UBC Theses and Dissertations

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UBC Theses and Dissertations

Optimal public policies in small open economies Turunen, Arja Helena 1985

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OPTIMAL PUBLIC POLICIES IN SMALL OPEN ECONOMIES  By A R J A HELENA TURUNEN  Lie.  B.Sc,  The U n i v e r s i t y  o f H e l s i n k i , 1976  M.Sc,  The U n i v e r s i t y  o f H e l s i n k i , 1978  Soc. Sc., The U n i v e r s i t y  of H e l s i n k i , 1981  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE  FACULTY OF GRADUATE STUDIES  (The Department of Economics)  We accept t h i s  t h e s i s as conforming  to the r e q u i r e d  THE  standard  UNIVERSITY OF BRITISH COLUMBIA  February  1985  ® A r j a Helena Turunen,  1985  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree at the  the  University  of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I  further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may department or by h i s or her  be granted by the head o f representatives.  my  It i s  understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be allowed without my  permission.  Department of The  ^ C o v ^  U n i v e r s i t y of B r i t i s h Columbia  1956 Main M a l l  Vancouver, Canada V6T  1Y3  Date  DE-6  (3/81)  1  written  -  i i -  ABSTRACT  Until trade  r e c e n t l y , proofs  have used  distribution  of  g o v e r n m e n t has instruments. these  the  assumption  i n c o m e by a  commodity taxes  can  Norman p r o o f  of can  gains  occur.  gains  the  t r a n s f er-s , i .e .,  the  be  the from  inadmissible.  It turns  from  purposes of  can  individual  trade  r e p l i c a t e d under  trade  that  this  out,  specific  transfer  situations.where  Dixit  and  Norman  alter  a l l domestic  (1980:  i s made w o r s e o f b y  however, that  trade  that and  t h e s i s i s to  when a v a r i e t y o f  tax  from  to  shows o n l y free  of gains  >alter  work h a s - b e e n d e v o t e d  are  trade.  the g a i n s  equilibrium  the  government can  l u m p sum  e n s u r e t h a t no  to f r e e  of  the  t h a t a government  from autarky  of  set of  However, r e c e n t  79-80) d e m o n s t r a t e  One  that  existence  at i t s d i s p o s a l a set of household  transfer instruments  will  e s t a b l i s h i n g the  this  the  not  Dixit  and  autarky  that p o s i t i v e  i n v e s t i g a t e the  and  moving  transfer  problem  instruments  are a v a i l a b l e . It policy  is fruitful  reform  to  question:  can  small  (differential)  trade  prohibitive tariffs  (differential) results  in a  question,  there  are  initial  Pareto  the government  i n the  net  balance of  i n the  production  trade.  gains  as  find  a  a  i n i t i a l international  commodity t a x to a n s w e r  structure, the  economy i s p r e s e n t e d  some v e r y  I n C h a p t e r 4,  trade  with a s u i t a b l e  In order  perturbations  from  home c o u n t r y  country's  s i d e o f an  t h a t , under  tariff  the  i n the  country's  improvement?  It i s established (differential)  problem of  which, accompanied  perturbation  strict  the  perturbation  a model f o r the  C h a p t e r 2.  regard  that  in  weak c o n d i t i o n s , improve the  i t i s shown t h a t  country's these  -  p r o d u c t i v i t y gains a  strict  Pareto  initial  i i i-  can be d i s t r i b u t e d  improving  to the consumers i n the economy i n  way by s u i t a b l y a d j u s t i n g  commodity tax r a t e s .  The p r i n c i p a l  r e s u l t s i s a d u a l i t y theorem:  accruing  and t a r i f f s  p r o p o r t i o n a l and uniform  Pareto  are g i v e n  reductions  toward u n i f o r m i t y i n the country's  improving  of t a r i f f s initial  tariff  Conditions  and t r a n s f e r p e r t u r b a t i o n s  to e x i s t  These  include  as w e l l as a change  tariff  to be able  i n i t i a l vectors of t a r i f f s  the v e c t o r o f commodity t a x e s .  perturbations i n  i n Chapter 7.  Next, the government i s assumed home country's  formulae f o r measuring the  from a change o f t a r i f f s .  Some examples o f s t r i c t commodity taxes  f o r e s t a b l i s h i n g these  Motzkin's Theorem.  Chapter 3 develops two approximative productivity gain  tool  the country's  structure. to a d j u s t  o n l y the  and lump sum t r a n s f e r s but not forstrict  Pareto  improving  are d e v e l o p e d .  In Chapter 9 i t i s shown that n e i t h e r the e x i s t e n c e o f s t r i c t gains  from trade under commodity t a x a t i o n or under lump sura compensation  n e c e s s a r i l y i m p l i e s the o t h e r . Examples o f s t r i c t t r a n s f e r s a r e given  and/or  taxes  also  taxes and  These i n c l u d e p r o p o r t i o n a l and movements toward u n i f o r m i t y i n  o f commodities i n consumption  in policy  The r o l e o f  recommendation  results  discussed. Chapter 11 develops s u f f i c i e n t  the home c o u n t r y ' s be  changes i n t a r i f f s ,  tax r a t e s f o r domestic and t r a d e a b l e commodities.  normality is  improving  i n Chapter 10.  reductions of t a r i f f s the  Pareto  strict  Pareto  conditions  for a perturbation i n  t a x s t r u c t u r e , which causes i n t e r n a t i o n a l t r a d e , to improving.  - iv -  I n Chapter 12 the g o a l of the government i s to choose a p o l i c y that reduces the l e v e l of economic i n e q u a l i t y a s s o c i a t e d i n i t i a l observed e q u i l i b r i u m i n the economy. i n e q u a l i t y reducing if  the p r e f e r e n c e s  perturbations  w i t h the  I t i s shown t h a t  i n commodity taxes and t a r i f f s  exist,  and i n i t i a l commodity endowments of the consumers  satisfy certain conditions.  - v -  TABLE OF CONTENTS Page ABSTRACT  i i  TABLE OF CONTENTS  v  LIST OF FIGURES  vii  ACKNOWLEDGEMENTS  viii  1.  INTRODUCTION  1  2.  PRODUCTIVITY IMPROVING CHANGES IN TARIFFS  6  2.1  Equilibrium for the Production Side of an Economy  6  2.2  Continuity of the Producers' Total Net Supply Functions  11  Local C o n t r o l l a b i l i t y of the Production Sector and Changes i n the Home Country's Net Balance of Trade  16  2.3  2.4  Existence of Productivity Improving  Changes i n  Tariffs  25  3.  HOW LARGE IS THE PRODUCTIVITY GAIN?  4.  STRICT PARETO IMPROVING CHANGES IN COMMODITY TAXES AND TARIFFS 4.1 A General Equilibrium Model..... 4.2 Existence of S t r i c t Pareto Improving Commodity Taxes and T a r i f f s 4.3  4.4  5.  34  46 46 Changes i n 53  Necessary Conditions f o r Pareto Optimality: Nonexistence of S t r i c t Pareto and Productivity Improving Tax and T a r i f f Perturbations  62  S t r i c t Pareto and Productivity Improving Changes i n Commodity Taxes and T a r i f f s When no Domestic Goods Exist  68  EXISTENCE OF STRICT GAINS FROM TRADE WHEN LUMP SUM TRANSFERS ARE NOT A FEASIBLE GOVERNMENT POLICY INSTRUMENT  72  - vi-  6.  '7. •  8.  EXISTENCE OF STRICT PARETO AND PRODUCTIVITY IMPROVEMENTS WHEN ONLY A LIMITED SET OF COMMODITY TAXES AND TARIFFS CAN BE PERTURBED  78  SOME PIECEMEAL POLICY RESULTS WHEN NO LUMP SUM TRANSFERS ARE USED AS GOVERNMENT POLICY INSTRUMENTS  85  PARETO IMPROVING POLICY PERTURBATIONS WITH LUMP SUM TRANSFERS  98  8.1  A Second Model f o r the P r o d u c t i o n  98  8.2  E x i s t e n c e of Constant U t i l i t y P r o d u c t i v i t y Improving Changes i n T a r i f f s  106  S t r i c t Pareto and P r o d u c t i v i t y Improving Changes i n T a r i f f s and Lump Sum T r a n s f e r s  113  Necessary C o n d i t i o n s f o r P a r e t o O p t i m a l i t y : Nonexistence o f S t r i c t P a r e t o and P r o d u c t i v i t y Improving T a r i f f and T r a n s f e r P e r t u r b a t i o n s  115  8.3  8.4  9.  10.  Side  of an Economy....  MORE ON GAINS FROM TRADE  124  PROPORTIONAL REDUCTIONS IN DISTORTIONS AND SOME PIECEMEAL  POLICY RESULTS  132  11.  COMMODITY TAXATION AS A CAUSE OF TRADE  160  12.  ECONOMIC  13.  INEQUALITY AND PUBLIC POLICIES  12.1  Existence  12.2  E x i s t e n c e of I n e q u a l i t y Reducing and W e l f a r e Improving P o l i c y P e r t u r b a t i o n s  CONCLUSIONS.  of I n e q u a l i t y Reducing P o l i c y P e r t u r b a t i o n s . . .  164 166 174  180  FOOTNOTES  184  REFERENCES  202  APPENDIX 1  204  APPENDIX 2  208  - vii -  LIST OF FIGURES  Figure 1  2  3  Page R e l a t i v e Numbers of Domestic Commodities and P r o d u c t i o n I n d u s t r i e s : C o n t i n u i t y of T o t a l I n d u s t r y Net S u p p l i e s  13  Rank of the M a t r i x Y and C o n t i n u i t y of T o t a l I n d u s t r y Net S u p p l i e s  15  S t r i c t P a r e t o Improving P e r t u r b a t i o n s Commodity Taxes  60  in Tariffs  and  4  E x i s t e n c e of S t r i c t  5  S t r i c t P a r e t o and P r o d u c t i v i t y Improvements i n T a r i f f s and T r a n s f e r s  122  E x i s t e n c e of S t r i c t Gains from Trade Under Commodity T a x a t i o n and Lump Sum Compensation  127  Normality Change  152  6  7  Gains from Trade  of Commodities  8  B-Optimality  and  9  P a r e t o Improving Perturbations  75  and the E f f e c t s  the H a t t a N o r m a l i t y  of a P o l i c y  Condition  and I n e q u a l i t y Reducing  154  Policy 165  - viii  -  ACKNOWLEDGEMENTS I thank the f o l l o w i n g persons and i n s t i t u t i o n s i n v a l u a b l e help and encouragement d u r i n g this  f o r their  the long process  of w r i t i n g  thesis: The  Department of Economics/U.B.C.  The  Y r j o Jahnsson  Foundation  My S u p e r v i s o r y Committee: Weymark, David  P r o f e s s o r s Erwin D i e w e r t , John  Donaldson  P r o f e s s o r s John Cragg, D. P a t e r s o n , K e i z o N a g a t a n i ,  J i m Brander,  W. Ziemba P r o f e s s o r A l a n Woodland P r o f e s s o r s Seppo Honkapohja, A a r n i Nyberg Doctor  Robert Hewko  My f r i e n d s :  A l e x i s Fundas, Jayne S t u e b i n g ,  S c o t t Schatz, Dan and  G l o r i a Gordon, M i c h e l and C a r o l e P a t r y , Andreas Phingsten, Roy  P a u l and Diana P r i c e , B e r y l and J i m Skinner,  D a h l s t e d t , Anders B a u d i n , Tapani  Luukkainen, L a u r i  J a u h i a i n e n , A l l a n Nurminen, Ken K r u e g e r , David Tomaso and Wanda D i C a r l o , Jason My s o u l b r o t h e r s : My f r i e n d  Pekka Turunen  I r j a Holopainen,  and h e r f a m i l y  My p a r e n t s , Aune and V e i k k o H a l t t u n e n : dedicated  The  My work, as always, i s  t o them.  K i i t t a e n omistan taman tyon My s k i l l f u l  F o l e y , L e o t i s Watson  Senyo A d j i b o l o s o o , C h r i s t i a n Jones  of many y e a r s :  My s i s t e r ,  Kinal,  typists:  vanhemmilleni.  T e r e s a P a t t e r s o n , Jeeva  A u d i o - V i s u a l S e r v i c e s , U.B.C.  Jonahs.  - 1 -  1.  INTRODUCTION  The  q u e s t i o n of,.,whether there are gains  interesting gains  one.  Until  from  r e c e n t l y , proofs establishing  from trade have used  the assumption  instruments. these  transfer  intruments  commodity taxes  can ensure that no  to f r e e  trade.  specific  been devoted  are i n a d m i s s i b l e .  79-80) demonstrate that a government  from a u t a r k y  transfers;  at i t s d i s p o s a l a set of household However, r e c e n t work has  the e x i s t e n c e of  that the government can  the d i s t r i b u t i o n of income by a s e t of lump sum government has  trade i s an  Dixit  t h a t can  individual  However, Kemp and  i . e . , the transfer  to s i t u a t i o n s where and  alter  alter  Norman (1980:  a l l domestic  i s made worse of by moving Wan  (1983) p r o v i d e  an  example of an economy which shows t h a t the a v a i l a b i l i t y of commodity instruments sufficient autarky  a l o n e (without  the use of lump sum  to ensure that a Pareto  to f r e e  trade.  t r a n s f e r s ) i s not  improvement w i l l  Thus the D i x i t  and  occur moving  One of are  not  that p o s i t i v e g a i n s w i l l  of the purposes of t h i s  from  Norman proof of the g a i n s  from trade shows o n l y that the a u t a r k y e q u i l i b r i u m may under f r e e t r a d e and  tax  be  replicated  occur.  t h e s i s i s to i n v e s t i g a t e the  the g a i n s from t r a d e when a v a r i e t y of tax and  transfer  problem  instruments  available. It  is fruitful  p o l i c y reform  to regard  question.  the problem of the g a i n s from trade as a  Suppose t h a t the i n i t i a l  autarky e q u i l i b r i u m i n  the home c o u n t r y i s a consequence o f the government t a r i f f p o l i c y ; i . e . , the i n i t i a l  tariffs  on  internationally  t r a d e a b l e commodities, i n a  - 2 -  c o u n t r y open f o r i n t e r n a t i o n a l resulting  producer  c o i n c i d e with government  p r i c e s faced by the domestic  trade p r o h i b i t i v e  tariffs  improvement  Pareto  the c o u n t r y ' s  initial  s t r u c t u r e — a policy  a l s o g a i n f u l l y opens the c o u n t r y f o r i n t e r n a t i o n a l  under what c o n d i t i o n s do s t r i c t commodity taxes  and t a r i f f s  trade.  Pareto  improving  In order  perturbations of  to answer t h i s q u e s t i o n , a  model f o r the p r o d u c t i o n s i d e o f an economy i s presented  constant is  terms, assuming  home c o u n t r y ' s  initial  net b a l a n c e  generated  tariff  t h a t , under some very weak c o n d i t i o n s on p e r t u r b a t i o n s i n the  s t r u c t u r e t h a t improve  In Chapter  p r o d u c t i v i t y g a i n s can be d i s t r i b u t e d  initial  It  the country's  o f t r a d e , i . e . , the n e t amount o f t r a d e a b l e s revenue  by the K i n d u s t r i e s .  Pareto  2.  t h a t t h e r e are K .  the p r o d u c t i o n s e c t o r s ' t e c h n o l o g i e s , t h e r e e x i s t  a strict  i n Chapter  r e t u r n s to s c a l e p r o d u c t i o n i n d u s t r i e s i n the home c o u n t r y .  e s t a b l i s h e d i n Theorem 2.1  initial  which  from trade thus becomes the f o l l o w i n g :  exist?  The model i s w r i t t e n i n d u a l i t y  I f this i s  the p o l i c y o f ( i n f i n i t e s i m a l l y )  tax and t a r i f f  problem o f the g a i n s  commodity  improvement, i . e . , i n a s t r i c t  f o r a l l households i n the economy?  p o s s i b l e , the government should adopt  The  Can the  which, accompanied with a  p e r t u r b a t i o n i n the home c o u n t r y ' s  tax s t r u c t u r e , r e s u l t s i n a s t r i c t  changing  production sector  the a u t a r k y e q u i l i b r i u m p r i c e s f o r t r a d e a b l e s .  (differential)  welfare  that the  f i n d a s m a l l ( d i f f e r e n t i a l ) p e r t u r b a t i o n i n the i n i t i a l  international suitable  t r a d e , a r e chosen to be such  improving  4, i t i s shown t h a t  these  to the consumers i n the economy i n  way by s u i t a b l y a d j u s t i n g t h e c o u n t r y ' s  commodity tax r a t e s .  The p r i n c i p a l  tool  forestablishing  these  - 3 -  results i s a duality theorem: Motzkin's Theorem.  The general  equilibrium model employed in this thesis i s similar to the model used in Diewert (1983b). Chapter 3 addresses the problem of measuring  the gain accruing  .from a productivity ( i . e . , net balance of trade) improving change of tariffs.  It i s shown that approximative formulae for the gain can be  found by applying Diewert's measurement of deadweight  loss methodology.  The proposed formulae are based on observable data and approximate  the  productivity gain to the second order. Chapter 5 discusses the gains from trade problem in more d e t a i l . In Chapter 6 i t i s assumed that only a subset of the country's initial  commodity taxes and t a r i f f s can be perturbed.  Chapter 7 gives  some examples of s t r i c t Pareto improving perturbations i n t a r i f f s and commodity taxes. the  These include proportional and uniform reductions of  home country's i n i t i a l  tariffs.  It i s also shown that a change  toward uniformity in the country's i n i t i a l  t a r i f f structure can be  s t r i c t Pareto Improving. In Chapter 8 i t i s assumed that the government can adjust the country's i n i t i a l vector of t a r i f f s and the i n i t i a l vector of household s p e c i f i c transfers. to be fixed.  The commodity tax rates in the economy are assumed  The problem considered i s : under what conditions are  there s t r i c t Pareto improving ( d i f f e r e n t i a l ) perturbations in the country's i n i t i a l  t a r i f f s and transfers?  As i n Chapter 2, the  conditions for s t r i c t Pareto improving t a r i f f and transfer changes to exist are developed by f i r s t considering the existence of productivity improving t a r i f f perturbations.  - 4 -  The  problem of the gains  the s u f f i c i e n t  from trade i s r e a d r e s s e d  c o n d i t i o n s f o r the e x i s t e n c e of s t r i c t  under commodity t a x a t i o n and under lump sum It  taxes  proportional  and  lump sum  gains  Pareto  t r a n s f e r s are g i v e n .  r e d u c t i o n s of t a r i f f s  and  role  9:  trade  compared.  under commodity  improving  other.  changes i n  These i n c l u d e  taxes, p r o p o r t i o n a l r e d u c t i o n s  e i t h e r domestic or t r a d e a b l e s commodity t a x e s , and u n i f o r m i t y i n the  from  compensation n e c e s s a r i l y i m p l i e s the  In Chapter 10 some examples of s t r i c t tariffs,  gains  compensation are  i s shown that n e i t h e r the e x i s t e n c e of s t r i c t  t a x a t i o n o r under lump sum  i n Chapter  tax rates f o r domestic and  of  movements toward  tradeable  commodities.  of n o r m a l i t y of commodities i n consumption i n p o l i c y  The  recommendation  results i s also discussed. C h a p t e r 11 c o n s i d e r s country's sum  e f f e c t s of a change i n the home  commodity tax s t r u c t u r e ( w i t h o u t  transfers).  equilibrium, be  the w e l f a r e  can  Pareto  (hence, w e l f a r e )  autarky  be i n t e r p r e t e d to  f o r a p e r t u r b a t i o n of t a x e s , which causes strict  or lump  e q u i l i b r i u m of the economy i s an  the c o n d i t i o n s g i v e n i n Theorem 11.1  sufficient  t r a d e , to be  I f the i n i t i a l  a change i n t a r i f f s  international  improving.  In Chapter 12 the main p o l i c y g o a l of the government i s assumed to  be  the  r e d u c t i o n of economic i n e q u a l i t y i n the s o c i e t y .  the a n a l y s i s i n t h i s chapter economic.inequality answered.  The  i s to o p e r a t i o n a l i z e the concept  i n such a way  tax and  tariff  i f the  initial  commodity endowments s a t i s f y  reducing  can  be  commodity  consumers' p r e f e r e n c e s  certain conditions.  of  of  that p r a c t i c a l p o l i c y questions  I t i s e s t a b l i s h e d that economic i n e q u a l i t y perturbations exist  goal  and  For example,  - 5 -  t h e r e must e x i s t a commodity w i t h r e s p e c t t o which the p r e f e r e n c e s of the " r i c h " and the "poor" i n the economy s i g n i f i c a n t l y d i f f e r . c a s e , a p r o p o r t i o n a l r e d u c t i o n of t a r i f f s can be made i n e q u a l i t y r e d u c i n g by s u i t a b l y p e r t u r b i n g the home c o u n t r y ' s structure.  commodity t a x  In this  - 6 -  PRODUCTIVITY IMPROVING CHANGES IN TARIFFS  2.1  E q u i l i b r i u m f o r the P r o d u c t i o n  The returns  production  S i d e o f an Economy  s i d e of an economy i s assumed to c o n s i s t of K 1  to s c a l e s e c t o r s , indexed k=l,...,K.  commodities, N of which are domestic (not  constant  There are N + M  internationally tradeable). N  The  p r i c e s o f the domestic goods are denoted by p e R  .  The  tradeables  M prices w e R  are  + +  i n t e r n a t i o n a l l y given.  Hence, the  country  in  question  is  assumed to be s m a l l . The technology of the kth i n d u s t r y ( o r producer) i s r e p r e s e n t e d by i t s 2 k k k k u n i t p r o d u c t i o n p o s s i b i l i t y set C , k=l,...,K. Thus, i f (y , f ) e C , 3 k k k T the v e c t o r y = (y^,«..,y^) of domestic ( n e t ) s u p p l i e s and the v e c t o r k f  / k  k  = (f ,...,f  T )  of (net)  used as an  tradeable The used if  i n s e c t o r k and  good i s an import unit production  input  ( o r with  sets C  respect  input  i s chosen as k  vectors  (y  n  producible  by  < 0,  if f  e [1,...,M], the  m  are d e f i n e d  with  respect  to some always  to some always produced o u t p u t ) . e x i s t s (e.g.,  the u n i t l e v e l ,  mth  k=l,...,K.  and  the  For  example,  l a n d ) , some amount of  set C  then c o n s i s t s of a l l  k , f ) that  the  s e c t o r can  produce using  one  C , k=l,...,K, i s assumed to be nonempty, c l o s e d  Define above.  s e c t o r k when i t  e [1,...,N], the nth domestic good  f o r s e c t o r k,  f o r s e c t o r k an always used input  this  set  input  are  I f y^ < 0,  o p e r a t e s at u n i t s c a l e . is  exports  z  e R  +  as  the  amount of the  produced output) i n s e c t o r k, k=l,...,K.  always used  u n i t of l a n d .  and  bounded  input  from  ( o r the  This v a r i a b l e gives  the  Each  always  - 7 -  scale  of s e c t o r k.  Using  set of s e c t o r k i s d e f i n e d i s a nonempty, c l o s e d According  the s c a l e z , the as T  k  k  =  k  {z C : z  total  k  production k  >_0}.  Each T ,  possibility k=l,...,K,  cone.  to d u a l i t y theory,: s i n c e the K p r o d u c t i o n  sectors  assumed to behave c o m p e t i t i v e l y , ;the u n i t p r o d u c t i o n  possibility  C  the  , k=l,...,K, can  e q u i v a l e n t l y be functions T  sectors' unit p r o f i t  T:  (2.1)  K  (p, w + T ) = max y \ f  T  represented  using  are sets  production  k5  {p y  defined  k  + (w +  x)  f o r k=l,...,K by  T  k  k  k  f : (y , f )  k  e  C },  k  M where T e R  i s a v e c t o r of trade  x) i n ( 2 . 1 )  (w + faced  the  traded net  vector  commodities  I f each i n t e r n a t i o n a l l y traded  good i s e i t h e r  by a l l s e c t o r s or produced as an output by a l l s e c t o r s ,  components of the v e c t o r  follows: i f f >  The  i s thus the v e c t o r of p r i c e s f o r t r a d e a b l e  by domestic p r o d u c e r s .  used as an i n p u t  taxes and/or s u b s i d i e s .  0 and  ,...,T^)  x^ > 0 (< 0) , net  good by s e c t o r k are  imports of good m  x = (T  may  exports  subsidized (taxed);  i n t o s e c t o r k are  taxed  be  i n t e r p r e t e d as  of the mth i n t e r n a t i o n a l l y if f  < 0 and m (subsidized) . 6  x > 0 « m In what  0), '  if  follows, x i s called  the  t a r i f f vector.  k=l,...,K, are w e l l - d e f i n e d bounded from above.  Furthermore, each u n i t p r o f i t  Assuming that the u n i t p r o f i t d i f f e r e n t i a b l e , the k coefficients y Lemma:  unit p r o f i t  f u n c t i o n s TT ,  s i n c e , by assumption, each set C  l i n e a r l y homogenous i n the p r i c e s (p, w +  continuously  The  i s closed  f u n c t i o n i s convex  x).  f u n c t i o n s n , k=l,...,K, are  s e c t o r a l price-dependent input  twice output  k and  f , k=l,...,K, can be determined u s i n g H o t e l l i n g ' s  and  and  -  (2.2)  y  (2.3)  f  The  = V TT (p, w +  k  k  k  vectors  p and  functions  w,  and  the  = p  L i n e a r homogeneity of  V TT (p, W + X)  T  +  k  T  = p y  k's by  the  scale z  total  k  k  respect  the u n i t  (  +  W  X)  net  to  the  profit  V TT (p, W + k  T  x)  w T  k  - (w + x ) f ,  k=l,...,K.  / k % k k i s p o s i t i v e ( z > 0 ) , the v e c t o r s y z and  domestic and  order  imply  p  If  first  the f u n c t i o n s ir , k=l,...,K, with  E u l e r ' s Theorem  k  K  k V TT ( p , w + x) denote w  respectively.  TT (p, W + X )  (2.4)  K  [y ,...,y ]  x ) , k=l,...,K; F = [ f , . . . , f ] .  k v e c t o r s V IT ( p , w + x) and p d e r i v a t i v e s of  1  =  1  = V Tr (p, w + w  partial  -  x ) , k = 1.....K; Y  P  k  8  export  supplies.  Hence, the  k k f z give  total  sector  profit  earned  sector k i s  k  (2.5)  ir z  Then, u s i n g o f the  k  T  k  = p y z  k  the m a t r i c e s  industry total  The  +  producers'  (w + x )  Y and  profits  T  k  S  S wp  ,  k=l  I  ' M P . PP  WW  I  V  w  +  T  >  k=l  X)z  k  I  k=l  vector  T Fz.  S i s defined  2 V  k=l K  Tf (p, W +  (row)  (w + x)  K  I  z  S k  the  s u b s t i t u t i o n matrix  2  pw  S  in (2.2)-(2.3),  T can be w r i t t e n as p Yz +  aggregate symmetric  PP (2.6)  k=l,...,K.  F defined  K S  k  f z ,  P  * <P.  w  +  w  T  )  z  2 V7 WW  TT (p, W + k  X)  Z  K  by  - 9 -  In ( 2 . 6 ) , the m a t r i x net  block  g i v e s the responses of the domestic  s u p p l i e s to changes i n domestic p r i c e s p, S  total  g i v e s the responses of pw  the domestic  total  and  gives  prices  (w,_+  profit  net s u p p l i e s to changes i n t r a d e a b l e s  the responses of the x) .  The matrix  total  net  S is positive  export  p r i c e s (w + x ) ,  s u p p l i e s to changes i n  semidef i n i t e ,  s i n c e the  unit  f u n c t i o n s tr , k=l,...,K, are convex i n p r i c e s ( p , w + x ) .  homogeneity of the u n i t (2.7)  T  [ p , (w +  profit  f u n c t i o n s and  T  T  ) ] S = 0.  N+M*  T h i s means that the producer  E u l e r ' s Theorem  Linear  imply  7  s u b s t i t u t i o n matrix  S has  at l e a s t one  zero  g  eigenvector The  which i s the v e c t o r of producer  p r i c e s (p, w +  e q u i l i b r i u m c o n d i t i o n s f o r the p r o d u c t i o n  x).  s i d e of the  assuming that each s e c t o r k, k=l,...,K, i s o p e r a t i n g  economy,  at a p o s i t i v e s c a l e ,  are:  A * *k w + x )z = y »  (2.8)  (2.9)  (2.10)  k  Tr (p*,  w +  x*)  =  0,  k =  I w V it (p , w + x ) z  1  , . . • ,K, = b  .  k=l  According  to ( 2 . 8 ) - ( 2 . 9 ) , at an e q u i l i b r i u m ( i n d e x e d  with an a s t e r i s k ) , A  s u p p l y of domestic commodities equals  an exogenously g i v e n y  net  A  (y  can  be,  f o r example, the consumers' net demand v e c t o r o r a v e c t o r of domestic goods  -  endowments) and  10  -  a l l i n d u s t r i e s make zero pure p r o f i t s .  Equation  * (2.10) d e f i n e s domestic  b  , the net  amount of f o r e i g n exchange earned by  the  producers.  The  N + K + 1 equations (2.8)-(2.10) endogenously determine  the  * equilibrium;vector  of domestic p r i c e s p  , the  equilibrium  * z  industry  scales  * , and  the  e q u i l i b r i u m net  balance of payments b  . The  exogenous v a r i a b l e s  * in  the model are  the net  output of domestic commodities, y  , the  constant  * international  p r i c e s w and  the  tariffs  t . I t i s assumed that  there  exists  an i n i t i a l e q u i l i b r i u m where (2.8)-(2.10) are s a t i s f i e d , and the v e c t o r s domestic p r i c e s and i n d u s t r y s c a l e s are s t r i c t l y p o s i t i v e , i . e . , * N * K * p e R,,, z e R , g i v e n y and the p r i c e s of t r a d e a b l e commodities T"  r  (w + X  TT  ).  For  the  * z  subsequent a n a l y s i s i t i s r e q u i r e d  that  the  endogenous p ,  * and  b  be  regarded as  (once c o n t i n u o u s l y  differentiable) implicit  * functions  of  the  p r i c e s (w + x ) .  The  conditions  that guarantee  the  e x i s t e n c e of these i m p l i c i t f u n c t i o n s can be d e r i v e d by t o t a l l y 10 d i f f e r e n t i a t i n g the model (2.8)-(2.10) at the i n i t i a l e q u i l i b r i u m :  (2.11)  B  where B  p  Ap*  + B  z  Az* + B, Ab* b  B  = P  =  = B  x  Ax*, '  Y  N  PP T w  T  =  -S -F  -1  w F  S wp  B  °K  °KxK  Y  and  A  to the  *  z (w + x ) and  I m p l i c i t Function *  Theorem, the  T  T  -w S  ft According  pw  functions  WW  *  p (w +  x ),  *  b (w + x ) around the  initial  equilibrium  exist  i f the  of  -  matrix  [Bp,  B,  B^]  z  is invertible.  11  Under t h i s  "k  d e r i v a t i v e s of evaluated [  v  v  at  v"  the  *  Jc  e q u i l i b r i u m are  the d i r e c t i o n a l  *  "k  and  determined by  A  T  b (w + the  )  matrix  v  Diewert and sufficient  supposition,  functions p ( w + T ) , z ( w + r )  the i n i t i a l 1  -  Woodland  conditions  (1977:  Appendix) show that n e c e s s a r y  f o r the m a t r i x  [B  , B p  (2.12)  rank Y = K (<  (2.13)  rank (S  z  , B, ] to be b  and  invertible  are:  N)  T  + YY )  =  N.  PP  It  i s assumed h e n c e f o r t h  that  (2.12)-(2.13) are  satisfied  at  the  equilibrium.  Economic i n t e r p r e t a t i o n s f o r assumptions (2.12) and  are  i n Sections  2.2  discussed  2.2  and  to develop an 11  economy's GNP  (2.14)  G(w  function  +  x, y*)  (2.13)  2.3.  C o n t i n u i t y o f the P r o d u c e r s ' T o t a l Net  In order  initial  Supply  Functions  i n t e r p r e t a t i o n f o r assumption (2.12),  / G(w  *\ + T, y ) must be  first  {  = max k  k  (y ,f )  • I y z k  k  k  k  I k  > 0  =  the  defined:  (w +  T)  T  f  k  k Z  :  1  e C ,  z  >. y*;  k = i . . . ,K}. f  k=l In  (2.14), the  maximized w i t h  p r o d u c e r s ' net  revenues from the  respect  constraint  to the  that  s a l e s of  the  tradeables  production  are  sectors, i n  -  the aggregate, (If  domestic  12  s u p p l y a predetermined  good n, n e [1,...,N],  -  amount y  i s an  o f domestic  commodities.  i n p u t , the p r o d u c e r s '  total  * demand f o r t h i s Using  f a c t o r must not exceed  the K a r l i n  (1959: p. 201)  Theorem,^ the concaveprogramming equivalent dual  (2.15)  Uzawa (1958:  problem  p. 34)  0.)  Saddle P o i n t  (2.14) can be w r i t t e n i n an  form:  + T , y*) = min  G(w  —  the g i v e n endowment - y^ >_  T  T)  k  {- p y * : - i r ( p , w +  >_ 0, k =  1,...,K}.  P>°N  In  ( 2 . 1 5 ) , the  as a net  s e c t o r earns  In order K = 3.  In t h i s  p r i c e diagram  (2.15).  For  the domestic  problem  (1982:  than K and  v e c t o r of domestic  d e p i c t e d i n the  commodity p r i c e s  assumption  s l i g h t l y modifying p.  the  factor  48).  area i n F i g . l a ) i s the f e a s i b l e y  no  ( 2 . 1 5 ) , c o n s i d e r an economy where N = 2  case, N i s l e s s  i n Woodland  that  profits.  F i g u r e 1 i s drawn by  the f i x e d  commodity v e c t o r y  w i t h r e s p e c t to the c o n s t r a i n t  particular  presented  shaded  using  p o s i t i v e (pure)  to i l l u s t r a t e  (2.12) i s v i o l a t e d .  The  c o s t s from  i n p u t are minimized  production  and  producers'  solution  figure,  1 3  set f o r  problem  the c o s t m i n i m i z i n g  i s r e p r e s e n t e d by  the p o i n t p^,  where  * the u n i t  profit  intersect.  1 4  level  curves  Clearly,  of the p r o d u c t i o n s e c t o r s ( a t f i x e d  there e x i s t  industry scales  z  1  > 0,  z  2  >  (w +  T  0,  y* = - V T ^ Z - V TT Z - V ir z i . e . , in F i g . la) a l l P P P t h r e e i n d u s t r i e s o p e r a t e at a p o s i t i v e s c a l e . However, i f the t a r i f f s T * are p e r t u r b e d , the p r o d u c e r s ' zero u n i t z  3  > 0 such  profit  that  curves  shift,  1  2  2  3  as d e p i c t e d In F i g . 2.1b).  3  As  a r e s u l t , one  of the  ))  -  Figure  13  -  1 - R e l a t i v e Numbers o f Domestic Commodities and I n d u s t r i e s : C o n t i n u i t y of T o t a l I n d u s t r y Net  Production Supplies.  -  three  i n d u s t r i e s i s l i k e l y to cease p r o d u c t i o n l b ) , s e c t o r 1 does not  Fig. since that  -  14  i t s unit profits  at  operate at the  these p r i c e s are  i f the number of p r o d u c t i o n  commodities N,  as  i n the  k k functions  y z  k and  new  at a p o s i t i v e s c a l e : cost minimizing  negative.  in  prices  p^,  Hence, i t can be  seen  s e c t o r s K exceeds the number of domestic  example above, the  producers'  total  net  supply  k  f z  , k=l,...,K, are  l i k e l y to be d i s c o n t i n u o u s  —  an  1 5  outcome to be avoided i f d i f f e r e n t i a l a n a l y s i s i s to be a p p l i e d . The assumption that K <^ N can thus be regarded as a c o n t i n u i t y c o n s t r a i n t on the  s e c t o r a l t o t a l net C o n t i n u i t y of  the  rank of  the  supply  functions.  sectoral total  the m a t r i x Y.  To  see  2 i s drawn assuming that  Figure  this,  there  domestic commodities with p r i c e s p = 2a)  the g r a d i e n t s  respect o f (y  1 2  , y ) is 1  change i n the  After  the  profit  consider  are  two  (p^, P2)  sectoral unit  (< K = 2 ) .  tradeables  level  hence, o n l y  At  Suppose now  functions Figure  production i n the  profit  curves;  the that  prices p the  0  a discontinuity in i t s total  net  i n d u s t r i e s and In F i g .  IT , k = 1 , 2 ,  x ) change.  respect  of the  with rank  shift  earn zero  Sector  1  will  i n the  at  The sectoral  in Fig.  2b).  pure p r o f i t s  and  c l o s e down,  supplies. 1  I f , however, the g r a d i e n t s  two  b o t h s e c t o r s are o p e r a t i n g  a p o s s i b l e outcome i s d e p i c t e d  s e c t o r 2 w i l l stay o p e r a t i v e .  on  2.  dependent, i . e . , the  p r i c e s (w +  o n l y s e c t o r 2 can  a l s o depends  economy.  functions  producer p r i c e s causes a  the change i n (w + x )  causing  supply  to the domestic goods p r i c e s are l i n e a r l y  a positive scale.  unit  of  net  unit profit  functions  to the domestic commodity p r i c e s are l i n e a r l y  ir  2  and  ir  with  independent, i . e . ,  - 15 -  -  the  1 2 rank of (y , y ) i s 2 (= K ) ,  intersect  both before  and  16  -  the s e c t o r a l zero u n i t p r o f i t  a f t e r a s m a l l p e r t u r b a t i o n i n the  curves  will  tradeables  ft p r o d u c e r p r i c e s (w + x ) . In t h i s case, despite  the  tradeables  p r i c e change.  f u n c t i o n s TT , k = 1,2,  are  twice  both s e c t o r s w i l l  Assuming  continuously  t o t a l net  supply  functions y z  the u n i t  operative  profit  differentiable,  k k sectoral  that  stay  the  k k and  f z , k=l,...,K,  are  continuous.  2.3  L o c a l C o n t r o l l a b i l i t y o f the P r o d u c t i o n Home C o u n t r y ' s Net  The  g o a l of t h i s  (2.13) and  V b(w x  s e c t i o n i s to p r o v i d e  be  the  net  It w i l l  be  required  that  o3M  *  seen that both assumptions (2.13) and  l o c a l c o n t r o l l a b i l i t y of p r o d u c t i o n of l o c a l  i n t e r p r e t a t i o n s f o r assumption  *  f u n c t i o n b (w + x ) .  + x*)  the  Balance o f Trade  *  It w i l l  Changes I n  f o r a r e s t r i c t i o n imposed l a t e r i n the a n a l y s i s on  b a l a n c e of trade  (2.16)  S e c t o r and  controllability  (2.16) are  i n the home c o u n t r y ,  r e l a t e d to  where the  concept  i s that d e f i n e d by G u e s n e r i e (1977) and Weymark  (1979). T L e t us s t a r t  by c o n s i d e r i n g  the assumption t h a t  the matrix  S  +  YY  PP is 2.3,  positive d e f i n i t e , this  supposition  1 6  i . e . , assumption  is sufficient  * the  implicit  well defined.  *  functions p ( w + x ) ,  (2.13).  (together  *  with  *  z ( w + x )  As  assumption (2.12)) f o r  * and  Hence, from the mathematical p o i n t  shown i n S e c t i o n  *  b (w + x ) to  be  of view, assumption  -  (2.13) i s needed to guarantee 1  functions  *  7  17 -  the e x i s t e n c e o f t h e i n v e r s e demand  A  *  *  p (w + x ) , the i n d u s t r y s c a l e f u n c t i o n s z (w + x ) , and the  *  *  net b a l a n c e o f t r a d e f u n c t i o n b (w + x ) around t h e i n i t i a l e q u i l i b r i u m (which solves (2.8)-(2.10)) .  ••  To g i v e an i n t u i t i v e meaning f o r a s s u m p t i o n (2'.13), i t i s n e c e s s a r y to b r i e f l y e x p l a i n the agenda f o r the f o l l o w i n g s e c t i o n s .  The p r o d u c t i o n  s i d e o f an economy, d e s c r i b e d by the model ( 2 . 8 ) - ( 2 . 1 0 ) , i s a n a l y z e d .  * The a u t h o r i t y c h o o s i n g  the exogenous v e c t o r o f t a r i f f s  t h e government o f the home c o u n t r y .  x  i s called  The government i s assumed  to have a  p o l i c y g o a l : t o improve the c o u n t r y ' s i n i t i a l n e t b a l a n c e o f t r a d e b by s u i t a b l y changing the i n i t i a l e q u i l i b r i u m t a r i f f s x , w h i l e m a i n t a i n i n g t h e aggregate domestic net s u p p l y a t i t s i n i t i a l l e v e l y . I n o t h e r words, t h e government i s assumed to s e a r c h f o r a p e r t u r b a t i o n o f the initial  tariffs x  such t h a t , a f t e r the c o u n t r y ' s p r o d u c t i o n s e c t o r has  * a d j u s t e d to the change i n the r e l a t i v e producer  p r i c e s (w + x ) , a h i g h e r  l e v e l o f n e t e x p o r t revenue i s a t t a i n e d w i t h o u t  sacrificing  i n i t i a l domestic  any o f the  * net s u p p l y y .  I t i s e v i d e n t t h a t the induced  change i n the r e l a t i v e producer p r i c e s  * (w + x ) w i l l g e n e r a l l y change both the p r o d u c e r s ' net s u p p l i e s .  domestic  and t r a d e a b l e s  Hence, to achieve i t s p o l i c y g o a l , the government must be  a b l e to i n f l u e n c e domestic  goods p r o d u c t i o n i n the home c o u n t r y i n such a  way t h a t , i n t h e a g g r e g a t e ,  the change i n the domestic  net s u p p l y i s zero  even though the s e c t o r a l net s u p p l i e s do not g e n e r a l l y s t a y a t t h e i r initial  levels.  Consider  the Guesnerie-Weymark d e f i n i t i o n o f l o c a l  c o n t r o l l a b i l i t y of production:  - 18  D e f i n i t i o n 2.1:  ( G u e s n e r i e (1977), Weymark (1979))  The government i s s a i d characterized  -  to have l o c a l  by the equations  c o n t r o l o f p r o d u c t i o n i n an economy  (2.8)-(2.10) , i f the rank o f the  s u b s t i t u t i o n m a t r i x S i s maximal (= N + M - 1) induce a  differential  production producer  possibility  so that  producer  i t is possible  to  change i n s u p p l i e s i n any d i r e c t i o n on the economy's f r o n t i e r by a s u i t a b l e d i f f e r e n t i a l  change o f  prices.  Returning  to the government's p o l i c y problem, i t seems c l e a r  government has can induce  local  c o n t r o l of p r o d u c t i o n at the i n i t i a l  that  i f the  equilibrium, i t  the p a r t i c u l a r kind of change i n the i n d u s t r y net e x p o r t s and i n  the domestic  net s u p p l i e s ( i f such a change e x i s t s )  that w i l l  leave  the  * aggregate  y  constant w h i l e , at the same time, the i n i t i a l  net b a l a n c e o f  * trade b  i s being  improved.  s u b s t i t u t i o n matrix  I t can a l s o be seen  that when the  producer  S i s of maximal rank (= N + M - 1 ) , assumption  (2.13)  18 is  satisfied.  (2.13) and  There  local  seems thus  to be a c o n n e c t i o n between  assumption  c o n t r o l l a b i l i t y o f p r o d u c t i o n i n the c o u n t r y .  