M i s a s t a b l e m a t r i x . It long  i s i n t e r e s t i n g t o note how  run  elasticities  if  we  different ignore  e x p e c t a t i o n s and simply take the reduced  a r e the e s t i m a t e s o f  our  specification  of  form parameters of i n p u t  60 demand as g i v e n f o r any e x p e c t a t i o n s . Write t h e reduced  form as  x,. = (I + M)x^_ + r , + c. t t-1 t-1  (3.26)  z  I g n o r i n g e x p e c t a t i o n s , a change i n c u r r e n t and f u t u r e i n p u t p r i c e s by Az would l e a d t o a change i n x  of  t"t"S  Ax  t + s  = (I + (I + M) + (I + M )  2  + ... + (I + M ) ) T A z , S  (3.27)  and as s -> oo, i f M i s s t a b l e t h i s converges t o  Ax^, = -M r A z . t+s  (3.28)  1  Taking  t h e reduced  estimated  form e s t i m a t e s  o f M and T from Table  3.4 t h e  l o n g run e l a s t i c i t i e s a r e  l o n g run e l a s t i c i t y w i t h r e s p e c t t o :  The  r  w  capital  -.035  +.039  labour  +.001  +.001  estimate  o f t h e l o n g run e l a s t i c i t y  with  respect  t o the r e n t a l  reduced  form o f t h e model, i s o n l y about one h a l f t h e s i z e o f t h e  estimate  r a t e , when e s t i m a t e d  of c a p i t a l stock  considering only the  when we account f o r e x p e c t a t i o n s as s p e c i f i e d i n e q u a t i o n  (3.21). We cannot measure whether t h i s d i f f e r e n c e i n t h e e s t i m a t e s  of  long  since  run e l a s t i c i t i e s  we  have  elasticity double  treating  with  the  i n an e c o n o m e t r i c  standard  respect  terms);  i n estimates i n  of  expectations  (in elasticity  difference answered  measure  of capital  when  number  no  i s significant  errors.  t o the rental as r a t i o n a l  whether  we  as q u a n t i t a t i v e l y  following  The  section  of  should  we  small  regard  this  i s perhaps  chapter.  s i m u l a t i o n s o f t h e model a r e r u n u n d e r b o t h methods; where forms a r e changed t o account  run  that  very  important this  long  rate  i s a  sense,  There reduced  f o r a new r e g i m e , a n d w h e r e t h e y a r e  not. Because both target  levels,  capital  and labour  i t i s useful  to  a r e slow know  t o adjust t o their  what  t h e medium  run  2 elasticities  are.  The m a t r i x  s l o w t o c o n v e r g e t o -M  (I +  ( I + M)  +  ( I + M)  +  . . .) i s  . From o u r e s t i m a t e o f M o f  -.1144  1.9576  -.0003  -.1883  M =  we f i n d t h a t  (1+  4.5158  2 s ( I + M ) + ( I + M ) + ... + ( I + M ) ) e q u a l s  19.3905 when s  -.0030  3.7838  62  6.4027  44.0446 when s = 10,  and  -.0068  4.7400  .5093  88.4639  -.0136  5.1697  that  V-l  -M  which i n d i c a t e s t h a t convergence t o l o n g run l e v e l s  i s v e r y slow.  Based on these c a l c u l a t i o n s ,  the f i v e -  elasticities,  expressed  levels  i n terms o f t a r g e t  and t e n - y e a r  so t h a t  they a r e more  easily  compared t o t h e l o n g run e l a s t i c i t i e s g i v e n above, a r e  five-year e l a s t i c i t y  ten-year  r  w  r  w  capital  -.037  +.002  -.052  +.004  labour  +.000  +.001  +.001  +.001  We a l s o e s t i m a t e d t h e model with s t a t i c achieved  by s e t t i n g  expected  future input p r i c e s ,  setting  a  equations parameters  elasticity  =  J ^ , which  0 and |3 e q u a l  o f (3.20) under  a r e then  static  i s like  expectations. This i s  an index  o f c u r r e n t and  e q u a l t o ^._^ (or, e q u i v a l e n t l y , by z  to the i d e n t i t y estimated.  expectations  matrix) .  The two  Estimates  of structural  are similar  t o those f o r  63 dynamic  expectations  relative  listed  magnitudes  are  i n Table  roughly  3.4;  the  no  same.  signs  change,  Static  and  expectations  estimates a r e :  P  .14155xl0~  (.00010)  3  P B  -.22744xl0~ -2 -.11599x10  a,  -3.5094 (2.9434)  2  2  P  3  -2.6673  (.92691)  B  2  12.536 (9.1424)  a„  -3471.0 (1.4765)  T h i s i m p l i e s a reduced  k  t  1  = -3664.545 + .878k  = 770.645 - 0 k _ t  which  i s similar  1  (compare w i t h Table  (.00086)  form f o r i n p u t demands o f :  , + 1.9611  t-1  + •7871 _ fc  to  (.00173)  those  1  , - 3725.996r , + 2.711w , t-1 t-1 t-1  + 2.711r _ t  obtained  3.4), although  1  with  + .219w _ fc  dynamic  the estimate  1  expectations  of the e l a s t i c i t y  of c a p i t a l demand with r e s p e c t t o t h e r e n t a l r a t e i s much g r e a t e r . The  likelihood  expectations  ratio  model a g a i n s t  statistic  f o r the 2  equation  the u n r e s t r i c t e d 2 equation  static  model i s  2 8.670,  which  marginal  i s distributed  %  with  3 degrees  s i g n i f i c a n c e o f t h e r e s t r i c t i o n s i s then  Finally,  we  given i n equation  turn  to estimates  o f freedom.  The  3.36%.  of the production  (3.1). U n r e s t r i c t e d e s t i m a t i o n y i e l d s  function,  64 y  I*  = 29417 - 2.0855k (126090)  + 2.92971  (4.5320)  +  .0001 ( k  1  )  (.0012)  The Wald x being  is  insignificance. on  the  The  usefulness  output  equation  demands and  1 t  _  1  )  with  2  s i g n s on of  this  together  input p r i c e s  with  of  yielded  four  results  adjustment  terms  suggesting  lend further  Restricted  the  (.0016)  freedom,  coefficients  model.  ^  (815.16)  degrees  the  ^  + 3238.8t.  2  f o r the j o i n t t e s t of b o t h  .0747  + .1322k 1  (.0166)  (.0118)  statistic  zero  -  2  L*  (.00004)  ~ .0026 (1  2  - .00201  2  ^  (91.362)  - \ _  t  - .0001k  ^  estimation  equation with  model  such  doubt of  of  the  input  a poor f i t of  the d a t a t h a t they are not worth r e p o r t i n g .  3.4  Simulations Table  using  3.5  the  g i v e s the  parameter  results  of a number of s i m u l a t i o n s made  estimates  of  the  4  equation  restricted  l i n e a r - q u a d r a t i c model p r e s e n t e d i n T a b l e 3.4. Although  the m e r i t s  of t h i s model as an e x p l a n a t i o n of the d a t a have been found t o dubious, doing  the s i m u l a t i o n s a t l e a s t i l l u s t r a t e the p r i n c i p l e s  policy  illustrate  analysis  the  with  empirical  a  rational  significance  expectations of  how  be  behind  model,  and  expectations  are  specified. For in  1975  labour path,  a l l the  simulations  making  long-range  i n these  estimates  the  in  Table  f o r e c a s t s of seems t o  lead  to  a  forecast  the  simply  f o r e c a s t s of l a b o u r are not  simulations  3.5,  value  we  capital  follow a  recorded of  imagine  labour  someone  stock  (since  predetermined  i n the t a b l e ; a l l i n 1984  of  3603,  65 w h i l e i t s a c t u a l v a l u e t u r n e d out t o be 3621) . Column A l i s t s the a c t u a l d a t a . Column B l i s t s a f o r e c a s t of c a p i t a l made i n 1975, standard using  u s i n g the model of Table  e r r o r s of the  the  f o r e c a s t s . The  method g i v e n  by  Judge  3.4,  standard  et.  al.  and  gives  e r r o r s were  (1988, pp.  the  found  764-67)  for  c a l c u l a t i n g the v a r i a n c e of f o r e c a s t s w i t h VAR(l) systems. The to  the  lowers  s i m u l a t i o n i n Column C i s the rental  r a t e on  r i n 1975  parameters of r the path past  capital  from  .1060  i n 1975  to  by  of a n e g a t i v e  a  .0954. The  factor  of  so  unchanged, t h e r e  even will  the  10%.  This  effects  on  f u t u r e r e n t a l r a t e depends on i t s  though  be  shock  underlying time-series  are l e f t unchanged. T h i s shock has two  of c a p i t a l . F i r s t ,  values,  result  the  time-series  parameters  some p e r s i s t e n t e f f e c t s  on  rental  are rates  from t h i s one-time shock. Second, s i n c e c a p i t a l demand responds t o the  shock t o the  on i t s own But  r a t e i n 1976,  p a s t v a l u e s , t h e r e w i l l be  matrix  -1) , the  M  effect  asymptotically to  i s "stable" of  this  to  capital  i n c r e a s e the  only  .02%  further persistent effects.  real in  capital  fact  the  very  that  the  between  long-term  1984,  greater  f i n d the impact e f f e c t ,  stock the  than  by  effect what  .04% of  over the  what shock  i t otherwise  0  tends  it is a  in  1976,  otherwise capital  would have  been.  completely.  s i m u l a t i o n i n Column D i s somewhat unusual.  t h e r e i s a permanent decrease of  the  eigenvalues  E v e n t u a l l y the e f f e c t s of the shock d i e out The  demand depends  zero.  would have been. By stock  (two  shock  Comparing Columns B and C we is  and  g i v e n the s t a t i o n a r i t y of r e n t a l r a t e s , and  adjustment and  rental  Here i n  1975  but the  path  l o w e r i n g the  1975  i n r e n t a l r a t e s of 10%,  wages i s l e f t unchanged. T h i s i s a c h i e v e d by  66 value C,  o f t h e r e n t a l r a t e by 10% d i r e c t l y ,  and i n changing  as we d i d i n s i m u l a t i o n  the b i v a r i a t e autoregressive  process  of the  r e n t a l r a t e and wages from  v  Z  t  l "  " 11 9  +  =  9  1 ' 2  t-1  • 2-  - 21 22-  V  9  9  to  9v„ z = t  6  11  t-1' 2-"  What  1  -  1 1 G  21  9  22  i s unusual about t h e s i m u l a t i o n  permanent  shift  i n the path  assume  firms  a r e unaware  place.  They  observe  realize  -"12  t h e change  i s not t h a t  of r e n t a l rates,  that  rental  the s h i f t  rates  i n regime;  but r a t h e r  i n t h e path  correctly,  i n their  there  but  that  has  they  is a we  taken  do not  minds each y e a r b r i n g s  a  s u r p r i s i n g l y low r e n t a l r a t e . This  simulation  i s presented  as a  contrast  Column E, which has t h e same permanent l o w e r i n g simulation  D,  (immediately)  but that  which  there  presumes  has been  that  a change  they had not a n t i c i p a t e d t h i s change a t a l l ,  t o t h e one i n  o f r e n t a l r a t e s as firms  do  realize  i n regime,  although  and r e s e t t h e i r  input  demand r u l e s a c c o r d i n g l y . Estimating  the r e v i s e d  follows.  