I n s p e c t i o n of assumption  (2.13) shows, however, that  the  producer  s u b s t i t u t i o n m a t r i x S need not be o f maximal rank f o r assumption be  satisfied,  Guesnerie  and  i.e., local  controllability  (2.13) to  of p r o d u c t i o n , i n the sense o f  Weymark, i s not n e c e s s a r y f o r (2.13) to h o l d .  A weaker  concept of c o n t r o l l a b i l i t y of p r o d u c t i o n i s i n o r d e r : Definition  2.2:  Domestic goods p r o d u c t i o n i n an economy d e s c r i b e d by the model is  said  to be l o c a l l y  c o n t r o l l a b l e around  the i n i t i a l  (2.8)-(2.10)  equilibrium  (which  - 19 -  satisfies  ( 2 . 8 ) - ( 2 . 1 0 ) ) , i f t h e r e e x i s t once c o n t i n u o u s l y  *  *  *  * *  differentiable  *  f u n c t i o n s p ( y , w + x ) and z ( y , w + x ) such that (2.8)-(2.10) i s satisfied  when the i n i t i a l  Here, l o c a l  v e c t o r of t a r i f f s  c o n t r o l l a b i l i t y of domestic goods p r o d u c t i o n i s d e f i n e d ^  that  there e x i s t  initial  is x .  ^  &  "ft  "fi  to mean  &  f u n c t i o n s " p ( y , w + x ) and z (y , w + x ) (around the  e q u i l i b r i u m ) which can be used  to s o l v e the a p p r o p r i a t e  * p e r t u r b a t i o n s i n the e q u i l i b r i u m domestic p r i c e s p and i n d u s t r y s c a l e s once a . s u i t a b l e ( n e t b a l a n c e o f t r a d e improving) change i n the t a r i f f s  * z , x  * has been e s t a b l i s h e d . (The change i n the v e c t o r y equilibrium  net s u p p l y of domestic commodities  i s zero i f the i n i t i a l  i s maintained).  Lemma 2.1: Suppose that assumptions has l o c a l  (2.12)—(2.13) are s a t i s f i e d .  Then, the government  c o n t r o l o f domestic goods p r o d u c t i o n i n the home c o u n t r y i n the  sense of D e f i n i t i o n 2.2.  Proof: Consider e q u a t i o n s (2.8)-(2.9) which endogenously  * p  *  determine domestic  prices  *  and i n d u s t r y s c a l e s z , g i v e n t a r i f f s  x  and a v e c t o r o f domestic n e t  A  supplies y .  Differentiating  (2.8)-(2.9) around  z, x, and y: A  (2.17)  B Ap P  where B = P  A  + B  A  Az z  = B Ay , y  and  B  PP  > z "  KxK  B = y  KxN  the i n i t i a l  v a l u e s o f p,  - 20  -  * It  regarded  and  domestic net s u p p l i e s y exists.  According  implicit  z  A  A  and  tariffs  [B , B 1 p z A p p e n d i x ) , the matrix  T , i f the i n v e r s e matrix  i f and o n l y i f (2.12)-(2.13)  Hence, the i m p l i c i t  are s a t i s f i e d , A  which  A  f u n c t i o n s p (y , w + x ) and  A  z (y , w + x ) e x i s t .  Using  f u n c t i o n s of the —J  A  has been assumed.  can be  A  to Diewert and Woodland (1977:  B 1 is invertible  o )  P  industry scales z  as (once c o n t i n u o u s l y d i f f e r e n t i a b l e ) A  [B  *  f o l l o w s that the domestic p r i c e s p  QED  Lemma 2.L, assumption (2.13) can be g i v e n an i n t e r p r e t a t i o n as a  sufficient  c o n d i t i o n f o r l o c a l c o n t r o l l a b i l i t y o f domestic goods  i .. 19 production.  Let  us now  t u r n to c o n s i d e r A  of  production  and  the net  balance  A  trade f u n c t i o n b (w + x ) .  (2.12)-(2.13) are s u f f i c i e n t shows that assumptions  tradeables  As shown i n S e c t i o n 2.1, for this  (2.12)-(2.13)  assumptions  f u n c t i o n to e x i s t . are a l s o s u f f i c i e n t  Diewert  (1983a)  f o r the economy's  A  GNP  f u n c t i o n G(w  Woodland (1982:  + x, y ) to be twice  continuously  p. 59) proves t h a t the producers' A  t r a d e a b l e commodities,  differentiable. aggregate net supply of  A  f(w + x , y ) , can be o b t a i n e d  as a v e c t o r of  20  p a r t i a l d e r i v a t i v e s of the GNP  (2.18)  f(w + x , y ) =  Then, a p p l y i n g assumptions  function:  T V TT k=l W  K  (p , w +  the r e s u l t s of Diewert  (2.12)-(2.13) ,  x )z  = V  G(w  + x , y ).  W  (1983a: pp. 189-190),  under  - 21 -  (2.19)  V^G(w  +  x*. y*) = [-  S  w  p  D  - F D ^ , 1^]  n  11 wp  12  hi where the. m a t r i c e s  (2.20)  and  a  and  D  ll  °12  D  21  °22  r  e  b l o c k s i n the symmetric i n v e r s e m a t r i x  S PP  -1  Y  D  KxK 2 The m a t r i x V ww  I„ M i s an (M x M) i d e n t i t y m a t r i x . J  * * G(w + x y,J y ) og i v e s  * the ( n e t ) export s u p p l y responses to changes i n p r i c e s (w + x ) h o l d i n g A  A  9  domestic net outputs c o n s t a n t a t y .  The m a t r i x  * positive  A  v G(w + x , y ) ww  is  *  s e m i d e f i n i t e s i n c e the GNP f u n c t i o n G(w + x , y ) i s convex i n  * p r i c e s (w + x ) . Because the GNP f u n c t i o n i s a l s o l i n e a r l y homogenous i n ft 2 ft ft (w + x ) , the m a t r i x V G(w + x , y ) has at l e a s t one zero e i g e n v e c t o r : ww  ft the v e c t o r of t r a d e a b l e s producer p r i c e s (w + x ) . The f o l l o w i n g lemma connects the net b a l a n c e of t r a d e f u n c t i o n * A 2 A A b (w + x ) and the m a t r i x V G(w + x , y ) . ww  Lemma 2.2:  ft ft The g r a d i e n t o f the n e t b a l a n c e o f t r a d e f u n c t i o n b (w + x ) w i t h r e s p e c t A  to  the t a r i f f v e c t o r x i s :  (2.21)  V  x  b*(w + x*) = - x * AT  = - x  T  T  [S - (S , F ) D ( S , F ) ] ww wp wp' ? WW  ,  *  *x  G(w + x , y ) ,  where the m a t r i x D i s d e f i n e d i n ( 2 . 2 0 ) .  '  J  - 22 -  Proof: After  differentiating  (2.8)-(2.10) a t the i n i t i a l  * * V b (w + x ) can be solved t  e q u i l i b r i u m , the g r a d i e n t  from:  * 0 PP (2.22)  KxK * T  V p  N  0 K  V z  -1  V b  a  X  pw  -F  *  -T F  *T„ S  T  T  wp  Since  *  T  AT  -T  -s  the i n v e r s e m a t r i x D e x i s t s ,  WW  (2.2) can be s o l v e d  f o r V b (w + x ):  T (2.23)  V b*(w + T * ) = - x* S + x* (S D + FD )S + x* (S D. T ww wp 11 21 pw wpl2 T  T  +  F  D 2  C o n s i d e r now the q u a d r a t i c n  T  0 1  0  ) F  2  ww  -I 1  T  f1  wp  wp  T T form [- S D,. - FD._, I.,] S [- S D., - FD.-., wp 11 12 M wp 11 12 L  - 23 -  (2.24)  [-  S  w  D  p  I ] S [-  - FDj ,  u  2  m  S D..S D..S + wp 11 pp 11 pw  S  w  p  D  - FD* , ! „ ]  u  S D..S D F wp 11 pp 12  T  1 0  - S D..S - S D F wp 11 pw wp 12  T  1 0  0  =  + FD^.S D,, S + FD^.S D F^ 12 pp 11 pw 12 pp 12  - S D..S - FD?„S + S wp 11 pw 12 pw ww  S D..S + FD? S D . F wp 11 pw 12 pp 12 0  T  2  T  T  1  - S D..S -S D.„F -S D..S wp 11 pw wp 12 wp 11 pw  - Fuf S + S 12 pw ww 0  - FD F 22 0 0  T  - S D. F wp 12  T  0  - S -D...S - FD^.S + S ww wp 11 pw 12 pw T  S - (S , F)D(S , F ) . ww wp wp  Hence, V b (w + x ) = - x  G(w + x , y ) u s i n g (2.19).  T  QED  U s i n g Lemma 2.2, assumption (2.16) can be g i v e n two i n t e r p r e t a t i o n s . one  hand, i f the government wishes to induce  an improvement  On  i n the home  * country's  initial  net balance  o f trade by changing  the t a r i f f s  x , i t seems  * natural  to r e q u i r e t h a t the g r a d i e n t V^b  2.4 i t w i l l  i s nonzero.  be shown that (2.16) i s a necessary  In f a c t ,  in Section  condition for s t r i c t  2 1  improvements i n b to e x i s t . On the other hand, assumption (2.16) i s r e l a t e d to l o c a l c o n t r o l l a b i l i t y o f the t r a d e a b l e s p r o d u c t i o n i n the  * c o u n t r y (keeping the m a t r i x  domestic  goods p r o d u c t i o n f i x e d  at y ) :  i f the rank o f  G(w + x , y ) i s maximal (= M - 1 ) , then, a c c o r d i n g to  D e f i n i t i o n 2.1, n e t export  production  c o n t r o l l a b l e , i . e . , any d i f f e r e n t i a l  i n the home c o u n t r y change i n net export  i s locally s u p p l i e s on the  * economy's p r o d u c t i o n can be induced  possibility  frontier  f o r tradeables ( a t fixed y )  by a s u i t a b l e change i n t a r i f f s  x*.  T h i s means  - 24 -  that  i fstrict  p r o d u c t i v i t y improving  d i r e c t i o n s of n e t export  production  * change e x i s t , they can be a t t a i n e d by p e r t u r b i n g appropriately. production? strict  the i n i t i a l  T  tariffs  Are there such d i r e c t i o n s o f change i n net t r a d e a b l e s  It will  be shown i n s e c t i o n 2.4 t h a t a n e c e s s a r y  p r o d u c t i v i t y improvements to e x i s t  condition for  i s that the " i n i t i a l  vector of  * tariffs i.e.,  x  i s nonzero and n o n p r o p o r t i o n a l  to the i n t e r n a t i o n a l  p r i c e s w,  the r e l a t i v e p r i c e s f o r t r a d e a b l e s a t home and abroad do not  coincide.  But i f the i n i t i a l  tariffs  x  a r e nonzero and n o n p r o p o r t i o n a l to  9  the  international  t r a d e a b l e s p r i c e s w, and i f t h e m a t r i x  ie  ic  T" G(w + x , y ) ww  ft ft is  o f maximal rank, the g r a d i e n t V^b (w + x ) must be nonzero, i . e . , (2.16)  is  satisfied:  (2.16) can be regarded  as a combined assumption  concerning  the e x i s t e n c e o f p r o d u c t i v i t y improving d i r e c t i o n s o f change i n the home c o u n t r y ' s n e t export s u p p l y and the economy's a b i l i t y t o a t t a i n them through d i f f e r e n t i a l changes i n the r e l a t i v e producer p r i c e s (w + x ) . It  should be noted,  production,  however, t h a t l o c a l  i n the sense of Guesnerie  c o n t r o l l a b i l i t y of tradeables  and Weymark, i s not n e c e s s a r y  * (2.16) to be s a t i s f i e d :  for  *  the g r a d i e n t V b (w + x ) can be nonzero even x  9 though the rank o f the m a t r i x  ie  ic  G(w + x , y ) i s l e s s than M - 1 (as long  ft as  the t a r i f f  vector x  2 * * V G(w•+ x , y ) ) . ww  i s not a zero e i g e n v e c t o r  In t h i s case,  the t r a d e a b l e s  of the m a t r i x production  possibility  ft frontier but,  i n the home c o u n t r y  around the i n i t i a l  (given a fixed  y ) i s r i d g e d and/or  e q u i l i b r i u m , at least  kinked,  some d i r e c t i o n s o f change i n  it 22 be a t t a i n e d by a d i f f e r e n t i a l the economy's n e t export supply ( t h a t can p e r t u r b a t i o n o f the t a r i f f s x ) e x i s t . F o r the c o n d i t i o n (2.16) to be  - 25 -  satisfied, that  these d i r e c t i o n s o f n e t export  the d i r e c t i o n a l d e r i v a t i v e o f the net balance o f trade f u n c t i o n i s  nonzero i n the c o r r e s p o n d i n g  2.4  supply change must a l s o be such  d i r e c t i o n o f change i n the t a r i f f s  x •  E x i s t e n c e o f P r o d u c t i v i t y - I m p r o v i n g Changes i n T a r i f f s  In  the p r e v i o u s s e c t i o n , the government's p o l i c y g o a l was d e f i n e d as  * follows:  find  a ( d i f f e r e n t i a l ) change i n the t a r i f f s  country's i n i t i a l  net b a l a n c e o f . t r a d e b  x  such  t h a t the home  i s improved, w h i l e keeping the  * domestic  goods net s u p p l y a t i t s i n i t i a l  level  g o a l of the a n a l y s i s i s to d e r i v e s u f f i c i e n t  y .  In t h i s  s e c t i o n , the  c o n d i t i o n s that make  p o l i c y g o a l f e a s i b l e , i . e . , the aim i s to develop minimal  this  sufficient  conditions f o r : "fi  (2.25)  ife  ife  .fe  t h e r e e x i s t Ap , Az , Ab , Ax and  Ab  > 0.  such  that (2.11) i s s a t i s f i e d  '  * A change o f x  f o r which (2.25) h o l d s i s c a l l e d  a productivity  improving  * change o f t a r i f f s  x .  Theorem 2.1 Suppose i ) rank Y = K < N, i i ) rank *^ — ' G(w + x*, y*) * 0^. tariffs  T  T  (S + Y Y ) = N, i i i ) x * V PP  Then, there e x i s t s a p r o d u c t i v i t y  2  improving  * x .  The proof o f Theorem 2.1 makes use o f two lemmas, which a r e established  i n Appendix 1.  w w . change i n  - 26 -  Lemma 2 .3: Any  N+K+l vector X e R satisfying  T T X [B , B ] = 0., „ must be p z N+K  the e q u a t i o n s  o f the form  (2.26)  T  X  T  T  = k [ ( p * + 6 ) , y , 1 ] , k e R,  where  (2.27)  T  «  T  = x* [S  D  w p  u  +  PDj ], 2  Y  T  T  - t* [  D  S w p  4- F D ^ ] .  1 2  Lemma 2.4: For  the v e c t o r  X solved  T  (2.28)  T  X B  = kx* V  G(w + T* , y * ) , k z R.  WW  T  Proof  2  i n Lemma 2.3,  o f Theorem 2.1:  * A sufficient to e x i s t  c o n d i t i o n f o r a p r o d u c t i v i t y improving  i s (2.25).  An e q u i v a l e n t  change i n t a r i f f s x  form f o r t h i s c o n d i t i o n can be d e r i v e d 25  u s i n g a theorem of a l t e r n a t i v e , M o t z k i n ' s Theorem:  (2.29)  t h e r e does n o t e x i s t  a vector  X e R  N + K + 1  such  T  t h a t X [ B , B , -B ] p' z' x J  -  °N+K+M>  X  \  < °'  On the c o n t r a r y , suppose such a X e x i s t s . solves  the e q u a t i o n s T  T  X that  T T T X [B , B ] = 0„,.,, must be o f the form X = p z N+K  k [ ( p * + 6 ) , y , 1 ] , k e R. be n e g a t i v e ,  By Lemma 2.3, a v e c t o r  T  For such a X, X B  fe  = -k.  k > 0 (and k may be chosen to be one).  T  Thus, f o r X B  By Lemma 2.4,  fc  to  - 27  T  XB  = T* V T  2  G(w  following  tariff  By assumption  T  i i i ) ,XB  * 0^,  a  two  propositions give  examples of  p r o d u c t i v i t y improving  changes.  Proposition If  y*).  QED  contradiction.  The  + x*,  -  2.1:  the assumptions of Theorem 2.1  of t a r i f f s  x  will  increase  domestic p r o d u c t i o n  the  are  s a t i s f i e d , a proportional  amount of  reduction  f o r e i g n exchange produced by  s e c t o r without d i m i n i s h i n g  the  net  supply  the  of domestic  commodities.  Proof ; A sufficient  condition  for a proportional  reduction  in t a r i f f s  x  to  be  p r o d u c t i v i t y improving i s : k  (2.30)  there  exist  Ap  > 0 and  Ax  *  Ab  k  , Az  k  , Ab  k  k  , Ax  such that  (2.11)  holds,  k  = -rx  , r >_ 0.  26  Applying  Motzkin's Theorem,  an e q u i v a l e n t  condition i s : N+K+l  (2.31)  t h e r e does not  exist a vector  X e R  such that  T X [B p  P r o c e e d i n g as 0^ , N+K 1T  are  where the  i n the  solved  proof  f o r X.  For  of Theorem 2.1, t h i s X,  T  XB  x  the  T e q u a t i o n s X [B^, B ]  x* = k x ^ V  p r o p o r t i o n a l i t y f a c t o r k must be  z  2  =  + x*, y * ) x * , ' T p o s i t i v e s i n c e X B^ < 0. Thus, ww  G(w  , B 1 = z  3  - 28  -  * k may  be  set equal  to one.  t a r i f f v e c t o r T i s not a 2 * * semidefinite matrix V G(w + T , y ) . ww  By assumption, the  zero e i g e n v e c t o r of the p o s i t i v e T* = x * V  T  T  Hence, X B X  2  G(w  + x*,  Proposition  y*)x* > 0.  '  WW  QED  J  2.2: *  Suppose t h a t x  >^ 0^ ( i m p o r t s  are taxed  the assumptions of Theorem 2.1 one  t r a d e a b l e such  productivity  and  exports  are s a t i s f i e d .  that lowering  the t a r i f f  are s u b s i d i z e d )  and  Then, there e x i s t s at  on  least  t h a t good l e a d s to a  improvement.  Proof : A  A c c o r d i n •go to Lemma 2.2, Al  x  1  0  A  NT ww  G(w  A  A  V xb (w +  A*£  x ) = -x  A  2 Vww  A  G(w  =-Vb(w+x)x x  J  Hence, '  9  A A  + x , y ) x  A  + x , y ).  >0  (the m a t r i x  T ww  A  G(w"+  A  x , y ) * J '  * is  positive  of  the m a t r i x  least  one  s e m i d e f i n i t e and, by assumption, G(w  + x , y )).  x  i s not a zero  I f the v e c t o r  of the components i n the v e c t o r  x  x  eigenvector  i s nonnegative,  G(w  at  + x , y ) must  be  ft positive.  P i c k any  *  tariff  x , m e [1,...,M], c o r r e s p o n d i n g  *  number ( = - 3b /3x  to the  positive  * ).  Then, l o w e r i n g  m amount of f o r e i g n exchange earned  the  tariff  x w i l l i n c r e a s e the m by the p r o d u c t i o n s e c t o r s , g i v e n that a  ft fixed  y  A policy  i s being  supplied.  QED  i m p l i c a t i o n of P r o p o s i t i o n 2.1  i s that small, competitive  c o u n t r i e s c o u l d improve t h e i r p r o d u c t i v i t y performance by p r o p o r t i o n a l l y  * reducing t h e i r trade b a r r i e r s .  If a l l t a r i f f s  x  are nonpositive  (imports  are  subsidized  holds:  and e x p o r t s are t a x e d ) ,  there  e x i s t s at l e a s t one t r a d e a b l e  such that an i n c r e a s e Assumption  in  iii)  (2.32)  T*V  i n Theorem 2.1  S.  i n terms o f the  Using ( 2 . 1 9 ) , the f o l l o w i n g  T  J  (2.32) can a l s o be w r i t t e n  T  x* V  *  Then, s i n c e  (2.35)  commodity m, m £ [1,...,M],  can be w r i t t e n  G(w + x*, y*) = T * [ - S D,, wp 11  ww  T  T  =  T  T  *  5 i s defined  T [ S  D  wp ll  the m a t r i x  *T  9  V  - FD?_ , 12'  +  F D  S  as  G(w + T*, y*) = [-<5, x* ] S  2  where the v e c t o r  (2.34)  to P r o p o s i t i o n  i s found:  The e q u a t i o n s  (2.33)  parallel  i s p r o d u c t i v i t y improving.  aggregate producer s u b s t i t u t i o n matrix expression  a result  T  -D..S 11 pw  T  by  ]  12 «  S i s positive semidefinite,  * * G(w + x , y )  * x =  T *T [- 6 , T ] S  > 0.  2.2  - 30 -  A s s u m p t i o n i i i ) can now be r e p l a c e d by:  (2.36)  the vector  [- 6 , x  eigenvector  If  ] i s not p r o p o r t i o n a l to any zero  of the producer s u b s t i t u t i o n m a t r i x  the o n l y zero e i g e n v e c t o r of the m a t r i x  S i s the v e c t o r of domestic  producer p r i c e s ( p * , w + x * ) , (2.36) i s e q u i v a l e n t  (2.37)  tariffs  x  S.  to e i t h e r  (* 0^) are not p r o p o r t i o n a l to the i n t e r n a t i o n a l  prices w  or  (2.38)  the v e c t o r o f producer p r i c e s f o r d o m e s t i c commodities p p r o p o r t i o n a l to the v e c t o r 5 d e f i n e d  i n (2.34).  Lemma 2.5 i n Appendix 1 e s t a b l i s h e s t h e e q u i v a l e n c e D i e w e r t (1983b:  i s not  p. 273) shows t h a t the v e c t o r ( p  o f (2.37) and ( 2 . 3 8 ) . + 6) i s t h e a p p r o p r i a t e  p r o d u c t i v e e f f i c i e n c y v e c t o r o f shadow p r i c e s ( f o r d o m e s t i c commodities) for  choosing  present  government p r o j e c t s when d i s t o r t i o n a r y t a r i f f s and t a x e s a r e  i n t h e economy.  Using  (2.37) and ( 2 . 3 8 ) , i t can be added t h a t *  the d o m e s t i c producer p r i c e s p  should be used as the shadow p r i c e s  * for  c o s t b e n e f i t a n a l y s i s o n l y i f i ) the i n i t i a l v e c t o r o f t a r i f f s  zero, i . e . x prices  = 0^, or i f i i ) x  x is  i s p r o p o r t i o n a l to the i n t e r n a t i o n a l  w.  Under the r a t h e r s t r o n g maximal rank s u p p o s i t i o n about the producer s u b s t i t u t i o n matrix  S, the seemingly c o m p l i c a t e d  a s s u m p t i o n i i i ) i n Theorem  - 31 -  2.1  simplifies  Furthermore, domestic  to an e a s i l y understood  i f the producer  goods producer  nonproportionality condition.  s u b s t i t u t i o n m a t r i x S i s o f maximal rank, the  s u b s t i t u t i o n matrix  which means t h a t assumption  i s o f f u l l rank (= N),  i i ) o f Theorem 2.1 i s s a t i s f i e d .  P r o p o s i t i o n 2.3: Suppose i ) rank Y = K. <_ N, i i ) rank S = N + M - 1, i i i ) the i n i t i a l v e c t o r ft of  T  tariffs  0^) i s not p r o p o r t i o n a l  to the i n t e r n a t i o n a l  p r i c e s w.  ft Then, t h e r e e x i s t s a p r o d u c t i v i t y improving x  If  may be chosen  change i n t  ; the change i n  to be a p r o p o r t i o n a l r e d u c t i o n .  there i s o n l y one aggregate  the rank assumption  p r o d u c t i o n s e c t o r i n the home c o u n t r y , even  on the domestic  P r o p o s i t i o n 2.3 may be erased.  goods net supply m a t r i x Y i n  The r e s u l t i n g  extremely  simple v e r s i o n o f  Theorem 2.1 r e v e a l s the b a s i c  economic c o n d i t i o n s which a r e s u f f i c i e n t f o r  productivity  changes to e x i s t :  improving  tariff  in  p r o d u c t i o n , and the r e l a t i v e producer  in  the home c o u n t r y must d i f f e r  additional  p r i c e s f o r t r a d e a b l e commodities  the i n t e r n a t i o n a l  s u p p o s i t i o n s i n Theorem 2.1 a r e needed  cases where the economy's t o t a l and  from  there must be s u b s t i t u t i o n  p r i c e s w.  The  to cover the more g e n e r a l  production p o s s i b i l i t y  s e t may be kinked  the number of p r o d u c t i o n s e c t o r s exceeds one. The  central  assumptions establish  r o l e o f the s u b s t i t u t a b i l i t y  i s emphasized  that  strict  changes t h a t cause  and n o n p r o p o r t i o n a l i t y  i n Theorem 2.2 and i n i t s c o r o l l a r y below:  p r o d u c t i v i t y improving  a strict  tariff  they  changes, i . e . , t a r i f f  i n c r e a s e i n the home c o u n t r y ' s i n i t i a l net  - 32 -  b a l a n c e of t r a d e b*, cannot e x i s t  i f assumption i i i ) of Theorem  2.1 i s  violated. Let  us f i r s t  consider  n o n e x i s t e n c e of s t r i c t conditions  the most g e n e r a l  p r o d u c t i v i t y improving, t a r i f f  perturbations.  These  can be regarded e q u i v a l e n t l y as 'necessary c o n d i t i o n s f o r  p r o d u c t i v i t y o p t i m a l i t y of the i n i t i a l  Theorem  necessary conditions f o r  equilibrium.  2.2:  A necessary c o n d i t i o n  for a s t r i c t  p r o d u c t i v i t y improving  tariff  change  to not e x i s t i s :  (2.39)  there T  X B  U  b  i s a vector  X e R  n  +  k  +  1 s u c  h  that  T  T  X [B , B , - B ] = 0 , , , , p z' x N+K+M  < 0.  Proof: A n e c e s s a r y c o n d i t i o n f o r p r o d u c t i v i t y o p t i m a l i t y o f the  initial  equilibrium i s :  1c  (2.40)  t h e r e do not e x i s t Ap  1c  , Az  1c  , Ab  ~k  , Ax  such t h a t (2.11) i s  * satisfied  and Ab  Using Motzkin's Theorem  > 0.  and the proof  (2.39) and (2.40) are e q u i v a l e n t .  Corollary Suppose  of Theorem  2.1, i t can be seen  QED  2.1: T  i ) rank Y = K < N, i i ) rank (S + YY ) — pp  = N, i i i ) x *  T  V  2  ww  that  - 33 -  it  k  X  + T , y ) = 0^. Then, the i n i t i a l  G(w  condition  for productivity  productivity  equilibrium  optimality  given  satisfies  the necessary  i n Theorem 2.2, and no s t r i c t T  improving d i r e c t i o n s of change i n t a r i f f s  exist.  Proof: Assumptions i ) - i i ) which s o l v e s  = k[(p  T  A B,  b  2.1 and Lemma 2.3 i m p l y that T  +6)  = - 1 < 0.  fj T , y , 1 ] , where k e R. T  Choose k = 1 so that T  2  By Lemma 2.4, A B = x * V x ww 1  T Hence, by assumption i i i ) , A B  C o r o l l a r y 2.1 i m p l i e s  that  G(w + x* y * ) . J  T = 0^ and (2.39) i s s a t i s f i e d .  (2.16) i s a n e c e s s a r y c o n d i t i o n T  productivity  improvements i n t a r i f f s  the g r a d i e n t  o f the net balance o f trade f u n c t i o n b ,  to e x i s t .  zero M-vector (and (2.12) and (2.13) are s a t i s f i e d ) ,  optimality.  satisfies  the n e c e s s a r y c o n d i t i o n  However, (2.39) i s not s u f f i c i e n t  to be a p r o d u c t i v i t y maximum, i . e . , such that &  balance of trade function  for  QED  strict  I n o t h e r words, i f •k  equilibrium  a vector A  T T the e q u a t i o n s A [B , B ] = 0„ , must be o f the form p z N+K  * A  of C o r o l l a r y  -k  -k  b (w + T ) , i s a  the i n i t i a l  (2.39) f o r p r o d u c t i v i t y f o r the i n i t i a l the maximum f o r  ft  b (w + x ) i s a t t a i n e d .  equilibrium the net  - 34  3.  HOW  -  LARGE IS THE PRODUCTIVITY GAIN?  It  was  shown i n the  previous  c h a p t e r t h a t , under c e r t a i n r a t h e r  weak c o n d i t i o n s , p r o d u c t i v i t y improving  changes i n t a r i f f s e x i s t .  How  i . e . , the  l a r g e are  these.productivity gains,  home c o u n t r y ' s i n i t i a l formulae f o r the  net balance of t r a d e b ?  increase  in b  can  A l l a i s - D i e w e r t method of measuring inefficient  schemes of  i s useful  to g i v e  production  the  i n the  e q u i l i b r i u m amount y*  (3.1)  b  o  value  b°  the maximal amount of f o r e i g n exchange  s e c t o r , under the  aggregate s u p p l i e s  at l e a s t the  >0  initial  s o l u t i o n b°  to  k=l,...,K|.  K  that s o l v e s (3.1)  i s at l e a s t as l a r g e as the v  Diewert p r o d u c t i o n as  the  the  , f ) e C  * observed net b a l a n c e of trade b ( =  defined  c o n d i t i o n that  r ^ T k k ^ k k * { £ w f z : E y z ^ y ,  z  The  the  1  of domestic commodities, as  = max (y  to  productivity gain,  a b r i e f review o f the d e r i v a t i o n of  production  sector  the  the deadweight l o s s due  the a p p r o x i m a t i o n s f o r the  Diewert (1983c) d e f i n e s to  Approximating  be developed u s i n g  A l l a i s - D i e w e r t deadweight l o s s measure.  available  the  taxation.  Before c o n s i d e r i n g it  improvements i n  loss pertaining  T £ k=l to the  w  f  *k *k 2 • z )•  initial  The  initial  Allais-  e q u i l i b r i u m i s then  -  (3.2)  ^ = b  Using Point  - b  >  b  o  =  where the v e c t o r p r i c e vector  be  r ^  { E  p > 0  K  - Uzawa (1958: p. 34)  i n an  k  TT (p,w)z  equivalent  k  T * - p y :  dual  Saddle  form:  i k=l,...,Kj,  k=l  N  of Lagrange m u l t i p l i e r s p i s the ( p r o d u c e r ) shadow  f o r domestic  i n the  written  min  z >_ 0  present  can  max  -  0.  the K a r l i n (1959: p. 201)  Theorem, (3.1)  (3.3)  35  economy.  commodities when no The  first  order  tariff  d i s t o r t i o n s are  necessary conditions  for  (3.3)  are:  k  (3.4)  Tr (p,w)  (3.5)  Z k=l  It  =  k=l,...,K,  0,  k  V Tr (p,w)z  k  =  y*.  P  i s assumed  that a unique s o l u t i o n z e R  , p e R  ++ (3.5) be  e x i s t s , and  regarded  solution  -  implicit  f o r (3.4)  Let (3.4)  as  us now  (3.5)  that  -  the  industry scales  functions  (3.5).  of  initial  -  domestic p r i c e s p  exogenous (w,  y*)  around  can the  3  compare the o p t i m a l  to the  the  z and  f o r (3.4)  ++  reference  equilibrium  distorted equilibrium  satisfying  characterized  by  the  equations  (2.8) - (2.9).  distorted  equilibrium  (3.6)  ir (p(0,  (3.7)  E k=l  If  =  = 0,  Diewert  k  k  (3.4) - (3.5),  t h e same o p t i m a l  A(C) =  that  solves  the A l l a i s  (3.8),  Z k=l  w  T  V  k  equilibrium  and i f £ = 1, ( 3 . 6 ) - ( 3 . 7 ) d e f i n e s  each e q u i l i b r i u m  defines  (  (2.8) -  objective exchange  p  ( 5 ) ,  as  (3.9)  = A(0) - A ( l ) .  W  +  indexed  function  that  as the the  initial  (2.9).  o f t h e economy  the A l l a i s - D i e w e r t  be w r i t t e n  ^  f o r which  ,. .. , K,  aggregate) produces at the e q u i l i b r i u m  Using  to the  P  n e t amount o f f o r e i g n  (3.8)  c a n be mapped  V 7 r ( p ( 5 ) , w + T * ) z ( 5 ) = y*.  observed e q u i l i b r i u m For  equilibrium  a g-equilibrium  k=l  0, ( 3 . 6 ) - ( 3 . 7 ) d e f i n e s  equations  the  by d e f i n i n g  w + x*0  k  5  The o p t i m a l  A(Q  by 5 ( 0 <_ 5 <_ 1 ) , f o r t h e economy a s  the p r o d u c t i o n s e c t o r indexed  ( i nthe  by  T*?)^?). * 1  l o s s m e a s u r e A^ d e f i n e d  i n (3.2) can  - 37  Next, the Taylor A"(0)(1  Series  value  A( 1 ) i s approximated  expansion of A around  - 0)2.  Using  -  this  5 = 0 ,  f o r m u l a , the  using  d e r i v a t i v e of  function at  objective  I t can  be  order  - 0) +  Allais-Diewert loss  A"(0).  Allais  second  A(0) + A'(0)(1  becomes a p p r o x i m a t e l y - A ' ( 0 ) - 1/2 the  the  AL  shown that  5 = 0 ,  1/2  A'(0),  the  i s zero  and  hence,  (3.10)  A  L  = -1/2  A"(0),  5  where (3.11)  In  -A"(0) =  T  [p'(0) ,  T  S  c  P'(0) * T  >  0.  ( 3 . 1 1 ) , p ' ( 0 ) denotes the d e r i v a t i v e s of the  p r i c e s p with respect be  T* ]  to 5, e v a l u a t e d  c a l c u l a t e d by d i f f e r e n t i a t i n g  s o l u t i o n of  (3.4) -  s u b s t i t u t i o n matrix undistorted  ( 3 . 5 ) and  by  S° i n ( 3 . 1 1 )  equilibrium;  the  at  5 = 0 .  domestic  commodity  These d e r i v a t i v e s  equations ( 3 . 4 ) -  ( 3 . 5 ) at  setting 5 = 0 .  The  producer  i s evaluated  the  optimal  at  p o s i t i v e semidefiniteness  of  S°  can  the  implies  t h a t - A " ( 0 ) >_ 0. Let gain  us  accruing  now  return  to the  problem of measuring  from a p r o d u c t i v i t y improving  the  perturbation  productivity of  tariffs  * x . 1,  Suppose that the new  undistorted vity  (after  the  initial  the  t a r i f f change) e q u i l i b r i u m by  e q u i l i b r i u m by 0.  g a i n , AQ,  t a r i f f distorted equilibrium  I t seems n a t u r a l  between the e q u i l i b r i a  1 and  2 by  2 , and  i s indexed the  to measure the the  difference  by  optimal producti-  - 38 -  (3.12)  A_  = b  2  - b  1  >  LJ  0,  —  where b l i s the observed  net b a l a n c e  ( f o r m e r l y denoted by b*)  and  the change i n t a r i f f s  ( f o r m e r l y denoted by  perturbation  Adding and  b^ i s the net b a l a n c e of t r a d e  tariffs T  i n the i n i t i a l  the g a i n AQ i s  improving,  of t r a d e i n e q u i l i b r i u m 1  1  nonnegative.  T*).  is strict  after  I f the  productivity  6  s u b t r a c t i n g the maximal net b a l a n c e  of t r a d e b°  on  the r i g h t hand s i d e of (3.12), the g a i n AQ can be w r i t t e n as  (3.13)  A  1  G  In ( 3 . 1 3 ) , A^  - (b° - b )  a c t u a l observed  l o s s between the o p t i m a l  equilibria.  = A°  + 1  2  - A°* .  ^ denotes the A l l a i s - Diewert  the o p t i m a l and Diewert  2  - (b° - b )  equilibria, and  new  and  (after  p r o d u c t i v i t y g a i n AQ has  The  p r o d u c t i v i t y l o s s between A^^  2  i s the A l l a i s  the change i n t a r i f f s )  thus been expressed  as  a d i f f e r e n c e of p r o d u c t i v i t y l o s s e s between the three e q u i l i b r i a and  0,  2. Applying  Diewert's  (3.11) to (3.13),  (3.14)  A  G  » I  change i n t a r i f f s  T  J  second order a p p r o x i m a t i o n  r u l e (3.10) -  the approximate g a i n a c c r u i n g from a s t r i c t  p r o d u c t i v i t y improving  1 T  T  1  T  T  S°  [ '(0) , T ]  S° [ ' ( 0 ) ,  T  P  2 T  is:  [ '(0) , T  [p'(0) , x ]  P  P  T  ]  2 T  -  T  ] , T  -  1  - 39 -  where x2 denotes  a t the e q u i l i b r i u m 2 .  the v e c t o r of t a r i f f s  In o r d e r to c a l c u l a t e  the g a i n a p p r o x i m a t i o n ( 3 . 1 4 ) , the  government must have i n f o r m a t i o n about  the aggregate  producer  s u b s t i t u t i o n m a t r i x S^, the domestic  price derivatives  depend on the net s u p p l y m a t r i c e s  and  p'(0) (which  a t the o p t i m a l 2  e q u i l i b r i u m ) , and the p o l i c y v a r i a b l e s  and x .  In p r a c t i c e ,  u n f o r t u n a t e l y , i t may be Impossible to form e s t i m a t e s f o r the unobserved m a t r i c e s S^, Y ° , and F^. thus c a l l e d  for.  Further approximations  to (3.14) are  One p o s s i b i l i t y i s , i f the o p t i m a l and observed  e q u i l i b r i a a r e not too f a r from each o t h e r , to r e p l a c e 1  the m a t r i c e s 1  S ° , F° and Y ° i n (3.14) by the observed m a t r i c e s S , F and Y^.  T h i s a p p r o x i m a t i o n , however, i n v o l v e s an e r r o r , the s i z e o f which  i s not known. Is t h e r e then any way o f measuring  the p r o d u c t i v i t y g a i n as  a c c u r a t e l y as i n ( 3 . 1 4 ) , u s i n g o n l y the observed e q u i l i b r i u m 1?  I t t u r n s out that  this  i n f o r m a t i o n at  i s indeed p o s s i b l e , i f the  f o l l o w i n g v e r s i o n o f the Q u a d r a t i c A p p r o x i m a t i o n Lemma i s employed.  Lemma 3.1:  (The Q u a d r a t i c Approximation Lemma; Diewert  For a twice c o n t i n u o u s l y d i f f e r e n t i a b l e  (3.15)  fU ) 1  T  - f(z°) = V f ( z ° ) ( z  +  1 (z  1  1  if  (3.16)  f(z) = a  T  Q  T  V  2  7.2.  and o n l y i f T  + a z + i z Az,  z e R^,  - z°)  - z°)  I  function f ( z ) ,  (1976: p. 1 1 8 ) ) .  f(z°)  1  ( z - z°)  - 40 -  where A^ = A, or i f and o n l y i f  f ( z ) - f(:z°) = 1 X  (3.17)  1  T  [Vf(z°) + V f ( z ) ] ( z  1  - z°).  Lemma 3.1 e s t a b l i s h e s two exact e x p r e s s i o n s f o r the change i n the v a l u e of  a quadratic function f ( z ) :  S e r i e s e x p r e s s i o n , while gradient  1  formula  o r d e r approximation  1  ( z - z°). (3.16),  t o the change f ( z ^ ) - f ( z ^ ) .  (3.17) a l s o p r o v i d e s an a p p r o x i m a t i o n  which, by r e w r i t i n g  to f ( z ^ ) - f ( z ^ )  (3.17) i n the form  1  0  0  fCz ) - f(z°) « I [Vf(z°) (z - z )] + I [ V f ( z ) ( z - z ) ] , T  can be g i v e n a new i n t e r p r e t a t i o n : in  of the average  the f u n c t i o n f ( z ) i s not q u a d r a t i c , i . e . , of the form  (3.15) g i v e s a second  (3.18)  order Taylor  (3.17) uses the i n n e r product  l/2[Vf(z°) + V f t z ) ] w i t h the d i f f e r e n c e  If  The  (3.15) uses the second  (3.18) y i e l d s a f i r s t  f(z^)  around  first  order approximation  1  the term  order approximation  the p o i n t z^, whereas the term  1  T  1  V f ( z ^ ) T ( z * - z^) to the change f ( z * ) V f ( z ^ ) ^ ( z ^ - z^) p r o v i d e s a  t o f ( z * ) - f ( z ^ ) around the p o i n t TS .  Thus, i n (3.18), the change i n the v a l u e of the f u n c t i o n f i s approximated 3.1  u s i n g the average  o f two f i r s t  shows t h a t , f o r a q u a d r a t i c f u n c t i o n ,  good an a p p r o x i m a t i o n e x p r e s s i o n (3.15); quadratic, second  the a p p r o x i m a t i o n (3.15).  Lemma  (3.18) p r o v i d e s e x a c t l y as  to the change f ( z ^ ) - f ( z ^ ) as the second  both a r e i n f a c t e x a c t .  o r d e r formula  order approximations.  order  I f the f u n c t i o n f i s not  (3.18) i s a p p r o x i m a t e l y  as a c c u r a t e as the  - 41  The  above r e s u l t  A^*^  and  objective second the  two  2  A^  around  (3.19)  A^  to  the  (3.20)  Hence, by  (3.21)  regarded  1  1  the  1  *™  as  = A(0)  the  the  approximation in (3.12).  changes i n the  the l o s s A^*^, using  e q u i l i b r i a 0 and  equilibrium  -^A.(l).  shows t h a t A*(0)  the  To  0,  (3.20),  = 0, which  to  1.  [ A ' ( l ) - A'(0)]  170)  not  average of  - A ( l ) - -A'(0)  average of (3.19) and  Allais  the  1,  around the  Using  l o s s measures  - A(l) * A'(l)  order,  Diewert (1983c: p.  (3.22)  around  equilibrium  - I  two  the o p t i m a l e q u i l i b r i u m , but  = A(0)  taking  A^  another  as a d i f f e r e n c e of  Let us approximate  order l o s s e s  first  A^  to d e r i v e  which themselves are d e f i n e d  order around  order,  and  be  function A(£).  first  be used  p r o d u c t i v i t y g a i n measure AQ d e f i n e d  formula f o r the ( 3 . 1 3 ) , AQ can  can  -  implies  the  first  - 42 -  S i m i l a r l y , f o r the measure  A ^  (3.23)  The  Z  . I  A'(2).  7  average l o s s measures (3.22) and (3.23) a r e a p p r o x i m a t e l y as  a c c u r a t e as the c o r r e s p o n d i n g formulae  second o r d e r measures d e r i v e d u s i n g the  '('3.10) - ( 3 . 1 1 ) .  A p p l y i n g (3.22) - ( 3 . 2 3 ) , the p r o d u c t i v i t y g a i n measure AQ d e f i n e d i n (3.13) can be approximated  (3.24)  Using  A  « I  [A'(l) - A'(2)].  the r e s u l t s i n Diewert  objective  (3.25)  G  by  (1983c) , t h e d e r i v a t i v e s o f t h e A l l a i s 1  f u n c t i o n , A ' ( l ) and A ( 2 ) , can be shown  T  A'(l) = [p'(l) , x  1 T  ] S  1  2 T  ] S  2  T  [p*(l) ,  T  1  T  ]  to e q u a l  T  and  (3.26)  T  A'(2) = [ p ' ( 2 ) ,  Formulae  (3.24)  p r o d u c t i v i t y gain the observed  AQ  producer  -  x  (3.26)  T  [p'(2) ,  2 T  T  T  ] .  9  p r o v i d e the second a p p r o x i m a t i o n  i n this section.  The measure  (3.24)  f o r the  depends on  s u b s t i t u t i o n m a t r i x S* and the observed net  - 43 -  output m a t r i c e s Y  and F .  I t a l s o depends on the c o r r e s p o n d i n g -  unobserved m a t r i c e s i n the new If,  however, the p e r t u r b a t i o n  to assume that also 1  S .  small. After  ( a f t e r the t a r i f f change) X* i s small,  in tariffs  the changes i n the producer 2  Hence, the m a t r i x S t h i s adjustment,  equilibrium.  i t i s reasonable  s u b s t i t u t i o n matrix  can be approximated  S^- a r e  by the m a t r i x  the p r o d u c t i v i t y g a i n measure  (3.24)  becomes  (3.27)  A  G  -  1 , ,,..T [P (1) ,  IT.  x  2  The a p p r o x i m a t i o n matrix  S  2  r  1  2 T  error involved  i n the second  smaller  than the e r r o r  instead  of the m a t r i x  ,,.*T  1T.T  [p'(D  s  x ]  T  -^[P'(2) ,  l  c  ] S  , x T  [ '(2) , P  ]  x ]V° 2T  in substituting  term of (3.27) i s l i k e l y  l  S  the m a t r i x to be  f o r the  considerably  i n ( 3 . 1 4 ) , i f , i n ( 3 . 1 4 ) , the m a t r i x  i s used  S^. 2  If  the net output m a t r i c e s a f t e r  can be assumed  to be c l o s e  F*, the d e r i v a t i v e s equilibrium  the t a r i f f change, Y  to the i n i t i a l  p'(l).  by the  In t h i s c a s e , the g a i n  (3.27) s i m p l i f i e s to  (3.28)  A  G  «  wp  +  X  p'(l)  IT S  WW  + p'(l)  x  1  -  and F ,  observed m a t r i c e s Y* and  p'(2) i n (3.27) can be r e p l a c e d  price derivatives  2  x  2T  (x  T  c  S  l ww  2,  1  x J.  -  x ) 2  initial measure  - 44  This approximation  -  depends o n l y on o b s e r v a b l e  variables.  1 1  I t can be seen from (3.28) t h a t the p r o d u c t i v i t y g a i n AQ i n c r e a s e s p r o p o r t i o n a l l y w i t h the p r o d u c t i o n S  1  wp  s u b s t i t u t i o n terms 1  and S* , i . e . , i f a l l the m a t r i c e s ww  ,. S and pw' wp  by a s c a l a r a > 0, t h e g a i n AQ i s r e p l a c e d by OCAQ:  ww  S^,  are multiplied r  t h e more  s u b s t i t u t i o n t h e r e i s i n the d o m e s t i c p r o d u c t i o n s e c t o r , t h e l a r g e r the g a i n s from p r o d u c t i v i t y i m p r o v i n g  t a r i f f p o l i c i e s a r e l i k e l y t o be.  I n the case o f p r o p o r t i o n a l changes i n t a r i f f s  T^, i t i s  p o s s i b l e to c a l c u l a t e the d e r i v a t i v e o f the approximate g a i n (3.28) w i t h respect T  2  to a p r o p o r t i o n a l i t y f a c t o r . 1  = k x , k e (0,1).  (3.29)  A  Using  - i [(1 - k) x Cr  Assume, f o r example, t h a t  (3.28), 1 T  S  1  p'(l) +  Wp  Z p»  (1 - k )  (1)  T  S  1  T  1  +  T  1 T  S  1  T  1  - k  2  T  1 T  S  WW  pw  T ].  1  1  WW  Hence,  (3.30)  If  ^dk- = T 2[ -  t  1  T  s  1  wp P ' ( D " P ' ( 1 )  T  S pw T 1  1  - 2k T  1 T  S ww T ] , 1  the producer s u b s t i t u t i o n m a t r i x f o r d o m e s t i c and t r a d e a b l e  commodities i s zero (S  = 0„ . , ) , the d e r i v a t i v e (3.30) becomes NxM '  pw  (3.31)  -JT-^ = -2k T dk  A  S T ww L  < 0. —  1  - 45  The  -  weak i n e q u a l i t y i n (3.31) f o l l o w s from the p o s i t i v e s e m i d e f i n i t e n e s s  o f the m a t r i x  ww  .  From (3.31) i t can be  p r o p o r t i o n a l i t y f a c t o r k decreases ( i . e . , of  tariffs  2.1,  x^,  which i s s t r i c t  seen that as  as the p r o p o r t i o n a l r e d u c t i o n  p r o d u c t i v i t y improving  becomes l a r g e r ) , the p r o d u c t i v i t y g a i n F i n a l l y , i t should  AQ d e f i n e d  be noted  AQ  to a g e n e r a l  e q u i l i b r i u m gain  accruing  from a p r o d u c t i v i t y improving  accurate  a p p r o x i m a t i o n of the g e n e r a l  could be adapted  s i d e , measure of  formula.  