First,  involved  a change  equation  (3.21)).  the  change  input  i n the  i n the values According  demand  to  path  equations of  rental  o f t h e parameters (3.20),  input  proceeds rates  as has  v and 0 (see  demands  depend  on  i n p u t p r i c e s through t h e s t r u c t u r a l parameters B, P, and A, which  67 do  not  change  parameters The  with  i n J ^ , which w i l l  relationship  (3.14),  t h e changes  (3.15),  between  J  in v  and  0,  and  through  the  change w i t h t h e changes i n v and 0. and v and 0 i s g i v e n  i n equations  and ( 3 . 1 6 ) . With t h e new v a l u e s o f v and 0,  new  v a l u e s o f a and 0 a r e i m p l i e d . The new v a l u e s o f a and /3 a r e  .0995  .3965  .0094  . 6430  .5909  ,8836  a =  Comparing t h e s e v a l u e s o f a and 0 t o Table 3.4 we see, as we would expect, no r a d i c a l The  second  changes.  s t e p i s t o i n c o r p o r a t e t h e new a and 0  into the  i n p u t demand e q u a t i o n s . The new i n p u t demand e q u a t i o n s w i l l  i n the  reduced form have d i f f e r e n t  terms,  and  the  constant  parameters  relating  levels  not  do  parameters  k  t  t  w  t  will  also  on t h e i n p u t p r i c e  change.  The  reduced  c u r r e n t i n p u t demand t o t h e p r e v i o u s  change,  as  they  depend  only  on  the  form year's  structural  P and B. The new reduced form o f t h e system i s  .886 0  r  terms  coefficients  1.958 .812  0  0  0  0  -720. 808 19.167 • 837  .148  670  -.006  1. 587  .981  •  k  -4206 .004  t-l  V i r  t-l  w  t-l  and t h e s i m u l a t i o n i n Column E i s based on t h i s  687 .044  +  system.  .060  - .039  68 The effect  change  i n the input  demand e q u a t i o n s  has a  substantial  on t h e r e s u l t s . Compare Columns B, D, and E. Columns D and  E i n v o l v e t h e same l o w e r i n g o f r e n t a l r a t e s . F o r 1976 s i m u l a t i o n D gives  a capital  stock  .04% h i g h e r  than  been,  but s i m u l a t i o n E has a c a p i t a l  i t would otherwise  stock  that  have  i s .12% h i g h e r  than i t would otherwise have been. The v a l u e s f o r 1980 a r e D: .18% h i g h e r and E: .48% h i g h e r . F o r 1984 we have D: .26% and E: .67%. Even respect in  though  i n this  to rental  model  capital  r a t e s , we f i n d  i s quite  that accounting  inelastic  with  f o r t h e changes  i n p u t demand r u l e s t h a t s h o u l d take p l a c e i f t h e change i n t h e  rental their  r a t e path input  effects While  i s r e c o g n i z e d by t h e f i r m  demand r u l e s  of the r e n t a l  leads  into  t o a d i f f e r e n c e i n the p r e d i c t e d  r a t e change by a f a c t o r  one might i n t e r p r e t  specify  and i n c o r p o r a t e d  this  result  the expectations process  o f around 2 o r 3.  as s u g g e s t i n g  that  can have l a r g e e f f e c t s ,  how we  we  must  keep i n mind t h a t t h i s p a r t i c u l a r model was r e j e c t e d by t h e data, and  that  we  should  r e s u l t s of t h i s  3.5  not form  general  conclusions  based  on t h e  chapter.  Conclusions The  demand, wanting  linear  quadratic  estimated  with  rational  Canadian  expectations  manufacturing  i n many r e s p e c t s . Some r e s t r i c t i o n s  model data,  of  input  is  found  i m p l i e d by t h e model  were a c c e p t e d by t h e data, o t h e r s were n o t . The model g e n e r a t e d  a  demand f o r c a p i t a l e q u a t i o n c l o s e t o t h a t o b t a i n e d by u n r e s t r i c t e d regression,  although  the r e s t r i c t i o n s  reduced  the e l a s t i c i t y  of  c a p i t a l w i t h r e s p e c t t o r e n t a l r a t e s . But a demand f o r l a b o u r t h a t is  perfectly  inelastic  with  r e s p e c t t o both  i n p u t p r i c e s must be  somewhat the  suspect.  wrong  alternative  sign. model  Also,  t h e adjustment  In  the  following  of  dynamic  performance o f t h e two models.  input  costs  f o r labour  were of  chapter  we  estimate  an  demand,  and  compare  the  70  TABLE 3.1 Estimate of U n r e s t r i c t e d Linear-Quadratic Demands Depend on C u r r e n t Input P r i c e s . Dependent v a r i a b l e k  t  t constant  w  Dependent v a r i a b l e 1  1.8416  (.24982)  -5814.4  (3724.3)  -2387.2  (2347.6)  -93.470  (179.06)  167.51  (107.49)  -2852 .3 (1035.1)  1066.6  (625.19)  standard e r r o r  a  173.08  t-1 constant  =  .67065 (.12288)  Dependent v a r i a b l e w  (.13746)  .95233  (1.8451)  -.00588  (.00411)  .93761  (.05524)  .05832  (.02916)  .17842  (.39143)  standard e r r o r  sigma  (.01080)  .71315  t-l  W  -.01311  115.97  Dependent v a r i a b l e r  r  Input  (.02120)  V i r  where  .89175  t-l  k  Model  29958. 8869.3 .03047 -.36046  l o g o f determinant  .00908  13449. .10051 •9.4383  .12193  .00008 .00021  .01487  o f sigma = 5.0322  a Standard e r r o r s o f parameter e s t i m a t e s i n p a r e n t h e s e s , b sigma i s t h e v a r i a n c e - c o v a r i a n c e m a t r i x o f t h e 4 e q u a t i o n system where t h e o r d e r o f t h e e q u a t i o n s i s , by dependent v a r i a b l e , k, 1, r, w.  71  TABLE  3.2  Estimate of Unrestricted Linear-Quadratic Demands Depend on Lagged Input P r i c e s . Dependent v a r i a b l e k  k  t-l  t-l constant  w  (.02172)  1.8431  where  Input  Dependent v a r i a b l e 1 -.01246  (.01049)  (.24106)  .65938  (.11636)  -4295.1  (3073.8)  -1542.7  (1677.4)  -47.842  (154.29)  170 .79 (78.072)  -3210.4  (860.60)  986.56  t-l  V i r  .89080  Model  standard error  a  181.02  (447.35)  102.49  Dependent v a r i a b l e r ^  Dependent v a r i a b l e w  r  .71315  (.13746)  .95233  (1.8451)  w  -.00588  (.00411)  .93761  (.05524)  constant  .05832  (.02916)  .17842  (.39143)  standard e r r o r  .00908  sigma  =  32769. 9783.5 -.46505 -2.9321  l o g of determinant  10504. -.06417 -9.4383  .12193  .00008 .00021  .01487  of sigma = 5.0314  a Standard e r r o r s of parameter e s t i m a t e s i n p a r e n t h e s e s , b sigma i s the v a r i a n c e - c o v a r i a n c e m a t r i x o f the 4 e q u a t i o n system where the o r d e r of the e q u a t i o n s i s , by dependent v a r i a b l e , k, 1, r , w.  72  TABLE  3.3  E s t i m a t e of R e s t r i c t e d L i n e a r - Q u a d r a t i c Model where Input Demands depend on C u r r e n t Input P r i c e s Technological P  18.449  P  3  B  2  a  2  (2001.7)  a  -65630. (7120600.) 298440.  (32379000.)  Parameters P  2  -320.94  B  1  -166.35 (18048.)  a  -522700. (56711000.)  -87360000. (9478100000.) Parameters  a  .11469  (.02207)  a  (S  .37882  (.08566)  0  jS  (34821.)  -.82642 ( 1 . 6 7 2 5 )  sigma  .52768  2  2  (S-.  36060. 10874. -.50141 -2.9683  b  of J  13923. -.10572 -9.5567  (.40223)  -.01073  (.00454)  .92099  (.07531)  .00008 .00020  .01568  l o g of determinant o f sigma = 5.4362 m a r g i n a l s i g n i f i c a n c e of r e s t r i c t i o n s = R e s t r i c t e d e s t i m a t e s of reduced k  t  1  .0257 form  = - 4 2 0 7 . 0 8 8 + .889k  , + 1.9291^ , - . 0 1 0 r + 0w^ t-1 t-1 t t = 8 1 0 . 9 3 3 - .001k + .7801 + Or + Ow *V  Iv  r  = .071 + . 6 5 2 r _  w  =-.084 + 1 . 3 8 2 r  t  1  Iv  - .007w _ fc  \~  Iv  1  , + .994w^ ,  a Standard e r r o r s o f parameter e s t i m a t e s i n p a r e n t h e s e s . b sigma i s the v a r i a n c e - c o v a r i a n c e m a t r i x of the 4 e q u a t i o n system where t h e o r d e r o f t h e e q u a t i o n s i s , by dependent v a r i a b l e , k, 1, r, w.  73  TABLE 3.4 E s t i m a t e o f R e s t r i c t e d L i n e a r - Q u a d r a t i c Model where Input Demands depend on Lagged Input P r i c e s T e c h n o l o g i c a l Parameters P  .28307x10  (.01858)  P  3  -.48440x10  (.32678) -2  P  3  B  2  a  2  -2.9801 (38.030) 15.830  B  (197.64)  -.24745x10  1  a  (.16848)  -6.8069 (464.49)  -3470.9 (46354) Parameters o f J  a  .11043  (.12705)  a  0  .39786  (.75508)  0  0  -.79212 (3.0780)  b  sigma  =  /3  35224. 10487. -.46782 -2.7036  13861. -.10919 -9.2650  .68720 (2.2692)  2  22  -.00993  (.01935)  .88276  (.51155)  .00008 .00021  .01530  l o g o f determinant o f sigma = 5.4367 m a r g i n a l s i g n i f i c a n c e o f r e s t r i c t i o n s = .0253 R e s t r i c t e d e s t i m a t e s o f reduced form k  = -4225.810 + .886k , + 1.9581,. , - 723.758r , + 20.140w , t t-1 t-1 t-1 t-1 1 = 687.078 - 0k + .8121 , + .801r , + .147w , t t-1 t-1 t-1 t-1 r  = .067 + • 6 7 0 r _  w  = -.039 + 1.428r  t  1  - •007w _ t  +  1  .981w  a Standard e r r o r s o f parameter e s t i m a t e s i n p a r e n t h e s e s , b sigma i s t h e v a r i a n c e - c o v a r i a n c e m a t r i x o f t h e 4 e q u a t i o n system where t h e o r d e r o f t h e e q u a t i o n s i s , by dependent v a r i a b l e , k, 1, r, w.  74  TABLE 3.5 F o r e c a s t v a l u e s o f t h e c a p i t a l s t o c k under v a r i o u s c o n d i t i o n s . Simulation Year  A  B  C  D  E  1975  18715 .3  18715 .3 (-)  18715 .3  18715 .3  18715 .3  1976  19404 .4  19229 .2 (187.7)  19236 .9  19236 . 9  19253 .2  1977  20004 .6  19721 .8 (390.5)  19733 .4  19737 .6  19768 .1  1978  20376 .6  20189 .1 (616.9)  20202 .2  20212 .7  20255 .8  1979  21013 .3  20628 .6 (839.0)  20641 .7  20659 .7  20713 .7  1980  21982 .7  21039 .1 (1044.5)  21051 .2  21076 .9  21140 .7  1981  23286 .0  21420 .1 (1228.7)  21430 .7  21464 .1  21536 .3  1982  23805 .9  21772 .0 (1390.3)  21780 .8  21821 .4  21901 .0  1983  23698 .8  22095 .4 (1529.8)  22102 .5  22149 .6  22235 .7  1984  23603 .4  22391 .6 (1648.9)  22396 .9  22450 .0  22541 .7  D e s c r i p t i o n of S i m u l a t i o n s A: A c t u a l data f o r c a p i t a l s t o c k  (machinery  B: S i m u l a t e d forecast using l i n e a r reduced form e s t i m a t e s g i v e n i n T a b l e standard e r r o r s i n parentheses.  