t o t a l general  Nonetheless, i t  e q u i l i b r i u m gain  change of t a r i f f s .  I f a more  e q u i l i b r i u m g a i n i s needed,  the p r o d u c t i v i t y g a i n presented  to the g e n e r a l  1 2  t h a t the p r o d u c t i v i t y g a i n measure  i s v a l u a b l e as a lower bound f o r the  method of measuring  by P r o p o s i t i o n  increases.  i n (3.12) i s o n l y a p a r t i a l , p r o d u c t i o n  g a i n as opposed  the  e q u i l i b r i u m context  Debreu-Diewert measure of deadweight l o s s d e f i n e d  the  in this section by  employing  i n Diewert  the  (1984).  - 46  4-  -  STRICT PARETO IMPROVING CHANGES IN COMMODITY TAXES AND  4.1  A G e n e r a l E q u i l i b r i u m Model  There are H consumers ( h o u s e h o l d s ) , economy.  The  expenditure net  n  preferences  level  n  ra (u ,  n  1  In a d d i t i o n to u^  h=l,...,H, depend on  the e x p e n d i t u r e  their  the minimum a given  functions  the N-dimensional v e c t o r of domestic  (p +  producer p r i c e s and  t i s a v e c t o r of taxes  The  as  by  n e g a t i v e l y ) of a c h i e v i n g  consumer p r i c e s q =  commodities.  by h=l,...,H, i n the  q, v ) , which are d e f i n e d  s u p p l i e s are indexed  u .  indexed  of the consumers are r e p r e s e n t e d  n  functions  cost ( f a c t o r  utility m,  TARIFFS  t ) e R^  expenditure  , where p i s the v e c t o r of domestic or s u b s i d i e s on  1  f u n c t i o n s m*,  domestic  h=l,...,H, a l s o depend  a v e c t o r of consumer p r i c e s f o r i n t e r n a t i o n a l l y  on  t r a d e a b l e goods equal  to  M v = (w + x + s) e R t a r i f f v e c t o r and tradeable  , where w i s the world  s i s a v e c t o r of taxes  price vector,  or s u b s i d i e s on  T i s the internationally  commodities.  The  h  expenditure  functions m ,  h=l,...,H, a r e assumed  to  twice c o n t i n u o u s l y d i f f e r e n t i a b l e .  They are a l s o concave and  homogenous i n p r i c e s ( q , v ) .  Shephard's Lemma, the  Hicksian  expenditure  h  fuctions  as f i r s t  order  n  h  h  q  h  tradeable  p a r t i a l d e r i v a t i v e s of  mt  x ( u , q , v ) = V m ( u , q , v ) , h=l  linearly  consumers'  (compensated) net demand f u n c t i o n s f o r domestic and  commodities can be d e r i v e d  (4.1)  Using  be  H,  the  - 47 -  and  (4.2)  h  h  h  e (u ,  h  q, v) = V m ( u , q, v ) , h=l,...,H. v  The consumer net demand m a t r i c e s  f o r domestic and t r a d e a b l e goods a r e  defined  u s i n g (4.1) and ( 4 . 2 ) :  (4.3)  X = [x*,...,x ]; E = [e^,...,e ].  H  H  The aggregate consumer s u b s t i t u t i o n m a t r i x  2  E (4.4)  E =  E qq qv E E vq vv  H E  h=l  In ( 4 . 4 ) , the m a t r i x  h  u V m qq „2 h V m vq  block E  E i s d e f i n e d by  q v  V m qv „2 h V m vv  , f o r example, g i v e s the  aggregate net demand responses to changes i n t r a d e a b l e s consumer p r i c e s f o r domestic commodities.  The o t h e r  b l o c k s of the s u b s t i t u t i o n m a t r i x  have analogous i n t e r p r e t a t i o n s . The matrix  E i s symmetric, negative h  expenditure  functions m ,  satisfies  (4.5)  T  T  [q ,v ] E =  0^  + M  s e m i d e f i n i t e ( s i n c e the  h=l,...,H, a r e concave i n p r i c e s ( q , v ) ) and  E  - 48 -  This  f o l l o w s because the e x p e n d i t u r e f u n c t i o n s m , h=l,...,H, a r e  l i n e a r l y homogenous i n p r i c e s ( q , v ) . E q u a t i o n s  (4.5) imply that the  rank, o f the m a t r i x E can be a t most N + M - 1. It may be that some o f the N domestic  commodities a r e producer  d u r a b l e s or s u p p l i e d i n d e p e n d e n t l y o f p r i c e s by the consumers. the n  t n  domestic  good  i s such a commodity.  Then, the n  column o f the m a t r i x Eqq c o n s i s t o f z e r o s .  t n  Suppose  row and  S i m i l a r l y , i f an  i n t e r n a t i o n a l l y traded good m, m e [1,...,M], does not e n t e r i n t o the p r e f e r e n c e s o f any consumer, or i f i t passes p r o d u c t i o n s e c t o r b e f o r e being a v a i l a b l e the m a t r i x E q  through  to consumers, the column m of 1  V  the domestic  i s a column of z e r o e s , and the m-^  row o f the m a t r i x  E i s a row o f z e r o e s . 1  Since the e x p e n d i t u r e f u n c t i o n s m^u* , q, v) , h=l,...,H, a r e n  nondecreasing measured  i n u , h=l,...,H, the u t i l i t y  o f each household  i n terms o f income, h o l d i n g the consumer  p r i c e s ( q , v) f i x e d at  some g i v e n l e v e l .  I f the consumer  equilibrium  ^fe & h h h ^ ^ (q , v ) , the e q u a t i o n s u = m (u , q , v ) ,  levels  prices  can be  h=l,...,H, d e f i n e the consumers' u t i l i t y  are f i x e d  (4.6)  and  i m p l i e s the f o l l o w i n g  V u h  h  h  m ( u * , q*, v*) = 1 ,  initial  1  l e v e l s u* i n the  neighbourhood of the i n i t i a l e q u i l i b r i u m . utilities  at t h e i r  T h i s money m e t r i c s c a l i n g o f  restrictions:  h=l,...,H,  2  (4.7)  h  h  V m ( u * , q*. v*) = 0, h=l,...,H. u u hh  In a d d i t i o n , Diewert  (1978:  s c a l i n g of u t i l i t i e s °  i s a p p l i e d , the m a t r i c e s '  can  be i n t e r p r e t e d  ordinary  demand  r  E and E qu vu  metric  defined  below  as the income d e r i v a t i v e m a t r i c e s of the consumers'  functions:  1  (4.8)  p. 146) shows t h a t , i f the money  H  E = [m ,...,m ] , qu qu . qu  h  2  h  h  where m = V m ( u * , q*, v * ) , h=l,...,H, qu qu  and  (4.9)  1  H  E '= [m ,...,m ] , vu vu vu '  h 2 h, *h * * where m =V m(u , q , v ) , vu vu  h=l,...,H.  The government i n the economy imposes t a r i f f s t a x e s both domestic and t r a d e a b l e government may g i v e transfers.  commodities.  on t r a d e a b l e s  Furthermore, the  the households ( p o s i t i v e or n e g a t i v e ) lump sura  The v e c t o r  of lump sum t r a n s f e r s i s denoted by g e R^.  With i t s income, the government buys domestic and t r a d e a b l e in  and  0 N 0 M amounts x e R and e e R . +  p u b l i c goods and s e r v i c e s  +  commodities  ,. These commodities a r e used to produce  f o r the p r i v a t e  J  sector.  The v e c t o r s  x^ and  - 50  e  u  -  are assumed to s t a y constant throughout  p u b l i c goods e x p l i c i t l y  appear  the a n a l y s i s  no  i n the model.  There are H + N + K + 1 e q u a t i o n s t h a t equilibrium  so that  characterize  an  (indexed w i t h an a s t e r i s k ) when the demand s i d e of the  economy i s taken i n t o  account:  (4.10)  h *h * '* *h m (u , q , v ) => g , h=l,...,H,  (4.11)  T f ( p * , w + x*) = 0, k=l,...,K,  (4.12)  ^ Z h=l  k  h *h * * 0 ^ m (u , q , v ) + x =  q  E k=l  ?£ Vn m ( W u* h, q * , v *^) + wT e h=l  T  ii i i \ (4.13)  V  w  V  k * * * k ir (p , w + x ) z ,  P  0  =  V  L w k=l  According  7 T ( p , W + T ) Z  - b .  to (4.10) - (4.13), the consumers ( h o u s e h o l d s ) equate  e x p e n d i t u r e s on domestic and incomes to t h e i r lump sum returns  V w  t r a d e a b l e commodities  revenues  to s c a l e p r o d u c t i o n s e c t o r s  (which may  factor  1  be z e r o ) , the K c o n s t a n t  earn zero (pure) p r o f i t s ,  aggregate net demand f o r domestic commodities s u p p l y , and  minus t h e i r  their  equals t h e i r  the b a l a n c e of payments net s u r p l u s equals b .  consumer  aggregate net  - 51  If budget  -  (4.10) - (4.13) are s a t i s f i e d ,  by Walras' Law,  the government  constraint  (4.14)  t*  T  x°  (t*  +  +  T  s * ) e°  +  * H T [ E h=l  =  •*T q  T  A  e  o . x +  T  t*  +  E x* h=l  K E f k=l  *  *T o . v e +  JE  h  -  h +  s*  T  ?  e* h=l  h  k  z j  *h g +  b  *  h-1  also  holds.  With a simple m a n i p u l a t i o n (4.14) can be r e w r i t t e n as:  *T ? * h . *T ? *ti *T J t E x +s E e + x [ E  ,. ... (4.15)  *h e -  r  h-1  h-1  h=l  x w e° j.+ = p* x ° + T  * *k E f  T  *k, z ]  k=l  ?E g *  h  u.+ b K *  h=l  This  form of the budget c o n s t r a i n t  implies  that  the government  e x p e n d i t u r e s on domestic and t r a d e a b l e commodities plus the lump sum transfers  forwarded  s u r p l u s must e q u a l  to the consumers the government  The exogenous v a r i a b l e s p r i c e s w which a l s o  plus  tax and  the b a l a n c e of payments net tariff  income.  i n (4.10) - (4.13) are the i n t e r n a t i o n a l  provide a p r i c e n o r m a l i z a t i o n 1c  1c  1c  government p o l i c y i n s t r u m e n t s g , t , s  i n the model, and the  1c  and  x •  The  endogenous  - 52 -  v a r i a b l e s , which are determined of  in ( 4 . 1 0 )  -  (4.13)  as i m p l i c i t  the exogenous v a r i a b l e s , are u* (household u t i l i t y  ( d o m e s t i c producer p r i c e s ) , trade  z*(industry  scales)  functions  l e v e l s ) , p*  and b* ( b a l a n c e of  net s u r p l u s ) . It  i s assumed ie  that an i n i t i a l  equilibrium  satisfying  (4.10)  -  ie  (4.13)  and (p , z ) »  (4.10)  -  (4.13)  0  N +  £  exists.  at the i n i t i a l  T o t a l d i f f e r e n t i a t i o n of  v a l u e s of the v a r i a b l e s  of the model  yields:  ie  (4.16)  AAu  ie  =  B  p  Ap  ie  z  +  B  Az  ie  +  where the m a t r i c e s A, Bp,...,Bg  A =  B, Ab  ie  +  b  -w  E vu  -E  T  -E  -w  qv  T  t  At  + S  E  VV  pw  + w  T  S  WW  • ie  ie  +  B  s  As  +  B  g  Ag  1  as f o l l o w s : *  N  PP  KxH  T  =  B  + S qq  KxH  B  ie  +  HxK  -E  w  x  AT  are d e f i n e d  -X qu  B  T  E + w vq  T  S  wp  T w F  °K  -1  ,  - 53  -X  -E  -E  -E  -  H qv  qq  NxH  0,  KxN  °KxN -w  E  KxH  -w  T  E  vq  vv  i  H  Using  * u  the  I m p l i c i t Funct i o n Theorem, the d e r i v a t i v e s of the  *  *  , p , z  * and  determined by  *  b  with  respect  the m a t r i x  4.2  Existence  *  *  to the exogenous x , t , s , and _  [A, -B , -B , -B, ] ' p' z' b  J  *  endogenous  of S t r i c t P a r e t o  1  [B x  , B_, t  g  are  B , B ]. s 2  Improving Changes i n Commodity Taxes  and  Tariffs  A s t r i c t Pareto and  tariffs  x i s one  improving change i n the which leads  (household) i n the economy. sufficient  conditions  tax/tariff  i n the home country's  precisely,  the problem Is the  starting  from an i n i t i a l  do  e x i s t Au  * there  (4.16) holds One restriction  and  A  »  0 , H  of the s u f f i c i e n t on  the  increase  g o a l of t h i s  s e c t i o n i s to develop  tariff  change i s r e q u i r e d  initial  net  following:  * , Ab Ab  * , At > 0,  Ag  =  * , Ax 0 ?  exist  to l e a d to  an  More  conditions, (4.10) -  (4.13),  * and  Ag  such that  5  R  conditions given  consumers' p r e f e r e n c e s  to  balance of t r a d e . under what  * , As  f o r each consumer  perturbation  e q u i l i b r i u m which s a t i s f i e s  * * , Ap , Az  u  to a u t i l i t y  f o r such a tax and  when, i n a d d i t i o n , the increase  The  commodity tax r a t e s ( t , s )  and  i n Theorem 4.1 on  their  below i s a  initial  commodity  - 54  endowments.  (4.17)  I t i s required  -  that  there i s no s o l u t i o n a > 0  U  T  to  T  T  a [X ,  E ]  H  This supposition i s s a t i s f i e d  = 0^ .. N+M tx  i f t h e r e i s some domestic good n, n e  [1,...,N], which i s i n net demand or i n net s u p p l y by every household, i.e.,  > 0 or < 0 f o r a l l h=l,...,H.  Alternatively,  to have an i n t e r n a t i o n a l l y traded commodity m,  m  e [1,...,M],  demand or i n net supply by every consumer; i n t h i s f o r a l l h=l,...,H.  i t is sufficient  case, e S  i n net 0 or < 0  Goods that are demanded or s u p p l i e d by a l l consumers  i n the economy are o f t e n c a l l e d D i a m o n d - M i r r l e e s Weymark (1979: pp. 176-177) shows that  goods.  the e x i s t e n c e of a  D i a m o n d - M i r r l e e s good ( o r a composite Diamond-Mirrlees commodity) i n the economy guarantees that some P a r e t o improving d i r e c t i o n s p r i c e changes assumption, degree  (i.e.,  commodity tax changes)  the household p r e f e r e n c e s might  Without  be c o n f l i c t i n g  this to such a  that no P a r e t o improvement through a p e r t u r b a t i o n of the  economy's i n i t i a l  commodity tax s t r u c t u r e i s p o s s i b l e .  (4.17) thus i m p l i e s without which  Theorem  s u f f i c i e n t homogeneity  Theorem 4.1  cannot be  Assumption  of consumer p r e f e r e n c e s  established.  6  4.1  Suppose ( i ) rank Y = K •<_ N, ( i i )  (iii)  exist.  of consumer  x*  T  V  2  WW  G(w + T*,  y*) * 0^ and M  rank  [S  T  p p  + YY ]  =  N,  ( i v ) t h e r e i s no s o l u t i o n a > 0„ ri  - 55 -  to  a [X , E ] = 0 ^ T  T  T  +  M  .  Then, a ( d i f f e r e n t i a l )  p r o d u c t i v i t y improving  strict  change i n the economy's  P a r e t o and  initial  commodity  tax s t r u c t u r e s e x i s t s , h o l d i n g the i n i t i a l  transfers  constant.  t a r i f f and  v e c t o r of lump sum  T h i s change i n taxes and t a r i f f s  a l s o improves the  home c o u n t r y ' s net balance of t r a d e .  Proof: A p p l y i n g Motzkin's Theorem, a s u f f i c i e n t  condition  for a strict 7  P a r e t o and p r o d u c t i v i t y improving  (4.18)  there i s no v e c t o r A  T  tax and t a r i f f  = [A  1 T  , A  that X [ B , B , B , B , B ] = 0 T  p  z  C o n s i d e r the equations equations  x  t  c o r r e s p o n d i n g to the m a t r i x B  [X  2 T  ,  A  3 T  , X ] e R  ^  t  4  h  +  n  +  k  +  1 s u c  h  , X [ A , -B ] > 0 ^ . T  fa  S u b t r a c t the N  from the f i r s t  N equations  = 0  4  N+K  PP ,T  (4.19)  ^  3 T  This implies  , A ]  T w S wp  Equations  , X  T T A [B , B , B ] = 0 „ , „. p z t N+K+N  c o r r e s p o n d i n g to the m a t r i x Bp.  (4.19)  g  2 T  change to e x i s t i s :  KxK T w F  have a l r e a d y been s o l v e d i n Lemma 2 . 3 .  Hence,  - 56  if  X  4  = k e R, X  2 T  = k(p* + 6 )  are those d e f i n e d i n ( 2 . 2 7 ) .  T  and  X  -  3 T  T  = k y , where the v e c t o r s 5 and  T i n e q u a l i t i e s X [A, -B,]  The  > 0„,.  T  imply  H+l  b  4 that X  = k ^ 0.  In o r d e r to d e t e r m i n e  i f (4.18) can be s a t i s f i e d ,  two  cases need to be c o n s i d e r e d .  (i)  X  4  = k =  Q:  I f k = 0, then a l s o X  2T  T = 0„ and N  X  3T  T = 0 ,. K  In o r d e r to  T  satisfy  ( 4 . 1 8 ) , the f o l l o w i n g must h o l d :  T  (4.20)  XA  = X  By a s s u m p t i o n ,  > ol, H  1 T  T  XB  = -X  1 T  X  T  t  = oJ, N  T  XB  = -X  1 T  E  T  =  0^. M  s  t h e r e i s no s o l u t i o n to (4.20) and hence no s o l u t i o n to  (4.18).  (ii)  X  4  = k > 0: T  Set k = 1. here  T  L e t us c o n s i d e r the e q u a t i o n s X B^ = 0^.  The  goal  i s to show t h a t these e q u a t i o n s cannot be s a t i s i f e d i f  T  A  1T  T  T  1  (p + 6 ) , y , 1] ( t h e v e c t o r X may be s o l v e d from the T T 1 e q u a t i o n s X [ B ^ , B ] =» ^N+M' ^ knowledge °f ^ * required for X  = [ X ,  U t  s  n o t  g  the proof o f the theorem).  (4.21)  T  XB  = -X  1 T  T  E  U s i n g the above d e f i n e d v e c t o r  - (p* + 6 ) K  T + w  T  /  T  E  q  + (p* + 6 ) v  ^  p  S  ww  = (p* + 6 )  T  S  T  pw  + Y F  T  T  + wS  WW  ,  T  w  S  X,  T  + y F  T  - w  T  E w  y  - 57 -  since  X B  = 0^.  g  T  (4.22)  Applying  T  X B  = -x* V T  Lemma 2.4, i t can be seen  G(w + x*, y * ) * 0 ,.  2  T  WW  QED  M  Theorem 4.1 shows that a s t r i c t  that  ;  .  p r o d u c t i v i t y improving  change i n  A  the  initial  equilibrium  tariffs  x  can be c o n v e r t e d  to a s t r i c t  *  improving change o f the t a r i f f s without three in  x  and commodity  a change i n the consumers' i n i t i a l  They imply that a s t r i c t  change, s t a r t i n g from the i n i t i a l distribute  tax r a t e s ( t , s ) ,  lump sum incomes.  The f i r s t  p r o d u c t i v i t y improving  equilibrium, exists.  these p r o d u c t i v i t y g a i n s  preferences The  In order to  the consumer  r o l e of assumptions ( i ) - ( i v ) i s f u r t h e r c l a r i f i e d , i f  In t h e i r  to some r e s u l t s o f Diamond and M i r r l e e s s  classic  paper, Diamond and M i r r l e e s c o n s i d e r ,  t h i n g s , the e x i s t e n c e  taxes i n a c l o s e d  economy.  of P a r e t o improving They show that  choice  lies  i f ( i ) a l l production  i n s i d e the home c o u n t r y ' s  and ( i i i ) a Diamond-Mirrlees good  change i n the economy's i n i t i a l argue f u r t h e r t h a t  production  e x i s t s , then a P a r e t o  i n the  equilibrium possibility Improving  commodity tax r a t e s i s p o s s i b l e .  i f , a l t e r n a t i v e l y , production  among  changes i n commodity  economy i s under d i r e c t government c o n t r o l , ( i i ) the i n i t i a l production  Pareto  and endowments, must be s a t i s f i e d .  Theorem 4.1 i s compared (1971).  tariff  to the households i n a s t r i c t  improving way, the f o u r t h assumption which i n v o l v e s  set,  *  assumptions needed to e s t a b l i s h the r e s u l t have been encountered  Theorem 2.1.  other  Pareto  A  They  i n the economy takes  - 58 -  place i n a private production starting  from the i n i t i a l  s e c t o r , Pareto-improving  equilibrium s t i l l  exist,  consumer p r i c e s i n the economy can be p e r t u r b e d  tax changes  i f the producer and  independently  from each  <  other  and the above mentioned assumptions Consider  international  now the e x i s t e n c e r e s u l t  f o r the home c o u n t r y .  expressed  as the s e t  — r T = {( E k=l K  (4.23)  k y ,  K  K  k k E f ) : E y k=l k=l  sum  of the domestic p r o d u c t i o n  T.  In F i g u r e 3, which i s drawn assuming  internationally  T^,  frontier  T  made  technology  can be  K  k E f k=l  < o}.  s e t i s then generated  by t h e  that there a r e two  (M = 2) and one consumer (H = 1)  PP' g i v e s  the domestic  f o r t r a d e a b l e s keeping  frontier  f o r the t o t a l  production  domestic goods net supply  by the domestic technology  technology  that a l l p r o d u c t i o n  c o n t r o l , and t h a t the l i n e  production  technology  sets  The l i n e denoted by w d e f i n e s the p r o d u c t i o n  Suppose f i r s t public  the curve  L e t us d e f i n e  t e c h n o l o g i e s T^, k=l,...,K and the s e t  The f r o n t i e r PP' i s g e n e r a t e d  k=l,...,K.  possibility  possibility  t r a d e a b l e commodities  the home c o u n t r y ,  constant.  production  < 0 , w  economy's t o t a l  possibility  i n Theorem 4.1.  This a r t i f i c i a l  The  in  ( i i ) - ( i i i ) are s a t i s f i e d .  trade as an a d d i t i o n a l p r o d u c t i o n  available  8  possibility  frontier  k — E T + T. k  i n the country  i s under d i r e c t  denoted by w i n F i g . 3 i s the r e l e v a n t  f o r tradeables.  Suppose f u r t h e r , that  - 59 -  the in  initial Figure  e q u i l i b r i u m i n the economy corresponds to the  3.  feasible  Since  the  production  argument, the  only  initial  production  choice A l i e s  possibility  set, using  the  commodity tax r a t e s  X  ( t , s ) and  c o u n t r y c o n s i s t s of K c a s e , the the  production  constant  production tariffs,  4.1,  and  the  only  economy.  assumption.  s e c t o r i n the home  to s c a l e i n d u s t r i e s .  the  generated by  In  the sum  change i n the  i n the  of  the initial  change i s such that  perturbation  this  the  taxes  new and  Assumptions ( i ) - ( i i i ) i n Theorem  balance of  trade This  i n F i g . 3,  improving t a r i f f  change moves the  toward the  consumer p r e f e r e n c e s  s i n c e , i n Theorem 4.1,  Assumption ( i v )  v e r s i o n of t h i s  i f the  f r o n t i e r PP'.  from A,  the  to e x i s t , i s that  P a r e t o improving  condition, exists.  choice  change i n  k=l,...,K, becomes a c o n s t r a i n t :  established after  on the  this  Since  k  T ,  tariffs  guarantee that a net  production  returns  choose a s t r i c t  choice,  lies  satisfies  PP'.  x  p o s s i b i l i t y f r o n t i e r PP'  commodity tax r a t e s and  the  Diamond-Mirrlees  the domestic p r o d u c t i o n  s e c t o r a l technologies  government can  4.1  tariffs  i s the weakest s u f f i c i e n t  Suppose then that  B  X  i s a Diamond-Mirrlees commodity i n the  i n Theorem 4.1  inside  c o n d i t i o n needed ;for a P a r e t o improving X  there  p o i n t s A and  point  change, t h a t  economy's  C along  the  curve  s a t i s f y Assumption ( i v ) i n Theorem  the  consumer and  producer p r i c e s of 9  commodities can be perturbed i n d e p e n d e n t l y of each o t h e r , the government can a d j u s t the i n i t i a l commodity tax r a t e s ( t , s )  ft simultaneously  w i t h a change i n t a r i f f s  improvement  is attained.  Theorem 4.1  generalized  open economy v e r s i o n of  x  can  the  so t h a t a s t r i c t thus be  Pareto  regarded as  r e s u l t s of Diamond and  a Mirrlees.  - 60 -  GOOD 2  F i g u r e 3 - S t r i c t Pareto Improving P e r t u r b a t i o n s Commodity T a x e s .  in Tariffs  and  - 61  -  F i n a l l y , one more comment should assumed  i n Theorem 4.1:  the commodity  was  tax r a t e s t * and s*.  government  i s able  Theorem 4.1  still  to  nothing  to adjust  be made about what was not  said  about the i n i t i a l  I t was o n l y assumed  a l l of these  h o l d , i f the i n i t i a l  values of  that the  tax r a t e s i f need be.  Would  tax r a t e s ( t * , s*) happened  be Diamond-Mirrlees o p t i m a l , i . e . , they maximize some s o c i a l  f u n c t i o n W(u) model  with r e s p e c t  (4.10) - (4.13)?  welfare  to the c o n s t r a i n t s of the g e n e r a l e q u i l i b r i u m  I t turns out t h a t o n l y the p r o p e r t i e s of the  initial  t a r i f f v e c t o r T* matter.  tariffs  x* and the commodity  I f , a t the i n i t i a l  e q u i l i b r i u m , the  tax r a t e s ( x * , s*) are  Diamond-Mirrlees o p t i m a l , then no s t r i c t  Pareto  and p r o d u c t i v i t y •k  improving  changes i n them e x i s t .  But i f the v e c t o r of t a r i f f s  a r b i t r a r y , Diamond-Mirrlees o p t i m a l i t y o f the commodity not  change the c o n c l u s i o n of Theorem 4.1.  taxes  a r e not o p t i m a l  Pareto initial  improving tariffs  practical  at the i n i t i a l  policies,  tax r a t e s does  e q u i l i b r i u m , there exist  need not be p e r t u r b e d  Ax  at a l l .  government can a t t a i n s t r i c t changing o n l y the commodity  Pareto taxes.  strict  = 0M i . e . , the Considering  t h i s means t h a t as l o n g as the i n i t i a l  r a t e s ( t , s ) i n the home c o u n t r y  is  Rather, i f the commodity  tax and t a r i f f changes w i t h x  x  commodity t a x  a r e not Diamond-Mirrlees o p t i m a l , the (hence, w e l f a r e ) A f t e r these  improvements by  improvement  p o s s i b i l i t i e s have been exhausted, the more complex p o l i c i e s i n v o l v i n g changes i n t a r i f f s  a r e needed.  - 62 -  A.3  Necessary Conditions  f o r Pareto Optimality;  Nonexistence of S t r i c t  P a r e t o and P r o d u c t i v i t y Improving Tax and T a r i f f Changes  Having e s t a b l i s h e d of  sufficient  conditions  s t r i c t , P a r e t o and p r o d u c t i v i t y improving  perturbations, exist.  i t i s natural  the  existence  t a r i f f and commodity tax  to enquire when these p o l i c y changes do not  A r e s u l t i n t h i s v e i n was a l r e a d y  Theorem 2.2, where i t was  implying  shown t h a t  established  strict  i n Section  productivity  2.4 as  improving  ft changes i n the i n i t i a l gradient respect  equilibrium  tariffs  o f the net balance of trade to the t a r i f f s  interpreted  to g i v e  T  i s zero.  are not p o s s i b l e  Theorem 2.2 can a l s o be  satisfy,  that  the i n i t i a l  i f i t i s a local  The p r a c t i c a l s i g n i f i c a n c e o f t h i s r e s u l t l i e s  search f o r d i f f e r e n t i a l  i f the  f u n c t i o n b (w + T ) w i t h  a s e t of n e c e s s a r y c o n d i t i o n s  e q u i l i b r i u m o f the economy must optimum.  x  productivity i n preventing  improvements i n the government t a r i f f  a  policy  when none e x i s t . In and  t h i s s e c t i o n , the most g e n e r a l  necessary conditions  p r o d u c t i v i t y o p t i m a l i t y of the i n i t i a l  Thereafter,  some s p e c i a l cases are  equilibrium  for Pareto  are e s t a b l i s h e d .  considered.  Theorem 4.2: A necessary condition improving  commodity tax and t a r i f f  necessary condition initial  for strict  Pareto and p r o d u c t i v i t y  changes to not e x i s t , i . e . , a  f o r Pareto and p r o d u c t i v i t y o p t i m a l i t y of the  equilibrium, i s :  - 63  -  .H+N+K+l  (4.24)  there i s a v e c t o r X e R  such  t h a t A [A, - B ] > 0  H+l '  N+K+N+M+M*  P r o o f : .. A necessary productivity  c o n d i t i o n f o r the n o n e x i s t e n c e  there do not e x i s t that  Using that  Theorem  9c  Au , Ap  (4.16) holds and Au  Motzkin's  and  9c  , Az »  0  9c  , Ab  H'  Although Pareto  Theorem 4.2  Ab  9c  9c  9c  , At , As , Ax  such  > 0.  advises  QED  the government  not to search f o r  and p r o d u c t i v i t y improvements when (4.24) i s s a t i s f i e d , i t  is  rather d i f f i c u l t  is  related  a special  to see from the statement  to Theorems  2.1 and 4.1.  kind of i n i t i a l  In order  o f the r e s u l t ,  how i t  to i n t e r p r e t Theorem  t a x r a t e s ( t , s ) and t a r i f f s  B-optimal  w i t h r e s p e c t to the  x, i f i t s o l v e s the n o n l i n e a r  commodity  programming  problem  max  5  u,p,z,b ,t ,s ,x  4.2,  e q u i l i b r i u m f o r the economy i s i n t r o d u c e d .  An e q u i l i b r i u m i s c a l l e d  (4.26)  and  the proof of Theorem 4.1, i t can be shown  (4.25) and (4.24) are e q u i v a l e n t .  strict  Pareto  improvements i s : 9c  (4.25)  of s t r i c t  T  {3 u: (4.10) - (4.13) h o l d , g -  constant}.  - 64 -  In ( 4 . 2 6 ) , (t,s,T)  the government i s assumed to choose the  to maximize a s o c i a l w e l f a r e  3 > 0H»  The  w e l f a r e weights 3 can  V^W(u) of some g e n e r a l  tax and  tariff  f u n c t i o n of the form W(u)  be  regarded  s o c i a l welfare  as  8 u, T  the g r a d i e n t v e c t o r  f u n c t i o n W(u).  T  =  rates  (The  function  —  S u i s thus a l o c a l restrict  the 3-optimum to be a c o m p e t i t i v e  (possibly  zero) v e c t o r of lump sum  corresponds (1978) trade  l i n e a r i z a t i o n of W(u)).  The  constraints in  equilibrium  transfers.  1 0  (4.26)  with a fixed (4.26)  Essentially,  to the ( c l o s e d economy) Diamond, M i r r l e e s ( 1 9 7 1 ) - Diewert  optimal tariffs  tax problem; the o n l y d i f f e r e n c e i s t h a t , i n ( 4 . 2 6 ) , T are a l s o assumed to be  s e t so as to maximize  the  social  welfare. If respect it  the i n i t i a l  e q u i l i b r i u m of the economy i s a 8-optimum w i t h  to the commodity tax r a t e s ( t * , s*)  and  tariffs  T*,  then  must n e c e s s a r i l y s a t i s f y : *  (4.27)  there do not that ( 4 . 1 6 )  exist  holds,  Au  *  *  , Ap  *  , Az  3 Au* > 0 ,  *  , Ab  , At  Ab* > 0 and  T  *  *  , As  , AT  such  Ag* = 0 „ . rt  If  (4.27)  maximum.  (4.28)  i s v i o l a t e d , the i n i t i a l  e q u i l i b r i u m cannot be a w e l f a r e  A p p l y i n g Motzkin's Theorem, ( 4 . 2 7 )  there e x i s t s a vector X e  -X\  > 0 and  T  X [ B , B,, p  H  N  +  K  +  1  R  B , t  +  can be w r i t t e n as  T  s  B^  u  c  =  h  that X A  = 3  oj^^.  T  (>  oJ),  Comparing (4.28) and equilibrium  by  and  -  (4.24), i t can  be  seen t h a t , i f the  initial  p r o d u c t i v i t y optimum, i t i s a l s o a  s o c i a l welfare  welfare  f u n c t i o n (X^A)u, where the v e c t o r  X is  (4.28).  Proposition If,  65  i s a P a r e t o and  maximum f o r the defined  -  4.1: at the  initial  e q u i l i b r i u m of  the economy, no  p r o d u c t i v i t y improving commodity tax  and  tariff  strict  Pareto  changes e x i s t ,  the  T e q u i l i b r i u m i s a welfare where the v e c t o r  maximum f o r the T  X solves  X [B  , B p  P r o p o s i t i o n 4.1 P a r e t o and  rates  It  welfare was  government can all if  B u.  f u n c t i o n which may  a  respect  In p a r t i c u l a r , i f ,  assumed to choose the  tax and  to be  or may  and  i f (4.28) i s  a welfare  not  tariff  have  maximum  socially  weights.  assumed i n the o p t i m i z a t i o n adjust  or some of the any  showing that  maximum w i t h  T  =  observed e q u i l i b r i u m i s r e v e a l e d  to a w e l f a r e  0.  T  (1960) type r e s u l t  form W(u)  A)u,  , B ] = ( £ . , „, „ , - X B , > x N+K+N+M+M b  s  country so as to maximize s o c i a l w e l f a r e ,  the  with respect acceptable  f u n c t i o n of the  the government i s not  i n the  satisfied,  i s a Negishi  f u n c t i o n (X T  , B t  p r o d u c t i v i t y optimum i s a l s o a w e l f a r e  to some w e l f a r e initially,  , B z  s o c i a l welfare  the  tariffs  tariffs  p a r t i c u l a r values  x  are  T i n any  way  the  deemed o p t i m a l .  But i f  f i x e d , i t would be u s e f u l to know,  of the i n i t i a l  (4.24) under the s u p p o s i t i o n  problem (4.26) that  that o n l y  vector the  of t a r i f f s  x  satisfy  domestic commodity tax  rates  - 66 -  t*  and s* can be chosen  Proposition Let commodity (ii)  optimally.  4.2: the i n i t i a l  tax r a t e s  e q u i l i b r i u m be S-optimal w i t h r e s p e c t  ( t * , s * ) , and suppose that  and ( i v ) of Theorem 4.1  the assumptions ( i ) -  Then, i f T  are s a t i s f i e d .  to the  *T  2 * V G(w + x , ww  T  ft  y ) = 0^,  the i n i t i a l  equilibrium  satisfies  P a r e t o and p r o d u c t i v i t y o p t i m a l i t y g i v e n  the n e c e s s a r y c o n d i t i o n f o r  i n Theorem 4.2,  and no  P a r e t o and p r o d u c t i v i t y improving d i r e c t i o n s of change i n the T* and commodity  strict  tariffs  taxes ( t * , s*) e x i s t .  Proof: B-optimality H  X e  a vector  +  N  +  K  R  of the i n i t i a l +  1  that T  solves V  |  u  c  h  t  h  a  T  J  t  the proof of Theorem  By assumption, T  Proposition trade  V WW  satisfied.  4.2  to the o p t i m a l i t y  conditions  T X B  x  X  equals n  G(w + x , y ) = 0,.. M given  above, a l s o  (4.24)  QED  implies  that  the g r a d i e n t  vector  ft ft ft>j> f u n c t i o n , V^b (w + x ) , which e q u a l s -x  must be nonzero f o r s t r i c t  *  exists  2.1, f o r a v e c t o r  Tlt  J  Then, i n a d d i t i o n  there  T  T T the e q u a t i o n s X [B , B 1 = 0. ,, the v e c t o r ^ p' z N+K'  G(w + x , y ) .  that  x [ A , - B j > 0* , X [B , B , b H+1 P z  WW  is  implies  T  s  T B_, B ] = 0 . , A p p l y i n g t s N+K+N+M ° T  equilibrium  *  taxes ( t , s ) and t a r i f f s  of the net balance of ^  ie ie G(w + x , y ) ,  P a r e t o and p r o d u c t i v i t y improvements i n  * x  to e x i s t .  - 67 -  The  assumption i n P r o p o s i t i o n 4.2 t h a t  2  *T  *  T  the v e c t o r  *  G(w + T , y ) equals zero can be r e p l a c e d  involving  by a c o n d i t i o n  the aggregate producer s u b s t i t u t i o n m a t r i x S:  [-S , T * ] S = 0 ^ , T  (4.29)  T  + M  12 where the v e c t o r  5 i s defined  i n (2.27).  Then, i f the m a t r i x S i s of  maximal rank ( = N + M - 1 ) , a s u f f i c i e n t that and is  condition  implying  (4.29) i s  the r e l a t i v e producer p r i c e s f o r t r a d e a b l e s i n the home c o u n t r y 13 * * abroad c o i n c i d e . I t f o l l o w s t h a t i f , i n i t i a l l y , x = 0^, or i f x  some m u l t i p l e  of the i n t e r n a t i o n a l p r i c e s w, no s t r i c t  p r o d u c t i v i t y improving from the i n i t i a l equilibrium  commodity tax and t a r i f f  equilibrium,  satisfies  are p o s s i b l e ;  (4.24) f o r P a r e t o and  = %  o r when x  of the i n t e r n a t i o n a l p r i c e s w, the i n i t i a l  f a c t be a p r o d u c t i v i t y maximum. influence  starting  Furthermore, i t can be argued, using the  programming problem ( 3 . 3 ) , t h a t when x multiple  perturbations,  i . e . , the i n i t i a l  the n e c e s s a r y c o n d i t i o n  productivity optimality.  Pareto and  Hence, f o r a s m a l l  i s some  e q u i l i b r i u m must i n c o u n t r y that  cannot  i n t e r n a t i o n a l commodity p r i c e s , zero t a r i f f s a r e P a r e t o and  p r o d u c t i v i t y o p t i m a l even though lump sum t r a n s f e r s a r e not a f e a s i b l e government p o l i c y instrument as l o n g can  be chosen o p t i m a l l y ,  as the domestic commodity t a x r a t e s  the producer s u b s t i t u t i o n m a t r i x S i s of  maximal rank, and assumptions ( i ) and ( i v ) o f Theorem 4.1 a r e s a t i s f i e d . The  above statement can a l s o be i n t e r p r e t e d  t o t a l production  result.  Assuming t h a t  as an e f f i c i e n c y o f  the producer s u b s t i t u t i o n m a t r i x  - 68 -  S i s o f m a x i m a l r a n k , and a s s u m p t i o n s ( i ) and ( i v ) o f T h e o r e m 4.1 h o l d , starting V  f r o m an i n i t i a l  *  *  b (w + x ) = x  x  strict  Pareto  *x  e q u i l i b r i u m that  satisfies  2 * * T V G(w + x , y ) * 0.,, t h e r e ww M y  (4.10) - (4.13)  e x i s t s a path of  and p r o d u c t i v i t y i m p r o v e m e n t s l e a d i n g  p r o d u c t i v i t y o p t i m u m , where t h e r e l a t i v e have been e q u a l i z e d .  This  Pareto  producer  t o a P a r e t o and  p r i c e s w and (w + x )  and p r o d u c t i v i t y o p t i m u m i s e f f i c i e n t k  with respect  (4.23). the  to the t o t a l  production  ( I n F i g . 3, t h e P a r e t o  first  best  terminology,  technology  E T  — + T defined i n  and p r o d u c t i v i t y optimum c o r r e s p o n d s t o  e q u i l i b r i u m (C,D).)  i t c a n be s a i d t h a t  with  Using  total  Diamond-Mirrlees  production  (1971)  efficiency i s  k desirable  if  (with respect  the producer  (i) * (t  to the t e c h n o l o g y  substitution matrix  4.4  country,  assumptions  and t h e c o m m o d i t y t a x r a t e s  to maximize  social  welfare.  S t r i c t P a r e t o and P r o d u c t i v i t y Improving Changes i n Commodity Taxes and  In  Tariffs  when no Domestic Goods E x i s t  the f o r m u l a t i o n  o f T h e o r e m 4 . 1 , i t was i m p l i c i t l y  some d o m e s t i c c o m m o d i t i e s e x i s t , note  + T) i n a s m a l l  S i s o f maximal rank,  and ( i v ) o f T h e o r e m 4.1 a r e s a t i s f i e d , * , s ) c a n be chosen o p t i m a l l y  Z T k  that  i . e . , N > 0.  i f a l l g o o d s i n t h e home c o u n t r y  assumed  I t i s tempting  that  to simply  are i n t e r n a t i o n a l l y  t r a d e a b l e , a s s u m p t i o n s ( i ) - ( i i ) o f t h e theorem c a n be erased  and t h e  - 69 -  result  is restored.  A c l o s e r i n s p e c t i o n of the theorem shows,  however, that a l t h o u g h the s u f f i c i e n t  conditions for s t r i c t  Pareto  p r o d u c t i v i t y improvements to e x i s t , when N equals z e r o , are v e r y to those g i v e n in. Theorem 4.1,  and  similar  the i n t e r p r e t a t i o n of the g e n e r a l  e q u i l i b r i u m model (4.10) - (4.13) changes i f N = 0. I f the number of p r o d u c t i o n s e c t o r s i n the model (4.10) equals  the number of domestic  (4.13)  commodities i . e . , N = K (> 0 ) , the  ft s e c t o r a l p r o d u c t i o n t e c h n o l o g i e s and determine (If  the e q u i l i b r i u m  *  prices p  the t r a d e a b l e s p r i c e s f o r domestic  the number of the p r o d u c t i o n i n d u s t r i e s  domestic  goods (K < N),  (4.12) a l s o a f f e c t  the domestic  the p r i c e s p •)  goods i n ( 4 . 1 1 ) .  i s less  k  in  than the number of  I f t h e r e are no  domestic  (w + T*)  k=l,...,K, i n s t e a d  determining .  of the p r i c e s  p*  ( 4 . 1 1 ) , i . e . , the p r o d u c t i o n i n d u s t r i e s do not g e n e r a l l y earn zero  p r o f i t s when N = 0. profits  The  e x i s t e n c e of these p o s s i b l y  creates K a r t i f i c i a l  imputed. be  15  market e q u i l i b r i u m c o n d i t i o n s  commodities i n the economy, the t r a d e a b l e s p r i c e s the i n d u s t r y (pure) p r o f i t s 7r ,  (w + T )  domestic  Hence, i n the end, N,  positive  f a c t o r s , to which the p r o f i t s  the number of domestic  commodities,  are must  positive. If  the o n l y domestic  to which the s e c t o r a l  commodities i n the economy are the K  positive profits  are being imputed, the  factors  domestic  net s u p p l y m a t r i x Y becomes an (K x K ) - i n d e n t i t y m a t r i x : each s e c t o r s u p p l i e s one  unit  of i t s "ownership",  commodities might be c a l l e d .  The  c o n d i t i o n can then be w r i t t e n as p  as the newly c r e a t e d  aggregate AT  producers' A  + (w + T )  ^  zero ^  domestic profit A  F = 0 , where p ,  - 70 -  the  price vector for the a r t i f i c i a l factors of production, gives the  sectors' pure p r o f i t s .  The producer substitution matrix S i s of the  form  (4.30)  S=  KxK  KxM  MxK  ww  3  The consumers hold endowments of industry ownership shares, denoted by x  H  h  h , h=l,...,H ( Z x, = 1 , for a l l k=l,...,K). h=l fc  Theorem 4.3: Let  the only domestic commodities  i n the home country be the  ownership shares i n the production sectors k, k=l,...,K. T T T there i s no solution a > 0. to a [X , E ] H 7  T = a,,.,, K+M  Then, i f ( i )  and ( i i ) T  *T T S * 0 . ww M W  there exists a s t r i c t Pareto and productivity improving change in the i n i t i a l equilibrium t a r i f f s x  and commodity tax rates ( t , s ) .  Proof: If Y = Ijrj^. the rank of the matrix Y i s K (= N), and the matrix T YY are  i s positive d e f i n i t e . satisfied.  *X  Since x  Hence, assumptions ( i ) - ( i i ) of Theorem 4.1 T  S * 0 , assumption ( i v ) of Theorem 4.1 i s ww M W  9  also s a t i s f i e d . (4.30).)  QED  r  1c  1c  (Note that V G(w + x , y ) = S , i f S i s of the form ww ww  -  Assumption ( i i )  71  -  i n Theorem 4.3 has a s i m i l a r  i n t e r p r e t a t i o n as  assumption ( i v ) has i n Theorem ( 4 . 1 ) ; i f o n l y the p r o d u c t i o n s i d e of the economy were c o n s i d e r e d , possible  then  to show t h a t t h i s  p r o d u c t i v i t y improving  proceeding  s u p p o s i t i o n i m p l i e s the e x i s t e n c e o f s t r i c t  changes i n t a r i f f s  g a i n s can be d i s t r i b u t e d  as i n Theorem 2.1, i t would be  x*.  to the households i n a s t r i c t  way, i f the consumer p r e f e r e n c e s  are s u f f i c i e n t l y  assumption ( i ) of Theorem 4.3 i s s a t i s f i e d . commodities are the ownership shares c o n d i t i o n i s e a s i l y met: k, k £ [ 1 , . . . , K ] , such  the i n i t i a l domestic direction.  