and equipment).  q u a d r a t i c model, with 3.4, s t a r t i n g a t 1975,  C: S i m u l a t e d f o r e c a s t , u s i n g t h e model o f B, with a o n e - o f f n e g a t i v e shock t o t h e r e a l r e n t a l r a t e o f c a p i t a l i n 1975 o f 10%. D: S i m u l a t e d f o r e c a s t , u s i n g t h e model o f B, w i t h a permanent l o w e r i n g o f t h e path o f t h e r e a l r e n t a l r a t e by 10%, b e g i n n i n g i n 197 5, where f i r m s do not r e a l i z e t h e r e has been a change i n regime. E: S i m u l a t e d f o r e c a s t , with a permanent l o w e r i n g o f t h e p a t h o f the r e a l r e n t a l r a t e by 10%, b e g i n n i n g i n 1975, where t h e reduced form parameters o f t h e l i n e a r q u a d r a t i c model have been a d j u s t e d t o r e f l e c t t h e change i n t h e path o f r e n t a l r a t e s ( i . e . where f i r m s do r e a l i z e t h e r e has been a change i n regime)  CHAPTER 4 E p s t e i n and Denny's Model An  a l t e r n a t i v e model  of  input  demand t o  that  estimated  i n Chapter 3, the model of E p s t e i n and  estimated  in this  chapter,  using  m a n u f a c t u r i n g s e c t o r from 1962 A value when  the  to  same data  Denny  function that  a  second  In t h i s c h a p t e r  the v a l u e  Canadian  1984.  order  i s c o n s i s t e n t with  f u n c t i o n a l form  approximation  to  an  arbitrary  the u n d e r l y i n g economic  of the  and  (1983), i s  from the  f u n c t i o n i s s a i d t o have a f l e x i b l e  i t provides  described  theory.  firm's cost minimization  1  problem  i s d e s c r i b e d by a f l e x i b l e f u n c t i o n a l form. Input demand f u n c t i o n s are  derived  input  f o r the  prices  and  case  the  case  b e i n g d e s c r i b e d by f i r s t The to of  the  of  firms of  having  firms'  static  forecasts  order autoregressive  input  f l e x i b l e f u n c t i o n a l form model of t h i s chapter  respects.  In  both  models  a s i n g l e output,  manufacturing  sector,  adjusting  levels  forecast  of  the  of  firms  use  input  p r i c e s are  there  inputs.  are In  capital  labour  to  exogenous t o  the  internal, both  models  and  convex we  costs  assume  of  firms  f o r e c a s t s are approximated i n the e s t i m a t i o n with  those  generated  models i s t h a t be  modelled  See Diewert  as  autoregressions the  linear  higher  (although  quadratic  order  (1974, p. 133)  rationally,  i n a number  rational  order  prices  chapter  i s similar  these  first  input  and  prices  where  by  future  real  of  processes.  l i n e a r q u a d r a t i c model of the p r e v i o u s  produce  expectations  a d i f f e r e n c e between  model a l l o w s  autoregressions  or E p s t e i n  expectations  i f desired,  (1981, p.  87)  the to  whereas  the model i n t h i s  chapter  does n o t ) .  One d i f f e r e n c e b e t w e e n t h e two m o d e l s i s t h a t w i t h a functional  the  form,  technology  maximization  by  definition,  are  those  (or cost  technology  of "gross  have been  subtracted)  costs  themselves.  adjustment whereas  costs  in  the  specification changes  the only  required  by  minimization) .  output"  restrictions (  the This  applies  and t o t h e s p e c i f i c a t i o n example,  were  assumed  flexible  in to  the  form,  both  and  not  to  quadratic and  the  the  necessarily additively  model  additive,  more  t h a t c o s t s be i n c r e a s i n g and c o n v e x w i t h  i n inputs,  profit  of the adjustment  quadratic  functional  on  adjustment costs  linear  be  placed  assumption of  ( i . e . output before  For  flexible  general  respect  to  separable,  is  used. The chapter time, To  other firms  while  major take  the l e v e l  i n the l i n e a r  our knowledge  have  difference  been  there  quadratic  a r e no  are  rational,  We  i n the  as g i v e n  model  existing  there 2  f u n c t i o n a l form i s f l e x i b l e .  the  of output  e m p i r i c a l l y implemented  expectations  given  i s that  are  of  this  a t any p o i n t  output  models  where  model  was  endogenous.  of the firm  output  is  that  endogenous,  adjustment costs,  and  see below t h a t t r e a t i n g output  g r e a t l y improves t h e f i t of t h i s model, r e l a t i v e  in  to that  the  as of  l i n e a r q u a d r a t i c model. Finally,  This  would  provides  t h e model of t h i s  certainly  the  expectations.  be  theoretical  a  chapter  worthwhile model  for  implicitly  allows  project.  Epstein  the  case  of  f o r the  (1981) static  c o n t r i b u t i o n of a m a t e r i a l i n p u t , while the l i n e a r q u a d r a t i c model had  o n l y the two  4.1  The  i n p u t s , l a b o u r and  capital.  Model  The  specification  i s taken  (1983) without a l t e r a t i o n . As y , inputs x constant  =  (k  1.)',  discount  directly  r e a l input p r i c e s z  rate  this  chapter  R.  Chapter  3.  purely  i s given  The  Both  Input  prices  t  the  c l o s e t o , but  capital  variable  normalize  y  i s very  input,  and  labour  materials,  q u a s i - f i x e d input  =  in  given not  are  v^_,  and  Denny  to  (r  w )', and  the  this  chapter  are  i n Table  identical  2.1,  whose  so  to,  quasi-fixed.  p r i c e s . The  z  from Define  price  technology  fc  we of  a  will a  firm  by  F( =  V V  ^-^  Vi ' 1  t -  x  v a r i a b l e cost f u n c t i o n i s given  c(y  Epstein  i n Chapter 3 denote output at time t  d e f l a t e d by the m a t e r i a l s p r i c e index m, in  from  f  V  x  t " t-i> x  =  m  i  n  {  v  v  t  :  by  y  t -  F(  V  V  x  t -  x  t - i  )  K  t (4.2)  We and  assume f o r now  output  relaxed.  to  Under  remain static  t h a t the  constant,  future costs:  infinite  problem  at  a time path of i n p u t l e v e l s i n order  to  we  set  this the  the p r e s e n t  will  prices be  horizon  although  input  later  expectations  time 0 as t h a t of choosing minimize over an  f i r m expects c u r r e n t  firm's  discounted  value  of  78  00  -Rt  rain X e • 0 t  [C(y, x , x ) + z'x  ]dt  (4.3)  X  subject to x  Each  time  - x L>  period  r e v i s e d , and Let  = x *-r  , L*  x  J-  expectations  of  input  the s o l u t i o n t o problem  V ( X g , y,  z)  be  the  value  and x  given,  \J  prices  (4.3)  of  L.  > 0 for a l l t.  and  output  are  i s recalculated.  the  problem  (4.3).  We  note  here t h a t below t e c h n o l o g y w i l l be d e f i n e d i n terms of the  form of  V.  partial  If  we  define  V  and  V  zx derivatives, a d j u s t i n g x,  respectively,  2x2  then  z) =  and  and  the  as d e r i v e d by E p s t e i n  x*(x , y, t t  Epstein  as  2x1  matrices  of  z  V'  1  zx  decision  rule  for  (1981) , i s  (x , y, t  z)[RV  (1983,  pp.  Denny  optimal  z  (x , y, t  651-2)  z) - x ] . t  list  the  (4.4)  properties  V  3 must s a t i s f y  i f C i s to s a t i s f y  certain regularity conditions.  A  p a r t i c u l a r s p e c i f i c a t i o n of V t h a t s a t i s f i e s those p r o p e r t i e s i s  There  are  six  conditions:  i n c r e a s i n g i n y and  x and  that  must  decreasing  a unique s o l u t i o n t o problem the  C  (4.3)  be  i n x,  positive, C must be  C  must  convex i n x,  must e x i s t f o r each  (x^, y, z ) ,  unique s o l u t i o n must have a unique steady s t a t e i n p u t i s g l o b a l l y stable,  and  f o r any  * vector  of  input  prices  time 0 i n problem  (4.3)  z  ( Q? x  YI  x  be  ) there  level  exists  x a  .* such t h a t  given  (x , y,  x  i s the  z ).  optimal  policy  at  79  V(x , y, z) =  [z' 1]  y/2 + ( z ' * "  1  + A')x x t (4.5)  + R  where $, and  and Q  1  (z *A /  + h) + Q'x /y + x'Q x /2y x t t xx t  1  are each 2 x 2 m a t r i c e s of parameters, <f>, A , A,  X X  X  a r e each 2 x 1 v e c t o r s o f parameters, and b and h are s c a l a r  parameters. Combining  the  w i t h the f l e x i b l e  solution  (4.4)  functional  form  of  the  dynamic  problem  (4.5), the o p t i m a l r u l e  (4.3) for x  fc  is  X  t  =  M  <  X  t - l  "  X )  (4.6)  '  where  M = R - *,  (4.7)  x(y, z) = - (R -tf) { R * [$z + _1  and  where  diagonal state,  R  is  (as  element  i n Chapter  e q u a l t o R.  or t a r g e t ,  The  <P]y  3)  +  a  A},  (4.8)  diagonal  vector  x  represents  demands f o r t h e q u a s i - f i x e d  f u n c t i o n o f t h e l e v e l o f output and o f i n p u t The which The  optimal rule  arises  (4.6) has  i n the l i n e a r  structural  parameters  a c r o s s the two models,  with  underlying  M  each  the steady  factors,  and  is a  prices.  a reduced form i d e n t i c a l  q u a d r a t i c model  however.  matrix  to that  (see e q u a t i o n are  clearly  (3.9)).  different  80 The r e s t r i c t i o n s cost minimization,  which V must  and which w i l l  satisfy  t o be c o n s i s t e n t  with  be t e s t e d i n the e s t i m a t i o n of  the model are  $ i s symmetric  (4.9)  M has 2 r e a l e i g e n v a l u e s  Condition  (4.10) was  (see c o n d i t i o n  also  between -1 and 0.  imposed on the l i n e a r  be e s t i m a t e d .  A final  step before  (4.6),  t  = e  y t  model  (m , x , x t  fc  where j r e p r e s e n t s  fc  (4.7), and (4.8)  e s t i m a t i n g the model  be t o i n c o r p o r a t e t e c h n i c a l change by changing  y  quadratic  (3.13)).  The i n p u t demand f u n c t i o n s i m p l i e d by will  (4.10)  (4.1) t o  - x ^ ) ,  the e x p o n e n t i a l  will  (4.11)  r a t e o f t e c h n o l o g i c a l change. -yt  In  the  analysis  following  reduced  above form  substitute parameters  y^e using  f o r y^. the  same  Defining notation  the as  E p s t e i n and Denny, E = R m ,  (4.12)  G = Rtf#,  (4.13)  the e s t i m a t i n g e q u a t i o n  f o r t h i s s t a t i c e x p e c t a t i o n s case i s  81 x t  /y  =  t  (i +  M  )  x t  _ i  y  /  +  [  E  z  t  t  +  G ] ( 1  +  y ) _ t  +  x  /  y  t  +  u  t ' (4.14)  where  is  a  random  error  vector.  Epstein  and  Denny  use  the  a s s u m p t i o n t h a t c h a n g e s i n x a f f e c t y o n l y a f t e r a one  period lag,  so  of  y  i s predetermined i n  econometric non-static demands  techniques.  Below,  expectations,  depending  (4.14).  on  we  This  allows  when t h e  consider  lagged  the  model  as  output,  use  i s estimated  alternatives and  standard  (ii)  (i)  input  with input  demands  d e p e n d i n g on p r e d i c t e d c u r r e n t o u t p u t w h e r e t h e p r e d i c t i o n i s made in  the  preceding  time  exogenous, although  we  this  a s s u m p t i o n may  be  with  non-static  period.  We  suspect.  expectations  prices this  are  data  set  H o w e v e r , e s p e c i a l l y when we  in this  model,  i t i s an  work  assumption  estimation.  Estimates Equation  likelihood Table restriction  * =  The  (4.14)  is  SHAZAM.  4.1  the  gives  (4.9)  the  model,  11  12  21  22  $ $  of  $ $  21  imposed  r a t e , R,  follow Epstein  12  non-linear  e s t i m a t i o n of  maximum  (4.14) w i t h  J  on  '  <P =  r  •*2  22'  the  the  Since  the  the  setting  V  \  which  imposed by  affect  Denny by  •  •  estimates,  restriction  does not and  the  notation i s  11  other  with  a s t a b l e adjustment matrix.  discount we  results  i m p o s e d . The  i s r e j e c t e d . The  t h a t M be  estimated  o p t i o n of  single restriction  <$2l'  of  input  found i n Chapter 3 that with  w h i c h m u s t be made f o r p u r p o s e s o f  4.2  assume  is  the  choice  -  =  model  of a  value  reduced form of R  =  .07.  is  Then  the the  82 e s t i m a t e d v a l u e of M i s  -.1212  1.5230"  .0420  -.9026_  M =  F  which has  eigenvalues  of  -.047  and  -.977,  satisfying  restriction  (4.10) . Epstein follows.  If  adjustment occurs of  and  labour  of  i n one  the  year.  and  and  data Denny  90%.  In  rates  reason  linear  rate  the of  adjustment The  but  stock  state  towards  elements  value,  occurs  labour the  annual  in  were  model  11%  and  quadratic s p e c i f i c a t i o n for  labour  the two  than  of  the level 90%  Using  U.S.  to  Chapter  predicts  1976  So  3, for  a much  the  models each p r e d i c t  of  parameters  21%.  does  as  level,  1947  adjustment  quadratic  adjustment  year.  M  state  state  observations  identical  linear  one  of  12%  i t s steady  in  o b t a i n the  adjustment  specification,  i t s steady  the  i s at i t s steady  for  of  interpret  If capital  the  respective  659)  at  capital  adjustment  Epstein  (p.  is  the  manufacturing  12%  Denny  more  the  of the  some  slower  flexible  same speed  of  for capital. reduced  form  parameters  implied  by  the  restricted  e s t i m a t i o n are  A  " -.2012  .0030'  E = -.0140  The  .0206  A  , -.0092  ' -2890.4  A  G =  ,  and  \ 319.82  .0899  d i a g o n a l elements of E are the own-price c o e f f i c i e n t s  c a p i t a l / o u t p u t and l a b o u r / o u t p u t r a t i o s ,  f o r the  so t h e i r n e g a t i v e s i g n i s  expected. The  impact  elasticities  generated  by  the s t a t i c e x p e c t a t i o n s  83 model a r e , c a l c u l a t e d at 1975 l e v e l s , as f o l l o w s :  s h o r t run e l a s t i c i t y with r e s p e c t t o :  The s h o r t Epstein  r  w  Y  capital  -.048  + .027  + .026  labour  -.018  -.439  + .615  run e l a s t i c i t i e s  and  Denny  (1983,  are v e r y p.  661),  t o those  i n particular  elasticities  (although  elastic  r e s p e c t t o the wage than they obtain)  with  our e s t i m a t e  similar  obtained  the  i s f o r a labour  by  own-price  demand more  and the  output  elasticities. To  calculate  expectations  model  the we  long note  run that  elasticities steady  for this  state  input  static  levels  are  g i v e n by  x(y,  z) = - M  {[Ez + G]ye  -rt  + X}  (4.15)  ( o b t a i n e d by r e w r i t i n g (4.8) i n terms o f reduced As we  d i d i n Chapter  mean t h e r e l a t i v e  3 we  will  take  the long  change i n the t a r g e t  that M i s a s t a b l e matrix,  parameters).  run e l a s t i c i t y  levels  r e l a t i v e change i n i n p u t p r i c e s o r i n output.  form  of i n p u t s  given  Since we have  by the same r e a s o n i n g  to a  found  i n Chapter 3 t h i s  method of c a l c u l a t i n g l o n g - r u n e l a s t i c i t i e s g i v e s t h e same r e s u l t s as i f we c o n s i d e r e d t h e l o n g run change i n a c t u a l v a l u e s . again  the 1975  levels  of c a p i t a l  and l a b o u r :  18715.3  and  Consider 3513.0.  84 Applying  equation  inputs at  1975  (4.15) we  t o be  find  35209.3 and  l a r g e r than the maximum l e v e l s 1984  sample.  quadratic  This  is  model of  the  Chapter  inputs  3 we  l o n g run e l a s t i c i t i e s ,  of  those  found t a r g e t  in  levels  two  values  observed i n our  surprising, since  l e a s t ) much c l o s e r t o a c t u a l l e v e l s . The  levels  4384.7; each of these  of  somewhat  target  1961  the  is to  linear  (at 1975  at  4  c a l c u l a t e d at 1975,  are:  l o n g run e l a s t i c i t y with r e s p e c t t o : r  These  y  capital  -0.561  -1.181  +2.327  labour  -0.225  -0.831  +1.415  values  found by  w  are  Epstein  a l l much and  larger  in  absolute  Denny. Note t h a t  the  terms  elasticity  than of  those capital  demand with r e s p e c t t o the wage changes s i g n from the s h o r t t o  Nickell in  (1985) p o i n t s out  models  of  necessarily choice it  involve  v a r i a b l e and  i s not  necessary target"  input  simply to  (p.  demand  optimal  with  "asymptotically i t s optimal  up  completely  s t r a t e g y f o r the  adjustment  costs  c l o s i n g the  gap  target  worth i n c u r r i n g the  catch 121).  t h a t the  value...given  the  firm  will  not  between  his  discounting  a d d i t i o n a l adjustment with  the  [perhaps]  costs  growing  85  long  run. This  (1.523)  i s due  to  two  f o r c a p i t a l with r e s p e c t  elasticity  of labour  factors:  the  large  t o lagged labour  demand w i t h  respect  coefficient  demand, and t h e  t o t h e wage. While t h e  impact e f f e c t o f a change i n t h e wage i s a movement o f c a p i t a l i n the  same d i r e c t i o n , t h e r e  opposite this  i s a large  d i r e c t i o n to that  change i n l a b o u r  o f t h e wage  response  of labour  change. A f t e r  one  i n the period  demand has a s u b s t a n t i a l e f f e c t on c a p i t a l  demand, r e v e r s i n g t h e o r i g i n a l e f f e c t o f t h e change i n t h e wage on c a p i t a l . The l a r g e c o e f f i c i e n t lagged labour  r e l a t i n g t h e demand f o r c a p i t a l t o  demand a l s o appeared  i n the l i n e a r  quadratic  i n Chapter 3 , both i n t h e r e s t r i c t e d and u n r e s t r i c t e d The  elasticity  of c a p i t a l with respect  was p o s i t i v e i n both t h e s h o r t not change because l a b o u r  estimation.  t o t h e wage i n t h a t  run and t h e l o n g  model  model  r u n . The s i g n d i d  was c o m p l e t e l y i n e l a s t i c w i t h r e s p e c t t o  the wage. Compared t o t h e e s t i m a t e s we o b t a i n e d l i n e a r quadratic  i n Chapter 3 with t h e  model, we f i n d with t h e f l e x i b l e  f u n c t i o n a l form  a demand f o r c a p i t a l t h a t i s much more e l a s t i c with r e s p e c t r e n t a l r a t e , and a demand f o r l a b o u r responsive  t o the  t h a t i s , u n l i k e i n Chapter 3 ,  t o changes i n t h e wage and i n t h e expected d i r e c t i o n .  Although i d e n t i c a l data a r e used t o e s t i m a t e t h e two models, i t i s not  c l e a r e x a c t l y what d i f f e r e n c e i n t h e models i s r e s p o n s i b l e f o r  the  s u b s t a n t i a l d i f f e r e n c e i n estimated  elasticities.  A s u r p r i s i n g r e s u l t i s that the e l a s t i c i t y capital  with  respect  t o t h e wage  t o the r e n t a l  i s greater  rate.  than  i t s elasticity  with  respect  352)  a l s o o b t a i n t h i s r e s u l t , a l b e i t i n a model where o n l y  i s t r e a t e d as a q u a s i - f i x e d f a c t o r .  Morrison  o f t h e demand f o r  and Berndt  (1981,  p.  capital  86 The returns  output to scale,  elasticities casting  suggest  some  doubt  implausible  decreasing  on t h e r e l i a b i l i t y  of the  other estimated e l a s t i c i t i e s .  E p s t e i n and Denny a l s o found  elasticities  t o be g r e a t e r than one i n t h e l o n g  run,  f o r both  factors  output  a l t h o u g h not t o t h e degree o b t a i n e d here. As  i n t h e model o f Chapter  slow,  so  i t  elasticities.  is  again  The slow  3, adjustment  instructive  convergence  to  to target  calculate  levels i s  medium  t o t h e l o n g run response  t h i s model i s f o r t h e same reasons  as i n Chapter  3, t h a t  run with  i s that  2 the m a t r i x  (I + ( I + M )  + (I+M)  + ...) i s slow t o converge t o  -M . The f i v e - and t e n - y e a r e l a s t i c i t i e s a r e 1  five-year e l a s t i c i t y r  w  y  capital  -0.139  -0.238  +0.502  labour  -0.059  -0.459  +0.696  ten-year e l a s t i c i t y r  We  which  with respect t o :  w  y  capital  -0.228  -0.437  +0.888  labour  -0.094  -0.538  +0.849  see t h a t  elasticities reached  with respect t o :  for capital,  even  after  10 y e a r s  neither the  w i t h r e s p e c t t o p r i c e s nor t h e output e l a s t i c i t y  one h a l f  of t h e i r  long  some r e s e a r c h e r s t r e a t  run v a l u e s . Even  as a variable  input,  have  l a b o u r demand, i s remarkably  87 (perhaps i m p l a u s i b l y ) The  slow t o a d j u s t t o i t s l o n g  l e s s o n i s t h a t l o n g run e l a s t i c i t i e s  great  care,  not  only  in this  study,  Denny's e s t i m a t e  is  and  similar  to  elasticities  ours,  data  rental  Mohnen,  v a r i a b l e cost  yield  would expect  and  Vigeant's  f u n c t i o n with  a s h o r t run e l a s t i c i t y  rate  estimate,  we  of  but  -0.098,  their  which  is  ours  manufacturing  elasticities  of l a b o u r demand with  is  smaller  than  ours  respect  U.S.  are  Bischoff  Pindyck  but  to  -0.784,  and  similar long  run  their  output  and  find  +1.031,  and  +  -0.005  and  run  of  respect cost  elasticity  to  the  function  i s -0.271,  impact  run  and  long  run  -0.118  elasticity  elasticity  is  l o n g run e l a s t i c i t i e s  +0.038  and  a  manufacturing  our  of the  are +2.339 and  and  (1983a)  Rotemberg,  -0.035,  and  of of  +1.713.  