These p r o d u c t i v i t y Pareto  improving  homogenous, i . e . ,  I f the o n l y  i n the p r o d u c t i o n  domestic  industries,  this  i t i s enough to have a p r o d u c t i o n s e c t o r  that x^ > 0 f o r a l l h=l,...,H.  tax r a t e t ^ i s a s t r i c t  Pareto  Then, l o w e r i n g  improving  tax change  - 72 -  5.  EXISTENCE OF STRICT GAINS FROM TRADE WHEN LUMP SUM TRANSFERS ARE NOT A FEASIBLE GOVERNMENT POLICY INSTRUMENT  I t has been shown t h a t , under c e r t a i n r a t h e r weak assumptions about the i n i t i a l  e q u i l i b r i u m , t h e government can cause a s t r i c t  p r o d u c t i v i t y and P a r e t o improvement by a d j u s t i n g tax  r a t e s ( t * , s*) and t a r i f f s  x* a p p r o p r i a t e l y .  the i n i t i a l  commodity  This general  r e s u l t has a perhaps s u r p r i s i n g a p p l i c a t i o n : now, i t i s e a s y . t o prove the e x i s t e n c e  o f p o s i t i v e g a i n s from t r a d e ,  even i f no lump sum  t r a n s f e r s may be used to r e d i s t r i b u t e consumer income. Suppose that the i n i t i a l e q u i l i b r i u m  of the economy i s an a u t a r k y  e q u i l i b r i u m , i . e . , t h e r e i s no i n t e r n a t i o n a l t r a d e . describe  autarky using  assumed that tariff  I n o r d e r to  the open economy model (4.10) - ( 4 . 1 3 ) , i t i s  the l a c k o f i n t e r n a t i o n a l t r a d e i s caused by the government  policy.  To t h i s end, t h e i n i t i a l  equilibrium  tariffs  x are  d e f i n e d as  * a x =w -w,  (5.1)  where w is  3  i s the autarky e q u i l i b r i u m vector  the observed i n t e r n a t i o n a l p r i c e v e c t o r . 3  (w + x*) = w .  of tradeables I t follows  T h i s means t h a t i f t h e t a r i f f s  prices  1  and w  from ( 5 . 1 ) t h a t  x*, d e f i n e d by  ( 5 . 1 ) , a r e used i n the open economy model (4.10) - ( 4 . 1 3 ) , t h e model 2 characterizes  an a u t a r k y  equilibrium.  - 73 -  In o r d e r  to apply Theorem 4.1,  i t i s assumed  ( i ) - ( i v ) of the theorem are s a t i s f i e d .  t h a t the c o n d i t i o n s  Then, there e x i s t s a  strict  ft Pareto  and p r o d u c t i v i t y improving  commodity allows  tax r a t e s ( t , s ) .  international  prohibitive  tariffs  change i n the i n i t i a l  tariffs  T  and  I t f o l l o w s t h a t , i f the government  trade by p e r t u r b i n g  the i n t e r n a t i o n a l  x*, i t can a l s o change the i n i t i a l  trade  commodity tax  r a t e s ( t * , s*) i n such a way t h a t a l l households i n the economy 3 s t r i c t l y b e n e f i t : s t r i c t g a i n s from t r a d e e x i s t . It should be noted, however, t h a t a p e r t u r b a t i o n o f the t a r i f f s  ft x  i s not always n e c e s s a r y  for strict  g a i n s from trade to e x i s t :  ft  the autarky  commodity  ft  if  Lf  tax r a t e s ( t , s ) are not 8-optimal,  strict  ft Pareto  improvements can be found  by changing  o n l y the tax r a t e s ( t ,  ft s ).  At the new  (perturbed)  l e v e l s o f the commodity  tax r a t e s the  ft initial thus  tariffs  x  are not i n t e r n a t i o n a l  trade p r o h i b i t i v e ;  be opened up, causing a p r o d u c t i v i t y and w e l f a r e  trade  will  improvement.  5  P r o p o s i t i o n 4.2 g i v e s n e c e s s a r y c o n d i t i o n s f o r Pareto and p r o d u c t i v i t y o p t i m a l i t y of the i n i t i a l e q u i l i b r i u m , when o n l y commodity tax and t a r i f f r a t e s are used as government p o l i c y i n s t r u m e n t s . In the present  c o n t e x t , P r o p o s i t i o n 4.2 may  conditions  f o r the nonexistence  be i n t e r p r e t e d to g i v e  of s t r i c t  gains  from t r a d e .  necessary For  ft ft example, i f the i n i t i a l and  i f the i n i t i a l  commodity  tax r a t e s ( t , s ) are g-optimal,  trade p r o h i b i t i v e  a r e p r o p o r t i o n a l to the i n t e r n a t i o n a l the net balance potential  tariffs  x* equal  zero ( o r i f they  p r i c e s w) so that the g r a d i e n t of  o f t r a d e f u n c t i o n b*(w + x*) i s a zero  M-vector,the  p r o d u c t i v i t y g a i n s a c c r u i n g from the p a r t i c i p a t i o n i n  - 74 -  international  t r a d e have a l r e a d y been exhausted  further s t r i c t The V xb  gains  i n i t i a l autarky e q u i l i b r i u m can a l s o s a t i s f y * 0 M,  W  international production  possibility  and  M  p r i c e s w.  and  x  i s not  In t h i s c a s e , frontier  Pareto  x*  commodity tax r a t e s ( t * , s*) F i g u r e 4.  There are  two  commodities, g i v e n a f i x e d  tradeables prices w a  t r a d e a b l e s are v ( = attains  3  by the curve  a  + s ).  u^.  (The  the curve The  production  possibility  net s u p p l y of domestic  The  initial  the producers  autarky face  the  p r i c e s the consumer  shows the economy's p r o d u c t i o n = 0 , M  exogenous i n t e r n a t i o n a l p r i c e s . tangent  The  u^.  p o i n t B i n F i g . 4a)  i n d i f f e r e n c e curve  tariffs  the consumer p r i c e s f o r  At these  under f r e e u n d i s t o r t e d trade (x  the  PP'.  i s denoted by A:  the u t i l i t y l e v e l The  and 4 b ) .  (= w + x * ) , w h i l e w  differential  t r a d e a b l e commodities i n the  frontier  i n F i g . 4a)  no  exist.  i n F i g u r e s 4a)  equilibrium  the  country's  p e r t u r b a t i o n s i n the  economies d e p i c t e d  goods y*, i s r e p r e s e n t e d  pf r otp o r t i o n a l to  s i n c e the home  p r o d u c t i v i t y improving  f o r these  no  the c o n d i t i o n  i s r i d g e d and/or k i n k e d ,  strict  Consider  and  from trade are p o s s i b l e .  (w + T ) = 0 M when T  and  i n autarky,  W  s  = 0 ). M W  The  Let us denote the  to the l i n e w (not  choice  l i n e w denotes  consumer  shown i n F i g . 4 a ) )  p o i n t C denotes the tangency p o i n t between the l i n e w  by and  u^.) f i r s t best autarky  the consumer and  producer  e q u i l i b r i u m i n F i g . 4a)  l i e s a t A' where  p r i c e s f o r t r a d e a b l e s c o i n c i d e (w  a  a  = v ).  - 75 -  GOOD 2  b) F i g u r e 4 - E x i s t e n c e of S t r i c t  G a i n s from  Trade  - 76 -  Because o f a government tax revenue requirement nondistortionary  tax i n s t r u m e n t s ,  a t A i s o n l y a second  the a c t u a l  from A, i . e . , from a u t a r k y ,  trade) e q u i l i b r i u m  the i n i t i a l  The s h i f t  I f , f o r example, the t a r i f f s  production w i l l  shift  i n d i f f e r e n c e curve The  i s accomplished  x  are reduced  tangent  best  (free  by s u i t a b l y  to z e r o , r e a c h the a  to the consumer p r i c e l i n e v(= w + s ) .  level  on t h i s i n d i f f e r e n c e curve can be higher or  ( i n the F i g . 4a) i t i s lower) than  level  u^', but i t i s not lower from  the f i r s t  to B, w h i l e the consumer w i l l  consumer's u t i l i t y  starting  But i f the  the government can move the toward  lower  If,  equilibrium  commodity tax r a t e s ( t * , s*) and t a r i f f s  ft ft fi  x .  autarky  at B and C i n such a way t h a t the consumer i n the  economy s t r i c t l y b e n e f i t s . perturbing  initial  best e q u i l i b r i u m , i . e . , u ^ < u ^ .  c o n d i t i o n s o f Theorem 4.1 a r e s a t i s f i e d , equilibrium  and the l a c k o f  the f i r s t  best a u t a r k y  than u ^ and not h i g h e r  the e q u i l i b r i u m w i t h zero t a r i f f s ,  can be changed f u r t h e r ,  the g a i n s from  u ^ and the i n d i f f e r e n c e curve  tangent  utility  than u^. the commodity  taxes  t r a d e ( t h e d i f f e r e n c e between to the l i n e v) can be made even  larger. It i s easy and  to see i n F i g . 4a) t h a t , i f the i n t e r n a t i o n a l  the a u t a r k y producer  t r a d e can be found  p r i c e s (w + x*) c o i n c i d e , s t r i c t g a i n s  o n l y by changing  s * ) , i f they a r e not B-optimal from  prices w from  the commodity tax r a t e s ( t ,  at the i n i t i a l  equilibrium  (starting  an e q u i l i b r i u m , where p r o d u c t i o n takes p l a c e a t B, and the consumer  attains  the u t i l i t y  level  corresponding  to the consumer p r i c e s v, the  consumer's w e l f a r e can be improved, i f the tax r a t e s s* (and t*) are perturbed  o p t i m a l l y ; i n the F i g u r e 4a) the consumer p r i c e l i n e v could  -  be r o t a t e d toward  77  -  the l i n e w as f a r as the government revenue c o n s t r a i n t  allows). F i g u r e 4b) A satisfies  illustrates  the n e c e s s a r y  the case where the a u t a r k y e q u i l i b r i u m at  c o n d i t i o n s f o r Pareto  o p t i m a l i t y g i v e n i n P r o p o s i t i o n 4.2 tariffs w:  T  are nonzero and  i s a zero 2-vector (2 x 2 ) - m a t r i x .  s i n c e , at A,  p r o d u c t i v i t y improving possible: The Chapter  9.  strict  distribute  + T*)  tariff  V  2 ww  G(w  * * + x , y ) is a J  (differential)  changes s t a r t i n g  g a i n s from i n t e r n a t i o n a l  at A  trade do not  Pareto  zero and  from A are exist.  e x i s t e n c e of the g a i n s from trade i s f u r t h e r d i s c u s s e d i n In that chapter, lump sum  a d m i s s i b l e , and when e i t h e r  tax and  strict  prohibitive  to the i n t e r n a t i o n a l p r i c e s  of trade f u n c t i o n b*(w  the m a t r i x  I t f o l l o w s that no  productivity  even though the t r a d e  nonproportional  the g r a d i e n t o f the net balance  and  t r a n s f e r s are assumed  the c o n d i t i o n s under which s t r i c t  commodity t a x a t i o n or lump sum the g a i n s  to the consumers, are  to be  g a i n s from trade  compensation i s used compared.  to  exist  - 78 -  6.  EXISTENCE OF STRICT PRODUCTIVITY AND PARETO IMPROVEMENTS WHEN ONLY A LIMITED SET OF COMMODITY TAXES AND TARIFFS CAN BE PERTURBED  Suppose t h a t , i n s t e a d o f being initial  able  to a d j u s t a l l the N + M  commdity tax r a t e s t * and s* a t w i l l ,  constrained Pareto  to p e r t u r b o n l y the N domestic  and p r o d u c t i v i t y improving  tax r a t e s t * .  changes i n t a r i f f s  possible?  In p a r t i c u l a r ,  distribute  the g a i n s from i n t e r n a t i o n a l  economy i n a s t r i c t  are the N domestic  Pareto  the government i s  improving  Are s t r i c t  and taxes  still  tax r a t e s s u f f i c i e n t to  trade  tb the households i n the  way?  Theorem 6.1: Suppose that ( i ) rank Y = K £ N, ( i i ) S definite, (iv)  ftT""  T V X  strict  ( i i i ) t h e r e i s no s o l u t i o n ry  ft  ft  rp  Pareto  + YY  T  is positive  to a > 0 t o a^X^ = 0^, H N  G(w + x , y ) * Of., and v) g  WW  p p  ft  M  and p r o d u c t i v i t y improving  = 0„. H  Then, t h e r e e x i s t s a  change i n t a r i f f s  x* and  commodity tax r a t e s t * .  Proof: It X e R  i ssufficient  H + N + K + 1  T X B^ =  T 0^.  such  to show t h a t t h e r e does n o t e x i s t T  that X [A,  Following  T  p  the proof of Theorem 4.1,  s o l v e s the N + K e q u a t i o n s k[(p  - 1^ ] > 0 ^ , X [ B ,  a vector  B^, B j =  0 ^ ,  the v e c t o r X t h a t  T T X [B , B 1 = 0„ must be o f the form p z N+K T  + 6 ) , y , 1 ] , k e R, i n i t s t h r e e l a s t  components.  Assumption  -  (iii)  i m p l i e s that  -  79  i f k = 0, no v e c t o r X s a t i s f y i n g  i n e q u a l i t i e s g i v e n above  the equations and  exists.  Suppose k = 1: T Consider  in  the equations  T  A  = 0^.  Using  the d e f i n i t i o n of  (4.16), these e q u a t i o n s can be w r i t t e n as  (6.1)  T  XB  = - X?E 1  x  T  + y F  T  - (p* + 6 ) ^  T  + w  T  S  T  T  E  - wE qv  + (p* + 5 ) . y  w  T  S pw  .  WW  T By assumption, X B  rri  (6.2)  rrt  - A,X q l  T = 0>  t  This i m p l i e s , using (4.16),  N  ^  rp  ^  - (p  ^  + 6)  E q qq  m  - w  that  ^  E v  q  = 0,  q  where  q* = (p* + t * ) .  Using  the homogeneity o f the e x p e n d i t u r e f u n c t i o n s , (6.2) becomes  (6.3)  T  T  [X E 1  + (p* + 6 )  T  E  + w  qv  T  E  vv  ] v* - X? g* = 0, 1  where v* = (w + s* + x * ) . S i n c e g* = 0 , H U  A  XB  v x  = [(p  (6.3) y i e l d s  + 6)  1  S  pw  T  XB  + Y F  s  v* = 0.  x  + wS  WW  ] v  It f o l l o w s , using (6.1), that  = -x  1  WW  G(w + x , y ) v .  - 80 -  By assumption, T  V  J  ww  G(w + x , y ) * 0..; hence, s i n c e v M '  T * X B^v * 0 and the v e c t o r  Theorem 6.1 tariff  shows that  and tax p e r t u r b a t i o n s  government i s c o n s t r a i n e d rates that  T X B^ cannot be z e r o .  t  strict  Pareto  are s t i l l  »  0 , M W  QED  and p r o d u c t i v i t y improving  p o s s i b l e even thought the  to adjust  only  the N domestic commodity tax  i n a d d i t i o n to the t a r i f f s  x .  Theorem 6.1 a l s o  the s e p a r a t i o n  o f the consumer and p r o d u c e r s e c t o r s  implies i s not  n e c e s s a r y f o r s t r i c t P a r e t o and p r o d u c t i v i t y improvements to e x i s t i n an open economy. rates and  1  ( I n Theorem 4.1, the assumption  ( t , s ) can be f r e e l y a d j u s t e d  producer p r i c e s can be perturbed  means t h a t  is  that  t r a n s f e r s at the i n i t i a l  to h o l d .  transfer vector g * * y ) v , where  qq  + w  are not much s t r i c t e r  than the supposition  the ( p o s s i b l y composite) D i a m o n d - M i r r l e e s good, that i s supposed  lump sum  E  the consumer  The o n l y a d d i t i o n a l r e s t r i c t i v e  to e x i s t , must be a domestic good.  result  that  i n d e p e n d e n t l y from each o t h e r . )  The assumptions of Theorem 6.1 assumptions o f Theorem 4.1.  that a l l commodity tax  T  Inspection *  The assumption that  e q u i l i b r i u m i s not n e c e s s a r y f o r the  of the proof  must o n l y be such t h a t  the v e c t o r  X^ s o l v e s  T  T  2  f o r Theorem 6.1  equilibrium  T T / * vT X^X + ( p + <5)  T T X B = 0 . t N XT  is sufficient  X^ g* * - x * V G(w + x*, y*) v* to be 1 ww  shows that the  T * *7J1 • * X g * -x v G(w + x , 1 ww  the e q u a t i o n s  T E = 0„ , i . e . , the e q u a t i o n s vq N  t r a n s f e r s at the i n i t i a l  there are no  satisfied.  Nonexistence of f o r the i n e q u a l i t y  - 81 -  Theorem 6.1 proves t r a d e can be d i s t r i b u t e d way a l s o t*,  that p r o d u c t i v i t y g a i n s from to the households  when the s e t o f c o n t r o l l a b l e  the s e t o f domestic  in a strict  tax instruments  commodity tax r a t e s .  s* i n the .home c o u n t r y g i v e s r i s e  complications.  First,  (free  p r e s e n t i n the system.  (6.4)  T  T  XB  = -X E  T  *T  the i n i t i a l  equation  + t*  commodity taxes  t h a t cannot  as f o l l o w s :  be a d j u s t e d , a r e  write  equation  T  ?  I + (x* + S * ) qv  ,  strict  Pareto  Yet,  FD?J 12  I qv  *.  to a zero  x* at the i n i t i a l  exist  identity;  equilibrium  and p r o d u c t i v i t y improving  tax r a t e s t * would s t i l l Theorem 6.1  T  £ - x* [S D,, + vv wp 11  x* a r e z e r o , but ( t * , s*) * 0"N+M>  (6.4) does not reduce  the t a r i f f s  *  T  V G(w + x , y ) • ww  tariffs  though  fixed  to some  i n the form  + x  Pareto  i s r e s t r i c t e d to  t r a d e ) a r e not g e n e r a l l y o p t i m a l f o r a  T h i s can be seen  x  If  improving  i n c o n t r a s t to the r e s u l t s o f the p r e v i o u s  s m a l l c o u n t r y , i f nonoptimal  (6.1)  Pareto  Y e t , e x i s t e n c e of f i x e d  tax d i s t o r t i o n s  s e c t i o n s , zero t a r i f f s  international  (assuming  i n o t h e r words, even  were z e r o , some  changes i n x  and commodity  t h a t the o t h e r c o n d i t i o n s o f  aresatisfied.) there i s a case where f r e e t r a d e , i . e . , zero  tariffs, i s  and p r o d u c t i v i t y o p t i m a l f o r a s m a l l c o u n t r y , i r r e s p e c t i v e of t h e initial  commodity t a x r a t e s s*.  T h i s o c c u r s when t h e r e i s no  s u b s t i t u t i o n i n consumption, i . e . , when I = 0 ( N + M ) X ( N + M ) • known t h a t taxes on goods i n f i x e d  3  I t i s well  demand a r e n o n d i s t o r t i o n a r y ; hence,  - 82 -  free  trade under these c i r c u m s t a n c e s The  above r e s u l t  that  free  i s optimal f o r a small country.  trade i s not g e n e r a l l y Pareto and  p r o d u c t i v i t y o p t i m a l f o r a s m a l l c o u n t r y i f some d i s t o r t i o n a r y the country e x i s t , can be rephrased  to express n o n d e s i r a b i l i t y o f t o t a l  p r o d u c t i o n e f f i c i e n c y under the presence form,  the statement  propositions  i s an a p p l i c a t i o n of Guesnerie's  then the e x i s t e n c e o f s t r i c t  improvements i n t a r i f f s  Pareto  Pareto  supposing  are a d j u s t a b l e :  and p r o d u c t i v i t y  t h a t assumptions  a sufficient  tariff  condition fora  i s that  k  ) i s nonzero.  k  k  improvement  i n ( t , s ) and x to e x i s t , m ( i ) , ( i i ) and ( i v ) of Theorem 4.1 a r e x  satisfied,  •m  and p r o d u c t i v i t y  I n t h i s c a s e , Theorems 2.1 and 4.1  k  y )  (1977) e a r l i e r  be a p p l i e d , i f a l l commodity tax r a t e s ( t * , s*) or at  the r a t e s t  strict  In t h i s  and commodity taxes when o n l y one i n i t i a l  * x , m e [1,...,M], can be v a r i e d . m  least  o f market d i s t o r t i o n s .  to an open economy.  Consider  can s t i l l  taxes i n  the d e r i v a t i v e  V  X  b  x  xT  x  *?  ( w + x ) ( = x  V  WW  m 2 & k (The n o t a t i o n V G(w + x , y ) refers ww .m  G(w + x , rh to the m  2 * * column o f the m a t r i x V G(w + x , y ).) ww •' J  * If in  addition *  in  o n l y one i n i t i a l domestic :  x  strict  to the t a r i f f s  commodity tax r a t e t , n e [1,...,N], n  x* i s v a r i a b l e , s t r i c t  Pareto  improvements  * and t  n  are not p o s s i b l e , but the government can s t i l l  w e l f a r e (and p r o d u c t i v i t y ) improvement  s o c i a l w e l f a r e f u n c t i o n W(u) = 8*^11.  generate  i n an exogenously  given  a  - 83 -  Theorem  6.2: Suppose that  j c. • definite,  ( i ) rank Y = K < N, ( i i ) S + YY — pp  , / j * *\ i_ and ( i n ) the v e c t o r  A  =V,  T  does not s o l v e welfare tax  0 j  +  K  +  1  ] [ A ,  i s positive  , ' „H+N+K+1 , ,. ,, A e R d e f i n e d by  T  (6.5)  T  -  B  - B , z  T T the e q u a t i o n s A B^ = 0^.  (B^.J"  Then, t h e r e  1  exists a  and p r o d u c t i v i t y improving change i n the t a r i f f s  strict  x* and the  * r a t e t , n e [1,...,N].  Proof: A sufficient improving  condition  for a s t r i c t  change i n the t a r i f f s  welfare  and p r o d u c t i v i t y  * * x and the tax r a t e t , n e [1,...,N] ,  to e x i s t i s :  * (6.6)  there  e x i s t Au  satisfied  * * * , Ap , Az , Ab  ^T 9c and 3 Au »  * * , A t ^ , Ax  such that  (4.16) i s  1  "J 9c 0 , Ab > 0. ri  E q u i v a l e n t l y , by Motzkin's  (6.7)  Theorem:  there must  not e x i s t a v e c t o r  x T [  <V.J  v  v  -  A e R  X  \  =  such that °M'  -  X  \  >  °-  A A = B  ,  -  By  the  [A,  Implicit  - B , - B P  Function - B  z >  Z  84  Theorem, the  -  inverse  ] * e x i s t s , and  matrix  hence (6.5)  d e f i n e s a unique X  n JL  for  the v e c t o r of w e l f a r e  vector  T  A B  Why improving can  be  i s not  T  zero;  weights  3 .  thus (6.7)  holds.  i s i t that o n l y a s t r i c t T  change i n t a r i f f s  found?  The  intuitive  and  By assumption, f o r t h i s  tax  explanation  the  QED  welfare the  A,  (not a s t r i c t rate t  , n  Pareto)  £ [1,...,N],  i s t h a t , to produce  improvements, the government needs a s u f f i c i e n t l y  can  Pareto  l a r g e number of f r e e  u  tax  instruments;  can  be  Pareto  f o r welfare  sufficient.  The  improvements, o n l y one  t e c h n i c a l reason  f o r the n o n e x i s t e n c e  improvements above i s that the v e c t o r  from ( 6 . 5 ) , cannot be determined  adjustable  A, which can T  from the e q u a t i o n s  be  A [B ,  B  p  tax  of  rate  strict  solved  (B  z >  > n  )]  T =  Cv,,Tr.i  N+K+l  of  strict  t h a t would have to be used Pareto  improvements i n T  i n order  *  to show the  existence  * and  s  t^, n e  [1,...,N].  * If free  a tradeable  tax i n s t r u m e n t ,  established. replaced  by  In t h i s  commodity tax  case,  the column (-B  s t a y the  same.  £ [1,...,M], were  a theorem analogous to Theorem 6.2 i n (6.5), ) s  would  r a t e s^, m  , but «m  the  column (-B  the other  )  could >  n  would  the  be be  assumptions of the  theorem  - 85 -  7.  SOME PIECEMEAL POLICY RESULTS WHEN NO LUMP SUM  TRANSFERS ARE USED  AS GOVERNMENT POLICY INSTRUMENTS  The p r e v i o u s s e c t i o n s have been e n t i r e l y existence  of s t r i c t  perturbations.  concerned  Pareto and p r o d u c t i v i t y improving  Thus f a r , v e r y l i t t l e  n a t u r e of the Pareto  t h a t , f o r example, a p r o p o r t i o n a l  equilibrium t a r i f f s  p r o d u c t i v i t y improving.  In t h i s s e c t i o n , i t w i l l  tax  i s also a s t r i c t  Pareto  improving  x* c o u l d  be  strict  be shown t h a t such a  Pareto improvement, i f the i n i t i a l  r a t e s ( t , s ) are a d j u s t e d a c c o r d i n g l y .  strict  the s p e c i f i c  p o l i c y changes.  r e d u c t i o n o f the i n i t i a l  reduction  government p o l i c y  has been s a i d about  and p r o d u c t i v i t y improving  It was e s t a b l i s h e d e a r l i e r  with the  commodity  Some o t h e r examples o f  t a r i f f and tax p e r t u r b a t i o n  p o l i c i e s are a l s o  given.  ft ft Consider  first  a r e d u c t i o n of the i n i t i a l ft  Cty, and an i n c r e a s e o f x  Theorem  when x _>  <^ 0^.  7.1; L e t assumptions  strict  x  i  ft  when x  tariffs  ( i ) - ( i v ) of Theorem 4.1  Pareto and p r o d u c t i v i t y improving  x* and commodity  taxes ( t * , s*) e x i s t s .  be s a t i s f i e d  change i n the i n i t i a l Then, ( i ) i f the  so that a tariffs  tariffs  ft ft ft x  are nonnegative,  to be a r e d u c t i o n nonpositive, increase  i.e., x  >_ 0^,  the change i n x  may be  (Ax* _< 0^) , and ( i i ) i f the t a r i f f s  i.e., x  (Ax* ^> 0^) .  <_ 0^, the change i n x  taken  x* are  may be taken to be an  - 86 -  Proof: 9c  Consider for  a strict  commodity  (7.1)  Pareto  taxes  there  the case ( i ) where  x  >^ 0^.  A sufficient  condition  and p r o d u c t i v i t y improvement i n t a r i f f s and  to e x i s t i s :  i s no v e c t o r  X e R  h  +  n  +  k  +  1 s  u  c  h  that  X [A, - B j > T  b xT[  V  V  V  V  =  The c o n d i t i o n (7.1) i s d e r i v e d Ax  *  °N K +  using  + N +  M'  \  A  > °MV  M o t z k i n ' s Theorem; the c o n s t r a i n t  T T < 0_, r e s u l t s i n the i n e q u a l i t y \ B > 0,, i n ( 7 . 1 ) . — M x — M  equations  (7.2)  T A B  *  x  (w + x ) = X  T  -E  F  +  qv  S  T  )  pw  T  T  -w  T  = X  +  (W  -E  T  E + vv  *  wS WW  T  *  *  X q + E s - g  *  *  *  E q + E s - S p qq qv pp T * -Yp T ' * T * T * wE q - wS p + wE s vq wp vv r  Consider the  - 87  -  The homogeneity o f the e x p e n d i t u r e and u n i t AT  equations q  AT  X+v  AT  profit  AT  E=g  and p  A  f u n c t i o n s , and the  T  T  Y + ( w + x ) F  = 0  yield  the l a t t e r  K.  X  X B  (w + T*)  T  (7.3)  x  A  form of X B (w + T ). x  = - X V 1  T  -  ,T *  = - X  using  T  XB  p  §  -  p* -  S pp  • ,T „T * X  p  X  1  F u r t h e r , because X B  T  (w + T*)  t  1  X* L  t  by assumption, ' 3  X E 2  q q  v  p* - w S  X M J  ,T_  -  T  = G\ . N  T  X B  (7.4)  T  S i n c e X [B , B ] = 0,_ t' s N+M  T  p  *  - w  T  Wp  E  p*  *  r V  Q  P  ,  oJT, N  =  T  T  = - X ^ g * = - x [ [ X q * + E v*] = X * X q * T  T  -  A*E v*.  T T The e q u a t i o n s X [B . B ] = 0 „ , i m p l y t s N+M w  T  (7.5)  XB  (w + x ) = (p  + 6)  r  X  T  + w E  using  w  v  E  qq  *  q  n  T  + wE  vq  q  n  + (p r  + 6)  T  n  functions m ,  h=l,...,H. X  1]  that  the v e c t o r  has a l s o been employed.  strictly *  positive *  G(w + x , y )  (»  0^)  X must be of the form [X^, (p  W  *  In  T T + 6 ), y ,  By a s s u m p t i o n , the v e c t o r (w + x*) i s and the v e c t o r  T X B^ i s nonzero ( s i n c e  X  * 0 ;  v  = 0,  the homogeneity of the e x p e n d i t u r e  (7.5) the f a c t  E  qv  see the p r o o f o f Theorem  4.1).  ^T x  Thus, the v e c t o r  _2 7^  - 88  -  T X  must c o n t a i n  (7.1)  at l e a s t one  n e g a t i v e (and  positive)  element, i . e . ,  is violated. *  ie  Consider now that  (w + x ) »  productivity  (7.6)  Oj^.  the  "  increase  V  >  0j  + 1  X  ,  same r e a s o n i n g as  T  <^ Oj^.  [ B  h  +  n  +  k  ,  vector x  for a s t r i c t  +  1  B ,  u  c  h  T X B  (7.6)  (except  B ,  Z  B ]  T  above, i t can  the  Pareto  and  that  -  G  be  0 j  +  K  +  seen that  T satisfying  i s such  to e x i s t i s :  s  P  The  condition  i n x*  vector X e R  t h e r e i s no  T  case where x  A sufficient  improving  * tA,  Using  the  N  +  M  <  t  Oj.  X  f o r any  T  X B ^ <^ 0^) , the  inequalities  T  X B  ,  inequalities  T < x —  must be v i o l a t e d .  M  Hence (7.6)  h o l d s , and  a strict  Pareto  and  * productivity ie  (t  improving  perturbation  i n the  tariffs  x  and  tax  rates  ie  , s ) exists.  QED  Theorem 7.1 tariffs,  is strict  shows that  a movement toward  Pareto and  productivity  c o u n t r y , i f a l l the adjusted  initial  s u i t a b l y and  satisfied. shows t h a t  commodity  also  an  improving  tax  rates  the o t h e r c o n d i t i o n s  However, a c l o s e r increase  of  inspection positive  free  of  of  tariffs  i . e . , zero  f o r a small  ( t , s ) can  the  the  trade,  theorem  proof of and  be  are Theorem  a reduction  7.1 of  2 negative can  tariffs  can  be  strict  Pareto  t h i s seemingly c o u n t e r i n t u i t i v e  and  productivity  r e s u l t be  improving.  explained?  The  How  economic  - 89 -  justification  seems to be that a movement toward e q u a l i z e d  producer p r i c e s f o r t r a d e a b l e s  at home and abroad, i . e . , toward  r e l a t i v e p r i c e s (w + T ) and w, i s s t r i c t  equalized  relative  Pareto and  p r o d u c t i v i t y improving under the c o n d i t i o n s o f "Theorem 4.1, whether the  * change i s accomplished "by reducing (T  (increasing)  _< 0^) or by i n c r e a s i n g ( r e d u c i n g )  the t a r i f f s  T  >_ 0^  them.  It may sometimes be d e s i r a b l e to choose a p a r t i c u l a r kind of reduction  i n the home country's  proportional politically is  strict  a uniform  or uniform fair.  Pareto  trade b a r r i e r .  reduction of t a r i f f s  may be c o n s i d e r e d  I t turns out that a p r o p o r t i o n a l r e d u c t i o n of t a r i f f s improving under the c o n d i t i o n s o f Theorem 4.1  reduction of t a r i f f s  requires  the r e s p o n s e s o f the domestic p r o d u c t i o n functions  For example, a  to changes i n t r a d e a b l e s  somewhat s t r o n g e r  but that  assumptions;  s e c t o r s ' net export  supply  producer p r i c e s must be such that a  ft decrease  i n each t a r i f f  home c o u n t r y ' s  x , m e [1,...,M], l e a d s  to an i n c r e a s e  i n the  net balance of t r a d e .  Theorem 7.2: L e t assumptions ( i ) - ( i i ) and ( i v ) of Theorem 4.1 be s a t i s f i e d . A  Assume f u r t h e r t h a t V b x  A  A T  (w + x ) = - x  0  A  A  T  T~ G(w + x , y ) < 0 . ww M W  Then, '  ft a strict  Pareto  improving  reduction of t a r i f f s  * change i n the i n i t i a l  x , accompanied  by a  *  commodity tax r a t e s ( t , s ) , e x i s t s , and the  * change i n the t a r i f f s  Ax  m  x  can be chosen to be a uniform  = -h, h > 0, m e [1,..,M].  H  reduction, i . e . ,  - 90 -  Proof:  T  •  It  i ssufficient  T  t h a t A [A,  cc- •  u  ,  to snow that  there  i s no v e c t o r  H+N+K+l  A e R  T  - B, ] > 0 * - , A [ B , B , B , B ] = O*^.^., b ..-H+l p z t s N+K+N+M  where h^ i s an M-vector c o n s i s t i n g of numbers h(> 0 ) .  such  T  X B h > 0, T M — M  As shown i n the  T X s a t i s f y i n g X [B , B , B , B ] = ° p' z t' s T T ^T 9 A 1c 0 ,„.„,„, the f o l l o w i n g must h o l d : X B = - T V G(w + x , y ) . Then, N+K+N+M T ww proof  of Theorem 4.1,  for a vector  J  L  XT  rj*  the  inequalities 1 8 ^ ) 0  b^ =  h »  b  0 . M  kind  (w + x ) h^ >_ 0.  But,  as -x  A  G(w + x , y )  by assumption, V^b  (w + x ) < 0^ and  assumptions i n Theorem 7.2 are a l s o s u f f i c i e n t  of r e d u c t i o n  improving.  in tariffs  interested  x* to be s t r i c t  Suppose that x* »  net imports are  other  can be w r i t t e n  "  2  Hence, V b*(w + x*) h „ < 0. QED x M  The  and  r  9c V  taxed.  i n bringing  for a third  Pareto and p r o d u c t i v i t y  i . e . , net e x p o r t s are  subsidized  In t h i s c a s e , the government may be  the t a r i f f r a t e s  words, a change toward u n i f o r m i t y  x* c l o s e r to each o t h e r ; i n i ntariff  r a t e s may be  desirable.  P r o p o s i t i o n 7.1: 1c  Let  the i n i t i a l  Suppose that  vector  of t a r i f f s  x  1c  be p o s i t i v e , i . e . , x  the assumptions of Theorem 7.2 are s a t i s f i e d .  exists a strict  Pareto and p r o d u c t i v i t y improving  ie  ie  Then,  x  tariffs  x* can be chosen to be a r e d u c t i o n  and commodity tax r a t e s ( t , s ) , and the change o f  uniform t a r i f f  structure.  0^  there  change i n the i n i t i a l  1c  tariffs  »  toward a (nonnegative)  - 91  -  Proof:  Define  x^ = ( x , . . . , x )  toward which the  7.2,  *  x*  uniform  are p e r t u r b e d .  set of The  tariffs,  d i r e c t i o n of  *  i s Ax  for a s t r i c t  *  (>_ 0 ) as Che  initial tariffs  * change i n x  T  = -(x  -  x) < 0^.  P a r e t o and  Using  the  proof  p r o d u c t i v i t y improvement  of Theorem to e x i s t when  *  Ax = -k(x - x ) , k > 0, i t i s s u f f i c i e n t to show that there i s no A „H+N+K+1 , , ,T . „ , . „T ,T „ „ „ « T vector X e R such that X [A, - B , ] > C, X [B , B , B . B ] b H+1 p' z' t s r  °N+K+N+M' and  x  T  b  ( t  t  T  *  " )  thus,  X B (x x  -  It  should  be  negative,  i . e . , net  under the c o n d i t i o n s and  T  assumption, X B  y  x) < 0„ when (x M  noted  -  t h a t , i f the  export are  = ?  t  x) > 0 . M  structure.  i n i t i a l vector  taxed and  of P r o p o s i t i o n 7.1,  net  consumer one tariff  rate  welfare  of  the  i s closely related  0^  exists a strict x*  x  is  subsidized, Pareto  toward a uniform  to a r e s u l t e s t a b l i s h e d  of  the l e v e l of the next h i g h e s t consumer.  To d e r i v e  admissible,  i n the home c o u n t r y .  t r a n s f e r s are  <  H a t t a showed t h a t , under c e r t a i n c o n d i t i o n s , i n a  t r a n s f e r s are  exist  + x*)  of t a r i f f s  imports are  there  producer economy, a r e d u c t i o n to  b*(w  5  P r o p o s i t i o n 7.1 H a t t a (1977b).  x  =  QED  W  p r o d u c t i v i t y improving decrease i n t a r i f f s  tariff  sum  B  > °-  r  assumed to be  commodity taxes are  present,  and  the  highest  t a r i f f r a t e improves  the  that lump  d i s t o r t i o n a r y commodity taxes  In P r o p o s i t i o n 7.1,  i n c o n t r a s t , no lump  available, distortionary (specific) and  one  (ad valorem)  h i s r e s u l t , H a t t a assumed  that no  by  the numbers of  consumers  and  sum  -  92  -  p r o d u c t i o n i n d u s t r i e s are not r e s t r i c t e d the g r a d i e n t of the net balance respect to  to t a r i f f s  x*  to one.  The  assumption t h a t  of t r a d e f u n c t i o n b*(w  i s n e g a t i v e at the  initial  + x*)  e q u i l i b r i u m appears  be a g e n e r a l i z a t i o n of a s u p p o s i t i o n used by H a t t a :  the h i g h e s t  G(w  the good  t a r i f f r a t e must be a s u b s t i t u t e i n p r o d u c t i o n * (If x »  t r a d e a b l e commodities.  0 , M W  + x , y ) to be p o s i t i v e , p o s i t i v e  with  *T  to the o t h e r  V  2 ww  terms i n the m a t r i x  V  f o r the v e c t o r x  with  ww  * G(w  *  + x , y ), i . e . , substitution Is  t h e r e anything more to be *  initial also  tax r a t e s t  > 0N»  and  s ?  For example, i s i s p o s s i b l e to  x* (>_ % )  S * > 0(4«  are b e i n g  reduced?  L e t us  lower suppose  ( T h i s means t h a t the government i s t a x i n g the consumers, whereas f a c t o r s of p r o d u c t i o n  the households are being  initial  about the changes i n the  commodity tax r a t e s i n the economy are p o s i t i v e , i . e . ,  commodities bought by by  said  ie  them, when t a r i f f s  that a l l i n i t i a l t*  i n p r o d u c t i o n , must dominate.)  subsidized.)  Then, assuming  that  the  e q u i l i b r i u m i s not a B-optimum w i t h r e s p e c t to the tax  t a r i f f r a t e s ( t * , s*, x*)  ( i n which c a s e , no  strict  Pareto  p r o d u c t i v i t y improvement i n them c o u l d e x i s t ) , a s t r i c t  Theorem  and  and  Pareto  p r o d u c t i v i t y improvement can be a t t a i n e d by s i m u l t a n e o u s l y * * s and x .  and  reducing t ,  7.2: Suppose that the ie  positive,  i.e., t  > 0 , N  initial  commodity tax r a t e s ( t * , s*)  ie  s  ie  > C^,  and  the t a r i f f s  x  sold  satisfy  are  - 93 -  x  T  x  _> 0^.  Assume i n a d d i t i o n that  x  x  ( t , s , x ) do not s o l v e  the problem  T  (7.7)  for  max {g u: (4.10) - (4.13) h o l d , g u,p,z,b , t , s , x  any S > 0g.  improving  Then, there  exists a s t r i c t  simultaneous r e d u c t i o n  in t , s  Pareto  and x  = 0 }, R  and p r o d u c t i v i t y  (i.e.,  At  <_ 0^,  As* < 0 , Ax* < 0 ) . M  M  Proof: A sufficient  condition for a s t r i c t  i n t * , s* and T* to e x i s t i s :  simultaneous r e d u c t i o n  (7.8)  there  X T [  Since  i s no v e c t o r  V  V  the i n i t i a l  satisfying  =  °N K>  X e  X T [  +  H  +  N  +  K  +  R  B  V  ^  x  s  ]  u  c  h  that  T  X [A, -  T  T  (  ,  Q  )  i  .  that  +  i s not a B-optimum, f o r any v e c t o r X T  X A > 0^ H  ] >°^ ^,  > °N+M+M'  X [ A , - B ] > O^., and X [ B , B ] = o 3 , „ , b H+1 p z N+K  inequalities  eR  V  equilibrium  T X [B , B , B ] i s not a zero t s x  X H+N+K+l  Pareto p r o d u c t i v i t y improving  (N+M+M)-vector.  X * O^..,,,,.. H+N+K+l  the v e c t o r  It follows  from the  Then, f o r any  - 94 -  1  (7.9)  X [B , B t  B]  s >  = x  x  W +  T -X -Z qq °KxN  J  T  -w.  W +  -E -E  T  qv °KxM  E -w E vq vv  -E -E F  T  qv  +  S  pw  T  T T -w E + w S WW vv  X  T  *  *  X P E  T  *  - g  * * p - S p qq pp T  *  -Y P X * x * w E p - w S p vq wp^ r  x x using  the e q u a t i o n s X B  *  = 0^ when g  p  * S i n c e , by assumption,  = 0^.  *  t  > 0 , s N  * > 0^ and (w + x ) »  0^, the v e c t o r  X X [B . B , B ] must c o n t a i n n e g a t i v e elements t s x  ( i t i s not zero  the i n i t i a l  Thus, (7.8) i s s a t i s f i e d .  &  e q u i l i b r i u m i s n o t a B-optimum) .  because  QED  C o r o l l a r y 7 .2 .1: * If t  > 0 , s N  H+N+K+l, a s t r i c t ii  reduction  > 0 , x M M  simultaneous it  In t , s  > 0 and rank — M M  Pareto and p r o d u c t i v i t y improving  it  and x  [B , B , B , B , B ] = p z t s x  exists.  - 95 -  Proof: If  rank [ B ,  B ,  p  equations X [ B , T  z'  \,  P  \,  B , B , B ] = H+N+K+l, the only t S T \,  = 0  \]  N  +  K  +  T T X A = OJJ and X B^ = 0, which c o n t r a d i c t X B T  U  b  < 0 i n (7.8).  If  < 0^, s  are  subsidized,  commodity  < 0^) so that  t  , s  tax r a t e s  M  is X = 0  R  m  +  Then,  r  T X A > 0^ and  ( t * , s*) are. n e g a t i v e  s o l d by the households are  the same a n a l y s i s  Pareto  This  M  import  improving  p o l i c y change  on consumer goods, reduced  production (e.g., l a b o r ) , smaller  as i n the p r o o f o f  and p r o d u c t i v i t y  (<_ 0 ) i s p o s s i b l e .  reduced s u b s i d i e s  +  the c o n d i t i o n s  and f a c t o r s of p r o d u c t i o n  7.2, that a s t r i c t and T  M  commodities bought by the consumers  taxed, i t can be shown, u s i n g Theorem  +  to the  QED  the i n i t i a l  (t  N  solution  increase i n  t r a n s l a t e s to  taxes on f a c t o r s o f  subsidies  and lower  export  t a x e s , i f such e x i s t . The c o n c l u s i o n only  claimed  that a simultaneous r e d u c t i o n  taxes and t a r i f f s strict  o f Theorem 7.2 may not seem so s u r p r i s i n g ;  i s desirable.  Pareto and p r o d u c t i v i t y  of (positive)  distortionary  Yet, i t i s not self-evident improvement  i t is  can be achieved  that a  through a  change i n ( t * , s*) and x*, without a change i n the i n i t i a l equilibrium and  vector  Ag* = 0 ^ ) .  7  of lump sum t r a n s f e r s  A c t u a l l y , according  ( i n Theorem  to Theorem  assumptions o f Theorem 7.2 a r e s a t i s f i e d ,  7.2, g* =  7.2, i f the  t h e r e e x i s t s an e n t i r e path o f  - 96 -  simultaneous  strict  Pareto and T  p o s i t i v e ( t , s ) and  p r o d u c t i v i t y improving  which, i n the l i m i t ,  lead  n o n d i s t o r t i o n a r y government which behaves l i k e  r e d u c t i o n s i n the  to a  an a d d i t i o n a l  private  3  production  s e c t o r i n the economy.  interpreting  One  t h i s r e s u l t as a p r a c t i c a l  s h o u l d , however, be c a u t i o u s i n p o l i c y recommendation:  g e n e r a l e q u i l i b r i u m model (4.10) - ( 4 . 1 3 ) , which i s used Theorem 7.2,  to d e r i v e  c h a r a c t e r i z e s a s t a t i c , p e r f e c t l y c o m p e t i t i v e economy,  where no e x t e r n a l i t i e s Furthermore,  the  i n consumption or p r o d u c t i o n are p r e s e n t .  the outcome of a s t r i c t  Pareto and  productivity  improvement g  may  be u n a c c e p t a b l e If  from an economic e q u a l i t y p o i n t of view.  the government i s r e s t r i c t e d JL  to a d j u s t a subset o f the  JL  commodity tax r a t e s ( t , s ) , r e s u l t s analogous still proved  (i)  be e s t a b l i s h e d . using s i m i l a r  A  If t T  > 0 , N l7  to Theorem 7.2  For example, the f o l l o w i n g  statements  techniques as i n the proof of Theorem  s  *  the i n i t i a l  tax r a t e s t  p r o d u c t i v i t y improving  ie  If t T  r e d u c t i o n of t  7  s  7.2:  the  strict  tariffs Pareto  T  .  ie  > 0„. and M  the i n i t i a l  tax r a t e s s  and  the  tariffs  ie  are not 3-optimal, t h e r e e x i s t s a simultaneous ie  and  (iii)  be  ie  and  ic  = 0. , N  and  a simultaneous A  (ii)  can  ie  = 0.. and M  are not 3 - o p t i m a l , t h e r e e x i s t s  and  can  p r o d u c t i v i t y improving  r e d u c t i o n of s  I f t * > 0, n e [1,...,N], t *  n  = O^,  1 0  and  s* = 0^,  T  *  strict  Pareto  .  g* = 0  H  and  the  - 97  -  * initial  tax r a t e  t  * and  n e x i s t s a simultaneous r e d u c t i o n of t  n  the t a r i f f s x  s t r i c t P a r e t o and  and x .  are not 3 - o p t i m a l , productivity  there  improving  - 98  8.  -  PARETO IMPROVING POLICY PERTURBATIONS WITH LUMP SUM  8.1  A Second Model f o r the P r o d u c t i o n  S i d e o f An  Economy  Consider a model of an economy's p r o d u c t i o n the domestic supply relations  (2.9),  side c o n s i s t i n g of  e q u a l s demand e q u a t i o n s ( 4 . 