and  +0.150,  -0.130,  who  and  Kokkelenberg  own-price  elasticities  and  Bischoff  long  run  respectively.  own-price  They  claim  long  elasticities  of  -2.927  find  l a b o u r have l o n g run output and  and  r e p o r t o n l y the  l a b o u r have output  r e s p e c t i v e l y . Kokkelenberg  0.780  with  to  long  are  c a p i t a l and  e l a s t i c i t i e s : c a p i t a l and of  estimates  r e s p e c t t o the wage are  Rotemberg  (1986) . Pindyck  +1.476  M  s t u d i e s of c o s t f u n c t i o n s where t h e r e are adjustment  run e l a s t i c i t i e s , of  long  e s t i m a t e ) . T h e i r s h o r t run and  l a b o u r w i t h r e s p e c t t o output  costs  their  annual Quebec  r e s p e c t i v e l y ; t h e i r estimate  with  Two  that  (1989)  s u b s t a n t i a l l y l a r g e r . T h e i r s h o r t run and capital  well; for  (although q u i t e c l o s e t o E p s t e i n  Denny's U.S.  -2.354,  as  adjustment m a t r i x  of c a p i t a l  corresponding  s u b s t a n t i a l l y l e s s than  and  here.  i n t e r p r e t e d with  i n others  of the  level  are a l s o somewhat m i s l e a d i n g .  Carmichael, translog  so  must be  but  example, E p s t e i n and  run  much  and lower  elasticities  elasticities these  are  of "of  88  r e a s o n a b l e magnitude"  (p.429).  T a k i n g a l l o f t h e above s t u d i e s t o g e t h e r we f i n d , even  over  a  fairly  disagreement our fall  restricted  class  o f models  on t h e magnitudes o f e l a s t i c i t i e s ,  estimates  of short  run and l o n g  our  estimates  are  much  of input  higher  demand e l a s t i c i t y  than  other  there  studies,  with  that  i s wide  and second,  run own-price  w i t h i n t h e range e s t a b l i s h e d by p r e v i o u s  first,  that  elasticities  s t u d i e s , but t h a t respect  especially  t o output  the  long  and Rotemberg,  run  elasticities.  E p s t e i n and Denny, and Pindyck  also  f i n d evidence  o f d e c r e a s i n g r e t u r n s t o s c a l e , but not t o t h e same  degree as our e s t i m a t e s .  4.3 The Model with Non-Static Expectations Now  we  change  t h e model  levels  somewhat by assuming  expect  output  and i n p u t  prices  rather  form e x p e c t a t i o n s by a p r o c e s s  simple  first  t o remain  firms  constant,  t h a t we can r e p r e s e n t  order autoregressions. R e c a l l  from Chapter  v a r i a b l e t o be f o r e c a s t f o l l o w s a f i r s t  one  o b s e r v a t i o n o f t h e v a r i a b l e , p l u s e s t i m a t e s o f t h e time of  the  variable,  T h e r e f o r e t h e reduced  are  required  although  be d i f f e r e n t . will  be  process,  make  from t h e s t a t i c  only series  forecasts.  expectations  t h e i n t e r p r e t a t i o n o f t h e reduced  Such w i l l be t h e case here;  reestimated,  to  with  form o f i n p u t demands when e x p e c t a t i o n s a r e  formed t h i s way w i l l not be d i f f e r e n t formulation,  order  but  3 that i f  the  parameters  do not  but  t h e reduced  the s t r u c t u r a l  different  estimated  values.  Correct  structural  parameters  i s important  form  parameters  identification  i f policy  form  will  (4.14)  will of  have the  a n a l y s i s using the  r e s u l t s o f t h e e s t i m a t i o n i s t o be immune from t h e Lucas  critique.  89 We  model t h e e v o l u t i o n of output  levels  and i n p u t p r i c e s as  follows:  log  r  fc  „  t  (y ) = a + l o g ( y _ ) , fc  f c  = u^expte^) - D / 9  + exp(9  1 1  = u (exp(fl ) - 1)/G 2  (4.16)  1  2 2  2 2  1 1  )r _ , t  + exp(e )w _ 2 2  (4.17)  1  t  (4.18)  l f  u where  exp(w)  ponfused method  with of  denotes e their  and  where  meaning  describing f i r s t  the  i n Chapter order  AR  9 3.  terms This  processes,  are  not  i s an but  to  be  unusual  i t allows  E p s t e i n and Denny t o i n c o r p o r a t e the AR parameters i n t o the i n p u t demand e q u a t i o n s Equation  i n a r e l a t i v e l y s t r a i g h t f o r w a r d way.  (4.14) can now be r e e s t i m a t e d , but with the  reduced  form parameters now b e i n g d e f i n e d by M = *[(R - G)*"  1  - I],  (4.19)  E = (R - a ) * * - tf($9 + 9$),  (4.20)  G = * [ ( R - a - 9)0  (4.21)  - $u],  A r e p l a c e d by R ^ ( R - 9 ) * A ,  (4.21)  - 1  where  I i s the i d e n t i t y m a t r i x ,  the d i a g o n a l and zeroes  9 i s 2x2 with  6 ^ and 9  2 2  along  o f f the d i a g o n a l , a i s 2x2 d i a g o n a l  each d i a g o n a l element e q u a l  t o a, / i i s 2x1 c o n s i s t i n g  of fx  with and  90 H^,  and A i s 2x1 c o n s i s t i n g o f parameters (1983, p. 664)). Note t h a t i f 9 ,  Denny if  expectations  are s t a t i c ,  E,  G,  to  estimate  (4.14)  (4.18)  were  a,  and  ( E p s t e i n and  and fi a l l e q u a l 0,  M,  and X  return  to  i.e. their  p r e v i o u s form. Attempts (4.17),  and  estimates run  were always  elasticities  simultaneously  not s u c c e s s f u l ,  with  (4.16),  i n particular  because  o f an u n s t a b l e M m a t r i x ,  would  which means  not be w e l l d e f i n e d , u n l e s s  long  the discount  r a t e R was a b s u r d l y h i g h . So we f o l l o w e d t h e method o f E p s t e i n and Denny  by e s t i m a t i n g  (4.16),  then  taking  parameters  these  re-estimating maximum  (4.14),  likelihood  therefore  be  taken  which  option when  (4.17), as  was of  and  (4.18)  given  estimated SHAZAM.^  s e p a r a t e l y , and  when with  Note  i t  came  to  the non-linear that  considering the estimates  care of  must  standard  e r r o r s o f t h e model. It  was a l s o n e c e s s a r y  to place r e s t r i c t i o n s  on * so t h a t we  would have a s t a b l e M. S i n c e i t was not f e a s i b l e t o r e s t r i c t M t o be  stable  restricting reduced  i n any  general  sense  ^ t o the p a r t i c u l a r  i n the estimation, values  which  would  we  give  tried us t h e  form f o r M t h a t we o b t a i n e d i n t h e r e s t r i c t e d e s t i m a t i o n  of t h e s t a t i c e x p e c t a t i o n s model. Results of the 5 equations, $  imposed, and w i t h  with t h e r e s t r i c t i o n  restricted to  T h i s method i s suggested by W a l l i s (1980) .  $  =  91  .4434 -1.2883 .0716  are  given  in  1.0040  Table  4.2.  The  likelihood  statistic  of  the  2 restrictions  i s 3.606, d i s t r i b u t e d  %  with  5 degrees  o f freedom,  so they a r e not r e j e c t e d . The  reduced  expectations  form e s t i m a t e  case  given  o f M i s t h e same as f o r t h e s t a t i c  above.  Other  reduced  form  parameters  i m p l i e d by t h e Table 4.2 e s t i m a t e s a r e " -.1702  A  +.0011 ,  +.0029  .0292  A  E =  G =  -.0093  .0756  and t h e v e c t o r which r e p l a c e s X from t h e s t a t i c  expectations  case,  as d e f i n e d by (4.21) i s  -3149.991 R *(R - 0)* A = 1  1  703.127  The  i m p l i e d v a l u e s o f impact e l a s t i c i t i e s , taken  a t 1975, a r e :  s h o r t run e l a s t i c i t y with r e s p e c t t o :  The  r  w  y  capital  -.044  +.011  +.039  labour  +.004  -.486  +.510  own-price  elasticities  expectations estimates Finding  the  e x p e c t a t i o n s case  are barely  different  from  the  static  the  non-static  above.  long  run  elasticities  i s somewhat c o m p l i c a t e d ,  in  s i n c e we must c a l c u l a t e  92 how  the  or  the  reduced form parameters E and level  permanently The be  not  see  run  have  elasticity  difficult  that  a  to  their  expected  demands are  of  growth  input  a f f e c t e d by  r e f e r to equation  Regarding  rate  demand  the  (4.15),  t o f i n d the l o n g run  respect  proportional  exponential  parameters  with  future  t o output  c a l c u l a t e . From e q u a t i o n  permanent  underlying form  output  paths  shifted.  long so  of  G change when i n p u t p r i c e s  increase a  are  level  turns  out  to  (4.16) above  we  in  y  unchanged.  So  unaffected  change  leaves the  the  reduced  (although  input  i n output) , and  we  which d e s c r i b e s the t a r g e t i n p u t  can  levels,  elasticities.  long  run  price  elasticities,  a  long  run  p r o p o r t i o n a l change i n i n p u t p r i c e s would i n v o l v e a change i n t h e fx parameters affect  the  from e q u a t i o n s  (4.17) and  (4.18).  reduced form parameters of the  This  vector  G;  (4.21). A change i n n changes G by a f a c t o r of and for  $ are g i v e n Epstein  direction.  and  This  the  expected  and  the  Denny's)  expected  signs  in  the  that  /n and  (so t h a t M  own  G  For  example,  fall  in  do make the linear  we  r e n t a l rates  move  in  where $  price elasticities  see has  in a  equation  Estimates  will  estimates  i s stable)  will  see  of  f o r our e s t i m a t e s  we  reduced form parameters t o take  i n p u t demands i f we we  I t t u r n s out  f u t u r e p r i c e s a c t u a l l y lowers  permanent  if  4.2.  i s s u r p r i s i n g . For  adjustment m a t r i x  adjusting  change.  i n Table  change would  are  o b t a i n the account  the  (and  the  and  *  same  $  have  negative  result  that  of a change i n  effects  simulations  of  the  below  price that  larger positive effect  a on  do not a d j u s t the reduced form parameters than ( c o r r e c t ) adjustment. The  q u a d r a t i c model of Chapter 3;  converse was the  the  intuition  case  behind  93 the p e r v e r s e From  result  o f t h e model o f t h i s c h a p t e r  equation  (4.15)  and  the  i s not o b v i o u s .  estimates  of  reduced  parameters from t h i s s e c t i o n we c a l c u l a t e t a r g e t l e v e l s and  labour  identical  in to  1975  of  the  35459.4  target  and  level  by  factors  of  about  2  and  of c a p i t a l  These  estimates  e x p e c t a t i o n s model, and a r e h i g h e r than time  4416.4.  from  are  nearly  the  the a c t u a l l e v e l s  1.5  form  for capital  static at that  and  labour  r e s p e c t i v e l y . The l o n g run e l a s t i c i t i e s a r e :  l o n g run e l a s t i c i t y with r e s p e c t t o : r  w  y  capital  -0.178  -0.971  +2.096  labour  -0.055  -0.545  +1.232  Each o f these e l a s t i c i t i e s i s s m a l l e r i n a b s o l u t e v a l u e than  those  o b t a i n e d from t h e s t a t i c e x p e c t a t i o n s model. Once again, since  we  should  t h e adjustment  ten-year  look  to long  a t t h e medium  run i s v e r y  run e l a s t i c i t i e s ,  slow.  The  five-  e l a s t i c i t i e s are  f i v e - y e a r e l a s t i c i t y with r e s p e c t t o : r  w  y  capital  -0.047  -0.216  +0.459  labour  -0.003  -0.248  +0.587  and  94  ten-year e l a s t i c i t y r  It  -0.376  +0.805  labour  -0.014  -0.311  +0.724  is  interesting i f we  of p r i c e s . reduced  would  changed) .  unadjusted  the  -  estimates  of  (although permanently long  run  o b t a i n e d from t h e reduced forms, a r e  (G unadjusted) w  with r e s p e c t t o :  y  -1.508  +2.104  labour  -0.166  -0.993  +1.237  noted  above t h e s e  when G i s a d j u s t e d t o account  a) y  do not change  o f output  -0.453  also  replacing y  run  i n t h e reduced  capital  reasons  long  a r e u n a f f e c t e d , because  o f growth  r  We  f o r t h e change  of  o f a g i v e n p r o p o r t i o n t o output  l o n g run e l a s t i c i t y  For  estimates  o f t h e i n p u t demands  i f the rate G  the  elasticities  parameters  change With  note  s h o u l d o c c u r with a change i n t h e l o n g run  The output  form  elasticities,  to  do not account  w i t h a permanent s h i f t they  y  -0.075  form parameter G t h a t  the  w  capital  elasticities  path  with r e s p e c t t o :  tried  are larger  i n absolute value  f o r changed e x p e c t a t i o n s .  e s t i m a t i n g t h e n o n - s t a t i c e x p e c t a t i o n s model  i n t h e i n p u t demand e q u a t i o n s with y^ ^ and w i t h  (which  than  c o u l d be  a proxy  f o r expected  y ) . There  (1 +  a r e no  clear  theoretical  another here. and  reasons  The  l o g s of the  -  (1  for  a)y  choosing  r e s p e c t i v e l y . There are no  specification  over  l i k e l i h o o d f u n c t i o n s from u s i n g y , t  are  t-1  one  236.359,  217.694,  changes i n the  and  219.556,  s i g n s of the  structural  parameters. Finally, imposing  we  the  quadratic  attempted  short  model of  run  estimating  Epstein  elasticities  Chapter  3 . We  we  are  and  Denny's  obtained  in  the  model linear  i n t e r e s t e d i n whether  this  r e s t r i c t i o n s u b s t a n t i a l l y lowers the goodness of f i t of the model. In  the  model  determined by  our  of  this  estimate  chapter, of  E,  the s t r u c t u r a l parameter $. We elasticity  (we  elasticity,  since that  the  adjustment  elasticity the  cannot  wage  of of  -.004  which  would e n t a i l Recall  i n turn  elasticities  are  i s determined  restrict  in  respect  long  t o the  3  we  run  on  obtained  rental rate a  by  run  restriction  Chapter  r e s p e c t i v e l y , and  +.004  the  a complicated  that  demand with  and  price  seek o n l y t o r e s t r i c t the s h o r t  simultaneously  process). capital  own  labour  and  demand  i n e l a s t i c with r e s p e c t t o both the r e n t a l r a t e and the wage. These are the  short  Chapter  3  price  the  index,  run e l a s t i c i t i e s rental rate rather  and  than  the  we  will  impose here.  Note t h a t  wage were d e f l a t e d by materials  price  the  index,  in  output so  the  comparison i s not p r e c i s e . Given our imposed v a l u e s  of R,  of $ of  -.0618  +.0028  -.0072  +.0033  f  i n o r d e r t o o b t a i n a v a l u e f o r E of  a,  and 6, we  impose a  value  96  -.0157  +.0004  0  0  and t h e e l a s t i c i t i e s g i v e n above. When t h e system i s e s t i m a t e d , t h e o n l y f r e e parameters a r e <f>, and A;  five  parameters.  parameters  i n a l l . There  The u n r e s t r i c t e d  238.16, and t h e r e s t r i c t e d  are eight  restricted  l o g of the l i k e l i h o o d  result  function i s  i s 188.84, so t h i s  restriction  i s c l e a r l y r e j e c t e d . S i n c e so many a s p e c t s o f t h e l i n e a r q u a d r a t i c model r e s u l t s were u n s a t i s f a c t o r y , we a r e not a l t o g e t h e r s u r p r i s e d t h a t t h e e l a s t i c i t i e s o b t a i n e d with t h a t model a r e r e j e c t e d by t h e model o f t h i s  chapter.  T h i s model with n o n - s t a t i c e x p e c t a t i o n s w i l l simulations  comparable  to  those  done  now be used f o r  i n Chapter  3  with  the  which  are  a l t e r n a t i v e model.  4.4 S i m u l a t i o n s w i t h N o n - S t a t i c E x p e c t a t i o n s Table  4.3  records  analogous  t o those  Column  lists  A  the r e s u l t s  i n Table  actual  of  3.5 from  data  simulations  the l i n e a r  for capital  q u a d r a t i c model.  i n the  manufacturing  s e c t o r . Column B g i v e s a l o n g term f o r e c a s t one might have made i n 1975  using  the f l e x i b l e  expectations  as  estimated  functional above.  form  At  model  with  1975, output  non-static i n 1984 i s  f o r e c a s t a t 88170.8 w h i l e t h e a c t u a l v a l u e was 76700.9, t h e r e n t a l r a t e i s f o r e c a s t a t .128 w h i l e t h e a c t u a l v a l u e s were .130 i n 1983 and  .146 i n 1984, and t h e wage  actual while  value  was 4.65. Labour  the a c t u a l  value  i s forecast  i s forecast  a t 4.43 w h i l e t h e  t o be 4062.1 i n 1984  was 3621.0. We see i n t h e t a b l e  that the  97 capital  stock  throughout  over-estimated with t h i s In rate  1975,  parameters  but  it  temporary  (column  B)  By  1984  forecast  rental  (column C) f o r e c a s t  and  demand  labour  even a t 1984  in  that  the  the e f f e c t s  the  rate  rental  f i ^ and  o f the shock  on the  10%  labour  quadratic  columns  from  the o n e - o f f  r a t e i s .12621. But  i s 4074.6,  their  capital  1975  i s 1.10%  on  i s .12775, and  base  t h e r e a r e s i g n i f i c a n t l i n g e r i n g e f f e c t s from t h e i n 1984  heavily  model  left  D  and  E  1984  h i g h e r than which  i t otherwise  i s 0.31%  i t s base  capital  only  h i g h e r than i t shock i n t h e  0.02%  h i g h e r than  listed  the  forecast  paths  the r e n t a l r a t e i s permanently  case  would  c o m p l e t e l y u n a f f e c t e d by the shock.  are  c a p i t a l s t o c k when i n 1975 by  underlying  and  and  depend  base case, and l a b o u r was In  rental  rate,  o t h e r w i s e would have been. R e c a l l t h a t t h e i d e n t i c a l linear  slightly  previous values,  Capital  been  been  has almost c o m p l e t e l y d i s s i p a t e d : the 1984  shock case  have  have  i s a 10% n e g a t i v e shock t o the  is  unchanged.  r e n t a l rate i t s e l f  shock.  would  of t h e time s e r i e s p r o c e s s o f the r e n t a l  are l e f t  case  1980s  model.  simulation C there  in  6^,  the  path.  In  column  E  the  of  the  lowered  reduced  form  parameters have been a l t e r e d as t h e y s h o u l d be g i v e n the change i n regime, w h i l e i n column path  of r e n t a l  rates  D the base case reduced form i s used.  i s lowered permanently  1975 v a l u e of r by 10%, from .1106  here by  cutting  t o .0950, and c u t t i n g n  by  The the 10%,  from .02819 t o .02537. The reduced form o f the r e n t a l r a t e p a t h i s changed  r  from  = .0254 +  .8071r _ t  1  98 to  r  =  t  As which  .0228 + .7069r  , . t-1  e x p l a i n e d above  the  i s part  change  in  changes  the  vector  G,  of the reduced form o f the i n p u t demand e q u a t i o n . G  changes from  ,0292 .0756  to  .0283 .0756  From e q u a t i o n (4.15), which d e s c r i b e s the t a r g e t l e v e l s of i n p u t s , we  see t h a t the t a r g e t l e v e l s x depend p o s i t i v e l y on G  elements input  of  -M  prices  actually  the  lowers  e s t i m a t e d own The  positive) ,  permanent  the  input  i n the  Comparing  f o r the  the  case  so  targets.  reduced  The  in  current  future  result  levels  rental  of  rates  surprises  because  are n e g a t i v e . form  forecasts  where  f o r given  reduction  price e l a s t i c i t i e s  change  effects. stock  ^ are  ( a l l of the  reduced  parameters  for form  demand e q u a t i o n s are a d j u s t e d i s 1.72%  1984  has  levels,  parameters  significant the  of  capital  the  input  h i g h e r than the base  case  f o r e c a s t , w h i l e when t h e s e parameters are not a d j u s t e d i t i s  3.12%  higher  than  labour  side;  the the  base  case.  column  E  The  difference  (adjustment  i s the  made)  1984  same  on  forecast  the is  4077.6,  0.38%  (adjustment  higher  than  t h e base  not made) f o r e c a s t  case,  i s 4092.2,  while  the  column  D  0.74% h i g h e r  than t h e  expectations  model  base c a s e .  