1 2 ) , the  and  the  net b a l a n c e of  trade  TRANSFERS  zero  profit  equation (4.13).  These  N+K+l r e l a t i o n s h i p s endogenously d e t e r m i n e the e q u i l i b r i u m v e c t o r domestic p r i c e s p*, and  the e q u i l i b r i u m v e c t o r  of i n d u s t r y s c a l e s  the home c o u n t r y ' s e q u i l i b r i u m net b a l a n c e of t r a d e b  exogenous t a r i f f s vector  x , the v e c t o r  of domestic commodity tax  rates  ( t * , s * ) , and  z*,  , given  of i n t e r n a t i o n a l p r i c e s w,  of  the  a fixed  a fixed  vector  ft o f household u t i l i t i e s u . production  sector d i f f e r s  production  sector  i s not  required  fixed u t i l i t y levels u equilibrium It  to s u p p l y  -  (2.10) i n that  a fixed vector  y*  consumers are assumed  the  of to be  kept  L  , h=l,...,H, which equal  l e v e l s of consumer  i s assumed  (4.13) e x i s t s .  model of the economy's domestic  i n s t e a d , the ft  initial  new  from the model (2.8)  domestic commodities but, at  The  the  welfare.  that a s o l u t i o n to the e q u a t i o n s  I t i s a l s o assumed  observed  that  the domestic p  (4.12), (2.9) and  and  the  A  industry scales positive. the  B  that  solve  ( 4 . 1 2 ) , (2.9)  D i f f e r e n t i a t i n g the  initial  (8.1)  z  p  s o l u t i o n p* »  Ap*  +  B  z  Az*  (4.13) are  e q u a t i o n s ( 4 . 1 2 ) , (2.9)  0 ,  z* »  N  + B, Ab* b  and  =  B  T  0 ,  AT*, '  K  b*  e R:  and  strictly (4.13) around  - 99 -  where B = P  S - E pp qq T Y w (S - E ) wp vq  KxK T w F  T  B  =  -S  + E  pw  T  -F  K -1  qv  T  T  -w (S  WW  - E ) vv  A  Applying b  the I m p l i c i t  can be regarded (w + T*)  prices  [Bp, B ,  F u n c t i o n Theorem, the endogenous p , z  as i m p l i c i t  (at fixed  and  f u n c t i o n s o f the exogenous t r a d e a b l e s  u*, t * and s * ) , i f the m a t r i x  B^] i s i n v e r t i b l e .  z  ie  Under t h i s A  A  d e r i v a t i v e s o f the f u n c t i o n s p ( w + e v a l u a t e d at the i n i t i a l s o l u t i o n by the m a t r i x [B , B , B , ] B . p z b x  s u p p o s i t i o n , the d i r e c t i o n a l A  A  T)  x ) , z(w+  A  A  and b (w + T  ),  to ( 4 . 1 2 ) , (2.9) and ( 4 . 1 3 ) , a r e g i v e n  - 1  Using can be seen  the r e s u l t s of Diewert that  and Woodland  the n e c e s s a r y and s u f f i c i e n t  [Bp, B , B ]  to be i n v e r t i b l e are (2.12) and  (8.2)  (S  z  b  rank  - E pp  solution  c o n d i t i o n s f o r the m a t r i x  + YY ) = N.  that (2.12) and (8.2) a r e s a t i s f i e d  to (4.12), (2.9) and  Assumption (8.2) has a s i m i l a r assumption  Appendix), i t  qq  H e n c e f o r t h , i t i s assumed initial  (1977:  (2.13).  Consider  (4.13).  at the  1  economic i n t e r p r e t a t i o n as  the f o l l o w i n g v e r s i o n of D e f i n i t i o n  2.2:  -  Definition  -  8.1:  Domestic goods p r o d u c t i o n around  100  the i n i t i a l  solution  i s said  to ( 4 . 1 2 ) ,  to be l o c a l l y  (2.9) and ( 4 . 1 3 ) , *  continuously d i f f e r e n t i a b l e A  A  A  which s a t i s f y  (4.12),  solution  initial  D e f i n i t i o n 8.1  tariffs  production  T  l e v e l s u*.  to ( 4 . 1 2 ) ,  i s simple.  I t i s assumed  that the  t h a t the amount of net f o r e i g n exchange earned by  i n d u s t r i e s , i n the aggregate,  In o r d e r  to a c h i e v e  i s i n c r e a s e d , w h i l e the  initial  t h a t as the t a r i f f s  x  equilibrium u t i l i t y  i t s t a r g e t , the government must be  a b l e to i n f l u e n c e domestic goods p r o d u c t i o n  i n the c o u n t r y  (and hence, the consumer  i n such a way  prices f o r tradeables  A  + s )) are perturbed  from t h e i r  change i n the consumers' w e l f a r e domestic goods p r o d u c t i o n t h i s end: A  initial  i s zero.  levels,  the  the f u n c t i o n s p  A  induced  L o c a l c o n t r o l l a b i l i t y of  i n the sense of D e f i n i t i o n 8.1 A  A  (2.9) and (4.13)  such a p e r t u r b a t i o n of the  consumers i n the economy are kept a t t h e i r  to  A  equilibrium values.  p o l i c y g o a l of the government i s to f i n d  (w + x  A  A  s  ( 2 . 9 ) , and (4.13) when the exogenous v a r i a b l e s u,  The i d e a behind  A  A  exist  f u n c t i o n s p (u , w + x , t , s ) and z (u ,  the i n i t i a l  x, t and s assume t h e i r  the  A  i f there  A  w + x , t , s ) around  initial  controllable  A  A  is sufficient  A  ( u , w + x , t , s )  A  and z  A  (u ,  A  w + x , t , s ) d e f i n e the a p p r o p r i a t e changes i n domestic goods p r i c e s A  and  i n d u s t r y scales corresponding  to g i v e n changes i n u  A  A  , x , t  and  s*.2 As i n Lemma 2.1, i t can be shown that Assumptions (2.12) and (8.2)  are s u f f i c i e n t  production  f o r l o c a l c o n t r o l l a b i l i t y o f domestic goods 3  1  i n the sense of D e f i n i t i o n 8.!. ' *  I f the consumer  - 101  s u b s t i t u t i o n m a t r i x f o r domestic  -  goods Eqq i s a zero (N x  N)-matrix,  c o n d i t i o n (8.2) c o i n c i d e s with c o n d i t i o n ( 2 . 1 3 ) , and c o n t r o l l a b i l i t y of domestic  goods p r o d u c t i o n i n the sense of D e f i n i t i o n  the c o n t r o l l a b i l i t y nonzero, (2.13).  concept of D e f i n i t i o n 2.2.  5  8.1  c o i n c i d e s with  I f the m a t r i x  however, c o n d i t i o n (8.2) i s l e s s r e s t r i c t i v e  than  Eqq i s  condition  6  As b e f o r e , the g r a d i e n t v e c t o r o f the net b a l a n c e of t r a d e function b analysis.  (w + T )  will  p l a y an important  Since the model c o n s i s t i n g  (4.13) d i f f e r s  from  the l a s t  (8.3)  V  row o f the m a t r i x  b*(w + x*) = w {[S  [S  where the symmetric  of the e q u a t i o n s ( 4 . 1 2 ) , (2.9) and  ww  - E ] vv J  1  - 1  [S wp  7  - E , F] D vq* 1  - E , F] }, qv  i n v e r s e m a t r i x D i s d e f i n e d by  —  (8.4)  B : x  T  wp  D = °11  °22_  e x p r e s s i o n f o r the  The g r a d i e n t V^b (w + x ) i s g i v e n  [B , B , B , ] p z b  T  x  i n the subsequent  the model (2.8) - ( 2 . 1 0 ) , a new  v e c t o r V^b (w + x ) must be found. by  role  —  S - E pp qq T Y  Y 0  KxK  -1  - 102  Assumptions  (2.12) and  In order hand  (8.2)  to develop  s i d e of ( 8 . 3 ) , a new  an  that the  interpretation  function, called  utility  tax a d j u s t e d b a l a n c e  (8.5)  B(w,  w + x, w +  guarantee  -  inverse matrix  f o r the m a t r i x  T + s, t) = max  T  h=l  ^  E  x  h  - x  ^h  (x^,  arguments.  i n d u s t r i e s can at  can  earn,  their  k  y z  k  -  the K a r l i n  initial  achieve  f o r e i g n exchange that tax  revenue,  equilibrium u t i l i t y  the  the  levels  net consumption v e c t o r s  utility  level  1  u*' ,  h=l,...,H.  l i n e a r l y homogenous i n i t s  (1959: p. 201)  - Uzawa (1958: p. 34)  Saddle  Q  P o i n t Theorem, a dual  k Z  k=l  net of government  represent  f u n c t i o n B i s convex and Using  Ev  :  the maximal net amount o f  e* ), h=l,...,H, than The  k  z >_ 0 „ } .  h ^h ; the s e t s M ( u ) i n (8.5) 1  f  N  when the consumes are kept u  T  IN  function B gives production  constant  0 . . , k ..k. _k , h h Jx. * h , >_ 0 , ( y , f ) e C , (x , e ) e M (u ),  h = l , . . . ,H,  private  H  t x  right  k=l  h=l  h=l  The  T  the  E (w + T )  T  {-w e° +  ?E .  H  exists.  i s defined:  z,Y,F,X,E  5  on  the home c o u n t r y ' s  of t r a d e f u n c t i o n B,  ' /( wJ. + T ^ +> s ) e - E .  D  the convex programming  problem  (8.5)  form which g i v e s an a l t e r n a t i v e e x p r e s s i o n  can be w r i t t e n i n  f o r the  f u n c t i o n B:  -  (8.6)  B(w,  w + x, w + x +  s,  103  -  t) = max  2.  z  £  Z k=l  /  .  N  (p, w +  x) z  the  functions  n^,  previously  defined  functions,  respectively.  v a l u e of p r i v a t e net  v a l u e of  household  and  economy's i n i t i a l  , s  tax  h=l,...,H, a r e  household  shows that the  at  the  B i s equal  t, w +  to  the  x) minus  p r i c e s (p, w) (p +  the  expenditure  p r i c e s (p, w +  prices  t = t , the  zero p r o f i t  (4.12) which are  and  , x ) , and and  and  first  net  the  minus the  net  x + s).  order c o n d i t i o n s  for  domestic s u p p l y e q u a l s demand  assumed  to be  satisfied  Hence, i f c o n d i t i o n s  tariff  variables  B(w)  the household  (4.12) can  domestic p r i c e s  p r i c e s w.  (8.7)  . w + x + s )  are  at  (2.12) and  f i x e d at  their  the (8.2) initial  «fe  e q u a t i o n s (2.9) and  °N  * and  equilibrium.  s a t i s f i e d , the <fe  the  *  become the  e q u a t i o n s (2.9)  at  m,  and  Problem (8.6)  p r o d u c t i o n valued  x = x , s = s  problem (8.6)  z  profit  production valued  public  n  k=i,...,K, and  sectoral unit  *  values (t  2.  ^ h *h Z m (u ,p+t, h=l  -  e x p e n d i t u r e s valued at  If  are  ^  p  °K»  TO e +  T 0, p x ].  -  In ( 8 . 6 ) ,  *  ir  r {-w  min  p  used  as i m p l i c i t  T h i s means that  = B(w,  be  w + x*,  utilities  x* +  f i x e d at u , the  to determine the functions  B becomes a f u n c t i o n  w +  are  s*,  t*) .  of the of  w:  industry  scales  international  - 104 -  The  g r a d i e n t o f the f u n c t i o n B can be c a l c u l a t e d by  differentiating  (8.8)  the o b j e c t i v e f u n c t i o n i n ( 8 . 6 ) :  V B(w) = -e  H E h=l  commodities  TT  (p(w) , w +  * T  k z (w)  )  ^ A A t , w + x + s ) .  produced by the constant  utility  to develop the H e s s i a n  B(w) i s d i f f e r e n t i a t e d  w  Using  obtained  k  V  i s the net excess s u p p l y v e c t o r o f i n t e r n a t i o n a l l y  In order  p r i c e s w.  ^ E k=l  V  w  V  +  h ^h Vm(u ,p(w) +  Thus, V B(w)  gradient  0  with  matrix  economy. o f the f u n c t i o n B the  respect  to the i n t e r n a t i o n a l  the formulae f o r the d e r i v a t i v e s V p(w) w  by d i f f e r e n t i a t i n g  the e q u a t i o n s  and V z(w) w  (4.12) and (2.9) w i t h  Q  to w, p and z:  (8.9)  V  2  ww  Hence, u s i n g  (8.10)  T  - [S - E , F] D[S - E , F] , wp vq' wp vq  (8.3),  b* (w + T*) = w  V  T  B(w) = S - E ww w  T  V  2  WW  B(w)  traded  respect  - 105 -  Lemma 8.1  The matrix  V  B(w) can be w r i t t e n  i n the form  WW  V^BCw) = [ - (  (8.11)  [S - Z]  S W P  -  E  V Q  )D  U  - PDj  2 f  I ] M  -D,. (S - Z ) - D._ F' 11 pw qv 12 M  where the m a t r i c e s  defined  and D^  are blocks  2  the m a t r i x  eigenvectors  i n Appendix 2.  Lemma 8.1 i m p l i e s  2 ** 10 V B(w) i s p o s i t i v e s e m i d e f i n i t e , but the zero ww  o f the m a t r i x  a r e g e n e r a l l y unknown.  however, that i f the i n i t i a l i.e.,  D  i n (8.4).  The proof o f Lemma 8.1 i s g i v e n that  i n the i n v e r s e m a t r i x  I t can be shown,  e q u i l i b r i u m commodity tax r a t e s are z e r o ,  * * TO t = 0^ and s = 0^, the f u n c t i o n B + w e , d e f i n e d  using  (8.5),  * must be l i n e a r l y homogenous i n the t r a d e a b l e s p r i c e s (w + x ) . 2 " means the matrix V B(w) must s a t i s f y t h e c o n s t r a i n t ww  (8.12)  if  v i  t* = 0  WW  N  T  B(w) (w + x * ) = 0 ,  and s* = Oj^.  M  This  - 106  8.2  -  Existence of Constant U t i l i t y P r o d u c t i v i t y I m p r o v i n g Changes i n Tariffs  The government's p o l i c y g o a l i s d e f i n e d as f o l l o w s : (differential)  change i n the home c o u n t r y ' s  find  i n i t i a l equilibrium vector  JU  of  tariffs  T  JU  that the c o u n t r y ' s  improved, w h i l e equilibrium  i n i t i a l net b a l a n c e  the consumers i n the economy a r e kept n  l e v e l s o f w e l f a r e u * , h=l,...,H.  k  there e x i s t  and  Ap  k  k  , Az , Ab  of trade b at t h e i r  is initial  More p r e c i s e l y , the  problem i s to determine the minimal s u f f i c i e n t  (8.12)  such  conditions f o r :  k  , Ax  such  t h a t (8.1) h o l d s  Ab* > 0. k  A p e r t u r b a t i o n of t a r i f f s  x  which s a t i s f i e s  (8.12) i s c a l l e d  a  * c o n s t a n t u t i l i t y p r o d u c t i v i t y improving  change o f t a r i f f s  T  .  Theorem 8.1: Suppose ( i ) rank Y = K < N, ( i i ) rank (S •  T 2 ~ (iii)  PP  + Y Y ) = N and qq  T  w V B(w) * 0 . ww  -  T  - E  M  p r o d u c t i v i t y improving  Then, t h e r e e x i s t s a c o n s t a n t change i n t a r i f f s  utility  x*.  Proof: The proof makes use of two p r e l i m i n a r y lemmas, the p r o o f s of which are g i v e n  i n Appendix 2.  a  107  -  Lemma 8 . 2 : Any v e c t o r  N+K+l  X e R  satisfying  the e q u a t i o n s  X [B , B ]  N+K  must be o f t h e form  (8.13)  X  T  =  k[(p*  T  T  + e ) , 8 , 1] , k e R,  where  *T (8.14)  wp' x  F] D.  i n Lemma 8 . 2 , the f o l l o w i n g  holds:  Lemma 8 . 3 : For  (8.15)  the v e c t o r  T  X B  =  -k w  T  X solved  V  x  Proof  2  B(w),  ww  k  e  R.  '  o f Theorem 8 . 1 : A sufficient  improving  c o n d i t i o n for a constant  change i n t a r i f f s  Theorem as i n the proof  x* to e x i s t  u t i l i t y productivity  is(8.12).  Using  Motzkin's  o f Theorem 2 . 1 , an e q u i v a l e n t c o n d i t i o n can be  derived :  (8.16)  t h e r e must n o t e x i s t  a vector  X e R  N +  K 1 +  such t h a t  -  X T [  On  V  the c o n t r a r y ,  -V  =  °N+K+M' * \ <  <>.  suppose such a A e x i s t s .  By Lemma 8.2, a v e c t o r  A that  T T the e q u a t i o n s A [B , B ] = 0„, , must be of the form • p z N+K  solves  T  f A  V  108 -  * = k[(p  T T + e) , 8 , 1 ] , k e R.  T F o r such a A, A B  = -k.  fe  Thus, f o r  T A B^ to be n e g a t i v e ,  .8.3,  = -w V  T  A B  T  T  2  k > 0 (and k may be chosen to be one).  B(w).  By assumption,  T  AB  * 0^, a c o n t r a d i c t i o n . M  X  WW  U s i n g Lemma  QED An  example of constant  provided  u t i l i t y p r o d u c t i v i t y improving  tariff  changes i s  by P r o p o s i t i o n 8.1.  P r o p o s i t i o n 8.1: If  the assumptions of Theorem 8.1 a r e s a t i s f i e d ,  a change of t h e  * tariffs  x  improving  i n the d i r e c t i o n o f the w o r l d p r i c e s w w i l l (keeping  be p r o d u c t i v i t y  the households i n the economy at t h e i r i n i t i a l  e q u i l i b r i u m u t i l i t y l e v e l s u*).  Proof:  * Let sufficient T  Ax  = rw, r > 0.  to show that there T  Using  the proof  i s no v e c t o r T  A [B  , B ] = oJ ,„, A B,_ < 0, A B w = 0. p' z N+K' b ' x  can  be seen t h a t f o r the v e c t o r  T  of P r o p o s i t i o n 2.1, i t i s  A e R  N + K +  ''  -  such  that  U s i n g Lemmas 8.2 and 8.3, i t ' &  N^*K.~H 1 A e R that s a t i s f i e s  T A [B , B ] = P z  -  T 0»,.„» N+K V  2  109 -  T the s c a l a r X B w must be equal x  T 2 ~ w V B(w)w. ww  The m a t r i x  T 2 ~ T B(w) i s p o s i t i v e s e m i d e f i n i t e and, by assumption, w V B(w) * 0 . M  T  Hence, .w V^B(w) w > 0.  QED  P r o p o s i t i o n 8.1 i m p l i e s that s m a l l , c o m p e t i t i v e improve t h e i r p r o d u c t i v i t y performance by s h i f t i n g  their  s t r u c t u r e toward the i n t e r n a t i o n a l p r i c e s w, without welfare  of their  c o u n t r i e s can tariff  s a c r i f i c i n g the  consumers.  Assumption ( i i i )  i n Theorem 8.1 may be w r i t t e n  i n an a l t e r n a t i v e  form which i n v o l v e s the producer and consumer s u b s t i t u t i o n m a t r i c e s and  E.  Using  the proof  S  o f Lemma 8.3, i n Appendix 2, the f o l l o w i n g  formula can be d e r i v e d :  (8.17)  T  2  T  w 7 B(w)w = w [-(S  - E  w  -D. . (S 11 pw  ) D  n  - F D ^ , ^ [ S - E]  - E ) - D. _F' qv 12  w  M  [-(p*  where the v e c t o r (iii)  +  E i s defined  £  )  T  T  , w]  (S - Z)  i n (8.14).  i n Theorem 8.1 can be r e p l a c e d by:  "(p w  + e)  Employing (8.17),  > o,11  assumption  -  (8.18)  the v e c t o r eigenvector  no  -  [-(p + e) , w ] i s not p r o p o r t i o n a l to any zero o f the s u b s t i t u t i o n m a t r i x (S - I ) .  I f t h e r e a r e no d i s t o r t i o n a r y commodity taxes a t the i n i t i a l ft ft e q u i l i b r i u m , i . e . , t = 0^, s = 0 and b o t h m a t r i c e s S and E a r e o f M >  12  maximal r a n k (= N + M - 1 ) , (8.18) s i m p l i f i e s t o :  (8.19)  the i n i t i a l v e c t o r of t a r i f f s  x  i s nonzero and not  p r o p o r t i o n a l to the i n t e r n a t i o n a l p r i c e s w  or  (8.20)  t h e v e c t o r o f d o m e s t i c producer p r i c e s p* i s n o t p r o p o r t i o n a l to t h e v e c t o r £ d e f i n e d  The  equivalence  i n (8.14).  o f (8.19) and (8.20) can be d e r i v e d u s i n g a s i m i l a r  c a l c u l a t i o n as i n Lemma 2.5, i n Appendix 1.  D i e w e r t (1983: p. 289)  shows t h a t the v e c t o r (p + e) i s t h e a p p r o p r i a t e p r o d u c t i v e  efficiency  v e c t o r o f shadow p r i c e s ( f o r d o m e s t i c goods) f o r e v a l u a t i n g government p r o j e c t s , when lump sum t r a n s f e r s a r e a v a i l a b l e f o r t h e p o l i c y government.  choosing  The c o n d i t i o n s (8.19) and (8.20) i m p l y t h a t the d o m e s t i c ft  producer p r i c e v e c t o r p  s h o u l d be used as t h e shadow p r i c e v e c t o r f o r  d o m e s t i c commodities o n l y i f no commodity t a x d i s t o r t i o n s a r e p r e s e n t i n the home c o u n t r y , and t h e i n i t i a l  tariffs  x* a r e zero o r p r o p o r t i o n a l  -  to  the i n t e r n a t i o n a l p r i c e s w. it  -  Ill  I f the i n i t i a l commodity  tax r a t e s  it  (t  , s ) differ  from z e r o ,  the a p p r o p r i a t e  shadow p r i c e s f o r  ie  domestic commodities a r e (p  + e) .  Using ( 8 . 1 9 ) , Theorem 8.1 may be w r i t t e n which i s p a r a l l e l  Proposition  to P r o p o s i t i o n  in a simplified  form  2.3.  8.2:  Suppose ( i ) rank Y = K_< N, ( i i ) rank S = rank Z = N+M-l, ( i i i )  * (t not  *  *  , s ) = 0. .,, ( i v ) the i n i t i a l v e c t o r N+M TI  proportional  constant  to the i n t e r n a t i o n a l p r i c e s w.  u t i l i t y p r o d u c t i v i t y improving  If  - m a t r i x , Theorems 8.1 and 2.1 c o i n c i d e :  improving  improving  i s nonzero and  Then, t h e r e e x i s t s a  change i n t a r i f f s  the consumer s u b s t i t u t i o n m a t r i x  productivity  T  of t a r i f f s  change o f t a r i f f s  x .  £ i s a zero (N + M) x (N + M) a constant  i s also a  utility  productivity  change o f t a r i f f s , when domestic goods n e t s u p p l y i s kept a t 13  its  i n i t i a l equilibrium  level.  Hence, a p p l y i n g  Proposition  2.1, the  JU  c o n s t a n t u t i l i t y p r o d u c t i v i t y improving taken to be a p r o p o r t i o n a l  reduction  change i n t a r i f f s  domestic commodity Consider  tax r a t e s  JU  (t , s ).  then the n e c e s s a r y c o n d i t i o n s  f o r constant u t i l i t y  p r o d u c t i v i t y o p t i m a l i t y of the i n i t i a l e q u i l i b r i u m . c i r c u m s t a n c e s , no s t r i c t in  tariffs  x* e x i s t .  can be  of x*, i r r e s p e c t i v e o f the JU  initial  x  Under  these  c o n s t a n t u t i l i t y p r o d u c t i v i t y improving  changes  -  112  -  Theorem 8 . 2 : A necessary condition optimality  (8.21)  of  the  for constant  i n i t i a l equilibrium  X e R  there e x i s t s a v e c t o r T  X [B  p >  B,  -B ]  z  = 0^  T  + K + M  ,  utility  productivity  is:  N  +  k  +  1 s u c  X \  <  h  that  0.  Proof: A necessary condition change i n T*  improving  for a s t r i c t  to not  constant  utility  productivity  exist i s :  ft ft ft ft (8.22)  t h e r e do  not  e x i s t Ap  , Az  , Ab  , AT  such that  (8.1)  is  ft  satisfied  and  Ab  Motzkin's Theorem g i v e s the  >  0.  equivalent  Suppose ( i ) rank Y = K < N, — T 2 ~ w V B(w)  T = 0 . M  necessary condition Theorem 8 . 2 ,  Then, the  The  ( i i ) r a n k (S  -  T  I  pp  + YY )  =  N,  qq  i n i t i a l equilibrium  and  f o r constant no  strict  proof of 2.1.1.  Corollary  utility  satisfies  productivity  constant u t i l i t y  d i r e c t i o n s of change i n t a r i f f s  Corollary  QED  the  w  WW  in  for ( 8 . 2 2 ) .  8.2.1:  Corollary  (iii)  form ( 8 . 2 1 )  T  8.2.1  optimality  productivity  improving  exist.  i s analogous to  the  given  proof  of  - 113 -  8.3  S t r i c t Pareto and Productivity Improving Changes In T a r i f f s and Lump Sum Transfers  In  this  distributing  section,  the problem to be c o n s i d e r e d  the g a i n s a c c r u i n g  improving change of t a r i f f s home c o u n t r y i s assumed  assumed  to the consumers.  utility  equilibrium  to be kept unchanged.  productivity  The government  to have lump sum t r a n s f e r  d i s p o s a l , but the i n i t i a l are  from a c o n s t a n t  i s that o f  commodity  i n the  instruments i n i t s  tax r a t e s  (t , s )  More p r e c i s e l y , the government's  p o l i c y problem i s :  (8.23)  find is  * * * * * * * * Au , Ap , Az , Ab , A T , At , As , Ag , such that  satisfied  *  and Au  »  0 , Ab  *  TT  *  > 0, At  Li *  *  *  = 0„ , As T  (4.16)  = 0,,.  M  M  Theorem 8.3: Suppose ( i ) rank Y = K < N, ( i i ) rank (S -  (iii)  T 2 T w V B(w) * 0 „ . ww M  PP  Then, t h e r e e x i s t s a s t r i c t '  T  - E + Y Y ) = N and qq Pareto and  JU  p r o d u c t i v i t y improving  change i n t a r i f f s  T  JU  and t r a n s f e r s g ,  w i t h o u t a change i n the home c o u n t r y ' s i n i t i a l Moreover, the change i n t a r i f f s country's i n i t i a l  and t r a n s f e r s  net balance of trade b .  commodity strictly  tax s t r u c t u r e .  improves the  - 114 -  Proof: Applying Pareto  Motzkin's Theorem, a s u f f i c i e n t  and p r o d u c t i v i t y improving  condition  t r a n s f e r and t a r i f f  for a s t r i c t  change to e x i s t  is :  (8.27)  there  i s no v e c t o r  X e R  H  +  n  +  k  +  1 s  u  c  h  that  X [B, B , B ] P z' g T  J  T N+K+H  T  L  and  B  T H+1  T  °TT,I >  ^  B  =  x  T 0,., where the m a t r i c s M  [B , B , B , B,] are those d e f i n e d P g o z  T T the e q u a t i o n s X [B , B , B ] = 0, , , . p z g N+K+H  Consider T  A  * [> ~t.]b >  °AI^ TT>  =  Tlt  rT  A  i n (4.16).  The e q u a t i o n s ^  T  X Bg = 0^ imply t h a t  the f i r s t  H components o f t h e v e c t o r  X must be  z e r o , i . e . , X. = 0 . Then, u s i n g Lemma 8.2, the o t h e r components o f the v e c t o r X c a n be s o l v e d . I t f o l l o w s that X = k[0„, ( p * + e ) , 9 , 1 ] , T  k  e R, where the v e c t o r s The  T inequality X B  £ and 9 a r e d e f i n e d  fe  > 0 implies  i t may be s e t to one.  Lemma 8.3 and the f a c t  that  by  the d i f f e r i n g  (4.16).)  But i f k = 0, then  that  T T X A > 0„. H  Consider now the e q u a t i o n s  TT  i n the p r e v i o u s  d e f i n i t i o n s o f the m a t r i c e s  T 2 ~ T By assumption w V ^ ^ w ) * 0 M  QED  B  T  Hence, k > 0  T T X B^ = 0^.  X, = 0 , i t can be seen that 1 H'  T 2 ~ w V B(w). (The change o f the s i g n ww  T  i n (8.14).  k >_ 0.  T T X A = 0 , which c o n t r a d i c t s the assumption H and  T  Using  T X B = x  formula i s caused  i n Lemma 8.3 and  - 115  Theorem 8.3  proves  that a constant u t i l i t y p r o d u c t i v i t y  change o f t a r i f f s can be c o n v e r t e d o f t a r i f f s and lump sum initial  utility  are e x a c t l y  assumptions  the home c o u n t r y ' s  needed f o r the r e s u l t  t a r i f f change to e x i s t .  f o r a constant In p a r t i c u l a r ,  on consumer p r e f e r e n c e s i s p r e s e n t i n Theorem  T h i s i s because a Diamond-Mirrlees  economy where household  8.4  The  specific  commodity always  t r a n s f e r s are  e x i s t s i n an  admissible.  l t f  Necessary Conditions f o r Pareto O p t i m a l i t y :  Nonexistence of  P a r e t o and  T r a n s f e r Changes  P r o d u c t i v i t y Improving T a r i f f and  C o n s i d e r f i r s t the most g e n e r a l n e c e s s a r y c o n d i t i o n s and  improving  P a r e t o improving change  the same as the c o n d i t i o n s  p r o d u c t i v i t y improving  no homogeneity assumption 8.1.  to a s t r i c t  t r a n s f e r s , w i t h o u t changing  commodity tax s t r u c t u r e .  to be e s t a b l i s h e d  -  p r o d u c t i v i t y o p t i m a l i t y of the i n i t i a l  government i s assumed to use lump sum  equilibrium  transfers  and  Strict  for Pareto  when the  t r a d e t a r i f f s as i t s  p o l i c y instruments.  Thorem  8.4: A necessary condition  for s t r i c t  P a r e t o and  productivity  improving  t a r i f f and  condition  f o r P a r e t o and p r o d u c t i v i t y o p t i m a l i t y of the  equilibrium  i s the  t r a n s f e r changes to not e x i s t , i . e . , a n e c e s s a r y  following:  initial  - 116 -  (8.25)  there  exists a vector  X e R  such that  X [A, -B. ] > O.V. , n+1  D  x  T  [  B  pV  B Z  »  B  g  ]  =  °N+K+H and  T  XB  = 0*. x M  Proof: If  the i n i t i a l  equilibrium *  there  must  not e x i s t Au  * satisfied  and Au  Motzkin's Theorem  result  and p r o d u c t i v i t y optimum,  *  * , Ax , Ag  such  that  (4.16) i s  * »  0  and Ab  > 0. Then (8.25) f o l l o w s ti as i n the proof of Theorem 4.2. QED  Although Theorem differential  i s a Pareto  * * * , Ap , Az , Ab  8.4  establishes conditions  by  using  under which  strict  P a r e t o and p r o d u c t i v i t y improvements a r e not p o s s i b l e , the  i s rather d i f f i c u l t  to i n t e r p r e t .  I t i s h e l p f u l to develop some  examples o f s i t u a t i o n s where (8.25) i s s a t i s f i e d . Consider f i r s t  8-optima w i t h r e s p e c t  to t a r i f f s  An e q u i l i b r i u m i s s a i d to be 8-optimal w i t h r e s p e c t t r a n s f e r s g, i f the v e c t o r s  x and g s o l v e  and t r a n s f e r s .  to t a r i f f s  the n o n l i n e a r  x and  programming  problem  (8.26)  max u,p,z,b,g,x  T  [8 u:  (4.10) - (4.13) h o l d , t =  constant,  s = constant] .  In ( 8 . 2 6 ) , the v e c t o r s welfare  f u n c t i o n W(u)  g and x a r e chosen so as to maximize T = B u, B > 0 , w i t h r e s p e c t U  n  the s o c i a l  to the c o n s t r a i n t s  - 117 -  of the g e n e r a l rates  e q u i l i b r i u m model (4.10) - ( 4 . 1 3 ) .  t and s are assumed Suppose now  that  8-optimum with r e s p e c t  to be kept at t h e i r  the i n i t i a l  The commodity  tax  previously set l e v e l s .  e q u i l i b r i u m of the economy i s a  to the t r a n s f e r s g  and t a r i f f s  x .  Then  necessarily,  * (8.27)  there must  not e x i s t Au  (4.16) i s s a t i s f i e d  As* -  T  Comparing  *  > 0, Ab  A  > 0, At  *  such = 0  •  >T  N  that  and  M  y i e l d s an e q u i v a l e n t  t h e r e must XA  T  and g Au  * * , Ag , Ax  0 .  Motzkin's Theorem  (8.28)  * * * , Ap , Az , Ab  exist a vector  = g , - X \ T  form f o r ( 8 . 2 7 ) :  X e R  T  > 0, X [ B , B , p  z  such B, g  B] T  -  that 0 ^ ^ .  (8.28) and (8.25), i t can be i n f e r r e d that i f the  initial  e q u i l i b r i u m of the economy i s a P a r e t o and p r o d u c t i v i t y optimum, i t must a l s o be a w e l f a r e  maximum with r e s p e c t  T W(u) = B u, where the welfare vector  X i s defined  i n (8.28).  to the s o c i a l w e l f a r e  weights B are equal  function  T to (X A) and the  T h i s r e s u l t corresponds to P r o p o s i t i o n  4.1.  ft Suppose not  the t a r i f f s  then that o n l y x*,  the domestic lump sum  can be chosen to maximize  social  t r a n s f e r s g , but welfare.  - 118 -  Theorem 8.5: Let  the i n i t i a l  e q u i l i b r i u m be a S-optimum with r e s p e c t  to the  t r a n s f e r s g*, and suppose that assumptions (2.12) and (8.2) a r e 9c  T  0  "  T  Then, i f V b (w + x ) = w V B(w) = 0,., no s t r i c t P a r e t o x ww M p r o d u c t i v i t y improving d i r e c t i o n s o f change i n t a r i f f s x and  satisfied. and  ie  transfers g  exist.  Proof: 8-optimality vector  X e  T = 0_ ^ . N+K+H TJ  IIT  H  +  N  +  K  +  o f the i n i t i a l  1  R  s  u  c  h  t  h  a  equilibrium implies  T  t  T  T  there  T  T  T X [B , B , B ] = • p z g  of Theorem  8.3 that  T the v e c t o r N+K+H'  0 . . , , , , „ ,  T X B  for a vector  x  equals H  T 2 T 2 ~ T w V B(w). S i n c e , by assumption, w V B(w) = 0.,, i t f o l l o w s ww ' ' ww M J  T T X B = 0,, and (8.25) i s s a t i s f i e d . x M  * * b (w + x ) zero? replaced  Using  follows  o f the net b a l a n c e o f trade  * (8.17), the e q u a t i o n V^b (w +  T  function * )  =  X °  c  a  n  b  e  M  T  T  [-(p* + e ) , w ] (S - Z) = ojj.  t h a t V b*(w + x*) equals zero T  aggregate s u b s t i t u t i o n m a t r i c e s matrices,  QED  by the c o n d i t i o n  (8.29)  It  that  r  When i s the g r a d i e n t  is a  X A = B (> 0 ) , -X B,_ > 0, X [ B , B , B ] H b p z g  I t was shown i n the proof  X, which s o l v e s  T  that  i . e . , there  at l e a s t  S and E a r e zero  i f both (N + M) x (N + M) -  i s no s u b s i t u t i o n i n consumption or p r o d u c t i o n  119  -  at  the i n i t i a l  equilibrium.  change of r e l a t i v e equilibrium Pareto the  and  Under these  circumstances,  supply and  net demand, and  p r o d u c t i v i t y improvements through  initial  tariffs  the v e c t o r  are p o s s i b l e .  (8.29), i t can be [-(p* + e ) ^ , w*-]  seen t h a t V b*(w + x*)  i s a zero e i g e n v e c t o r of the  because the zero e i g e n v e c t o r s of the m a t r i x  interesting  x* = 0^*  initial  Using  tax r a t e s ( t * , s*)  policy related  at  , s  *T  T ) * 0„, , N+M w  inferred E  and  +  S and  a s m a l l c o u n t r y , zero t a r i f f s  r  the v e c t o r  tariff  x ) * 0 , M W  and  changes e x i s t  are s a t i s f i e d ) .  and  E).  strict  (assuming  and commodity  consumer s u b s t i t u t i o n  , even i f lump sum  that  In other words,  ( f r e e t r a d e ) are not P a r e t o  the producer  are of maximal rank (= N + M - 1)  (S -  some  p r o d u c t i v i t y o p t i m a l i f t h e r e a r e d i s t o r t i o n a r y (nonoptimal) taxes i n the economy and  and  *  e q u i l i b r i u m , V b (w + x  p r o d u c t i v i t y improving  of  This implies that, '  T  and  one  from zero  E are of maximal rank,  the other c o n d i t i o n s of Theorem 8.3 for  differs  nonzero.  *  Pareto  Suppose  trade i s f r e e , i . e . ,  e ) , w j cannot be a zero e i g e n v e c t o r of the m a t r i x  Thus, at the i n i t i a l  are  Y e t , at  that, i f either E  be  qv  the v e c t o r e must be  l e a s t when the m a t r i c e s  [-(p  are z e r o .  equilibrium, international  ( 8 . 1 4 ) , i t can be  (S - E)  r e s u l t can be d e r i v e d .  qq *T;  matrix  to i n f e r when t h i s c o n d i t i o n might  the consumer s u b s t i t u t i o n m a t r i c e s  (t  strict  i s zero a l s o  T  satisfied  unknown, u n l e s s the i n i t i a l  hence no  1 6  In g e n e r a l , i t i s hard  t h a t , at the  differential  ( d i f f e r e n t i a l ) changes i n  (S - E ) .  l e a s t one  no  p r i c e s i n the home c o u n t r y can change the economy's  v e c t o r s of net  Using if  -  matrices  t r a n s f e r s could  be  - 120  -  employed as government p o l i c y i n s t r u m e n t s . regarded  T h i s r e s u l t can a l s o  as an e f f i c i e n c y o f p r o d u c t i o n r e s u l t :  t o t a l production  e f f i c i e n c y with r e s p e c t to the t e c h n o l o g y T + Z T k  d e f i n e d i n (4.23)  is.-nolt d e s i r a b l e i n a s m a l l c o u n t r y , i f t h e r e are nonoptimal taxes i n the system and if  lump sum  the m a t r i c e s S and  be  commodity  Z a r e of maximal rank, even  t r a n s f e r s c o u l d be chosen o p t i m a l l y to maximize  social  welfare. If,  however, the i n i t i a l  z e r o , u s i n g (8.19), f r e e optimal to  commodity tax r a t e s ( t , s ) are  trade can be s a i d  to be Pareto  f o r a small c o u n t r y , i f the lump sum  maximize s o c i a l w e l f a r e and  and  transfers g  productivity  are chosen  the s u b s t i t u t i o n m a t r i c e s S and  Z are of  18 maximal rank.  Under these c i r c u m s t a n c e s , t o t a l  production e f f i c i e n c y  _ k w i t h r e s p e c t to the t e c h n o l o g y T + Z T i s d e s i r a b l e , and k appropriate  shadow p r i c e s f o r t r a d e a b l e s f o r c o s t b e n e f i t a n a l y s i s  the i n t e r n a t i o n a l  prices  production e f f i c i e n c y i s  i n a small c o u n t r y , even though the commodity tax s t r u c t u r e i n  the home c o u n t r y a r b i t r a r y . Z =  are  w.  There i s a l s o a c a s e , where t o t a l desirable  the  (N+M)  X  the m a t r i x V  2  B(w)  i  , e  *>  C o n s i d e r an i n i t i a l  t h e r e i s no  substitution  c o i n c i d e s w i t h the m a t r i x  WW  e q u i l i b r i u m where  v  ?  i n consumption.  G(w  Then,  * * + T , y ) defined  WW  * in  (2.19).  I t f o l l o w s that  the v e c t o r o f producer 2 ~ must be a zero e i g e n v e c t o r of the m a t r i x V B(w). ww  *  p r i c e s (p , w + x ) * Hence, i f x = 0.. ,  M  - 121  -  or i f x  i s p r o p o r t i o n a l to the i n t e r n a t i o n a l  * V b (w + x  * T ) (which  equals  T  w  V  p r i c e s w,  2 ~ B ( w ) ) m u s t be a z e r o ww ie  i r r e s p e c t i v e o f the i n i t i a l  the  curve  that there  keeping  first  the F i g . 5a).  at  A and  a r e two  net s u p p l i e s of o t h e r  p r i c e s (w +  improving choice if  x  + s ) .  the p o i n t  international  trade  and  i s free.  since  the commodity t a x r a t e s s  prices  Consider  equilibrium, production  takes  level  Pareto  place  u^- a t  and p r o d u c t i v i t y  t r a n s f e r s moves t h e economy's  B, w h i c h i s t h e p r o f i t m a x i m i z i n g  C, w h i c h c o r r e s p o n d s  a t B and  commodities constant.  A strict  point  equilibrium  f o r these  *  change o f t a r i f f s  toward  frontier  i n t h e economy a t t a i n s u t i l i t y  * consumer  Figure 5 i s  c o m m o d i t i e s i n t h e economy;  possibility  At t h e i n i t i a l  t h e one c o n s u m e r  ie  i n F i g u r e 5.  tradeable  PP' d e n o t e s t h e p r o d u c t i o n  goods  M-vector,  commodity t a x r a t e s ( t , s ) .  The d i s c u s s i o n a b o v e i s I l l u s t r a t e d drawn assuming  the g r a d i e n t  The c o n s u m e r  i s shifted  to the consumer's f i r s t  best  c a n n o t be p e r t u r b e d ,  C c a n n o t be r e a c h e d .  (w + s*) t h a t w o u l d be o b s e r v e d  This  production  output toward  choice the  optimum.  the f i r s t  Yet, best  i s b e c a u s e the consumer  i f tariffs  were r e d u c e d  to  z e r o , do n o t g e n e r a l l y s u p p o r t t h e f i r s t b e s t i n d i f f e r e n c e c u r v e u*-" a t C. H e n c e , t o t a l p r o d u c t i o n e f f i c i e n c y c a n n o t be P a r e t o and p r o d u c t i o n optimal  f o r the country. I n s t e a d , t h e r e e x i s t s some t a r i f f v e c t o r , o denoted by x i n F i g . 5a), which I s Diamond-Mirrlees o p t i m a l given the JL  existing  tax d i s t o r t i o n s  equilibrium * (w +  s  +  i s denoted  ° x) a t D.  s .  The c o r r e s p o n d i n g  by D i n F i g . 5 a ) .  The u t i l i t y  level  second-best  The c o n s u m e r  faces  the p r i c e s  a t t a i n e d by t h e consumer  at  these  -  122  -  •G O O D1  F i g u r e 5 - S t r i c t Pareto and P r o d u c t i v i t y Improvements i n T a r i f f s and Transfers.  - 123  p r i c e s i s a t l e a s t as h i g h as  -  but g e n e r a l l y l e s s than the  first  p best l e v e l u .  * In F i g . 5 b ) , the case where t  * = 0\ and s T  = O  N  S t a r t i n g from the i n i t i a l e q u i l i b r i u m at A, assuming S and E a r e o f maximal rank, i t i s p o s s i b l e t a r i f f s and  w  i s depicted.  M  .  that both matrices  to a d j u s t the i n i t i a l  t r a n s f e r s so t h a t the f i r s t b e s t e q u i l i b r i u m at B and  C  ( t o t a l production efficiency) i s attained. In F i g . 5 c ) , the consumer has L-shaped i n d i f f e r e n c e c u r v e s , i . e . , ^  =  <  "'(N+2)x(N+2) *  at B and  ^-  so  under these c i r c u m s t a n c e s , the f i r s t  C can be r e a c h e d .  equilibrium  T h i s i s because, a t C, the consumer  (w + s*) are a support v e c t o r of the i n d i f f e r e n c e c u r v e  u^.  prices  - 124 -  9.  MORE ON GAINS FROM TRADE  Assuming that the i n i t i a l international  v e c t o r of t a r i f f s  x* equals the  3  trade p r o h i b i t i v e  t a r i f f v e c t o r (w '- w) d e f i n e d i n  ( 5 . 1 ) , Theorem 8.3 may be a p p l i e d to show the e x i s t e n c e o f s t r i c t from  trade when the government uses lump sum t r a n s f e r s as income  redistribution discussion  instruments.  i n Chapter  taxes a l o n e (without the  gains  strict  should  lump sum compensation) can be used  from trade always e x i s t  in international  two p r o p o s i t i o n s ?  Do  when o n l y commodity t a x a t i o n i s  a d m i s s i b l e , i f the c o n d i t i o n s i m p l y i n g  the e x i s t e n c e o f s t r i c t  under lump sum compensation are s a t i s f i e d ? gains  to the  to d i s t r i b u t e  a c c r u i n g from the p a r t i c i p a t i o n  What i s the c o n n e c t i o n between these gains  be compared  4, where i t was e s t a b l i s h e d t h a t a l s o commodity  p r o d u c t i v i t y gains  trade.  This r e s u l t  Or v i c e v e r s a :  gains  do  strict  from trade under lump sum compensation always e x i s t , i f the  sufficient  c o n d i t i o n s implying  the e x i s t e n c e o f s t r i c t  gains  under  commodity t a x a t i o n are s a t i s f i e d ? Kemp and Wan i n t h e i r mimeo that  sufficient  or not household  (9.1)  1  analyze  conditions for s t r i c t specific  gains  the p r o d u c t i o n p o s s i b i l i t y smooth and s t r i c t l y  iii.  questions.  They c l a i m  from trade to e x i s t  whether  t r a n s f e r s are a d m i s s i b l e a r e :  i . each t r a d e a b l e good i s produced ii.  these  i n the home c o u n t r y ,  s u r f a c e o f the home country i s  concave,  a l l consumers i n the economy have l o c a l l y u n s a t i a t e d preferences  i n autarky.  - 125 -  The  goal  of t h i s chapter i s to g e n e r a l i z e  and s t r e n g t h e n  t h i s Kemp and  Wan r e s u l t . B e f o r e the g e n e r a l i z e d necessary  p r o p o s i t i o n can be e s t a b l i s h e d , i t i s  to understand, why s t r i c t  gains  from t r a d e may not e x i s t i f  lump sum compensation.is not p o s s i b l e (Kemp and Wan p r o v i d e of a s i t u a t i o n where t h i s i s the c a s e ) . under c e r t a i n c i r c u m s t a n c e s , s t r i c t unless  an example  I t can a l s o be shown t h a t ,  gains  from trade do not e x i s t ,  commodity t a x a t i o n i s a f e a s i b l e government p o l i c y o p t i o n ( a n  example i s o f f e r e d l a t e r  i n this  section).  