4.5 Comparing t h e Performance o f t h e Models Both  the  linear  quadratic  Chapter 3 and t h e f l e x i b l e  rational  f u n c t i o n a l form model of t h i s  of  chapter  seek t o e x p l a i n t h e m a n u f a c t u r i n g s e c t o r ' s demand f o r c a p i t a l and labour.  The  quadratic index,  a r e non-nested;  model  input  prices  whereas  with  the  parameters function  models  of the value required  recall  that  were d e f l a t e d w i t h  flexible  function  deflating  functional  i n the  linear  an output  price  form  of a r e s t r i c t e d  quasi-fixed  input  model  the  variable prices  cost  with  a  v a r i a b l e i n p u t p r i c e index. A l s o , t h e r a t e o f output i s assumed t o be  determined  explanatory former  exogenously  i n the  v a r i a b l e i n the input  output  i s endogenous  latter,  and  appears  demand equations,  and does  not p l a y  a  as  an  while  i n the  part  i n the  estimation. We with  now  a  test  specifically demand  examine  the r e l a t i v e  proposed  their  equations  by  performance  Davidson  "P t e s t " . U s i n g t h e i r  of the l i n e a r  quadratic  and  o f t h e two MacKinnon  notation, model  models (1981) ;  l e t the input  o f Chapter  3 be  w r i t t e n i n t h e form  x  and  t  = f (X , t  the input  |3) + e  demand  Q t  ,  equations  model be w r i t t e n i n t h e form  (4.22)  of the f l e x i b l e  functional  form  100  = g(  x t  where x and  Z  . K) + e  z t  l t  ,  (4.23)  i s t h e v e c t o r o f i n p u t demands  (capital  and l a b o u r ) , X  a r e t h e d a t a used t o e s t i m a t e t h e two models, and |3 and y  fc  are t h e parameters t o be e s t i m a t e d . F o l l o w i n g Davidson and MacKinnon, d e f i n e  A  f  A f c  = f ( X , /3),  (4.24)  = g ( Z , V),  (4.25)  t  and  <3  T  t  where  A  A  /3 and  y  restricted  are  dynamic  the  maximum  likelihood  e x p e c t a t i o n s models,  given  estimates  of the  i n Tables  3.4 and  4.2 r e s p e c t i v e l y . We  test  hypothesis  x  where  fc  b  - f  the n u l l  hypothesis  (4.22)  against  the a l t e r n a t i v e  (4.23) by e s t i m a t i n g t h e r e g r e s s i o n  t  = a(g - f ) + x b + e . t  t  i s a vector  t  (4.26)  fc  o f unknown  parameters and  i s an  error  term.If the n u l l hypothesis i s t r u e , . t h e true value of a i s zero. If  the a l t e r n a t i v e  hypothesis  i s true, the estimate  of a  should  converge a s y m p t o t i c a l l y t o one. The e s t i m a t e  o f a i n (4.26) does not have an a s y m p t o t i c a l l y  v a l i d s t a n d a r d e r r o r u n l e s s we  a l s o i n c l u d e on the r i g h t - h a n d s i d e  a v e c t o r r e p r e s e n t i n g the d e r i v a t i v e s  a t each o b s e r v a t i o n of  the A  p r e d i c t e d values  of the model with  in  MacKinnon's  Davidson  and  r e s p e c t t o i t s parameters:  n o t a t i o n . I f the  model  f (X , /3) i s  A  linear,  then  t  F  is  X^_.  The  linear  quadratic  a l t h o u g h with n o n - l i n e a r r e s t r i c t i o n s , The  elements of X  are the  (note t h a t  index i n Chapter  r  3 ) . The in  w  and  so we  right  l i n e a r q u a d r a t i c model of Chapter constant  F  A  A  series f  and g  Tables  3.4  and  estimated  separately for capital  and  linear,  i n c l u d e X^_ i n (4.26) .  , 1  are d e f l a t e d  given  is  hand s i d e v a r i a b l e s  3: k  estimates  model  , r. ,, w with  an  output  Equation  labour,  by  the  , and a  were generated  4.2.  of  price  from  the  (4.2 6)  was  ordinary  least  squares. F o r c a p i t a l the e s t i m a t e of a is. 0.807, with a t - s t a t i s t i c 1.419  (there are  estimate  of  a  17  is  degrees  1.095,  s t r o n g evidence t h a t we  of  with  freedom), a  and  t-statistic  s h o u l d r e j e c t the model  f o r the of  labour  9.934.  This  (4.22), the  of the is  linear  q u a d r a t i c model. This  does  not  mean  we  should  necessarily  accept  a l t e r n a t i v e model. To t e s t the model of t h i s chapter, the functional flexible  form, we  would have t o c o n s t r u c t the t e s t  functional  form  was  the  null  hypothesis.  flexible  so t h a t We  the  do  the the  following regression:  x  " t g  t  =  a(f  t  " t  g )  +  t  G d  +  V  (4.27) A  where v  i s an  represents  the  error  term,  derivatives  d of  i s a vector the  of parameters,  p r e d i c t e d values  of  and  G  fc  y)  102 with w t  respect  y (1.01911)  to t  t  capital from  and  Table  ,  the  y  parameters,  (1.01911) ,  4.2.  and  _t  labour equations, Input  namely a  r^y^(1.01911)  constant,  in  both  r ^ and  w^  are here  is  - t (1.01911)  (  -  t  -  1  (E  )  e q u a t i o n - t (1.01911)  (  are  the  -  t  derivatives  r y  1 1  _  t  1  + E^w  t  + G^y  y  )  ( E ^ r ^ + E^w^  )  of  the  predicted  and  f  of  the  the  labour  ) , where  2  these  flexible  f u n c t i o n a l form model with r e s p e c t t o y, and where E^_. and G^, 1, 2 are from the reduced  i =  form e s t i m a t e s g i v e n above.  F o r c a p i t a l the e s t i m a t e of a i s 2.545, with a t - s t a t i s t i c 0.2 97  (there are  estimate 5%  of  a  critical  15  is  degrees  of  -790. 97,  value  of  r e j e c t the f l e x i b l e  y  equation  i n the  + Gy values  of  d e f l a t e d by  m a t e r i a l s i n p u t p r i c e index. A l s o i n c l u d e d i n the c a p i t a l  ,  the  where 1.01911 i s the e s t i m a t e  prices  t  the  freedom),  with  a  f o r the  t-statistic  t-statistic  f u n c t i o n a l form  and  is  of  1.753,  so  i n the presence  of  labour  the  -1.408.  The  we  fail  of the  to  linear  q u a d r a t i c model as an a l t e r n a t i v e h y p o t h e s i s . Finally, performance by quadratic  model  we  ask  whether  E p s t e i n and  appears  Denny's model  regressing  of  gk  and a c o n s t a n t . The  = -2825.4 + -344y ,  from  the  of  output  linear  by  values  inclusion  the  begin  chapter, on output  the  to  superior  e x p l a n a t o r y v a r i a b l e i n the former but not the l a t t e r . We predicted  to  relative  be  an  the  solely  to  as  g^_,  i s due  what  model  this  r e s u l t s are, f o r c a p i t a l  (4.28)  and f o r l a b o u r  gl  f c  = 2475.5 +  .018y . t  (4.29)  103  We  t r y Davidson  model  using  alternative still  and MacKinnon's  P test  the p r e d i c t e d v a l u e s hypothesis,  reject  the n u l l  of the l i n e a r  of the above  r a t h e r than hypothesis  g^. The  quadratic  equations  results  of the l i n e a r  as the  are that  we  q u a d r a t i c model,  even when s e t a g a i n s t s i m p l y t h a t p a r t o f the p r e d i c t e d v a l u e s o f Epstein The  and  estimate  t-statistic t-statistic in  Denny's of of  a  model  f o r the  2.346,  and  explained capital for  equation  labour  of 7.954. T h i s suggests  by  a  current  output.  is  with  a  with  a  is  2.219,  t h a t the i n c l u s i o n  one model but not t h e o t h e r e x p l a i n s a t l e a s t  the d i f f e r e n t  .113  of  output  t o some degree  l e v e l s o f performance of t h e models.  I t i s a l s o the case, however, t h a t the l i n e a r q u a d r a t i c model is  r e j e c t e d by t h e E p s t e i n and Denny model even when we  for  the  effects  variable. could  be  of  the  The e s t i m a t e d thought  inclusion residuals  of as t h e p a r t  of  output  of e q u a t i o n s  using  the r e s i d u a l s  results  were  from  an e s t i m a t e  r e g r e s s i o n (4.2 6) (4.28)  and  f o r a of  (4.29)  .114  an  explanatory  (4.28) and  of t h e demands  l a b o u r e x p l a i n e d by t h e "non-output v a r i a b l e s " Denny model. We performed  as  discount  (4.29)  for capital  of the E p s t e i n and  f o r c a p i t a l and l a b o u r i n place  of g . The  i n the c a p i t a l  equation,  with a t - s t a t i s t i c  of 2.461, and .759 i n the l a b o u r e q u a t i o n ,  a  10.270.  t-statistic  of  Again  we  reject  and  the  linear  with  quadratic  model.  4.6  Conclusions The dynamic  i n p u t demand model  has been e s t i m a t e d  f o r both  static  o f E p s t e i n and Denny  (1983)  e x p e c t a t i o n s and f o r r a t i o n a l  104 expectations,  where  by  rational  parameters  d e s c r i b i n g the  and  were i n c o r p o r a t e d  output  general  the  satisfied,  model and  Elasticities researchers,  in  although  implausibly  (large)  estimates  In  input a  of  that  input  the  prices  demand e q u a t i o n s .  symmetry of  the  range  of  estimates  output  expected  was sign.  of  other  elasticities  were of  non-nested h y p o t h e s i s  t e s t s between the model of t h i s c h a p t e r  and  the  model  quadratic  of  according  In  restriction  were  of  addition,  mean  results  linear  high.  well;  estimates  the  we  s e r i e s processes  i n t o the  performed  parameter  were  time  expectations  Chapter  3,  to  the  model i s r e j e c t e d , but the model of t h i s chapter We  found  a  paradoxical  result  when  the  linear  quadratic  i s not.  i t came  to  noting  how  t a r g e t l e v e l s of i n p u t demand changed when the time s e r i e s path  of  input  of  prices  Chapter 3 we prices  on  changed. While found t h a t  the  with  i g n o r i n g the  t o underestimate  of  this  chapter  c a p i t a l ' s own ignoring  this  overestimate  the e l a s t i c i t y . The  3  so  and  to  4 are the  little  different  changes  specification  linear effect  reduced form parameters of  one  and  the  of  about the  with  i n long  expectation  run  quadratic  model  of the path  of  input  i n p u t demand would  price elasticity, effect  would  lead  i n the model  cause  one  to  r e s u l t s of the models of Chapters regards  to  estimated  elasticities  formation,  s i z e or even the  that  s i g n of the  when we i n the effects  the Lucas c r i t i q u e i n dynamic models of f a c t o r demand.  