L e t us f i r s t c o n s i d e r the  Kemp-Wan example i n more d e t a i l . Kemp and Wan assume that country. are a l s o  there  The consumers i n e l a s t i c a l l y two producers m a n u f a c t u r i n g  fixed  c o e f f i c i e n t s technologies.  N=K  = 2, M = 2 ,  a r e two consumers i n the home supply  two t r a d e a b l e  In the present  (t  , s ).  To  answer the q u e s t i o n ,  Do s t r i c t  theorem i s s a t i s f i e d ,  gains  using  H = 2,  initial  but that values  e x i s t i n these c i r c u m s t a n c e s ? Assumption ( i ) of the  s i n c e the producers i n the economy supply Also  separate  assumption ( i i ) i s s a t i s f i e d , s i n c e the  i s p o s i t i v e d e f i n i t e when N = K and the rank o f the m a t r i x  Y equals K.  In order  must be c a l c u l a t e d .  2  qv  from t r a d e  from t h e i r  Theorem 8.3 i s a p p l i e d .  commodities and N = K = 2.  £  notation:  that lump sum t r a n s f e r s a r e a d m i s s i b l e ,  the commodity tax r a t e s cannot be changed  T  commodities  There  £ = 0- _ , S = 0. , . qq 2x2 4x4  Suppose f i r s t  matrix YY  two primary f a c t o r s .  - On i> 2x2  to confirm  assumption ( i i i ) ,  By assumption,  E  = O;,^*  and i t can thus be seen that  the m a t r i x V B(w) ww * ^ ^ i m p l i e s that s  2 ~ T 7 B(w) = E and w E * ww vv vv  - 126 -  0^.  Hence, a l l the assumptions of Theorem 8.3, are s a t i s f i e d  strict  gains  from i n t e r n a t i o n a l trade  Consider lump sum It  exist.  then the case where o n l y commodity  t r a n s f e r s , can be perturbed  and  from t h e i r  tax r a t e s , but not  initial  autarky  levels.  i s immediately obvious that assumption ( i i i ) of Theorem 4.1 i s  violated. to  T h i s i s because the producer  a zero  (4 x 4 ) - m a t r i x ,  Hence, a c c o r d i n g (differential)  s u b s t i t u t i o n matrix  S i s assumed  which i m p l i e s t h a t v G(w + T , y ) = CL „. ww Zxz  to P r o p o s i t i o n 4.2,  i n t e r n a t i o n a l trade,  p e r t u r b a t i o n of the t a r i f f s  x , i s not  caused by a  strictly  gainful. How  can these d i f f e r i n g  c o n c l u s i o n s be explained?  Consider  the  F i g u r e 6. F i g u r e 6 r e p r e s e n t s a two t r a d e a b l e economy, where (keeping  At  possibility frontier  domestic goods net supply  the a u t a r k y x*),  the p r o d u c t i o n  commodities one consumer for tradeables  f i x e d ) i s g i v e n by the curve  e q u i l i b r i u m A, the producers f a c e  the t r a d e a b l e s  PP'.  At  p r i c e s (w +  whereas the consumer p r i c e s f o r t r a d e a b l e a r e (w + x* + s * ) .  these  p r i c e s the consumer a t t a i n s the i n d i f f e r e n c e curve  At possiblity  the a u t a r k y frontier  equilibrium point  for tradeables  the economy's  i s kinked.  u\  production  T h i s means t h a t no  ft differential net  change i n the autarky  supply of tradeables  no s t r i c t production  increase  1  from A. *  tariffs  x  can s h i f t  Then, a c c o r d i n g  the  producers'  to P r o p o s i t i o n 4.2,  i n the net amount o f f o r e i g n exchange earned by the  industries i s possible.  I t f o l l o w s , t h a t the w e l f a r e o f the  -  127  -  GOOD 2  ure 6 - E x i s t e n c e of S t r i c t Gains from Trade Under Commodity T a x a t i o n and Lump Sum Compensation.  -  consumer cannot be s t r i c t l y s  (and t ) .  commodity ., .  128 -  improved by changing the commodity  I n o t h e r words, s t r i c t  taxation  tax r a t e s  g a i n s from trade under  do not e x i s t .  I f , however, the i n i t i a l  equilibrium  lump sum t r a n s f e r g f o r  •the consumer can be a l t e r e d , i . e . , lump sum t r a n s f e r s  are a f e a s i b l e  government p o l i c y i n s t r u m e n t , the s i t u a t i o n of the consumer i n F i g . 6 can  be improved.  are  perturbed  Suppose, f o r example, that  toward the t a r i f f s  x depicted  tariffs T  the i n i t i a l  i n F i g . 6.  As a  consequence, the producers aggregate n e t s u p p l y of t r a d e a b l e s does not change from i t s i n i t i a l indifference  l e v e l but the consumer i s moved a l o n g h i s 7  curve u^-. I f , s i m u l t a n e o u s l y w i t h the t a r i f f  consumer i s g i v e n a t r a n s f e r i n the d i r e c t i o n of Ag consumer i s made s t r i c t l y  better  off.  i n F i g . 6, the  Hence, i n t e r n a t i o n a l  caused by a p e r t u r b a t i o n i n the i n i t i a l if  change, the  tariffs  trade,  x* i s s t r i c t l y  gainful  lump sum compensation i s a v a i l a b l e . It  can be concluded  that  the Kemp-Wan example i s based on the  e x i s t e n c e of a " s u b s t i t u t i o n gap" between the Theorems 4.1 and 8.3. A c c o r d i n g to the l a t t e r strict  P a r e t o improvement p o s s i b l e  productivity But also are  result, substitution  i n consumption can make a  even though a s t r i c t  P a r e t o and  improvement i n the sense of Theorem 4.1 does not e x i s t .  the " s u b s t i t u t i o n gap" between Theorems 4.1 and 8.3 can be employed i n another f a s h i o n . satisfied;  Suppose that  a l l the c o n d i t i o n s of Theorem 4.1  i n p a r t i c u l a r , the p r o d u c e r s u b s t i t u t i o n m a t r i x S i s not  a zero (N + M) x (N + M) - m a t r i x (S * °(N+M)x(N+Mp• addition,  that  the consumer s u b s t i t u t i o n  matrix  b  u  t  E i s such  s u  PP  o s e  that  »  i  n  -  (9.2)  w  Then,  T  2  V B(w) = o J . ww M  assumption  ( i i i ) of Theorem 8.3 i s v i o l a t e d  Theorem 8.5, no s t r i c t tariffs  129 -  P a r e t o improving  ft ft x and t r a n s f e r s g  exist.  The  ( d i f f e r e n t i a l ) . p e r t u r b a t i o n s of  In o t h e r words, s t r i c t  from trade under lump sum compensation international  and, • a c c o r d i n g to  gains  a r e not p o s s i b l e a l t h o u g h  trade under commodity t a x a t i o n would  be s t r i c t l y  gainful.  s u b s t i t u t i o n e f f e c t , which i n the Kemp-Wan example gave e x i s t e n c e of  the s t r i c t  gains from trade under lump sum compensation,  can thus work  adversely. The Kemp-Wan example and the example above demonstrate n e i t h e r the e x i s t e n c e of s t r i c t  gains  t a x a t i o n or the e x i s t e n c e of s t r i c t compensation  from trade under  gains  result  commodity  from trade under lump sum  n e c e s s a r i l y i m p l i e s the o t h e r .  Theorems 4.1 and 8.3, a g e n e r a l  that  Yet, with  the help of  can be e s t a b l i s h e d .  Theorem 9.1: I.  Suppose t h a t the c o n d i t i o n s of Theorem 8.3 a r e s a t i s f i e d so  that s t r i c t if  g a i n s from trade under lump sum compensation  a > 0  L  H  S  strict  t o a [X , gains II.  that  rp L  strict  Then,  T ( i ) rank ( p p + YY ) = N, ( i i ) there Is no s o l u t i o n  i n autarky, rp  exist.  rp  rp  E ] = 0 ^ ,  AT  and ( i i i ) T  9  A  A  T  G(w + T , y ) * 0*,  from trade under commodity t a x a t i o n e x i s t . Suppose t h a t the c o n d i t i o n s of Theorem 4.1 are s a t i s f i e d so g a i n s from trade under commodity t a x a t i o n e x i s t .  Then, i f  -  T 2 ~ T w V B(w) t 0 at the i n i t i a l ww M W  trade  autarky  under lump sura compensation  The  130 -  equilibrium, s t r i c t  gains  exist.  c o n d i t i o n s ( i ) - ( i i i ) i n the f i r s t  part of Theorem 9.1 a r e  the g e n e r a l i z a t i o n s o f the t h r e e Kemp-Wan assumptions ( 9 . 1 ) . Wan suppose that autarky  be  the consumer p r e f e r e n c e s  are u n s a t i a t e d  Kemp and  around the  e q u i l i b r i u m and t h i s assumption i s i m p l i c i t l y p r e s e n t  Theorem 9.1. produced  condition  But the Kemp-Wan requirement  i n autarky  from  that each t r a d e a b l e good must  i n the home c o u n t r y  that the same K p r o d u c t i o n  also i n  seems to t r a n s l a t e to the  sectors  that operate In a u t a r k y , o  operate also a f t e r i n t e r n a t i o n a l trade  has become p o s s i b l e .  Kemp-Wan assumption  production  that  the country's  i s g l o b a l l y smooth and s t r i c t l y i n Theorem 9.1: a r e allowed  kinks  but, a t the autarky  * balance o f trade prohibitive  i n the p r o d u c t i o n  T  frontier  to assumption ( i i i ) possibility  e q u i l i b r i u m , the g r a d i e n t  frontier  o f the net  *  f u n c t i o n b ( w + T ) with  tariffs  possibility  concave g e n e r a l i z e s  and r i d g e s  The  respect  to the trade  must be nonzero.  Assumption ( i ) i n the p a r t I o f Theorem 9.1 i s needed when the numbers of p r o d u c t i o n  s e c t o r s and domestic commodities do not c o i n c i d e  as i n the Kemp-Wan c a s e . local in  The r o l e o f t h i s assumption i s to guarantee  c o n t r o l l a b i l i t y b f domestic goods p r o d u c t i o n  the sense o f D e f i n i t i o n 2.2.  Theorem 9.1 ensures that  Assumption ( i i ) i n the P a r t  some s t r i c t  consumer p r i c e changes ( s t a r t i n g This condition i s s a t i s f i e d  i n the home  Pareto  country  I of  improving d i r e c t i o n s o f  from the autarky  equilibrium)  exist.  i n the model of Kemp and Wan but they do not  -  131  -  c l e a r l y s t a t e i t as a n e c e s s a r y c o n d i t i o n f o r s t r i c t g a i n s from when commodity  taxations  i s used to r e d i s t r i b u t e consumer  income.  The second p a r t of Theorem 9.1 emphasizes the f a c t that g a i n s from trade trade that  under lump sum compensation need not e x i s t  under commodity  taxation i s s t r i c t l y gainful.  the l i k e l i h o o d of t h i s a b n o r m a l i t y i s s m a l l : T  vector w  V  2 B(w) cannot be a zero M - v e c t o r ww  x  *  /  0 ) are not p r o p o r t i o n a l  strict  whenever  I t seems, however,  f o r example, the  i f both the s u b s t i t u t i o n  m a t r i c e s S and Z are of maximal rank, and the i n i t i a l tariffs  trade  equilibrium  to the i n t e r n a t i o n a l p r i c e s  w.  9  - 132  10.  -  PROPORTIONAL REDUCTIONS IN DISTORTIONS AND  SOME PIECEMEAL POLICY  RESULTS  )•  The goal of the analysis in this chapter i s to develop some  examples of s t r i c t Pareto and household s p e c i f i c lump sum instrument.  productivity improving policy changes  transfers are an admissible  whe  policy  The results found are often similar in nature to those  presented in Chapter 7, where lump sum some differences do occur.  transfers were not allowed, but  F i r s t , the existence  of a r b i t r a r y tax  d i s t o r t i o n s in the economy makes i t harder to e s t a b l i s h e x p l i c i t policy recommendations—often to derive a r e s u l t , i t i s necessary to  *  assume either that t  = 0„, s N  *  = 0  ,  W  M  (tax d i s t o r t i o n s at the  initial  equilibrium do not e x i s t ) , or that there i s no substitution in consumption between some or a l l commodities. sum  On the other hand, i f lum  transfer changes are possible, more r e s u l t s , where the changes in  the policy instruments are either proportional variables toward uniformity,  or s h i f t the policy  can be proved.  Let us start by considering  a s h i f t toward the international  * prices w i n the i n i t i a l equilibrium t a r i f f s x—a  policy that was  to be constant u t i l i t y productivity improving in Proposition 8.1. lump sum  transfers may  shown If  be employed, this policy can be converted to a  s t r i c t Pareto improvement.  Theorem  10.1: Suppose that the assumptions of Theorem 8.3  there exists a s t r i c t Pareto and  are s a t i s f i e d .  Then  productivity improving change in the  -  initial x  equilibrium tariffs  133 -  and t r a n s f e r s , and the change i n t a r i f f s  can be chosen to be a movement toward  the world  p r i c e v e c t o r w.  Proof:  * Let  Ax  = rw, where r > 0.  A sufficient  condition for a s t r i c t  ft Pareto  and p r o d u c t i v i t y improving  change i n t a r i f f s  x and  ft t r a n s f e r s g to e x i s t i s :  (10.1)  t h e r e does not e x i s t T  X [A,  -B ] > o g b  A vector X satisfying  rjt  rp  the form X  a vector T  + 1  , X [B , B , B ] = O £ p  the e q u a t i o n s >£.  X e ^H+N+K+l  rp  z  g  Hence, choose k=l.  Using  T  + k + h  , - X B.w > 0.  T T X [B , B , B ] = 0„. , must be o f p z g N+K+H T  1 T T  rp  = k[0 , ( p + e ) , 0 , 1 ] , k e R. H  T T X [A, -B ] > 0 imply k > 0. b H+1 —  ^at  Since  The i n e q u a l i t i e s  T T X A > 0 , k must be p o s i t i v e , H  T T 2 Lemma 8.3, X B^ = w  B(w) .  (The  change  T 2 the s i g n o f the v e c t o r w V B(w) o c c u r s because the m a t r i c e s B ° ww x  in  defined  i n (8.1)  T 2 ~ T w V B(w) * 0 . ww M M  and (4.16) have o p p o s i t e  signs.)  By assumption,  T T 2 ~ I t f o l l o w s t h a t , -X B w = -w V B(w) w < 0 s i n c e m a t r i x ' x ww  9 V^ B(w) i s p o s i t i v e w  semidefinite.  QED  Theorem 10.1 i s a c o u n t e r p a r t o f Theorem 7.1.  A c c o r d i n g to  Theorem 7.1, i f the commodity tax r a t e s ( t , s ) can be a d j u s t e d , a strict  Pareto  and p r o d u c t i v i t y improving  p o l i c y i s to reduce the i n i t i a l  -  ft ft x . In Theorem  (nonnegative) t a r i f f s i s not a r e d u c t i o n gap between  134 -  10.1, the change i n t a r i f f s x  as such, but i t i s a r e d u c t i o n  the domestic and i n t e r n a t i o n a l  tradeables.  Hence, a l s o  i n the d i s t o r t i o n a r y  producer p r i c e s f o r  i t results i n a reduction  o f the home c o u n t r y ' s  trade b a r r i e r .  * A reduction  o f the i n i t i a l  equilibrium  tariffs  x , accompanied by  ft a perturbation and  o f the t r a n s f e r s  productivity  commodity consumer  g , may be shown to be a s t r i c t  improvement, i f e i t h e r  Pareto  t h e r e a r e no d i s t o r t i o n a r y  taxes i n the economy, i . e . , t  = 0^ and s  = 0^, o r i f the  s u b s t i t u t i o n m a t r i x E i s a zero (N+M) x (N+M)-matrix.  Theorem 10.2: Suppose assumptions ( i ) - ( i i i ) of Theorem 8.3 are s a t i s f i e d . Then, i f (a) t * - 0  N  and s* = 0  M  or (b) Z = 0  (  n  +  m  )  x  (  n  +  m  )  ,  there e x i s t s a  ft strict  Pareto and p r o d u c t i v i t y  improving  change i n t a r i f f s  x and  ft transfers  g .  The change i n t a r i f f s may be taken to be a  proportional  reduction. Proof: A sufficient productivity  condition  improvement  where the I n e q u a l i t i e s  * (a) rp  ^  If t  rp  = [0 , (p  profit  £  f o r the e x i s t e n c e o f a s t r i c t  in tariffs  x  and t r a n s f e r s g  X -X B^w > 0 are r e p l a c e d  Pareto and  i s (10.1),  X * by X B^x > 0.  * = 0„, s N  = 0 , and the v e c t o r W  M  X i s o f the form  rp  + e ) , 8 , 1 ] , using  andexpenditure f u n c t i o n s ,  (4.16) and the homogeneity o f the u n i t i t can be seen  that  - 135 -  T  (10.2)  T T  X B = 9 F x  T  *T  + e (S - E ) - x pw qv  *  The v e c t o r s e and 8 are d e f i n e d i n (8.14).  *  A  T  S + (x + s ) ww  Using  these  E . vv  definitions  *  when t  = 0., and s N  (10.3)  A B_ = x x  +  = 0.., M  1  [(-E V + S D vq 12 wp 12  + FD ) F 22  1 0  <-vql  0 0  E D  +  S  D  wp H  vq  +  F  °12  ) (S  wp -  = - * [-( wp " V l l " 12' ^ [-(S - E ) D. - FD* T ] wp 11 12 M T  T  S  T  = -x* V  2  D  F D  V <  "  (S  ww~  W  " V  S  T  B(w) .  WW  The l a s t derive  e q u a t i o n above i s d e r i v e d u s i n g Lemma 8.1.  the q u a d r a t i c form i n (10.3), the f o l l o w i n g  i n v e r s e m a t r i x D have been used: (S  pp  - £ ) D. = qq 11  T  D,.(S - E ) D , „ = - D „ „ , D,, 12 pp qq 12 22 11  T  2  A  -x \TWW(w) x  /VT  J^  N  J^  w  N  J^  G(w + x , y ) f o r any t  JU  l^T  e R  and s  V  2 ~ B(w) equals ww  0  9c  9c  — x V ^ G(w + x , y ) x  the m a t r i x JU  rri  w  e-R .  In t h i s c a s e , X B x  .  WW  A  X B x* = X  < 0.  (b) I f E = 0 ,„, ,„ , the m a t r i x (N+M)x(N+M)  9c*T  T  = -x*V B(w), i t f o l l o w s that w w '  *  *T_9  O  p r o p e r t i e s o f the  D,., D.,(S - E ) D. = 0. 11 11 pp qq 12  S i n c e , u s i n g (10.3), X B » * & > • ' »  v  In order to  =  X  9c  < 0.  QED  L e t us now assume that a l t h o u g h the commodity home country are a r b i t r a r y at the i n i t i a l  t a x r a t e s i n the  equilibrium,  the government i s  -  136 -  *  able It  to a d j u s t  was  them i n a d d i t i o n to the t a r i f f s  x  it  and t r a n s f e r s g .  shown i n Theorem 7.2 t h a t , under c e r t a i n c o n d i t i o n s ,  taxes and t a r i f f s  can be s i m u l t a n e o u s l y  P a r e t o and p r o d u c t i v i t y improvement. sura t r a n s f e r s to be a d m i s s i b l e , reduction  reduced This  commodity  to produce a  strict  r e s u l t d i d not r e q u i r e  lump  but i f they a r e , the simultaneous  i n taxes and t a r i f f s may  be chosen to be a p r o p o r t i o n a l  reduction.  Theorem  10.3: Suppose  satisfied.  that assumptions ( i ) - ( i i i ) of Theorem 8.3 a r e  Then, there  exists a strict  * improving tax  *  *  change i n t , s , x  r a t e s and t a r i f f s  Pareto  and p r o d u c t i v i t y  * and g  , and the change i n the commodity  can be chosen to be a p r o p o r t i o n a l  reduction.  Proof: A condition s u f f i c i e n t and  p r o d u c t i v i t y improving  to imply the e x i s t e n c e  proportional  reduction  of a s t r i c t  i n commodity  taxes and  tariffs i s :  (10.4)  there  * * * * * * * e x i s t s Au , Ap , Az , Ab , Ax , At , As  * such that (4.16) i s s a t i s f i e d ic  -rt  •&  , At  The M o t z k i n d u a l  x  x  = - r t , As  equivalent  Au  x  = - r s , r > 0.  to (10.4) i s :  * and Ag  * »  0 , H U  Ab  * > 0, Ax  Pareto  =  -  (10.5)  there i s no v e c t o r X e R  =  As 0  N+K+H  , X [A, - B ] b  > 0  +  n  +  k  +  R  +  H  and  T X [A, - B ]  > 0  b  T 0 , 1].  For  (10.6)  A [B  this  T H  X, u s i n g  +  1 s u c  R  +  1  ,  shown b e f o r e , the v e c t o r s a t i s f y i n g  T N  0  h  -  137  must be of  X [B , t  the  , B , B ] P z g  >  B ] T  form X  T  X [B ,  , B^] =  p  [0  =  T  , (p  0.  ^  E)  +  T  ,  , B , B ]  T * F T  w  T  expenditure  B  the e q u a t i o n s  E  equations  that X [B  (4.16),  i^2' ^3»  The  T  h  X B^ = 0  qq X  p  +1  I p vq  *  qv  w + S  + w  T  pw  T  X E w + wS vv  ww  T  *  T N  and  the homogeneity of the u n i t  f u n c t i o n s have been used  d e f i n i t i o n o f the v e c t o r  X,  to d e r i v e ( 1 0 . 6 ) .  profit  and  Applying  now  the  - 138 -  (10.7)  A [B , B t  G  ,  ,  B ]  *  T  , * (p +  +  the e q u a t i o n x  been employed.  (10.8)  Applying  A [B , B , B ] i  t  A  S p PP  * T E) S  T  + wS  where a l s o  *  = (p + e)  T  *  pw  *  x  + w  L  l  T  Z  e)  * T ,  w + wS  W  F = e Y derived  Z w qv  T T . * BY (p +  +  T  p wp  T  + (p + e)  WW  from (B5) i n Appendix 2 has  the e q u a t i o n s (B5) i n Appendix 2,  ,  *  T  = (p + e)  ,  *  S p PP  . Ti  *  *  *  T  + (p + e)  .  *  T  + ( p + e ) S x + (p + E) pw ^ r  9c  9c V 1  (p  + e) + x S  T  + wZ  7  r  9c  T  T  (Z - S ) qq pp L  9c V  T  p* + w Z w wp vv r  *  i  + w S  x WW  T  *  T  [(p* + s ) , w ] (Z - S) p  T  2  -w V B(w)w, ww  9c  (p + e) + p S (p + e) wp pp  ( p * + e) + w S  vq  Z w qv  + e w  - 139  using -w  T  V  Lemma 8.1.  2 B(w)w ww  Since,  must be  tax  and  starting  tariff  reductions financed  assumption w  negative.  Theorem 10.3 satisfied,  by  implies  t h a t , i f the  from an  initial  structures,  lump sum  T 2 V B(w) ww  T * 0 , M  there  taxation.  assumptions of  coincides points  D.  first  best  I f the  initial  with t  to  the  T  be  commodity  government i s an  * = 0. , N  the  autarky  *  s  = 0  W  and  x  =  M i n F i g . 3 by  0 M W  the  assumptions of Theorem 10.3 shifted  to t o t a l  t e c h n o l o g y T + ET k  commodity tax and  defined  tariff  are  production i n (4.23) by  * p r o p o r t i o n a l l y reducing  are  tariff  equilibrium  e q u i l i b r i u m depicted  the domestic economy can  theorem  to a n o n d i s t o r t i o n a r y  In other words, i f the  e f f i c i e n c y with respect  the  e x i s t s a path of tax and  f i n a l equilibrium  with the  C and  satisfied,  the  scalar  equilibrium with a r b i t r a r y  * equilibrium,  the  W  QED  which, i n the l i m i t , l e a d s  by  -  rates  *  t , s  * and  x .  *  P r o p o r t i o n a l r e d u c t i o n s i n the i n i t i a l commodity tax r a t e s t ft ft and/or s w i t h o u t changes i n the t a r i f f s x w i l l a l s o u s u a l l y be s t r i c t P a r e t o and  p r o d u c t i v i t y improving.  H e s s i a n m a t r i x has the  constant  to be d e f i n e d .  utility  tax  adjusted  To develop p r e c i s e  results, a  C o n s i d e r problem (8.6) b a l a n c e of  trade  which  new  defines  function  ft ft ft ft B(w, the  w+x,w+x zero p r o f i t  and  These e q u a t i o n s are  + s , t ) .  The  first  order conditions  s u p p l y equals demand equations (2.9) assumed  to be  satisfied  at the  f o r (8.6) and  initial  are  (4.12).  equilibrium  - 140 -  of  the economy and they can thus be used ft  /  domestic p r i c e and i n d u s t r y  to d e f i n e  scale vectors  the e q u i l i b r i u m ft  p  and z  as i m p l i c i t  ft ft ft functions  o f the v a r i a b l e s w, x , t  and s .  e q u a t i o n s (2.9) and (4.12) a t the i n i t i a l  D i f f e r e n t i a t i n g the  equilibrium:  * S - £ pp qq  (10.9)  Y  Ap Az  T  AT  +  0KxK  Y  qq  E - S qv pw  *  At  +  As qv  KxN  KxM  E q u a t i o n s (10.9) determine the d e r i v a t i v e s o f the i m p l i c i t  functions  ft ftftft ft ( w + T , t , s )  p (w + x , t , s ) and z  with respect  to the exogenous  ft ft ft commodity  tax r a t e s ( t , s ) and the t a r i f f s  x  at the i n i t i a l  equilibrium.  ft ft ft ft Differentiate respect  (10.10)  * * * to x , t and s .  V B = T  (10.11)  the f u n c t i o n  T  *  E  h=l  n  h  *h  V m (u  + s , t )  (8.6),  K k * * * k E V TT (p , w + x ) z k=l  V B = x  Using  B(w, w + x , w + x  *  , q , v ),  ^ E h=l  h * h * * V m (u , q , v ), T  with  - 141  (10.12)  h  h  V B = - E V m ( u * , q* , v*) . h=l S  S  The second order  p a r t i a l d e r i v a t i v e s of the f u n c t i o n B w i t h r e s p e c t  ie  T  ie  to  ie  the  tariffs  can  be c a l c u l a t e d by d i f f e r e n t i a t i n g  using  -  and tax r a t e s ( t , s ) at the i n i t i a l  equilibrium  e q u a t i o n s (10.10) - (10.12) and by  (10.9) :  (10.13)  V B =  V  2  TT  B, V  2 V" B,  tT ' V  2  ST  B, V  S wp  2  B, V  2  Tt '  TS  2  2  B,  tt ' 2  St  B  B  B, V  2  SS  B  -E  S wp  , 0 NxK  -E  , 0. MxK  -E  vq  , -E qq  -E  w  qq -E  - E , -E , -E vv vq vv  qv'  tS  - E , F vq  -E  S ww  vq  qv , -E  w  - E , F vq  qq  , 0. NxK  vq  , 0. MxK  The i n v e r s e m a t r i x D i n (10.13) i s d e f i n i t e d i n ( 8 . 5 ) .  Since  B is a  o  convex f u n c t i o n of i t s arguments, the m a t r i x  V B i s positive  semidefinite. C o n s i d e r now,  f o r example, a p r o p o r t i o n a l  reduction  i n the  * tradeables  taxes s .  perturbation vector  One o f the s u f f i c i e n t  to be s t r i c t  conditions  for this  Pareto and p r o d u c t i v i t y improving  *T 2 * T s V B i s nonzero, i . e . , s ss  7  T  v B * 0 : ss M W  policy  i s that the  the i n i t i a l  v e c t o r of  -  tradeables  taxes s  142  must not be a zero  -  eigenvector  of the m a t r i x  v B. ss  0  Theorem  10.4: Suppose  T  that ( i ) rank Y = K < N and ( i i ) rank (S - £ + — pp qq  = N at the i n i t i a l  equilibrium.  YY )  Then,  * T 2 T I f T = C- and w V B(w) * 0... t h e r e e x i s t s a s t r i c t N ww M Pareto and p r o d u c t i v i t y improving change i n the i n i t i a l commodity taxes (I)  T  ft ft ft ft ft*  (t  , s ) and t r a n s f e r s g .  The change i n ( t , s ) may  chosen to be a p r o p o r t i o n a l (II) ftrp  and  s  T  2  V B ss  M  * oJ or i f £ M —  qv  = 0„ NxM M  J  ss  improving  reduction.  I f t * = 0 „ , T* = 0 and s * N' M  2  V  be  B * 0 , there M' W  exists a strict  change i n the i n i t i a l  change i n the t r a d e a b l e s  Pareto and p r o d u c t i v i t y J  tax r a t e s s* and t r a n s f e r s g*.  The  taxes s* may be taken to be a p r o p o r t i o n a l  reduction. (III) ftrj,  t  B t  T  V  2  B * tt  C-3 N  or £ = 0„ „ and qv NxM  Y  2  ^  I f s* = 0 „ , T* = 0^ and t * M M  t  *  there  change i n the i n i t i a l  exists a strict  Pareto  and p r o d u c t i v i t y improving  ft ft tax r a t e s t and t r a n s f e r s g .  The change i n  ft t  may be chosen to be a p r o p o r t i o n a l  The proof Appendix 2. the commodity  of Theorem 10.4  The s u f f i c i e n t tax r a t e s t  i s rather  conditions and/or s  p r o d u c t i v i t y improving g i v e n not much i s r e q u i r e d  reduction.  and hence d e f e r r e d  for proportional  to be s t r i c t  i n Theorem  concerning  tedious  to  reductions i n  Pareto and  10.4 a r e r e s t r i c t i v e :  the s u b s t i t u t i o n m a t r i c e s  although  S and £ ( t h e  - 143 -  vectors  w  r p 2 ~  7 B(w), s ww  * T 2  V B and t ss  *T 2 V B depend on the m a t r i c e s ss  S and  * E) , i n t e r n a t i o n a l trade  * of  i s assumed  to be f r e e , i . e . , x  *  the tax r a t e s ( t , s ) are a l s o assumed  these c o n d i t i o n s  = 0^, and some  to be i n i t i a l l y z e r o .  a r e not s a t i s f i e d , a p r o p o r t i o n a l  reduction  and/or s*, even i f accompanied  with a change i n household  t r a n s f e r s , may cause a w e l f a r e  and p r o d u c t i v i t y l o s s .  If  of t *  specific  Theorem 10.5: T  Suppose ( i ) rank Y = K < N, ( i i ) rank (S - £ + Y Y ) = N, and — pp qq (iii)  the i n i t i a l  (10.14)  equilibrium  tariffs  x* a r e such that  V b*(w + x*) (w + x*) < 0. x  Then, there  exists a strict  P a r e t o and p r o d u c t i v i t y deproving change o f  commodity tax r a t e s ( t * , s*) and lump sum t r a n s f e r s g*, where the change i n ( t * , s*) i s a p r o p o r t i o n a l  reduction.  Proof: A sufficient A  condition f o r a s t r i c t A  deprovement i n ( t , s ) and g a proportional  (10.15)  Pareto and p r o d u c t i v i t y  A  A  A  to e x i s t , when the change i n ( t , s ) i s  reduction, i s :  t h e r e does not e x i s t a v e c t o r  X e H+N+K+1 R  g  u  c  h  t  h  a  t  T  A [A,  -  B ]  X [B ,  0  B]  t  A vector  <  B  +  1  X  >  >  s  satisfying  ;  the  T  [B  -  144  ,  B ,  P  Hp  *TI  form X  T X [A, - B^] Theorem  (p  rrt  +  =  G  0  T  +  K  +  H  ,  0.  T e q u a t i o n s X [B  JL  = k[0„, bl  B ]  Z  p the  -  e)  T , B , B ] = 0„ ' , 'must be z g N+K+H  of  frt  , 9 , 1], k  e R.  The i n e q u a l i t i e s  T < 0  H  i m p l y k < 0;  choose k = -1.  Then, u s i n g  the  proof  of  10.4,  T O  X [B , B ] i  (10.16)  = w  (;  *  V  B(w)  (w +  T ),  WW  U s i n g (8.10) and  A  X [B ,  (10.17)  T  (10.14),  = V^b  G  Theorem 10.5 initial  Je  B ]  shows t h a t , under v e r y  t a r i f f vector  reduction P a r e t o and  Jc  x  i s a r b i t r a r y and  of the commodity tax  &  (w + T )(w + x ) < 0.  r e a s o n a b l e c o n d i t i o n s , i f the nonzero, a  r a t e s ( t , s ) may  p r o d u c t i v i t y deprovement i n s p i t e of the  t r a n s f e r s can  be  QED  proportional  lead fact  used to r e d i s t r i b u t e consumer income:  to a  strict  that lump i t i s only  sum  -  required  that  the  it  can  -  producer p r i c e weighted net  A  derivatives  145  b a l a n c e of  A  b (w + x ), m=l,...,M, sum ra strong  conditions  to a n e g a t i v e  If  the  be  shown that movements toward u n i f o r m i t y  taxation)  trade  of Theorem 10.4  i n p o s i t i v e commodity taxes a r e  are  number,  slightly  modified,  ( a t a lower l e v e l  s t r i c t P a r e t o and  of  productivity  improving.  Proposition  10.1:  Suppose that that  the c o n d i t i o n s  by:  w  T 3  strict  ww  B(w)  P a r e t o and  a n d  g  The  AT  ss  , (II) s  »  0  10.1,  t  are  2  V  ss  satisfied,  B and  V  2 tt  B are  , and  except replaced  T  9  V^B tt  < 0... N  Then, there '  changes i n (I) ( t , s ) A  (III) t  exist »  A  »  0.  T  and g  , where  the  i n the commodity tax v a r i a b l e s are chosen to be movements  sufficient  10.1  of  taxation.  i s given  conditions  the r e d u c t i o n  consumer c l o s e d  a s i m i l a r kind  i n Appendix  for s t r i c t  in P r o p o s i t i o n  w e l l known r e s u l t of H a t t a (1977a).  In a one  AT  B < 0„, M'  and g  U  improvements to e x i s t g i v e n  conditions,  B(w),  A  of P r o p o s i t i o n  The  2 ~ ww  p r o d u c t i v i t y improving  a lower uniform l e v e l  proof  V  T  9  v  A  perturbations toward  the m a t r i c e s  < Of., s M  A  C-W™  on T  9  V  Assumptions of Theorem 10.4  Hatta  of the h i g h e s t  P a r e t o and 10.1  are  productivity  c l o s e l y r e l a t e d to a  showed t h a t , under c e r t a i n  tax  economy i s w e l f a r e  2.  r a t e (on improving.  a domestic good) In  Proposition  of p o l i c y recommendation r e s u l t i s achieved  open economy without r e s t r i c t i n g  i n an  the numbers of consumers, producers  and  - 146  commodities 2 V  tt  -  (except that K_<  N).  The assumptions about the m a t r i c e s  B in Proposition  10.1  a r e g e n e r a l i z a t i o n s o f the Hatta °  2 B and V  ss  assumption  that  substitutable  the good  with the h i g h e s t  ( d o m e s t i c ) tax rate must be  f o r a l l o t h e r ( d o m e s t i c ) commodities  i n the economy.  This  JL  can be seen as f o l l o w s :  suppose, f o r example, *T  using t  *T  (E  (10.13), the c o n d i t i o n t qq  + E  " T D, , £ ) > 0 .For qq 11 qq N M  off-diagonal i.e.,  2 ^  commodities must  q q  t  ^  *  s  0 . N  equivalent  to  must be s u f f i c i e n t l y  i n consumption  positive,  of the domestic  dominate.  Hatta r e s u l t  i s not s t r a i g h t f o r w a r d  and  the above  s i n c e H a t t a used ad  valorem (not s p e c i f i c ) commodity tax r a t e s i n h i s model. one o f the assumptions condition:  Then,  to be p o s i t i v e , the ~ '  Y e t , the c o n n e c t i o n between P r o p o s i t i o n 10.1 mentioned  »  T B  t t  the l a t t e r v e c t o r  terms i n the m a t r i x E  i n a sense, s u b s t i t u t i o n  that  the sum  used i n h i s theorem  was  Furthermore,  the H a t t a n o r m a l i t y  of the consumer demand income d e r i v a t i v e s weighted 3  by the c o r r e s p o n d i n g producer p r i c e s must be p o s i t i v e . s u p p o s i t i o n was recommendation  needed  i n P r o p o s i t i o n 10.1  model employed  by H a t t a i s examined  L e t us f i r s t  such  or i n the p o l i c y  r e s u l t s o f the p r e v i o u s c h a p t e r s .  the d i f f e r e n c e s between P r o p o s i t i o n 10.1  No  and  In order to r e s o l v e  the r e s u l t s o f H a t t a , the  in detail.  w r i t e the Hatta model i n the form (4.10) - (4.13).  The home c o u n t r y i s assumed  to be c l o s e d  with N+l  domestic  commodities.  There are N p r o d u c e r s , each one o f whom s u p p l i e s one o f the N commodities, indexed by n e [1,...,N], a fixed  The  r e s o u r c e which i s used as an input  commodities.  The f i x e d  [N + l ]  t  h  domestic good i s  to produce the other domestic  r e s o u r c e s e r v e s as the numeraire commodity i n  -  the model w i t h a p r i c e denoted regarded  transfers  on  the N domestic  -  by w = 1 (hence,  as a " t r a d e a b l e " good).  taxes t £  147  The  the f i x e d  government imposes ad  commodities and  g i v e s lump  g e R to the s i n g l e consumer i n the c o u n t r y .  equilibrium  o f the H a t t a economy i s c h a r a c t e r i z e d  (10.18)  m(u  (10.19)  ir (l,  (10.20)  V m(u*,  (10.21)  N Z n=l  (10.22)  q* =  factor is  by  valorem sum  The  initial  the e q u a t i o n s :  , 1, q ) = g ,  n  p ) =' P  q  ~  a n  = 0. n=l,...,N,  1, q*) = y* ,  V TT (1, p ) y  [1 + t*]  =  N Z ay n=l  = r = V m( u  , 1, q ) ,  p*.  ft In  (10.22),  the m a t r i x  [1 + t ] i s an (N x N)-matrix  with d i a g o n a l  * elements  equal  to (1 + t ) , n=l,...,N. n  The  o t h e r elements  of the  ft matrix  [1 + t ] are z e r o . According  domestic  to (10.18),  the consumer's e x p e n d i t u r e s on  commodities minus h i s revenue from  selling  the N  the f i x e d  factor  - 148 -  equal  h i s lump sum  income g .  The producers' u n i t p r o f i t  a r e of the form (10.19), where the number input  coefficient  fixed  f a c t o r as an i n p u t .  clearing supply  f o r producing  conditions:  i s the n  one u n i t of the n  t n  t n  functions  sector's  output u s i n g  the  The e q u a t i o n s (10.20) - (10.21) are market  consumer demand f o r the N commodities equals  y , and the producer demand f o r the f a c t o r equals  their  i t s fixed  k  supply  r.  The i n d u s t r y output l e v e l s y  f o r the N p r o d u c e r s .  serve  as the s c a l e v a r i a b l e s  T h i s c h o i c e o f the p r o d u c t i o n  because each p r o d u c t i o n  scale i s permissible  s e c t o r s u p p l i e s o n l y one o u t p u t .  The 1+N+N+l e q u a t i o n s (10.18) - (10.21) determine endogenously k  the e q u i l i b r i u m u t i l i t y  level  of the consumer  k  k  p r i c e s p , the e q u i l i b r i u m output l e v e l s y instruments, g*.  u , the e q u i l i b r i u m  given  the exogenous v a r i a b l e s a  and one of the p o l i c y = ( a ,...,a^), r , t  D i f f e r e n t i a t i o n o f (10.18) - (10.21) at the i n i t i a l  (which s o l v e s  (10.23)  (10.18) - (10.21)),  A Au* = B  Ap* + B P  —  where A =  y  yields:  Ay* + B At* + B Ag* • ' t g 6  —  1  -x [i A  + t ]  CL N E qu 0  -E [1 + t ] qq  , B = ' y  T N °NxN I., N T -a  and  equilibrium  - 149  -  B = g  The  1  n o r m a l i t y c o n d i t i o n g i v e n i n Hatta (1977a) i s a  E  >  0.  qu Using  the e q u a t i o n (10.19),  (10.24)  It  E  P  >  this  c o n d i t i o n can be w r i t t e n i n the  form  0.  qu  i s easy to see  t h a t (10.24) i s s a t i s f i e d ,  n=l,...,N, are normal, i . e . , i f a l l  i f a l l commodities  elements of the v e c t o r E q  n, are  U  positive. The  f o l l o w i n g theorem shows the c o n n e c t i o n between the  n o r m a l i t y assumption  Theorem  and  the r e s u l t s  o f the p r e v i o u s  Hatta  chapters.  10.6; If  equations transfer g  the i n i t i a l  equilibrium  i n the economy that s a t i s f i e s  (10.18) - (10.22) i s a g-optimum with r e s p e c t to the  the initial  , the Hatta n o r m a l i t y c o n d i t i o n (10.24) i s s a t i s f i e d .  -  150 -  Proof: Suppose the i n i t i a l * transfer g .  the  equilibrium  Then, there  I+M+M+I  T  R  such that  exists a vector  = 1>0,  X A = B  H = 1 , B can be s e t to equal 1.) X^ = 0.  i s a 8-optimum with r e s p e c t to X T T T T X = [X^, X^, X^, X^]  T  e  T  X [B , B , B ] = 0„ .. P y S N+N+l T  (Since  L X TL  The e q u a t i o n s  XB  =0  imply  that  Hence,  (10.25)  [X*, X*, xj]  N  = 0 N+N  NxN  -E  [i + t ] qq  o.N  It  follows  t h a t X^ = X. a  3  4  T  -a  and X^ - \li E [1 + t*] = 0^. 2 3 qq N  X T T T * Then, X = ka , X = ka E [1 + t ] . 3 2 qq  k £ R.  T becomes ka E =1. qu  Hence, k * 0.  t h a t , i n f a c t , k > 0.  Set X. = k,  4  The e q u a l i t y J  T X A =1  In a d d i t i o n , i t can be i n f e r r e d '  ( T h i s can be seen by r e p l a c i n g  the e q u a l i t y  (10.21) with a weak i n e q u a l i t y (<). A l t h o u g h , a t the i n i t i a l (10.21) i s s a t i s f i e d  equilibrium,  p o s s i b i l i t y of f r e e d i s p o s a l nonnegativity  constraint  f o r the f i x e d  on the l a s t  when the Kuhn-Tucker c o n d i t i o n s X £  1  +  N  R  +  N  +  1 >  a  r  e  derived  as an e q u a l i t y , a l l o w i n g the f a c t o r r imposes a  component  o f the v e c t o r X  T T X A = 1, X [B , B , B ] = 0„ „ , , p z' g N+N+l'  f o r the problem maxfu:  (10.18) -  rn  hold}).  S e t t i n g k = 1, X A = 1 (> 0) i s e q u i v a l e n t  (10.21)  rrt  to a  E  q  u  = 1 > 0.  - 151  Thus, u s i n g a (10.19), satisfied.  -  the Hatta n o r m a l i t y c o n d i t i o n (10.24) i s  QED  Why  i s i t that n o r m a l i t y of goods i n demand and  the i n i t i a l  equilibrium  are connected?  c o n s i d e r F i g u r e 7 presented  by H a t t a  There are two goods,  In order  to answer the q u e s t i o n ,  (1977a).  X £ , i n the economy.  and  d e f i n e s the c o u n t r y ' s p r o d u c t i o n p o s s i b i l i t y  frontier  given  initial  the amount of the f i x e d  S - o p t i m a l i t y of  factor r .  The  The  f o r x^ and equilibrium  economy i s at x^ where the consumer a t t a i n s the u t i l i t y the good X£ i s taxed  At x^,  at a h i g h e r l e v e l  Suppose the tax r a t e f o r X2 i s reduced equilibrium decomposed original  shifts to two  to x^. parts:  The  (income consumption) c u r v e . production move from  possibility x^ to x^  change from  first,  i n d i f f e r e n c e curve  to x 4  and  than  line  level  the good  PP' X2 f o r the  xP.  x^.  as a consequence, the x^  to x^  may  be  the consumer i s moved along h i s *  and  then, up  i to x 1  along h i s Engel  If the consumer's Engel curve c u t s the  f r o n t i e r PP'  from  must be a w e l f a r e  the  inside  to the o u t s i d e , the  improvement.  When does the Engel curve have the form  required?  Consider  the  * T  Hatta n o r m a l i t y ' c o n d i t i o n . consumer's t r a n s f e r i.e., the  If p  income r a i s e s  E  qu  > 0,  an  i n c r e a s e i n the  the v a l u e of h i s t o t a l  h i s E n g e l curve must cut the p r o d u c t i o n p o s s i b i l i t y inside  to the o u t s i d e .  consumption, frontier  Hence, the n o r m a l i t y c o n d i t i o n i n the  recommendation r e s u l t of H a t t a (1977a) i s needed  to guarantee  from policy  that a  p o l i c y change ( h e r e , a r e d u c t i o n of the h i g h e s t commodity tax r a t e ) i s welfare  improving.  -  x  152  -  2  F i g u r e 7 - N o r m a l i t y of Commodities and the E f f e c t s o f a P o l i c y  Change.  -  If,  153  -  on the o t h e r hand, the i n i t i a l  a B-optimum w i t h r e s p e c t consumer's Engel  e q u i l i b r i u m i n the economy i s  to the t r a n s f e r g*, i t can be seen t h a t the  curve must cut the p r o d u c t i o n  the i n s i d e to the o u t s i d e .  I f t h i s were not  possibility frontier  the c a s e , the  from  initial  e q u i l i b r i u m c o u l d not be a welfare-maximum: i t would be p o s s i b l e to reduce the consumer's lump sum a new  income and move along h i s E n g e l curve  e q u i l i b r i u m w i t h a h i g h e r l e v e l of consumer w e l f a r e .  The  to  shift  would a l s o be p r o d u c t i o n a l l y f e a s i b l e because the change would l e a d to an e q u i l i b r i u m i n s i d e the economy's p r o d u c t i o n p o s s i b i l i t y s e t . s i t u a t i o n i s depicted  i n F i g u r e 8.  The  p o i n t x^ i n F i g . 8 cannot be *  B-optimum w i t h r e s p e c t It is  can  present  5  to the t r a n s f e r g .  thus be concluded  t h a t the H a t t a n o r m a l i t y c o n d i t i o n which  e q u i l i b r i u m of the H a t t a economy i s a w e l f a r e maximum w i t h  to the lump sum  transfer g .  