elasticities change end of  we  the know  heeding  105  TABLE  4.1  R e s t r i c t e d E s t i m a t e s o f F l e x i b l e F u n c t i o n a l Form Model with Expectations  Static  Parameter E s t i m a t e s 11 21  *1  11 22  .19120 (.03325)'  12  -1.5230  (.34998)  .97260  (.07449)  -.04199  (.02292)  -2890.4  (560.18)  319.82  (451.98)  18.372  (11.877)  2.1133  (1.1012)  -1.3022  (1.2419)  .02566  (.00719)  22  -25.405 (15.504) -.19199  12  (.10463) .11811x10  -4  sigma .13303x10  .55536x10  -6  l o g l i k e l i h o o d f u n c t i o n = 234.4712 unrestricted  0  l o g l i k e l i h o o d f u n c t i o n = 238.1624  m a r g i n a l s i g n i f i c a n c e of r e s t r i c t i o n =  .007  a s t a n d a r d e r r o r s of parameter e s t i m a t e s i n p a r e n t h e s e s . b sigma i s the v a r i a n c e - c o v a r i a n c e m a t r i x of the 2 e q u a t i o n system where the o r d e r o f the e q u a t i o n s  i s , by  dependent v a r i a b l e ,  1/y. c u n r e s t r i c t e d e s t i m a t e does not impose $  0  = $  .  k/y,  106  TABLE 4.2 Restricted Estimates of F l e x i b l e Autoregressive Expectations.  Functional  Form  Model  with  Exogenous Parameters  H 9  1  11  a  .04456 (.01842) --  9°  (.14197)  .04146  (.01018)  3 4 6  ^  .32748  (.18897)  -.06908  (.05060)  2  0  Technology Parameters A <P 11 »22  -856.05  (73.123)  235.89  (76.501)  1.2184  (.25150)  .76094  (.06895)  -1.0356  (.23971)  -.09908  (.03174)  -.06883  (.00835)  .01911  (.00249)  ,11497x10 sigma ,15097x10  12 -4 -5  .55177x10  log l i k e l i h o o d function unrestricted  = 236.3593  l o g l i k e l i h o o d function  = 238.1624  m a r g i n a l s i g n i f i c a n c e o f r e s t r i c t i o n = .607 a s t a n d a r d e r r o r s o f parameter e s t i m a t e s i n p a r e n t h e s e s . b sigma i s t h e v a r i a n c e - c o v a r i a n c e m a t r i x o f t h e 2 e q u a t i o n system where t h e o r d e r o f t h e e q u a t i o n s  i s , by dependent v a r i a b l e ,  k/y,  l/y. c unrestricted  e s t i m a t e does not impose $  f o r $ used t o ensure a s t a b l e  1 2  =  adjustment m a t r i x .  o  r  t  h  e  v  a  l  u  e  s  107  TABLE 4.3 F o r e c a s t v a l u e s o f t h e c a p i t a l s t o c k under u s i n g t h e f l e x i b l e f u n c t i o n a l form model.  various  conditions,  Simulation Year  A  B  C  D  E  1975  18715 .3  18862 .2  18945 .1  18945 .1  18903 .6  1976  19404 .4  19611 .9  19750 .9  19771 .2  19692 .4  1977  20004 .6  20325 .8  20507 .6  20562 .5  20447 .2  1978  20376 .6  21018 .1  21232 .2  21332 .9  21182 .0  1979  21013 .3  21696 .4  21933 .9  22089 .4  21903 .7  1980  21982 .7  22366 .7  22620 .9  22838 .0  22618 .2  1981  23286 .0  23034 .5  23299 .7  23584 .0  23330 .7  1982  23805 .9  23704 .5  23976 .1  24331 .7  24045 .6  1983  23698 .8  24380 .8  24655 .3  25085 .3  24766 .8  1984  23603 .4  25066 .8  25341 .4  25848 .2  25497 .9  Description of Simulations A: A c t u a l data f o r c a p i t a l s t o c k B: Simulated forecast using model, w i t h parameter e s t i m a t e s 1975.  (machinery  and equipment).  flexible from T a b l e  functional form 4.2, s t a r t i n g a t  C: S i m u l a t e d f o r e c a s t , u s i n g t h e model o f B, w i t h a o n e - o f f n e g a t i v e shock t o t h e r e a l r e n t a l r a t e o f c a p i t a l i n 1975 o f 10%. D: S i m u l a t e d f o r e c a s t , u s i n g t h e model o f B, with a permanent l o w e r i n g o f t h e p a t h o f t h e r e a l r e n t a l r a t e by 10%, b e g i n n i n g i n 1975, where f i r m s do not r e a l i z e t h e r e has been a change i n regime. E: S i m u l a t e d f o r e c a s t , with a permanent l o w e r i n g o f t h e p a t h o f the r e a l r e n t a l r a t e by 10%, b e g i n n i n g i n 1975, where t h e reduced form parameters o f t h e model have been a d j u s t e d t o r e f l e c t t h e change i n t h e path o f r e n t a l r a t e s ( i . e . where f i r m s do r e a l i z e t h e r e has been a change i n regime).  108 CHAPTER 5 Conclusion In t h i s t h e s i s we from 1961  t o 1984  have used data  to estimate  demand, each of which t o of  rational  different respect  to  their  uncertainties  of  the  prices.  about  expectations  The  most  of  this,  other  the  i s wide disagreement,  importance  investment.  We  the  we  question.  expectations  of  the  have  rental  here  or  at  cost  attempted  Estimates  model,  rate.  and  because  ways  rental the  as  of  a  specified  Estimates  demand w i t h r a t e i n the  P test  of  of  of  least  other  modelling rental  of to  capital  obtain  much g r e a t e r l o n g run.  of Davidson and  on  model seemed t o outperform  about  business  results  on  this  and  (1985),  respect  using  on  rational  Yatchew  with  model  elasticity  confusion  quadratic  i n Epstein  a  to  the  flexible  (1983), were of a  with  respect  to  When the models were compared  MacKinnon  the  f o r e s t i m a t i n g dynamic  linear  dynamic  on  uncertainty,  quite i n e l a s t i c a  with  investment.  f u n c t i o n a l form, as d e r i v e d by E p s t e i n and Denny capital  demands  have o n l y succeeded i n i n c r e a s i n g our  were of a demand f o r c a p i t a l rental  input  quite  components of the  question using r e c e n t l y d e r i v e d techniques models, but  hypothesis  no d e f i n i t i v e c o n c l u s i o n s c o u l d be reached  e f f e c t s of the r e n t a l r a t e of c a p i t a l on There  of  appropriate  and  the  models generated  elasticities  about t a x p o l i c y  c o s t of c a p i t a l ,  incorporates  alternative  Because  the  manufacturing  d i s t i n c t models of dynamic i n p u t  some degree  expectations.  estimates  two  from Canadian  (1981), E p s t e i n and  the  using  Denny's  t h e l i n e a r q u a d r a t i c model, although  we  109 were g i v e n  some r e a s o n  inclusion not  o f output  to believe  that  as an e x p l a n a t o r y  this  might  v a r i a b l e i n t h e former but  i n the latter. When t h e s e t w o m o d e l s w e r e u s e d t o e s t i m a t e  change  i n the time  series  where  the simulations  Lucas  critique  functional the by  be due t o t h e  of  path  of the rental  done  i n such  were  econometric  policy  a  the effects of a rate  way  of  as t o a v o i d  evaluation,  the  term c a p i t a l  stock  than  t h e change  t h e l i n e a r q u a d r a t i c model. Each o f these e s t i m a t e s different  from  elasticities  from  expectations.  Estimates  the  the  predictions  estimates  of  an even  adjusted  wider  input  range  but  the  of  two  the  predicted  i n t u r n was corresponding  models  with  t h e .warning o f t h e L u c a s  of estimates.  demand r u l e s  We  f o r various  L u c a s w o u l d s u g g e s t we s h o u l d results,  flexible  static  o f t h e e f f e c t s o f a change i n t h e p a t h o f  r e n t a l r a t e w h e r e we i g n o r e d  gave  the  f o r m m o d e l ' s p r e d i c t i o n was o f a much l a r g e r c h a n g e t o  medium a n d l o n g  quite  capital,  the direction  found  that  rental rate  critique  whether regimes  we (as  do) made a l a r g e d i f f e r e n c e t o o u r of  the bias  from  not  making  the  a d j u s t m e n t s was a m b i g u o u s . These input  r e s u l t s suggest  demands when t h e r e  that  that  still  a r e adjustment  much more t o b e d o n e i n t h i s many q u e s t i o n s  we  area.  know costs,  very  and t h a t  L i s t e d below  estimate  estimate  a  about  there i s  a r e some o f t h e  remain.  What a r e t h e e f f e c t s o n o u r r e s u l t s o f u s i n g to  little  model  of  a  "representative  i n Chapter 4 i s a t l e a s t p o t e n t i a l l y  firm"?  aggregate The  model  data we  consistent with the  110 aggregation  of  consistent  of  set  d i d not t e s t  the  consistent (1985)  number  aggregation  a l t h o u g h we Whether  a  linear with  suggests  firms, out  according  i n Blackorby  the  Lucas  quadratic  respect  model  to  t h a t the  critique.  conditions for  and  Schworm  whether these c o n d i t i o n s were estimated  aggregation  biases  from  is  I t would be  in  of  satisfied. 3  is  clear.  Geweke  i n such  models  i g n o r i n g the  interest  (1982),  Chapter  less  aggregation  might e a s i l y be as l a r g e as any b i a s e s from of  to  to  warnings  find  whether  t h i s i s t r u e with a c t u a l d a t a . Can  more  developed? with  We  rational  interesting mentioned  above  expectations,  endogenous output (1987) suggest  specifications  has  that  never  that a  a  of  dynamic  flexible  be  model  of  functional  been e s t i m a t e d .  i t would a l s o  technology  of  the  firm  form,  and  B e r n s t e i n and  interest  be  Nadiri  to incorporate  v a r i a b l e u t i l i z a t i o n r a t e s i n dynamic models, which c o u l d have the e f f e c t of e n d o g e n i z i n g the d e p r e c i a t i o n r a t e of Finally,  although  this  a n a l y s i s of m a n u f a c t u r i n g future  attempt  t o use  thesis  has  i n Canada and  been  purely  a  positive  rental  rates,  we  could i n  improved e s t i m a t e s  to  Woodward  (1974) and Kesselman, W i l l i a m s o n ,  appropriate  to help  answer  of the p r i v a t e s e c t o r ' s  response  questioned  tax p o l i c y  some normative  whether  tax  incentives for  device  for  dealing  with  capital.  and Berndt  investment  unemployment  questions. 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