But, as can be r e c a l l e d ,  e a r l i e r p o l i c y r e s u l t s given i n t h i s chapter i n i t i a l e q u i l i b r i u m o f the economy b e i n g v e c t o r of t r a n s f e r s g*. understood? without  any  commodities? in  a  i n many e a r l i e r p o l i c y recommendation r e s u l t s i s s a t i s f i e d i f  the i n i t i a l respect  The  How  can  How  were not c o n d i t i o n a l on  the  a 8-optimum w i t h r e s p e c t to  the  a r e the e a r l i e r r e s u l t s then to be  they i m p l y t h a t some w e l f a r e improvements do  n o r m a l i t y c o n d i t i o n imposed on the d o m e s t i c or The  the  answer i s r a t h e r s i m p l e .  exist  tradeable  I f the n o r m a l i t y c o n d i t i o n  the H a t t a economy i s v i o l a t e d , the i n i t i a l e q u i l i b r i u m cannot be  8-optimum w i t h r e s p e c t to the lump sum some s t r i c t P a r e t o e x i s t s , without  transfer g .  a  T h i s means t h a t  Improving p e r t u r b a t i o n o n l y i n the t r a n s f e r g*  a change i n the commodity t a x r a t e s t .  However, once  - 154  x  -  2  i  Figure 8 - B-Optimality  and  the H a t t a  Normality  Condition.  - 155 -  all  such Pareto  resulting  Improvement  p o s s i b i l i t i e s have been exhausted, the  e q u i l i b r i u m i s a B-optimura  w i t h r e s p e c t to g , the Hatta  n o r m a l i t y c o n d i t i o n i s s a t i s f i e d , and then, f u r t h e r Pareto can be a t t a i n e d by p e r t u r b i n g some o f the other p o l i c y (commodity taxes) together with the lump sum earlier  theorems  instruments  t r a n s f e r g , as the  demonstrate.  The H a t t a model (10.18) - (10.22) i s a v e r y much v e r s i o n of the model (4.10) - ( 4 . 1 3 ) . appropriate (10.24). +  simplified  Hence, the n o r m a l i t y c o n d i t i o n  f o r the g e n e r a l model i s not the simple Hatta  I t was  X s ^ N K+l +  R  a s s o c  improvements  shown i n the proof o f Theorem 8.1 t h a t  xated  assumption  the v e c t o r  with a B-optimum with r e s p e c t to the lump  transfers g  i n the economy d e s c r i b e d by (4.10) - (4.13) must  proportional  T to the v e c t o r X =  T [0 , (p  T inequalities  X  be  T + e) , 9 , 1], where the  H  v e c t o r s e and 9 a r e d e f i n e d i n ( 8 . 1 4 ) .  sum  For t h i s v e c t o r X, the  T A > 0  t r a n s l a t e to H  (10.29)  (p* + e )  T  £  + w qu  T  I > 0* . vu H  which i s the g e n e r a l i z e d H a t t a n o r m a l i t y , c o n d i t i o n - (4.13).  I f (10.29) i s observed  to be v i o l a t e d , the  e q u i l i b r i u m o f the economy i s not a B-optimum transfers g  f o r the model  (4.10)  initial  with r e s p e c t to the  and s t r i c t P a r e t o and p r o d u c t i v i t y improving changes i n *  only transfers g  exist.  The c o n d i t i o n (10.29) d i f f e r s (10.24) i n that  from  the Hatta n o r m a l i t y c o n d i t i o n  the t r a d e a b l e s income d e r i v a t i v e s  E  v u  (weighted  by the  -  international  p r i c e s w)  income d e r i v a t i v e s E q instead to be to  initial  initial  transfers g*.  6  lump sum  i n s t r u m e n t , the  (H+N+K+l) v e c t o r  the  equations X  generalized  T  [B  p  X  economy are  +  +  e)  e) can be  shown respect  normal.  f e a s i b l e government p o l i c y  of Lagrange m u l t i p l i e r s a s s o c i a t e d  rp  =  r a t e s ( t * , s*) rp  [X^, (p  +  6)  T  , y ,  by  is  1 ] , where X s o l v e s  In t h i s c a s e , the '  T  c o n d i t i o n , implied  with  rp  T , B , B . B ] = 0. , . z t' s N+K+N+M  normality  the domestic  B-optimura, (10.29) i s  to the commodity tax  to the v e c t o r  Also  shadow p r i c e s (p  p r i c e s (p  not a  rn  proportional  the  Hence, at a  t r a n s f e r s are  B-optimum with r e s p e c t  formula.  e q u i l i b r i u m i s a 8-optimum with  i f a l l commodities i n the  If  the  are weighted by  u  nonnegative i f the  satisfied  a  are added to  ft ft producer p r i c e s p . The  of the  the  -  156  the  inequalities X  T  T A > 0 , H  is  (10.30)  If  X? + 1  (p* +  5)  T  E  qu  + w  T  E  vu  >  (10.30) i s v i o l a t e d , s t r i c t P a r e t o and  * perturbations Let  i n only  us now  oL H  p r o d u c t i v i t y improving  *  (t , s ) e x i s t .  return  to c o n s i d e r  Proposition  (1977a) r e s u l t a c c o r d i n g  to which a r e d u c t i o n  tax  consumer economy i s w e l f a r e  rate  in a closed  one  of  10.1 the  and  the  highest  Hatta domestic  improving.  - 157 -  Theorem 10.7:  ( H a t t a (1977): Theorem  Suppose that  1)  the model (10.18) - (10.22) i s used  an economy where the tax r a t e t  (»  0 ) , n e [1,...,N], on the n  * domestic commodity i s the h i g h e s t o f a l l tax r a t e s t . domestic sense  commodity  i s substitutable  to c h a r a c t e r i z e  Then, i f the n  f o r a l l o t h e r domestic goods i n the  that  T  (10.31)  where (E  p* (£  qq  ^ [p ])  [p*])  «n  > 0,  denotes  the n  th  ^ column of the (N x N ) - m a t r i x E [p ] qq  * there e x i s t s a s t r i c t  w e l f a r e improving change o f the tax r a t e t ^ ,  * n e [1,...,N], and t r a n s f e r g . reduction  toward  * The change In t ^ may  be taken to be a  the l e v e l o f the next h i g h e s t domestic  tax r a t e  * t  t  , ( ^ 0 ) , n' e [ l , . . . , n - l ,  n+l,...,N].  Proof: A sufficient above form to e x i s t  (10.32)  condition  for a s t r i c t  i n the Hatta economy i s :  there i s no v e c t o r X e  AT[  w e l f a r e improvement o f the  V V V  =  °5« l» +  1  +  N  +  N  +  1  R  s u c  ^V'n  h  h  t  h t  X  a  > °'  W  h  e  T  r  A > 0,  e  -  Using  the proof  OXT.,,.1  =  N+K+l  a  T  158 -  A satisfying  of Theorem 10.6, the v e c t o r  A = [ 0 , a  and A A > 0 must be of the form  T A (  , 1]... Hence, i t s u f f i c e s to show that  B t  ).  E  qq  < 0.  n  T A [B , B , B ] p y © [1 + t ] ,  Using  (10.23),  (10.19) and (10.31) ,  A (BJ = - a ( Z [ p * ] ) < 0. t •n q q ^ « n T  (10.33)  T  QED  In h i s v e r s i o n of Theorem 10.7, H a t t a assumes that condition  (10.24) i s s a t i s f i e d .  This  condition  holds,  the n o r m a l i t y  i f the i n i t i a l  ft equilibrium  i s a 3-optimura w i t h r e s p e c t  i s not the c a s e , then there g* ( A t * = 0, n = l , . . . ,N) . n  to the t r a n s f e r g .  a r e some w e l f a r e  Theorem 10.7 can be regarded  improving  highest  domestic  changes i n o n l y  as a s p e c i a l case o f P r o p o s i t i o n  10.1, s i n c e a movement toward uniform domestic the  If this  tax r a t e , i f a l l other  t a x a t i o n means  reducing  domestic commodity tax r a t e s  equal some common r a t e t (>_ 0 ) . The apparent d i s s i m i l a r i t y o f the *T 2 * 7 c o n d i t i o n t (^tt ^«n ^ ^ * P r o p o s i t i 10.1 and the c o n d i t i o n B  *X  n  0 1 1  *  p  (Z  of  the d i f f e r e n c e between s p e c i f i c  q q  [p  derives  ]).  n  > 0 i n Theorem 10.7 can be e x p l a i n e d  an e x p r e s s i o n  perturbation the w e l f a r e  and ad valorem  f o r the consumer w e l f a r e  i n the economy's commodity change depends on the s i g n s  by t a k i n g  taxation:  account Hatta  change caused by a  tax s t r u c t u r e , and the s i g n o f of the v e c t o r  p*^ E  q q  [p*];  8  - 159 -  Chen, using  the homogeneity c o n d i t i o n  a t an a l t e r n a t i v e  form f o r the v e c t o r  EqqU + t ] p  p  E  = 0 , Hatta N  which can be  arrives  interpreted  qq to  imply the s u b s t i t u t a b i l i t y  of the n  other domestic goods, as assumed  t n  domestic commodity  by H a t t a i n h i s theorem.  9  for a l l  - 160 -  11.  COMMODITY TAXATION AS A CAUSE OF INTERNATIONAL TRADE  In  a paper on t h e r e l a t i o n s h i p  international is  1  trade M e l v i n  caused by a consumption  between commodity  g i v e s - an i n t e r e s t i n g tax, the country  t a x a t i o n and  statement:  imposing  "When  t h e t a x may be made  w o r s e o f f b y t r a d e so t h a t a p r o h i b i t i v e  tariff  Two q u e s t i o n s  how d o e s a c o m m o d i t y t a x  "cause"  immediately  t r a d e , and s e c o n d l y  change i n t h e c o u n t r y ' s In  order  equilibrium.  tariffs  T* a r e t h o s e  with respect  first,  commodity t a x s t r u c t u r e ,  The a u t a r k y  (international  d e f i n e d i n (5.1).  to the i n i t i a l  are perturbed  welfare  by a  reducing?  trade  The t a r i f f s  prohibitive)  x* a r e d e f i n e d  e q u i l i b r i u m c o m m o d i t y t a x r a t e s and t h e  from t h e i r  s i m u l t a n e o u s l y changing  t r a d e , induced  p r o b l e m , c o n s i d e r a n economy a t a n  t r a n s f e r s i n t h e home c o u n t r y .  variables  w o u l d be a p p r o p r i a t e . "  when i s i n t e r n a t i o n a l  to answer t h e f i r s t  autarky  initial  arise:  trade  Hence, i f any o f these  autarky l e v e l s ,  the t a r i f f s  policy  without  x*, t h e r e l a t i v e  producer  and  c o n s u m e r p r i c e s i n t h e economy c h a n g e , c r e a t i n g a p o s s i b i l i t y f o r international tax  trade; adjustments  i n t h e home c o u n t r y ' s  o r t r a n s f e r v e c t o r s cause i n t e r n a t i o n a l  the autarky p r e s e r v i n g trade b a r r i e r Whether or  the trade  transfers i s strict  t r a n s f e r s g*•  changing  x .  and p r o d u c t i v i t y  commodity  improving  taxes  depends on the  c o m m o d i t y t a x r a t e s ( t , s ) and  I f , f o r e x a m p l e , ( t * , s*) a r e B - o p t i m a l  (with  * respect  commodity  trade by i n d i r e c t l y  i n d u c i n g change i n the a u t a r k y  Pareto  properties of the i n i t i a l  autarky  to the c o n s t r a i n t s :  (4.10) - (4.13) h o l d , x = x  * and g = g ) ,  -  no  161 -  p e r t u r b a t i o n of o n l y ( t , s ) (which would cause  t r a d e ) can be s t r i c t initial  Pareto o r w e l f a r e i m p r o v i n g .  autarky e q u i l i b r i u m  international S i m i l a r l y , i f the  i s a S-optimum w i t h r e s p e c t to the t r a n s f e r s  ft g  (assuming that  the home c o u n t r y ' s commodity tax s t r u c t u r e i s  ft f i x e d ) , no change o f the t r a n s f e r s g w e l f a r e improving. *  alone can be s t r i c t  On the other hand, i f the i n i t i a l  Pareto and  commodity t a x  *  r a t e s ( t , s ) a r e a d j u s t a b l e but a r b i t r a r y i n a u t a r k y , s t r i c t and  p r o d u c t i v i t y improving p e r t u r b a t i o n s i n them e x i s t ;  Pareto  t r a d e caused by  these changes i n the home c o u n t r y ' s commodity t a x s t r u c t u r e i s w e l f a r e improving.  Theorem 11.1: Suppose that i n autarky ( i ) rank Y = K _< N, ( i i ) rank (S  T  + Y Y ) = N, ( i i i ) t * > 0 , s* > 0 , g* = 0 , ( i v ) ( t * , s*) are not  p p  N  8-optimal  T  X [B , p  R  T with r e s p e c t to any s o c i a l w e l f a r e f u n c t i o n W(u) = B u,  B > 0 „ , i . e . , any v e c t o r X e tl and  M  B ] = 0^ z  + R  H  +  N  +  R  K  +  1 t  h  does n o t s a t i s f y  a  t  s o  X [ A , - B j > 0^ , b H+1 T  lves  T  X [ B , B ] = 0^ t  g  +M  Then, t h e r e  ft ft exists a strict i.e.,  At  Pareto  <^ 0 , As N  and p r o d u c t i v i t y  improving  reduction of ( t , s ) ,  _< 0^,without a change i n the autarky  the autarky tax r a t e s ( t  tariffs  T . If  , s ) are n o n p o s i t i v e , there e x i s t s a s t r i c t  ft ft Pareto  and p r o d u c t i v i t y improving  ft in  the i n i t i a l  tariffs  x .  i n c r e a s e o f ( t , s ) , without  a change  - 162 -  Proof: Suppose that  t  > 0.., s  —  „H+N+K+1  no v e c t o r  X e R  T  N  > 0,,.  such t h a t  I t s u f f i c e s to show t h a t  >T .  „  r  .  Using  (4.16),  f u n c t i o n s , and assumption  (iii),  B ] >^ OJJ+M* g  T X [B , L  (11.1)  „T  .T  „  r  .  T  X [A, - B, ] > 0 , X [B , B ] = 0 „ „ , b H+l p z N+K T T 1 1  the homogeneity  o f the e x p e n d i t u r e s  T T T B ] ( q , V ) ' = 0. 1  t  s  By assumption, (q,v)  »  0  and the v e c t o r  T X [B  t >  B ] i s nonzero.  T X [B , B ] must c o n t a i n n e g a t i v e t s T T i n e q u a l i t i e s X [B . B 1 > 0„ must be v i o l a t e d . t s — N+M  follows  that  the v e c t o r  * If vector  X e  < 0„ and s  — H  +  N  < 0,,, i t s u f f i c e s to show t h a t  N  +  K  +  —  M  1  R  T s  T T X [B^, B ] _< 0ji ^« g  u  c  h  t  h  a  t  X  [ A , - B, ] > 0*  elements, i . e . ,  +  there  T  p  Using ( 1 1 . 1 ) and a s i m i l a r r e a s o n i n g  Theorem 11.1 i m p l i e s t h a t  i n n e t demand ( s u p p l y )  Z  nonoptimal  N+tv  as above,  QED  the government i n a small  c o u n t r y , where lump sum t r a n s f e r s are not a d m i s s i b l e , o f the i n i t i a l  i s no  , X [B , B ] = 0 ^ „ , rl+1  T T i n e q u a l i t i e s X [B , B ] < 0 „ must be v i o l a t e d . t s — N+M  reduction  It  *  t  D  the  v  there i s  T  X [B^,  the  M  —  autarkic  can f i n d  such a  taxes ( s u b s i d i e s ) on the commodities  by the consumers that  balance of trade  is strictly  improved  i s made s t r i c t l y  better of.  Equally  the c o u n t r y ' s i n i t i a l net  and every household i n the economy w e l l , i f the c o u n t r y ' s  commodity  -  tax  163  -  s t r u c t u r e i n autarky i s such that goods i n net demand by  consumers are  subsidized  taxed, there  and  the  e x i s t s some s t r i c t  f a c t o r s s o l d by Pareto and  the  the  households  are  p r o d u c t i v i t y improving 3  reduction  i n the l e v e l s of these s u b s i d i e s  The  example, on  b e g i n n i n g of Melvin  are  based  i s in fact  using  sets.  lump sum  taxes.  h i s statement c i t e d  s  = 0 ). M  countries  with  11.1.  identical  In both c o u n t r i e s , government  expenditures  t r a n s f e r s without commodity t a x a t i o n  Because the demand s i d e s of the  W  to be  countries  tax  coincide.  It follows  i n the  two  f u n c t i o n W(u)  of Theorem 11.1  that  However, i t can  r a t e s ( t * , s*)  s o c i a l welfare  is violated.  the  (i.e.,  economies  be  autarky t a r i f f s shown that  T  = 1 u if x H  ^  IL  = 0 . M  Furthermore, using  8 = l j j ) , no  strict  the  Pareto and  tax  rates (t  zero  commodity the  P r o p o s i t i o n 4.2,  i t can  * , s ) are  B-optimal  p r o d u c t i v i t y improving d i r e c t i o n s of  * * change i n the commodity tax r a t e s ( t , s ) and M e l v i n ' s example, the consumers i n the on a t r a d e a b l e  T* i n each  Hence, assumption ( i v )  * = 0^ and  the  are  two  8-optimal f o r  c o u n t r i e s must be  * seen that when T  (for  two  s i m i l a r , the autarky e q u i l i b r i u m p r i c e s i n the  c o u n t r y equal z e r o .  tax  the  * = 0„, N  assumed  be  in  c l o s e l y r e l a t e d to Theorem  a world c o n s i s t i n g of two  possibility  financed  * t  t h i s chapter,  considered  production  which M e l v i n  and  * the  tariffs  T  exist; in  c o u n t r y which imposes a nonzero  good are a c t u a l l y made worse o f .  -  12.  164 -  ECONOMIC INEQUALITY AND PUBLIC POLICIES  The a n a l y s i s i n the p r e v i o u s c h a p t e r s has been e n t i r e l y w i t h the e x i s t e n c e of s t r i c t P a r e t o  improvements.  A strict  concerned  Pareto  improvement i s , of c o u r s e , a l s o a s t r i c t w e l f a r e Improvement f o r any increasing Blackorby  s o c i a l w e l f a r e f u n c t i o n , but i t c o u l d be argued, f o l l o w i n g and Donaldson (1977: p. 374), that a c c e p t i n g P a r e t o  improvements as s o c i a l w e l f a r e improvements may " a l l o w too much inequality  to creep  i n t o a s o c i a l arrangement."  Consider F i g u r e 9.  Suppose t h a t a t the I n i t i a l e q u i l i b r i u m i n a two consumer economy, the households  1 and 2 a t t a i n  the u t i l i t y l e v e l s  ( u ^ , u^)  d e p i c t e d by the p o i n t S i n the f i g u r e , and t h a t the u t i l i t i e s u^ and are f u l l y of  comparable.  1  The p o i n t S' g i v e s the symmetric d i s t r i b u t i o n  ( u ^ , u ) w i t h r e s p e c t to the equal u t i l i t y l i n e u^ = u »  from  2  2  S, a s t r i c t P a r e t o  instruments  shifts  improving  change i n the government  the consumers to a new u t i l i t y  d i s t r i b u t i o n p o i n t i n the cone ASB.  (real  T h i s p o l i c y change i s a l s o  to move i n the cone ASC, and ( r e l a t i v e ) i n e q u a l i t y  causes  are b e i n g  a shift  shifted  i n the cone CSB.  i n the consumer u t i l i t i e s  i n e q u a l i t y r e d u c i n g , but not P a r e t o The primary assumed  the consumers  increasing,  A policy  if  the  change t h a t  i n the cone S'SA i s ( r e l a t i v e )  improving.  p o l i c y g o a l of the government, i n t h i s c h a p t e r , i s  to be to reduce  home c o u n t r y .  policy  income)  ( r e l a t i v e ) i n e q u a l i t y r e d u c i n g , i f the p e r t u r b a t i o n causes  households  Starting  the i n i t i a l  l e v e l of economic i n e q u a l i t y  The aim of t h i s chapter  i n the  i s to o p e r a t i o n a l i z e the concept  - 165 -  Figure  9 - Pareto Improving Perturbations.  and  I n e q u a l i t y Reducing  Policy  - 166  -  of economic I n e q u a l i t y , i . e . , t o d e f i n e i t i n a way a n a l y s i s of p r a c t i c a l p o l i c y q u e s t i o n s : commodity tax and simple  that allows  the  .when do i n e q u a l i t y r e d u c i n g  t a r i f f p e r t u r b a t i o n s e x i s t and when, i f e v e r ,  p o l i c i e s (e.g., p r o p o r t i o n a l reductions  are  of t a r i f f s ) i n e q u a l i t y  reducing?  12.1  E x i s t e n c e of I n e q u a l i t y Reducing P o l i c y P e r t u r b a t i o n s  I t w i l l be assumed i n t h i s s e c t i o n t h a t each household  h,  h=l,...,H, i n the economy possesses n o n n e g a t i v e endowments of domestic —h and  i n t e r n a t i o n a l l y t r a d e a b l e c o m m o d i t i e s , denoted by c  (>_ 0^)  respectively.  —h 0> 0 ) N  and  I t i s a l s o assumed t h a t i f , f o r example, the  d nth  d o m e s t i c commodity i s a l a b o r s e r v i c e s o l d by the consumer h, the element c* i n h i s endowment v e c t o r c* i s e q u a l to z e r o . The reasons f o r the n 1  1  l a t t e r assumption are t w o f o l d :  f i r s t , the government i s not l i k e l y  to  o b t a i n c o r r e c t i n f o r m a t i o n about the consumers' l a b o r s k i l l s t h a t i t needs i n o r d e r to tax t h e i r a b i l i t i e s  and  consumers' l a b o r endowments are a l l o w e d ,  secondly,  i f taxes on  i t i s p o s s i b l e that  the  the  i n d i v i d u a l s i n the s o c i e t y are f o r c e d to donate t h e i r time to the g o v e r n m e n t — a n outcome t h a t c o r r e s p o n d s to s l a v e r y and  i s obviously  o  incentive i n The  compatible.  equations  d e s c r i b i n g the i n i t i a l e q u i l i b r i u m i n the economy  q  are the f o l l o w i n g :  (12.1)  h * * * *h m (u^, 3 » v ) = g +q <  *T — h c + v  *T  —h d , h=l,...,H,  -  (12.2)  E h=l  h  V m (u* q  q*, v*) + x ° =  h  ^  k  E k=l  k  (12.3)  Tr (p*,  (12.4)  w  T  w + T*) = 0,  •  E h=l  h  *  *  ^  +  _r)  E h=l  C ,  k=l,...,K,  *  *  Tn  =  V  V  w V  1  C = [c ,...,^];  *  *  Tr (p , w +  x ) z  * k  + w  W  us denote the m a t r i c e s of consumer  Differentiation  * k  X ) Z  P  T  (12.5)  *  V TT (p , W +  V m (u , q , v ) + w e  E k=l  Let  167 -  D E  of the model  T  E h=l  —h * d - b .  endowments by  1  [d ,...,^].  (12.1) - (12.4) a t the i n i t i a l  equilibrium  yields:  (12.6)  where  AAu* = B Ap* + B Az* + B,Ab* + B At* + B As* + B Ax*, p ^ z b t s x •  - 168  —  A =  —  I H  , B  -X -Z  =  p  I qu  W  =  PP  Y  L  -X  —T + C + S  qq  -w  vu  t  T  T  °KxH  B  T  —T + C  B  =  T  Z vq  + w  T  —T + D  -E  -Z  T  S  wp  > B  -Z qq  -Z  °KxN T „ -w vv  r  The m a t r i c e s B It  and B, b  z  i s assumed that  e q u i l i b r i u m i s denoted  by  u  W  of t h i s  as  u  i  = rr H  +  qv  T  S  pw  T Z + w S vv WW  i n (4.16).  f u n c t i o n at the  social  initial  ( u ).  D e f i n e the household average r e a l income u  _*  —T + D  the government agrees on the form of a  the g r a d i e n t  8 = V  -w  T  i n (12.6) are those d e f i n e d  w e l f a r e f u n c t i o n W(u);  (12.8)  -E  qv  KxN T -w vq  (12.7)  -  h Z m h=l  * * * (u, , q , v ). ^  ,at the i n i t i a l  equilibrium  - 169 -  Next, d e f i n e the Kolm (1969) equal  *  equivalent  *  *  \i_ as the s c a l a r K(u ) which s a t i s f i e s W(K(u K(u  ) gives  of the r e a l income v e c t o r  * w  u  )1^) = ^ ^ ^ '  ^he s c a l a r  the amount of r e a l income which, i f a t t a i n e d by each  i n d i v i d u a l i n the economy, would g i v e r i s e  to the same l e v e l of s o c i a l  ft w e l f a r e as the a c t u a l observed v e c t o r of r e a l income u . Assuming t h a t the s o c i a l w e l f a r e f u n c t i o n W(u) i s c a r d i n a l l y s c a l e d , the e q u a t i o n  (12.9)  is  W(u*)  = K(u*)  s a t i s f i e d at the i n i t i a l  measure of i n j u s t i c e ,  equilibrium.  Hence, u s i n g  the Kolm  (1969)  a monetary measure of economic i n e q u a l i t y a t the  ft initial  e q u i l i b r i u m , I , can be d e f i n e d :  (12.10)  I * = u* - W(u*).  A r e d u c t i o n o f economic i n e q u a l i t y i s d e f i n e d *  in the  the measure I initial  *  (Al  commodity  5  < 0).  When can such r e d u c t i o n s  tax rates ( t * , s*) and the t a r i f f s  government p o l i c y instruments?  * an e q u i v a l e n t  t o mean a r e d u c t i o n  form - I  * = W(u  L e t us r e w r i t e  be reached  using  x* as the  the e q u a t i o n  (12.10) i n  _*  ) - u .  Then, i t can be seen that a  r e d u c t i o n of economic i n e q u a l i t y i s e q u i v a l e n t  to an i n c r e a s e i n the  -  initial  value  of the  170  s o c i a l welfare  -  f u n c t i o n W(u)  = W(u)  _ the  sufficient The  conditions  - u.  What  are  *  f o r an improvement i n W(u  ) to e x i s t ?  problem i s to determine the minimal s u f f i c i e n t  conditions  for:  k  (12.11)  there  e x i s t Au  (12.6) h o l d s , JU  7  Theorem  u  W(u  ) = (8  k  , Ap 3  cn  k  T  , Az  Au  k  , Ab  > 0 and  , At Ab  np  rri  - 1 / H ) , (B R  T  T  , Ax  such T  that  =  *  Og).  12.1:  T  - C ),  initial  (E  T  T  - D )]  equilibrium.  e q u a l i t y improving s*)  does not in  , As  k  >_ 0, where 8  * oJ. , M  ( i v ) x*  ( i i ) rank ( S  T  V  INTiM  (t*,  k  rx%  Suppose that ( i ) rank Y = K <_ N,  B [(X  k  (without  Then, there  change i n the  can  be  G(w  + YY )  + x*,  y*)  = N, ( i i i )  * 0^ at  WW  tariffs  x and  a change i n the lump sum net  welfare  and  commodity tax  t r a n s f e r s g*)  economic rates  which  balance of t r a d e .  chosen to be a p r o p o r t i o n a l  The  change  reduction.  Proof: Motzkin's Theorem y i e l d s an e q u i v a l e n t  the  M  exists a strict  reduce the c o u n t r y ' s i n i t i a l  tariffs  2  T  p p  c o n d i t i o n to  (12.11):  -  (12.12)  171 -  there must  not e x i s t a v e c t o r  XT[  V  V  V  T  XB,_  B  s  ]  = °N+K+N+M'  A e R  X  T  B  h + n + k +  1  X  x " °M>  s u c  T  A  h  that  -  ^>  < 0.  D  T The i n e q u a l i t y X B  fa  < 0 i m p l i e s X^ _> 0. Set ^  k(p  X^ and X^ of the v e c t o r  + 6) , X^ => k y » where  I f k = 0, X  2  = 0  N  and X  3  As i n the proof  T T X [B , B , B . B ] = ()„,.,,„,„ can be used p z t s N+K+N+M  of Theorem 2.1, the equations  to s o l v e the components  = k > 0.  the v e c t o r s  = 0 .  X when X^ = k:  6 and y a r e d e f i n e d  Then, the equations  R  T X^ =  i n (2.27).  i n (12.12) s i m p l i f y  to  (12.13)  X  1  1  By assumption, there  Suppose k > 0.  1  - C ]  Then, the equations  the m a t r i x  By assumption, T *  T  V  2  ww satisfied. proportional  = O j , - X j [E  i s no s o l u t i o n X^ =  G(w + x , y ) = 0 ' w h e r e M (2.12).  1  = B , - Xj [ X  P r o p o s i t i o n 2.1 reduction.  QED  g  T  - D ] =  0*.  to (12.13).  T x X B^ = 0^ are e q u i v a l e n t  V ww  to x  *T  G(w + x , y ) i s d e f i n e d i n J  G(w + x*, y*) * 0^, i . e . , (12.12) i s M  shows that  Ax  may  be chosen to be a  2  -  Corollary  172 -  12.1.1:  If  rank  [X  T  —T T - C , E  T  D ] = H (_< N + M), c o n d i t i o n ( i i i ) i n  Theorem 12.1 i s s a t i s f i e d .  Proof:  If  T  —T T - C , E  is  A., = 0. H*  rank [X T N+M  Hence, s i n c e 8  Theorem 12.1 must be s a t i s f i e d .  similar  instruments): sufficient  x* to e x i s t ,  are the a d m i s s i b l e government p o l i c y  5  p r o d u c t i v i t y improving  and assumption  to the households  income toward e q u a l i t y .  economy i n t o  interpretation. three c l a s s e s :  p e r t u r b a t i o n i n the  ( i i i ) ensures  from a p r o d u c t i v i t y improving  interesting  P a r e t o improvements  assumptions ( i ) , ( i i ) and ( i v ) i n Theorem 12.1 are  for a strict  distributed real  conditions that  i m p l y i n g the e x i s t e n c e of s t r i c t  (when commodity taxes and t a r i f f s  accruing  (iii) in  the e x i s t e n c e of improvements i n economic e q u a l i t y are v e r y  to those  tariffs  T assumption H'  QED  Theorem 12.1 shows that the s u f f i c i e n t guarantee  0  i n a way  t h a t the gains  p e r t u r b a t i o n of x that s h i f t s  Assumption  can be  the d i s t r i b u t i o n of  ( i i i ) can a l s o be g i v e n an  L e t us p a r t i t i o n  the households  i n the  those w i t h a p o s i t i v e w e l f a r e weight  8 ,  ^h those w i t h a zero w e l f a r e weight 8 , and those w i t h a n e g a t i v e w e l f a r e h / weight 8 . The f i r s t c l a s s of consumers i s c a l l e d "the poor" and the A  last  "the r i c h . "  F o r assumption  ( i i i ) i n Theorem 12.1 to be  satisfied,  -  it  is sufficient  internationally and  173  -  to assume that there e x i s t s  some good  (domestic  or  t r a d e a b l e ) i n ( n e t ) demand ( o r supply) by a l l "the  i n ( n e t ) supply  ( o r demand) by a l l  the p r e f e r e n c e s of "the r i c h " and  "the  rich."  7  poor"  In o t h e r words,  "the poor" must s i g n i f i c a n t l y  from each o t h e r a t l e a s t i n the case of a commodity.  Under  differ  this  ft s u p p o s i t i o n , a p r o p o r t i o n a l r e d u c t i o n of the  tariffs  x  can be made  8  economic i n e q u a l i t y  reducing.  Using P r o p o s i t i o n 4.2,  i t can be  seen  that s t r i c t  welfare  and  ft economic e q u a l i t y improving X  commodity taxes  directions  of change i n the t a r i f f s  x  X  ( t , s ) cannot  exist,  i f the i n i t i a l  equilibrium is  a 8-optimum w i t h r e s p e c t to the commodity tax r a t e s ( t * , s*) and  * g r a d i e n t of the net zero.  One  balance  can thus conclude  of trade f u n c t i o n , that zero t a r i f f s  commodity tax r a t e s are f r e e l y a d j u s t a b l e and m a t r i x S i s of maximal rank ( = N + M -  the  *  V b (w +  x ), equals  ( f r e e t r a d e ) are not  P a r e t o but a l s o " e q u a l i t y o p t i m a l " f o r a s m a l l c o u n t r y , i f the  If  and  the producer  only  domestic  substitution  1).  the government i n the home c o u n t r y can a d j u s t the  initial  X  v e c t o r of lump sum of  Theorem 8.3,  t r a n s f e r s g , i t can be shown that the assumptions  which imply  the e x i s t e n c e of s t r i c t  Pareto  improving  ft ft p e r t u r b a t i o n s i n the t a r i f f s sufficient  for a s t r i c t  x  and  w e l f a r e and  9 t r a n s f e r change to e x i s t .  t r a n s f e r s g , are equality  Improving  also tariff  and  -  12.2  174  -  E x i s t e n c e of I n e q u a l i t y Reducing and W e l f a r e Improving P o l i c y Perturbations  By  comparing  Theorem 8.3,  which guarantee the e x i s t e n c e  perturbations x*,  the assumptions of theorem 12.1  i n the commodity tax  the c o n c l u s i o n s  of s t r i c t  strict  Pareto  strict  r e d u c t i o n i n economic i n e q u a l i t y ,  be  1 0  and  and  Pareto  e q u a l i t y improving? improving  And  improvement does  when are  a  not.  1 1  A  both  commodity tax and  tariff  functions  W(u)  12.2:  V  2  ww  G(w  I.  (12.14)  and  A  f o r both s o c i a l w e l f a r e  Suppose that ( i ) rank Y = K _< N, T  (i) a  W(u)?  Theorem  x*  confirmed:  the government f i n d changes i n ( t , s , x ) that are  perturbations welfare and  tariffs  ( i i ) a strict inequality  A  Pareto  the  of  improving  ( i f i t e x i s t s ) does not n e c e s s a r i l y imply  r e d u c t i o n can e x i s t even i f a s t r i c t  When can  Pareto  r a t e s ( t , s ) and  drawn from F i g u r e 9 can  improvement  to the c o n d i t i o n s  + T*  J  y*)  Then, i f B  t h e r e i s no  * Cv M  T  ( i i ) rank ( S  at the i n i t i a l  = V W(u*) u  = (B  s o l u t i o n a > C>  H  T  + YY  p p  ) = N, ( i i i )  equilibrium,  - lJ/H) H  T  to a [ X  T  and  T  - C,  E  T  T  - D]  =  ojJ  +M>  -  (12.15)  there  ( r  T  r > 0^ t o  i s no s o l u t i o n  T  ^  +  T  )  [X -  there e x i s t s a s t r i c t  T  C , E  Pareto  175 -  T  -D  T  =0^  ]  + M  ,  and e q u a l i t y improving  change i n the  ft ft ft initial  commodity  tax r a t e s ( t , s ) and the t a r i f f s  T  (without a  ft change i n the i n i t i a l  v e c t o r of lump  sum t r a n s f e r s g ) .  p e r t u r b a t i o n does not reduce the l e v e l  The p o l i c y  of the economy's i n i t i a l net  ft  balance  II.  of trade  b .  If  (12.16)  there (r  i s no s o l u t i o n r > 0 t o  T  g +  6 ) [X - C , E T  - D ] = 0^  T  T  + M  ,  and  (12.17)  sHx -  c\  V-  C  1  6  there e x i s t s  a strict  T  E  ,  T E  1  - V ] * 0TmM,  - D  welfare  T  1  commodity  +M  and e q u a l i t y improving A  initial  *0l ,  p e r t u r b a t i o n of the  A  A  tax r a t e s ( t , s ) and the t a r i f f s  x  (without a  A\  change i n the i n i t i a l  v e c t o r of lump  sum t r a n s f e r s g ) •  does not reduce the l e v e l of the economy's i n i t i a l  The change  n e t balance  of trade  176  -  -  Proof: (I) A s u f f i c i e n t  c o n d i t i o n f o r a Pareto  and e q u a l i t y improving  change i n ( t * , s*) and x* to e x i s t i s :  JU  (12.18)  there e x i s t  JL  JU  Jf  is  satisfied,  U s i n g Motzkin's  JU  Au , Ap , Az , Ab  yu  JL  JL  Ab  > 0, Au —  ju  , At , As , Ax *Srp  »  such  that (4.16)  Jf  0„ and tr Au H  > 0.  Theorem, an e q u i v a l e n t c o n d i t i o n f o r (12.18) can be  derived: 1 (12.19)  t h e r e i s no v e c t o r X e R  V  Using  - °N K N M M' +  +  +  X T  B  +  T X B^ _< 0, X^ = k _> 0.  b  T  T  ?  A  -  V  l  +  7  ?  2 '  <V  V  I f k = 0,  X  2  = 0  and X  N  >  <W  of Theorem  3  = 0 . R  It  F o r (12.19) to be s a t i s f i e d ,  there must not  a s o l u t i o n to  rp  (12.20)  Since  A  T  f o l l o w s t h a t X A = X^. exist  < °«  Then, as shown i n the proof  2.1, X^ = k ( p * + 6 ) , X^ = k Y « T  T that X [B , B , B„, B , p z ' t' s '  such  rp  rp ^rp  rp 1  x{ = 7f + V ' g , - xj[x  rri  - c\  rp  E  rri  - TJ ]  (v , v ) > 0 , (12.20) i s e q u i v a l e n t to 1 2. H*r1  = o  N + M  , (^, 7 ) > o 2  R + 1  .  -  (12.21)  177 -  T  X^ = v* (> 0 )  or X* = v* + 8 ,  R  V  > 0 ,  l  and  R  Using assumptions (12.14) - (12.15) and (12.21), i t can be seen that there  i s no s o l u t i o n to (12.20). T  I f k > 0, choose k - 1. assumption.  Hence  (II)  Now  Then, X B  there  T  V  2  G(w + T * , y*) * o3  by  (12.19) i s s a t i s f i e d .  i t i s required  * (12.22)  = x*  that  *  *  *  *  *  yu  (4.16) i s s a t i s f i e d  ^ rp!  and Ab  > 0,  *  As , A T  e x i s t Au , Ap , Az , Ab , At ,  that  rfl yu  JU  3 Au  such  > 0, 3 Au  > O.  Equivalently,  (12.23)  there must not e x i s t a v e c t o r  XT[  v  V  x  V.  8 + v  V  2  V  V  X e R  = °N K N M M> +  +  +  +  x  2  = 0  satisfied,  there  N  +  \  such  < °- ^  that  =  2  <^ 0 i m p l i e s  ^  = k > 0.  T 2  X  K 1  8 , (v , v ) > 0 .  P r o c e e d i n g as above, X  I f k = 0, X  H + N +  and X  3  = 0» K  T  Then, X A = ^ .  must not e x i s t a s o l u t i o n to  F o r (12.23) to be  - 178 -  (12.24)  rp  ^rp  "TP  1  A| = v ^  1  + v ^ , -AjtX  Since ( v ^ , v ) > 0 , 2  (12.25)  2  • Aj = 6  rp  T  [X  T  2  rp  T  -  rp  rp  rp  1  - C , E  rp  rp  - D j = 0*  1  + M >  (vj,  > 0,,.  (12.24) i s e q u i v a l e n t t o  + v g , v  rp  -A*  rp  _> 0 o r A^ =  2  rp  T  C, E  rp  V j  B  T  T  + 3 ,  > 0, and  rp  - T) ] = 0 j  + M  .  U s i n g assumptions (12.16) - (12.17) and ( 1 2 . 2 5 ) , i t can be seen t h a t t h e r e i s no s o l u t i o n to ( 1 2 . 2 4 ) .  X T *T J2 The e q u a t i o n s A B = 0 s i m p l i f y t o x v x M ww  I f k > 0. choose k = 1. *  X  A  G(w + x , y ) = 0^. violated.  Corollary  W  U s i n g assumption ( i i i ) ,  QED  12.2.1:  I f rank [ X  T  T  - C, E  T  - DT]  = HC< N+M),  (12.22) of Theorem 12.2 a r e s a t i s f i e d . (<_ N+M)  these e q u a t i o n s must be  and 8 > 0 , H  c o n d i t i o n s (12.21) T  T  I f rank [ X - f J , E  T  - DT]  = H  c o n d i t i o n s (12.16) - (12.17) are s a t i s f i e d .  Proof: T  T  I f rank [ X - C , T  equations A [ X  T  -  E  T  E  T  - Dj  - D1]  =  = H (<_ N+M),  the o n l y s o l u t i o n f o r the  is A = 0 .  Hence, (12.14) i s  -  satisfied.  179  -  Then, s i n c e 3 * - r , r > 0„, t h e e q u a t i o n s ( r + 3 ) H  [X  T  T  not a l l 0  If  T  - C , E h  - D j = 0^  (= B  T  h  + M  have no s o l u t i o n r > 0 .  h  ( I f E B = 1, h= 1  - 1/H) can be n e g a t i v e . )  T  rank [ X - C , E  T  T  - D ] = H, c o n d i t i o n (12.17) must be s a t i s f i e d . •p  rp  "p  F u r t h e r m o r e , i f rank [X - C , E (12.16) must be s a t i s f i e d  rp  - D ] = H  ( s i n c e the v e c t o r  and 6 > 0 , c o n d i t i o n Tjr  ( r B + B) cannot be z e r o ) .  QED  Assumption (12.14) i n P a r t I o f Theorem 12.2 i s the g e n e r a l i z e d D i a m o n d - M i r r l e e s c o n d i t i o n t h a t i s needed f o r a P a r e t o i m p r o v i n g d i r e c t i o n o f change i n the i n i t i a l Condition in  to e x i s t .  (12.15) guarantees t h a t a P a r e t o Improving d i r e c t i o n of change  the i n i t i a l  commodity tax and t a r i f f  i n e q u a l i t y reducing. that, according  r a t e s can be made economic  C o r o l l a r y 12.2.1 i m p l i e s t h a t the rank c o n d i t i o n  to C o r o l l a r y 12.1.1 i s s u f f i c i e n t  improvements to e x i s t ,  strict  commodity t a x and t a r i f f  i s also s u f f i c i e n t  t o ensure the e x i s t e n c e ft ft ft  P a r e t o and e q u a l i t y i m p r o v i n g p e r t u r b a t i o n Condition  for equality of a  in (t , s , T ).  (12.17) i n P a r t I I o f Theorem 12.2 c o r r e s p o n d s t o  a s s u m p t i o n ( i i i ) i n Theorem 12.1.  Condition  (12.16) imposes a  restriction gradient  on the w e l f a r e w e i g h t s B ( w h i c h are determined by the * V^VKu ) ) . C o r o l l a r y 12.2.1 shows t h a t (12.16) i s s a t i s f i e d  a t l e a s t i f 3 > 0„ and the m a t r i x rank.  [X - C, T  T  E  T  1  - IF ] i s of f u l l ( r o w )  -  13.  180  -  CONCLUSIONS  One  of the famous problems i n the area of i n t e r n a t i o n a l  t h e o r y has been the q u e s t i o n of the g a i n s from t r a d e : f o r a l l the consumers i n an a u t a r k i c c o u n t r y is  opened up  does t h i s It  for international  trade and  to b e n e f i t i f the  i s well-known that f r e e i n t e r n a t i o n a l  the home country has  lump sum  Is i t s t i l l  instruments  set f o r the  at i t s d i s p o s a l , i t  i n the economy w i l l  be b e t t e r of ( o r at  trade than  i n autarky.  r e d i s t r i b u t i o n of Income i s i n a d m i s s i b l e ?  p o s s i b l e to show t h a t a l l consumers i n an a u t a r k i c c o u n t r y  can b e n e f i t from f r e e i n t e r n a t i o n a l textbook  provided a r e s u l t  trade?  according  Dixit  and  Norman (1980) i n  to which the a u t a r k y  l e v e l s of a l l households i n the home economy can be r e p l i c a t e d f r e e trade i f the government can f r e e l y a d j u s t the c o u n t r y ' s tax s t r u c t u r e .  However, D i x i t and  Norman d i d not  from trade would o c c u r , i . e . , that the s h i f t would produce a s t r i c t This  ( o r at  Hence, i f the government i n  not worse o f ) under f r e e i n t e r n a t i o n a l But what i f lump sum  trade e n l a r g e n s  consumption p o s s i b i l i t y  transfer  can ensure t h a t every household  Pareto  thesis started  conditions for s t r i c t  welfare under  commodity  show t h a t s t r i c t  from a u t a r k y  to f r e e  gains trade  improvement.  as an attempt  to f i n d  the  sufficient  g a i n s from trade to e x i s t when o n l y commodity  t a x a t i o n i s used to i n f l u e n c e the d i s t r i b u t i o n of income. is  country  i f so, under what c o n d i t i o n s  consumers i n a p r e v i o u s l y a u t a r k i c c o u n t r y .  their  i s i t possible  occur?  l e a s t , does not reduce) the f e a s i b l e  least,  trade  approached as a p o l i c y reform  question:  when can  The  problem  the government  find  - 181 -  s t r i c t Pareto improving ( d i f f e r e n t i a l ) perturbations  i n the c o u n t r y ' s  commodity tax and t a r i f f s t r u c t u r e i f the i n i t i a l a u t a r k y  t a r i f f s are  defined  prohibited?  to be such t h a t i n t e r n a t i o n a l trade i s j u s t being It, t u r n s out t h a t under some weak c o n d i t i o n s  C h a p t e r 2 o f the t h e s i s ) on the p r o d u c t i o n production  (established i n  technologies  s e c t o r s and on the i n i t i a l v e c t o r of t a r i f f s  changes i n the economy's t a r i f f v e c t o r  of the d o m e s t i c there e x i s t  such  t h a t the amount of f o r e i g n  exchange earned by the domestic p r o d u c e r s i s i n c r e a s e d .  As an example,  the p e r t u r b a t i o n of_ t a r i f f s can be chosen t o be a p r o p o r t i o n a l reduction. I n Chapter 4 of the t h e s i s the q u e s t i o n increase  i n the amount of f o r e i g n exchange earned by the domestic  p r o d u c e r s t o the consumers i s c o n s i d e r e d . on the p r e f e r e n c e s satisfied;  I t t u r n s out t h a t a c o n d i t i o n  and i n i t i a l endowments of the households has to be  t h i s c o n d i t i o n i s a g e n e r a l i z a t i o n of the D i a m o n d - M i r r l e e s s  assumptions that a good i n n e t demand o r s u p p l y exists.  of r e d i s t r i b u t i n g the  by a l l households  I t can be shown t h a t i f the above mentioned c o n d i t i o n on the  consumer p r e f e r e n c e s domestic production  and the p r e v i o u s l y developed c o n d i t i o n s on the technologies  improving perturbations  a r e s a t i s f i e d , then s t r i c t  Pareto  i n the c o u n t r y ' s i n i t i a l e q u i l i b r i u m t a r i f f s and  commodity tax r a t e s e x i s t .  I n p a r t i c u l a r , i f the i n i t i a l e q u i l i b r i u m i s  an a u t a r k y  e q u i l i b r i u m , these c o n d i t i o n s are s u f f i c i e n t f o r s t r i c t  from trade  to e x i s t .  gains  In Chapter 3 of the t h e s i s the problem of a p p r o x i m a t i n g the s i z e of the p r o d u c t i v i t y g a i n a c c r u i n g  from a p r o d u c t i v i t y i m p r o v i n g  tariff  - 182 -  change i s a n a l y z e d .  The measures developed  provide production side approximations t r a d e when the i n i t i a l l y Having  i n t h i s chapter a l s o  f o r the s i z e of the g a i n from  t r a d e p r o h i b i t i v e t a r i f f s are b e i n g p e r t u r b e d .  e s t a b l i s h e d the e x i s t e n c e of s t r i c t P a r e t o  improving  t a r i f f and commodity t a x p e r t u r b a t i o n s , the next q u e s t i o n i s : what k i n d of  examples of these p o l i c y changes can be found?  I n Chapter 7 of the  t h e s i s i t i s shown t h a t , f o r example, r e d u c t i o n s of p o s i t i v e t a r i f f s and i n c r e a s e s of n e g a t i v e t a r i f f s , u n i f o r m r e d u c t i o n s o f t a r i f f s toward  and changes  u n i f o r m i t y i n the c o u n t r y ' s t a r i f f s t r u c t u r e can be s t r i c t  i m p r o v i n g i f the c o u n t r y ' s i n i t i a l  Pareto  commodity tax r a t e s can be f r e e l y  varied. In Chapter 8, lump sura t r a n s f e r s a r e assumed to be a v a i l a b l e and s u f f i c i e n t c o n d i t i o n s f o r s t r i c t P a r e t o i m p r o v i n g t a r i f f and t r a n s f e r p e r t u r b a t i o n s to e x i s t are developed.  These p r o d u c t i o n s i d e c o n d i t i o n s  a r e a l s o s u f f i c i e n t f o r s t r i c t g a i n s from t r a d e to e x i s t i f lump sum t r a n s f e r s can be used t o r e d i s t r i b u t e consumer income. In  Chapter 9, the e x i s t e n c e r e s u l t s under d i f f e r e n t  assumptions  about the a v a i l a b i l i t y of government p o l i c y i n s t r u m e n t s a r e compared. An i n t e r e s t i n g p r o p o s i t i o n d e r i v e d i n t h i s s e c t i o n i s t h a t s t r i c t from  gains  trade need n o t e x i s t under lump sura compensation even i f s t r i c t  g a i n s under commodity t a x a t i o n would be p o s s i b l e . Chapter  10 c o n t a i n s some examples of s t r i c t P a r e t o  commodity t a x , t a r i f f and t r a n s f e r p e r t u r b a t i o n s : toward  improving  these i n c l u d e changes  i n t e r n a t i o n a l prices i n t a r i f f s , simultaneous p r o p o r t i o n a l  r e d u c t i o n s of commodity taxes and t a r i f f s , and changes toward u n i f o r m i t y i n commodity t a x e s .  The r e s u l t s c o n c e r n i n g the changes  toward  -  183  -  u n i f o r m i t y i n commodity t a x a t i o n g e n e r a l i z e the e a r l i e r H a t t a  (1977a)  r e s u l t a c c o r d i n g to which a r e d u c t i o n i n the h i g h e s t commodity tax r a t e i n a c l o s e d one  consumer economy (where lump sum  admissible) i s welfare In  improving.  the l a s t c h a p t e r of the t h e s i s the government i s assumed to  s e a r c h f o r p o l i c y reforms country.  t r a n s f e r s are  t h a t improve economic e q u a l i t y i n the  I n o r d e r to c o n s i d e r a c t u a l p o l i c y p e r t u r b a t i o n s , a r e d u c t i o n  i n economic i n e q u a l i t y i s d e f i n e d to correspond  to an improvement i n an  e s p e c i a l l y defined social"welfare function.  One  i n e q u a l i t y r e d u c i n g commodity tax and  changes to e x i s t t u r n s out  to  tariff  be a c o n d i t i o n on the consumer p r e f e r e n c e s :  w i t h r e s p e c t to which "the r i c h " and  of the c o n d i t i o n s f o r  t h e r e must be a good  "the poor" are on the o p p o s i t e  s i d e s of the ( n e t ) market. In  the f u t u r e , the same methods t h a t , i n t h i s t h e s i s , have been  used to a n a l y z e p o l i c y reforms to  i n a s m a l l open economy c o u l d be a p p l i e d  study m u l t i c o u n t r y t a r i f f and o t h e r p o l i c y agreements.  a l s o want to develop  One  might  ( p r o d u c t i o n s i d e ) a p p r o x i m a t i v e measures f o r the  g a i n s a c c r u i n g to the c o u n t r i e s t h a t agree to implement b e n e f i c i a l p o l i c y reforms.  A n o t h e r i n t r i g u i n g d i r e c t i o n of reseach would be to  combine the t h e o r i e s o f o p t i m a l tax r e f o r m s , i m p e r f e c t i n f o r m a t i o n and uncertainty.  I n t h i s c a s e , the g o a l of the a n a l y s i s would be to f i n d  p o l i c y r u l e s f o r a government ( o r a f i r m ) a c t i n g under u n c e r t a i n t y about the o t h e r economic a g e n t s ' g o a l s and c h a r a c t e r i s t i c s .  Of c o u r s e ,  one  c o u l d a l s o i n v e s t i g a t e the c o n d i t i o n s f o r g a i n s from the t r a d e to e x i s t under these c i r c u m s t a n c e s .  I n a many c o u n t r y c a s e , one  c o u l d compare  p o l i c i e s (and the g a i n s a c c r u i n g from them) under i m p e r f e c t i n f o r m a t i o n to  those under c e r t a i n t y and  cooperation.  -  184  -  FOOTNOTES Chapter  2;  1.  I f some of the p r o d u c t i o n s e c t o r s e x h i b i t d i m i n i s h i n g returns to s c a l e , new domestic commodities that correspond to the ownership shares of these s e c t o r s are added to the model. The new f a c t o r s absorb the pure p r o f i t s earned by the d i m i n i s h i n g r e t u r n s to s c a l e sectors.  2.  The t e c h n o l o g i e s of the p r o d u c t i o n i n d u s t r i e s are d e s c r i b e d u s i n g t h e i r u n i t p r o d u c t i o n p o s s i b i l i t y s e t s s i n c e the t e c h n o l o g i e s of the s e c t o r s are assumed to e x h i b i t constant returns to s c a l e . I f the t o t a l p r o d u c t i o n p o s s i b i l i t y s e t s of the i n d u s t r i e s were used, the s e c t o r a l t o t a l p r o f i t f u n c t i o n s would assume only two v a l u e s , zero and i n f i n i t y . Each p r o f i t f u n c t i o n would thus be d i s c o n t i n u o u s and n o n d i f f e r e n t i a b l e .  3.  A l l vectors denotes the  4.  There are s e v e r a l ways of d e f i n i n g the s c a l e of a p r o d u c t i o n sector. I f the s e c t o r k, k=l,...,K, produces only one output, the s c a l e of the s e c t o r k i s the amount of output produced i n that s e c t o r i n each time p e r i o d . I f j o i n t p r o d u c t i o n i s present i n s e c t o r k, the s c a l e of the s e c t o r k, k=l,...,K, can be d e f i n e d as i n Ch. 2 by u s i n g an always needed i n p u t or a l t e r n a t i v e l y by employing u n i t s of v a l u e added (Woodland (1982 : p. 135)).  5.  For u n i t p r o f i t f u n c t i o n s and (1977: pp. 377-378).  6.  I f some s e c t o r s produce i n t e r n a t i o n a l l y traded good m, m e [1,...,M], w h i l e other s e c t o r s u t i l i z e good m as an i n p u t , i t may be n e c e s s a r y to r e d e f i n e good m as two s e p a r a t e commodities: an i n p u t good and an output good. I t i s assumed that each technology s e t C , k=l,...,K i s such that good m i s e i t h e r produced or used as an i n p u t , but not both. Thus a f t e r r e d e f i n i t i o n , each i n t e r n a t i o n a l l y traded good w i l l be e i t h e r produced ( o r not u t i l i z e d at a l l ) by each s e c t o r or used as an i n p u t by each s e c t o r .  i n t h i s t h e s i s are d e f i n e d as transpose (row) v e c t o r of x.  column v e c t o r s ;  d u a l i t y , see Diewert and  x^  Woodland  k  7.  Notation:  O^+M  denotes an  (N+M)  - vector  of  zeroes.  8.  I f a l l p r o d u c t i o n s e c t o r s have a common always used input (or an always produced o u t p u t ) , the matrix S w i l l have a zero row and column corresponding to t h i s commonly a p p l i e d i n p u t . Then, the rank of the matrix S i s at most N+M-2.  9.  The c o n s t a n t r e t u r n s to s c a l e assumption t o g e t h e r with c o m p e t i t i v e p r o f i t m a x i m i z a t i o n i m p l i e s t h a t a l l p r o d u c t i o n s e c t o r s earn zero pure p r o f i t s i n e q u i l i b r i u m . I f the e q u i l i b r i u m t o t a l p r o f i t s were  -  185 -  p o s i t i v e i n some s e c t o r , the p r o f i t s of t h i s s e c t o r could be d r i v e n to i n f i n i t e l y simply by i n c r e a s i n g the i n d u s t r y ' s s c a l e ( s i n c e the s e c t o r ' s technology i s CRS). Hence, the zero p r o f i t c o n d i t i o n s (2.9) c o n s t r a i n the e q u i l i b r i u m to the f i n i t e s c a l e case. 10.  Notation: Ap* denotes an i n f i n i t e s i m a l change i n p*, u s u a l l y denoted by dp*. O ^ K (KxK)-matrix of z e r o e s . i  s  a  n  11.  The p r o p e r t i e s Ch. 3.7).  of the GNP f u n c t i o n are g i v e n  12.  A s p e c i a l case of problem (2.15) i s p r e s e n t e d i n Woodland (1982: Ch. 3 ) . There, i t i s assumed that each p r o d u c t i o n s e c t o r s u p p l i e s o n l y one t r a d e a b l e good u s i n g domestic commodities as i n p u t s . I n t h i s case, assuming that the s e c t o r k, k=l,...,K, produces the th k k t r a d e a b l e , s e c t o r k's. u n i t p r o f i t f u n c t i o n i s TT (p, w + x) = (w^ + x^) - c ^ ( p ) ,  where c^(p)  i n Woodland  (1982:  i s s e c t o r k's u n i t c o s t f o r  th p r o d u c i n g the k tradeable. The problem (2.15) can then be i n t e r p r e t e d as m i n i m i z i n g the i n p u t c o s t under the c o n s t r a i n t th the producer p r i c e f o r the k  13.  tradeable,  that  (w^ + x ^ ) , does not  exceed the s e c t o r k's u n i t cost c ^ ( p ) . The net i n p u t endowment v e c t o r -y i s drawn i n the diagram p° as the o r i g i n . convexity  taking  14.  C o n v e x i t y of the u n i t p r o f i t f u n c t i o n s i m p l i e s u n i t p r o f i t l e v e l curves - i r ^ ( p , w + x ) = 0.  of the  15.  A l l the three p r o d u c t i o n s e c t o r s can s t a y o p e r a t i v e i n s p i t e o f the change i n (w + x*) i f the change i s v e r y s p e c i a l , i . e . , i t i s n e c e s s a r y that a l l the u n i t p r o f i t l e v e l curves c o r r e s p o n d i n g to the new producer p r i c e s f o r t r a d e a b l e s i n t e r s e c t a t p'.  16.  The m a t r i x S  T p p  + YY  i s positive definite,  f o r example, ( i ) i f T  S = 0.. „ and N = K ( i f rank Y = K(= N), the m a t r i x Y Y i s pp NxN * p o s i t i v e d e f i n i t e ) , or ( i i ) i f rank S = N+M-l, which i m p l i e s rank S = N. Proof f o r ( i i ) : suppose on the c o n t r a r y that  that  p p  rank S  = N - l when rank S = N+M-l.  Then, there  PP rri  rn  x * 0 , such that x S = 0„. N pp N XT  rri  Choose y J  rti  N  exists x e R , ' rri  = (x , 0„). M  I t follows  T t h a t y Sy = 0; hence, y i s a zero  eigenvector k  of the p o s i t i v e k  s e m i d e f i n i t e S. Because both the v e c t o r s (p , w + x ) and (x, Oy) a r e zero e i g e n v e c t o r s of the m a t r i x S (and the v e c t o r s are  -  186 -  x  not  l i n e a r l y dependent s i n c e  M  (w + T ) £ R  ), rank S i s at most  T T  N+M-2, which c o n t r a d i c t s the v e c t o r  that  rank S = N+M-l.  17.  Note that vector.  18.  I f rank S = N+M-l, then rank S  19.  Note a l s o that c o n t r o l l a b i l i t y of p r o d u c t i o n i n the sense of Guesnerie and Weymark i m p l i e s c o n t r o l l a b i l i t y of domestic goods p r o d u c t i o n i n the sense of D e f i n i t i o n 2.2, but not c o n v e r s e l y .  20.  The v e c t o r o f domestic goods p r i c e s p* a p p r o p r i a t e i n (2.18) i s the one that s o l v e s problem (2.15). T h i s p* i s the shadow p r i c e v e c t o r c o r r e s p o n d i n g to the domestic goods c o n s t r a i n t y •  21.  If V  x  x  y  the assumption  can be regarded as a net domestic demand  p p  = N and (2.13) i s s a t i s f i e d .  T  x  b (w + x ) = 0 , one cannot determine i f the f u n c t i o n b x M W  * (w + x ) i s i n c r e a s i n g , d e c r e a s i n g x  r  x  r*  b (w + x ) = 0 , b  or s t a t i o n a r y .  Yet, i f  x  cannot be s t r i c t l y  increasing i n i t s  argument a t (w + x ) . 9  22.  k  k  I f the m a t r i x G(w + x , y ) i s a zero (MxM)-matrix, the c o n d i t i o n (2.16) i s v i o l a t e d . Under these c i r c u m s t a n c e s , no i n f i n i t e s i m a l change i n x can change the p r o d u c t i o n s e c t o r s ' aggregate p r o d u c t i o n c h o i c e ( f o r t r a d e a b l e s ) ; even i f p r o d u c t i v i t y improving d i r e c t i o n s of change i n t r a d e a b l e s supply ( g i v e n a f i x e d y ) would e x i s t , they could- not be a t t a i n e d through d i f f e r e n t i a l p e r t u r b a t i o n s i n the p r i c e s (w + x * ) . N  23.  Notation:  For a vector x e R , x »  0^ means that each component  of x i s p o s i t i v e ; x _> 0^ means that each component of x i s nonnegative; x > 0^ means that x >^ 0^ but x * 0^.  24.  When x e R, x  » 0 i s e q u i v a l e n t to x > 0. Note that the e x i s t e n c e o f a p r o d u c t i v i t y improving t a r i f f change i s e q u i v a l e n t to the e x i s t e n c e of an A l l a i s p r o d u c t i o n g a i n . (Diewert (1983c) d e f i n e s the A l l a i s p r o d u c t i o n l o s s as the e x t r a amount of f o r e i g n exchange the producers could earn by having an o p t i m a l i n t e r n a l rearrangement of p r o d u c t i o n i n s t e a d o f one d i s t u r b e d by the e x i s t e n c e of d i s t o r t i o n a r y taxes and t a r i f f s . ) Another p o s s i b l e way of d e f i n i n g a p r o d u c t i v i t y improving change i n tariffs x  -  187 -  JU  is B  to r e q u i r e  Ap* + B p z  that  there e x i s t  J^  y^  J^  yu  Ap , Az , Ab , Ax , Ay  Az* + B, Ab* + B Ay* = B b y T  such  that  Ax* and Ab* > 0, Ay* > 0 „ , ' ^ N  J  T where B^ = [ - > - T  25.  ^ ^ K '  N  *  This  approach  corresponds  Debreu measure o f p r o d u c t i v i t y l o s s d e f i n e d M o t z k i n ' s Theorem:  i n Diewert  Let  * 0.  ,  (i)  and Q  Q^p »  (ii)  be g i v e n m a t r i c e s w i t h  3  to the (1983c).  Then, e i t h e r  0, Q P >^ 0, Q p = 0 has a s o l u t i o n or 2  + a Q 2  3  T  +  2  = 0 ,  > 0, a.^ !> 0, has a s o l u t i o n , but  never b o t h . Motzkin's Theorem and o t h e r theorems of a l t e r n a t i v e i n Mangasarian (1969: pp. 17-37).  are d i s c u s s e d  In o r d e r to show the e q u i v a l e n c e of (2.25) and (2.29), P  T  T  T  - [Ap* ,  T  Az* ,  [B , B , B, , -B ] . p z ' b x T such  must not e x i s t  26.  Choose p ]  T  , «  T  (0  Q  1  a  4  a  1  e R  N  +  K  1  +  =  [  ,  3  i=l,...,4,  = 0.  1 ]  that  X X * a„, a~x = - a . < 0. 3 3 4 — such  2  p  z  l ^ ®' p  Q  ,  E  q  u i v a  1, 0 ^ ] , Q  -  3  a, e R , a, e R 1 ' j  lently,  z  N  +  K  +  there  oJj , +K+1+M  T  [ B  u  = [0  p' z ' V  "V  B  V ^  e R, 0 ^ e R a\  c^B  B  M  ,  U  Then  '  l  (  N  +  R  )  ,  1,  t h e r e m u S tn 0 t  e R such that  oc > 0, 2  + <* = 0, - a ^ + a* = (£, 2  = - 0 ^ < 0, otj^B^ =  o£[B , B ] = 0^ , Z  + K  a, e R 1  Hence, t h e r e must, not e x i s t  N+K '  x  t h e r e must riot e x i s t v e c t o r s a^,  that a F [ B , B ] = o 3 , „ , 1  l x K  <^[B , B , - B ] =  ' 31 "  Equivalently,  such  0  T  p  3  ,  Ab* , A x * , r] , Q  > 0, and oc*[B , B j = 0 ^ ,  X * a x +  l x N  and (2.29) f o l l o w s .  °Mx(N+K+l)'  a  a  T  Az* ,  LK 1 M>  32 +  such t h a t = a  +  Q  - [0  l  °N+K+1+M'  =  T  = [Ap* ,  °(N+K+1) x l ' exist  3  Choose X  °lxN' ° ' 1 2 -  Q  Then, t h e r e must not e x i s t ' T  that CJQJ^ + "30.3  = ~a^< 0.  T  Ab* , A x * ] ,  choose  < 0, OTB X * < 0. l  b  '  I  T  —  N+K+l  Take X  T  T  = tx . 1  1  -  27.  is a  i n the magnitude of x . m  ft  ft  m  V  b (w + x ) * 0„ i s a l s o one of the s u f f i c i e n t c o n d i t i o n s f o r x M s t r i c t p r o d u c t i v i t y improving t a r i f f changes to e x i s t .  Chapter 1.  -  Note that an i n c r e a s e i n a n e g a t i v e x , m e [1,...,M], decrease  28.  188  3:  See Diewert  (1983c) f o r more d e t a i l s . *k  *k  *k  2.  The i n i t i a l e q u i l i b r i u m s o l u t i o n (y , f , z ), k = l , . . . , K, i s f e a s i b l e f o r the problem (3.1) but not n e c e s s a r i l y o p t i m a l .  3.  The YY^  4.  A l t e r n a t i v e l y , A(O g i v e s the net v a l u e of i n t e r n a t i o n a l l y traded goods produced by the e n t i r e p r o d u c t i o n s e c t o r , when the goods are v a l u e d at the i n t e r n a t i o n a l p r i c e s w.  5.  Diewert  6.  I f the d i r e c t i o n a l d e r i v a t i v e of the net balance of trade f u n c t i o n i n the d i r e c t i o n of the t a r i f f change i s s t r i c t l y p o s i t i v e , then f o r a s m a l l f i n i t e change of t a r i f f s i n t h i s d i r e c t i o n , the net b a l a n c e of t r a d e f u n c t i o n must be i n c r e a s i n g .  7.  The formula (3.23) c o n t a i n s an abuse of n o t a t i o n . The term A'(2) r e f e r s to the d e r i v a t i v e of the f u n c t i o n A(£) a t 5 = 1, where the v a l u e C = 1 corresponds to the new ( a f t e r the t a r i f f change) e q u i l i b r i u m , p r e v i o u s l y indexed as the e q u i l i b r i u m 2.  8.  See Diewert (1983c): pp. 169-170. Set 5 = 1 . Using formulae (28) and (30) i n Diewert (1983c), the e x p r e s s i o n (3.25) f o l l o w s . Note t h a t ( 2 8 ) , i n Diewert (1983c), c o n t a i n s a typo: the next to kT k l a s t term i n (28) should be x y ( ? ) z£ ( ? ) .  9.  The same abuse of n o t a t i o n as encountered p r e s e n t here. See f o o t n o t e 7.  10.  i m p l i c i t functions exist i s positive definite.  (1983c:  pp.  i f ( i ) rank Y = K _< N and  (ii) S  p p  +  169-170).  i n formula (3.23) i s  The government might be able to g i v e e s t i m a t e s f o r the net output m a t r i c e s Y^ and that determine the p r i c e d e r i v a t i v e s p ' ( 2 ) .  -  11.  Note that  i f N = K,  12.  Note that  if S  = 0 pw  Chapter  p'(l)  T  = x  189  1 T  -  F^Y  1 1  l  )  .  2  „, NxM  a l s o A„ = (1 - k ) G  XT  T  1 T  S  1  T  1  > 0. —  ww  4:  1.  Consumer p r e f e r e n c e s are assumed to be holds as an e q u a l i t y .  2.  I f some p u b l i c goods are i n p u t s i n t o the p r o d u c t i o n p r o c e s s , the p o s s i b l e nonzero pure p r o f i t s g e n e r a t e d by these f a c t o r s can be imputed to domestic f a c t o r s of p r o d u c t i o n c r e a t e d f o r t h i s purpose.  3.  & £ & The i m p l i c i t f u n c t i o n s u , p , z , and [A, - B , -B , - B, ] i s i n v e r t i b l e . p z b  nonsatiated  so that  (4.10)  & b  exist  4.  Note t h a t i n order to d e r i v e ( 4 . 1 6 ) , the (4.11) and (4.12) has been changed.  order  5.  The  strict  i f the  of the  matrix  equations  ft P a r e t o improvement i s r e q u i r e d  to be  (Au  »  0  instead ti  ft o f Au > OJJ) i n order to a v o i d problems w i t h a c t u a l ( f i n i t e ) changes i n u t i l i t i e s , as e x p l a i n e d i n Diewert (1978: pp. 154-155). Assuming that ( i ) the i m p l i c i t f u n c t i o n s that determine the endogenous v a r i a b l e s as f u n c t i o n s of the exogenous v a r i a b l e s of the model e x i s t i n the neighborhood of the i n i t i a l e q u i l i b r i u m , and t h a t ( i i ) the changes i n the exogenous v a r i a b l e s have been n o r m a l i z e d by r e q u i r i n g the sum of the changes squared to be one, ft ft ft ft the changes Au , Ap , Az and Ab may be i n t e r p r e t e d as d i r e c t i o n a l  ft ft ft ft d e r i v a t i v e s of the i m p l i c i t f u n c t i o n s u , p , z and b g i v e n w, * * * * *h x , t , s and g . I f f o r some consumer h, h e [1,...,H], 9u ft ft ft ft ft ft (w , T , t , s , g ) / 3 t = 0, m e [1,...,N], the consumer's u t i l i t y *h * u may a c t u a l l y decrease i f the tax r a t e t i s p e r t u r b e d from i t s n  from i t s i n i t i a l e q u i l i b r i u m v a l u e by a s m a l l f i n i t e amount. *h, * *h However, i f 3u /3t > 0, the f u n c t i o n u i s increasing in t at n n * t . n 6.  A condition sufficient T  T  T  a [X , E ]  = 0^  + M  to imply t h a t there  i s to r e q u i r e  that  i s no T  rank [ X ,  s o l u t i o n a > 0^ T  E ]  = H.  to  This  assumption means that the number of domestic commodities (N) p l u s the number of t r a d e a b l e goods (M) must be equal or g r e a t e r than the number of households (H) ( i . e . , H < N+M). I n t u i t i v e l y , i n order to  -  190 -  produce s t r i c t P a r e t o improvements, the government must have a s u f f i c i e n t l y l a r g e number of f r e e tax instruments (commodity taxes) in i t s disposal. 7.  Choose p  T  T  T  T  = [ A p * , A z * , Ab*,  At* ,  T  u  T °Hx(N+K+l+N+M+M) ' 12 ]  Q  =  -B,, -B„, -B , -B 1. b t s x a  J  [  T  As* , A T * ] , Q . . = [ I , T  °H+N+K'  There must  °N+M+M * ]  not e x i s t  Q  a  B  +  «  3  = 0» «  2  > 0 , H  a  3  =  [  A  '  B  "V  ~ z'  a, e R^+N+K+I 1  £ R such that of A + C L = 0?;, a?[-B , -B 1 z H l p  O-NVK+N+M+M' " l b  3  > 0.  , -B z  ^H  2  '  , -B , -B ] = t s x  Choose X  T  = - c £  a  n  d  (4.18) f o l l o w s . .8.  9.  By changing the producer p r i c e s , the government can induce a change i n p r o d u c t i o n that corresponds to the change i n consumer demand caused by a P a r e t o improving tax change. T h i s i s because a l l the commodity adjusted  i n Theorem 4.1.  tax r a t e s ( t * ,  s*) can be + x  I f some producer p r i c e (w m  ), m  *  m e [ 1 , . . . , M ] i s changed, the t a x r a t e can be p e r t u r b e d so t h a t the e f f e c t of the t a r i f f change on the consumers i s z e r o . T h i s k i n d of s e p a r a t i o n of p r o d u c t i o n and consumption s e c t o r s i s not present Theorem 6.1 i n Chapter 6. 10.  The consumers and producers i n the economy choose t h e i r net demands and ( n e t ) s u p p l i e s by maximizing u t i l i t y and p r o f i t s , respectively. The government can i n f l u e n c e these ( n e t ) demands and (net) s u p p l i e s i n d i r e c t l y through changes i n r e l a t i v e p r i c e s o b t a i n e d by imposing taxes and t a r i f f s on commodities. The government cannot d i r e c t l y choose the consumers' ( n e t ) demand v e c t o r s or the producers' ( n e t ) s u p p l i e s .  11.  The e q u i l i b r i u m that s o l v e s (4.26) i s u s u a l l y a second best e q u i l i b r i u m , s i n c e the lump sum t r a n s f e s g have not been chosen optimally. Only by a c c i d e n t — - i f g happened to be f i x e d at the o p t i m a l l e v e l — c o u l d the e q u i l i b r i u m be a f i r s t best optimum.  12.  See S e c t i o n  13.  See (2.37) i n S e c t i o n  14.  I f N = 0, c o n t r o l l a b i l i t y of domestic goods p r o d u c t i o n cannot cause problems and assumption ( i i ) i s thus not needed. I f N = 0, d i s c o n t i n u i t y of s e c t o r a l net s u p p l i e s cannot occur i f the u n i t p r o f i t f u n c t i o n s T T ^ ( W + T * ) , k=l,...,K, are twice c o n t i n u o u s l y differentiable. Hence, assumption ( i ) i s not needed.  2.4. 2.4.  -  15.  191  -  T h i s i s a v e r s i o n of the McKenzie F a c t o r Theorem; McKenzie (1955).  Price Equalization  Chapter 5: 1.  3  The p r i c e s w are obtained by s o l v i n g the f o l l o w i n g g e n e r a l e q u i l i b r i u m model: (i) m (ii) (iii) (iv) (v) (vi)  h  h  autarky  h  ( u , q, v) = g , h=l,...,H k  7r ( p , w + x) = 0, k=l,...,K EV  1  v  EV^ m x =  1  m*  (u^ , q, v) + e^ = E V  h  0  h  ( u , q, v) + x ° = I V  TT  ( p , w + x) z  k  i r ( p , w + x) z  k  K  w  k  p  M  Wj = 1 ( p r i c e  normalization).  2.  Note t h a t the t a r i f f s x* do not g e n e r a t e income to the government. Hence, the a u t a r k y government budget c o n s t r a i n t i s satisfied.  3.  Note that i n order to reach p o s i t i v e g a i n s from trade x need not be reduced to zero ( f r e e t r a d e ) .  4.  ( t , s ) are not g-optimal, i f they do not s o l v e  ie  the  tariffs  ie T  max {B u: (4.10) - (4.13) h o l d , u,p,z,b,t,s constant}.  x* = (w  a  the problem - w), g* =  5.  Welfare e f f e c t s of i n t e r n a t i o n a l t r a d e that i s caused by changes i n the c o u n t r y ' s commodity tax s t r u c t u r e a r e f u r t h e r d i s c u s s e d i n Chapter 11.  6.  T h i s change i n v o l v e s a s e r i e s o f i n f i n i t e s i m a l toward x* = Oj»i.  7.  F i n i t e s t r i c t P a r e t o and p r o d u c t i v i t y improving changes In taxes and t a r i f f s do e x i s t at A.  Chapter  changes of x  6:  1.  Note t h a t the v e c t o r s* i s kept f i x e d . Hence, a change of t a r i f f s x* a f f e c t s both consumers and p r o d u c e r s .  2.  A n e c e s s a r y c o n d i t i o n f o r x to be P a r e t o and p r o d u c t i v i t y o p t i m a l i s t h a t the v e c t o r X^B_ i s zero; see P r o p o s i t i o n 4.2.  ft  - 192 -  3  -  I  F  1  =  T  °(N M)X(N M)' +  H  E  E  ^  +  A  T  L  O  N  S  *  - I = <>£•. X X T  °l i»piy  \ -  T h e r e I s no p o s i t i v e s o l u t i o n t o t h e s e e q u a t i o n s , ,  rri  A, = 0 „ . 1 H T  *  T  T h e n , A A = ( p + 5)  =0... I t i s known t h a t M  scaling  of u t i l i t i e s ) .  rr%  ~  = 0 «  This  M  B  p  Hence,  rpi X  N+K+N  '°M4.V4. .  =  r  E + w E = p qu vu  A B  b  T  E i f vu  E = 1„ ( m o n e y vu H  for this  A, A A > 0 , X [ B , B , B ]  V  T  metric  J  T  T  R  JU  JL^rri =  x  £ + w qu  E + w qu  rrt  K ^ 0 and  but choose  * T  T  p  JU  z  t  r^i  G(w + T , y ) = ww  oA M s i n c e  * T  M  Pareto  i s sufficient  to imply  that  no ( d i f f e r e n t i a l )  and p r o d u c t i v i t y i m p r o v e m e n t s e x i s t ,  starting  strict  from the T  h  e  i n i t i a l e q u i l i b r i u m , i f T = C> and E = ° ( ( ) • initial e q u i l i b r i u m thus s a t i s f i e s a n e c e s s a r y c o n d i t i o n f o r P a r e t o and p r o d u c t i v i t y o p t i m a l i t y . F u r t h e r m o r e , t h e i n i t i a l e q u i l i b r i u m must be. a p r o d u c t i v i t y maximum s i n c e , u n d e r f r e e t r a d e , t h e amount o f f o r e i g n e x c h a n g e e a r n e d by t h e p r o d u c t i o n s e c t o r i s m a x i m a l ( s e e p r o b l e m ( 3 . 3 ) ) , and i f the consumer s u b s t i t u t i o n m a t r i x E i s a z e r o m a t r i x , e x i s t e n c e o f n o n z e r o commodity t a x e s does n o t i n f l u e n c e t h e c o n s u m e r n e t demand y i n ( 3 . 3 ) . M  4.  N + M ) x  N + M  See T h e o r e m 6.1. A s u f f i c i e n t condition implying assumption ( i i i ) i n T h e o r e m 6.1 i s t h a t r a n k X = H. T h i s means t h a t H _< N, i . e . , t h e number o f v a r i a b l e t a x i n s t r u m e n t s must be a t l e a s t a s l a r g e a s t h e number o f c o n s u m e r s ( o r c o n s u m e r g r o u p s ) i n t h e economy. T  5.  A sufficient  c o n d i t i o n f o r a s t r i c t P a r e t o p r o d u c t i v i t y and it it i n x and t , n e [ 1 , . . . , N ] , t o e x i s t i s : t h e r e  improvement  A e  H  +  N  +  K  +  1  x [ A , - B. ] > oL, , A [ B , B , ( B ) ] T  s  R  u  c  h  t  h  a  i s no  t  T  '  b  H+l '  p  z'  t *n  T  = oJ. .. , A BT = o M j. N+K+l' Vj  C h a p t e r 7: 1.  I f a g o o d m, m e [ 1 , . . . , M ] , T  it  i s a net export  f o r s e c t o r k, k = l , . . . , K ,  it  > 0 ( T < 0 ) means t h a t p r o d u c t i o n o f t h e g o o d i s s u b s i d i z e d m — m — ( t a x e d ) , w h e r e a s i f t h e good i s a n e t i m p o r t f o r s e c t o r k , * * k=l,...,K,  j>  0  (  T m  N . O )  means t h a t  the net i m p o r t  Is being  taxed  (subsidized). JL  2.  JU  C o n s i d e r a n i n c r e a s e i n T when x >^ Oj^. A s u f f i c i e n t c o n d i t i o n f o r a s t r i c t P a r e t o and p r o d u c t i v i t y i m p r o v e m e n t is:  to e x i s t  -  there  does not e x i s t a v e c t o r X l  193 -  A e R  such that  B  A [A, - B ] > b  X  °H+1' t p> V V V " °N+K+N+M' \ < °M' U s i n g the same reasoning as i n the proof o f Theorem 7.1, c o n d i t i o n can be shown to be s a t i s f i e d .  this  •k  For a p r o p o r t i o n a l r e d u c t i o n i n t a r i f f s x to be s t r i c t P a r e t o and p r o d u c t i v i t y improving, i t i s s u f f i c i e n t to show that there i s no v e c t o r A e such that A [A, - B j > 0?; . , A [ B , B , B , b H+1 p z t H  +  N  +  K  +  1  T  T  R  B ] = 0 ,„,. ,„, A B x s N+K+N+M' x M  > 0. —  7  For a v e c t o r  rp  A satisfying °  A [B , B , p' z'  X T  A  J  rp  B  0  A  . B ] = 0.,^..^.., the s c a l a r A B x e q u a l s -x V G(w + x , y ) t' s N+K+N+M' x ww '• * 2 / * *v x . T h i s number i s n e g a t i v e , s i n c e the matrix V G(w + x , y ) i s ww A *. T H  positive  s e r a i d e f i n i t e and,  J  by assumption, V^b  (w + x ) * 0^.  ft Note that  i f x^ _< 0, m e [1,...,M], t h i s  change amounts to an  * increase  i n the a b s o l u t e  value  of x . m  The assumption  that  ft ft V^b  (w + x ) i s negative  can be i n t e r p r e t e d as f o l l o w s : i f  ft x >_ 0 (net e x p o r t s are s u b s i d i z e d and n e t imports a r e t a x e d ) , s u b s t i t u t i o n i n p r o d u c t i o n o f t r a d e a b l e s must dominate i n the sense t h a t x V G(w + x , y ) > CL,. In t h i s case, the ww M '  * uniform reduction  of x  i s equivalent  to a uniform r e d u c t i o n of  ft s u b s i d i e s f o r net expors and taxes f o r net imports. Ifx 0^ (net e x p o r t s are taxed and net imports s u b s i d i z e d ) , complementarity i n p r o d u c t i o n must dominate i n the sense that A P o at x T* A x V G(w + x , y ) < 0... Then, the u n i f o r m r e d u c t i o n o f x ww ' M ' amounts to a u n i f o r m i n c r e a s e i n the e x p o r t taxes and to a u n i f o r m i n c r e a s e i n the Import s u b s i d i e s . r  J  * Use  the proof A  A  of P r o p o s i t i o n A  A  A  7.1 and note that when ( x - x) > 0 , A  A  A  T*  V b (w + x , y ) (x - x) < 0 i f V b (w + x , y ) < CT. x x M  -  194  -  6.  I t can a l s o be shown that a s t r i c t P a r e t o improving i n c r e a s e i n p o s i t i v e t * , s and T * and a s t r i c t P a r e t o improving decrease i n n e g a t i v e t * , s* and x* e x i s t .  7.  In Chapter 8 i t w i l l be shown t h a t i f lump sum t r a n s f e r s are a v a i l a b l e , the simultaneous r e d u c t i o n i n taxes and t a r i f f s can chosen to be a p r o p o r t i o n a l r e d u c t i o n . A  8.  If t  A  = C> , N  A  s  be  A T  = 0^,  =  x^ = 0^,, e^ = 0^,  a n M  f\  AT'  c o n s t r a i n t becomes p  °  x  d  g  =  0 ,  the government budget  R  (~\  TP  + w e  =0,  which can o n l y h o l d i f e i t h e r  or i f the government behaves l i k e a p r i v a t e  p r o d u c e r w i t h the government budget c o n s t r a i n t s e r v i n g the r o l e of the zero p r o f i t c o n s t r a i n t . T h i s i m p l i e s that some of the components of x^ and e^ must be n e g a t i v e , c o r r e s p o n d i n g to i n p u t s bought to produce o t h e r domestic or t r a d e a b l e goods. 9.  10.  Government p o l i c y c h o i c e s and Chapter 12. Notation:  Chapter  t = -n  economic i n e q u a l i t y are d i s c u s s e d i n  [t, , ... , t 1 n-1  t n+1  ,t  1 T  ]  T  e R  N  1  .  N  8: T  1.  Assumption (8.2) i s s a t i s f i e d ,  i f f o r example, the m a t r i x S  +  YY  PP i s p o s i t i v e d e f i n i t e , i . e . , (2.13) i s s a t i s f i e d , or i f the m a t r i x Z i s of f u l l rank N. T h i s o c c u r s when the matrix Z i s of maximal qq rank (= N+M-l). A  A  the changes i n u , t  A  2.  Note that  and  s  are going to be  3.  D i f f e r e n t i a t e (4.12) and (2.9) around the i n i t i a l v a l u e s of the variables. The assumptions (2.12) and (8.2) are n e c e s s a r y and A  sufficient A  f o r the i m p l i c i t  A  z ( u , w + x , Lemma 2.1. 4.  A  A  A  A  zero.  A  A  f u n c t i o n s p (u , w + x , t , s ) and  A  t , s ) t o  exist.  Compare t h i s  to the proof of  Note that l o c a l c o n t r o l l a b i l i t y of p r o d u c t i o n i n the sense of D e f i n i t i o n 2.1 i s s u f f i c i e n t to imply l o c a l c o n t r o l l a b i l i t y of domestic goods p r o d u c t i o n i n the sense of D e f i n i t i o n 8.1 ( i f rank S = N+M-l, the rank of the m a t r i x S must be N). However, l o c a l c o n t r o l l a b l i t y of p r o d u c t i o n i n the sense of D e f i n i t i o n 2.1 i s not n e c e s s a r y f o r l o c a l c o n t r o l l a b i l i t y of domestic goods p r o d u c t i o n i n the sense of D e f i n i t i o n 8.1. p p  -  195  -  5.  I f there i s no s u b s t i t u t i o n i n consumption of domestic goods, the requirement that the consumers are kept at t h e i r i n i t i a l u t i l i t y l e v e l s means t h a t the net supply of domestic commodities must stay at the i n i t i a l y . Hence, D e f i n i t i o n s 2.2 and 8.1 of l o c a l c o n t r o l l a b i l t y of domestic goods p r o d u c t i o n coincide.  6.  The  matrix £ q q  7.  See  the  8.  The s e t s C , k=l,...,K, and M (u ), h=l,...,H, are assumed to be c l o s e d and convex; i n a d d i t i o n , the S l a t e r c o n s t r a i n t q u a l i f i c a t i o n c o n d i t i o n i s assumed to be s a t i s f i e d . ( I f the u t i l i t y f u n c t i o n s of the consumers are quasiconcave and continuous from above, the s e t s M , h=l,...,H, are convex and c l o s e d . I f there i s a f e a s i b l e s o l u t i o n f o r (8.5) such that the i n e q u a l i t y c o n s t r a i n t s are s t r i c t l y s a t i s f i e d , the S l a t e r c o n d i t i o n i s s a t i s f i e d . )  9.  Note that  -  is positive  proof of Lemma  the  f u n c t i o n B(w)  2.2.  conditions to be  also s u f f i c i e n t  semidefinite.  (2.12) and  (8.2)  twice c o n t i n u o u s l y  f o r the  function  V B w  are  sufficient  differentiable. to be  once  for  the  They  are  continuously J  differentiable. 10.  T h i s c o u l d be seen a l s o by n o t i n g that the f u n c t i o n B(w) must be convex i n p r i c e s w, as the f u n c t i o n B i s convex i n i t s arguments.  11.  The weak i n e q u a l i t y f o l l o w s the m a t r i x (S - E ) .  12.  I f both m a t r i c e s S and  from the  positive semidefiniteness  k  the m a t r i c e s S and k  E are  of maximal rank and  of  k  t  = 0„, s = N E have a common zero e i g e n v e c t o r , i . e . , the  0 , M W  k  v e c t o r (p , w + T ). the matrix (S - E ) .  This vector  i s also  the  zero e i g e n v e c t o r  of  13.  The e q u i v a l e n c e of these concepts i s easy to v i s u a l i z e i f i t i s remembered that a zero consumption s u b s t i t u t i o n m a t r i x E corresponds to L-shaped i n d i f f e r e n c e curves i n the two commodity case.  14.  Theorem 8.3 c o u l d be i l l u s t r a t e d u s i n g F i g . 3. The P a r e t o i m p r o v i n g change i n t a r i f f s and lump sum t r a n s f e r s s h i f t s the economy's p r o d u c t i o n choice toward the p o i n t C, whereas the consumer i s moved toward D.  15.  See  C o r o l l a r y 8.2.1.  -  196 -  16.  Y e t , f i n i t e P a r e t o and p r o d u c t i v i t y improving changes i n t a r i f f s and t r a n s f e r s may e x i s t .  17.  I f rank E = N+M-l, then rank E E  18.  qq  = N.  Assuming that N >_ 1,  must be nonzero.  Hence, e * 0 „ i f t ' N  *  *  *  If t  * 0„. N  = 0.., s = 0.,, x = 0 and lump sum t r a n s f e r s a r e chosen N M M o p t i m a l l y , the i n i t i a l e q u i l i b r i u m i s a p r o d u c t i v i t y maximum, i . e . , the f i r s t best e q u i l i b r i u m the economy can a t t a i n under i n t e r n a t i o n a l trade. W  C h a p t e r 9: 1.  2.  Kemp and Wan (1983). T T T Suppose E = C) „ and x = (x, , x „ ) i s an a r b i t r a r y nonzero (N+M)qq NxN 1 2 T T T T vector. Then, x Ex = x„ E x. + x. E x„ + x„ E x„ < 0, s i n c e 2 v q l l q v 2 2 w 2 — ' the matrix E i s n e g a t i v e s e r a i d e f i n i t e . I t i s a l s o known that T x„ E x„ < 0, because E i s negative semidefinite. I f the m a t r i x 2 vv 2 — vv £  q v  the 3.  were nonzero, the v e c t o r x c o u l d i n e q u a l i t y x^ Ex _< 0.  If E qv  = 0 „, then E 2x2 w o  which i m p l i e s  v  be chosen so as to v i o l a t e  Hence, E q = % M « V  X  * = 0 „ . I n the Kemp-Wan example, s* = 0„ 2 ' 2  E (w + x ) = 0 „ . vv 2  r  The i n t e r n a t i o n a l  r  trade  * prohibitive  t a r i f f s x i n the Kemp-Wan economy a r e n o t p r o p o r t i o n a l T to w. Then, w E must be nonzero because the m a t r i x E i s of vv vv maximal rank (=1) i n t h i s example. 4.  The matrix V  w  f(w + x , y ) =  G(w + x , y ) i s a zero  (2x2)-  m a t r i x at A. 5.  I t i s assumed t h a t the tax r a t e s ( t , s ) are chosen DiamondM i r r l e e s o p t i m a l l y w i t h r e s p e c t to the c o n s t r a i n t s that * * x = x (= w - w) and g = c o n s t a n t . Otherwise, a w e l f a r e improving change i n o n l y ( t , s ) would e x i s t . a  -  6.  197 -  To be exact, one must assume that the i n i t i a l t r a n s f e r s g a r e not Diamond-Mirrlees o p t i m a l w i t h r e s p e c t to the c o n s t r a i n t s x = x (= w - w) and t = t , s = s . a  7.  There must be s u b s t i t u t i o n i n consumption, i . e . , E * ^(N+2)x(N+2)*  8.  See the rank Y = K (< N) assumption i n Theorem 8 . 3 . I f each p r o d u c t i o n s e c t o r s u p p l i e s o n l y one good, as i n the Kemp-Wan example, the requirements t h a t each t r a d e a b l e good i s produced i n a u t a r k y and that each s e c t o r operates a t a p o s i t i v e s c a l e i n a u t a r k y are e q u i v a l e n t .  9.  T 2 For w V  " T T T B(w) = 0.. to be s a t i s f i e d the v e c t o r [-(p + e) , w ] WW M i \r / > J must be a zero e i g e n v e c t o r o f the matrix (S - E) (see 8 . 1 8 ) . I t f o l l o w s t h a t , s i n c e the m a t r i c e s S and -E a r e p o s i t i v e s e m i d e f i n i t e , T T the v e c t o r [-(p + e) , w ] must be a zero e i g e n v e c t o r o f S and E. * * I f , however, the v e c t o r s w and x a r e n o t p r o p o r t i o n a l , x t 0^, and  the m a t r i c e s T  ([p ,  T  (w + x * ) ]  S and E have only S = 0jJ T  the v e c t o r  +M  ,  one zero  eigenvector  T  T  [ ( p + t ) , (w + x* + s ) ]  E = 0^  +M  ),  then  T  [-(p + e) , w ] cannot be a zero e i g e n v e c t o r  of S o r E.  Chapter 10: 1.  I f x* = C^j, t h i s  c o n d i t i o n cannot be s a t i s f i e d  since  V b*(w + x*)w = w V B(w)w > 0 i f V b*(w + x*) * 0, ,. x ww x M T  2.  F o r example, i f t  T  2  »  0^,  there  e x i s t s a uniform l e v e l of  domestic commodity taxes t (> 0^)  such that a small  of the i n i t i a l tax r a t e s i n the d i r e c t i o n and p r o d u c t i v i t y i m p r o v i n g .  *  perturbation  _  ( t - t) i s s t r i c t  Pareto  3.  T h i s c o n d i t i o n a l s o appears i n H a t t a and Fukushima (1979: p. 509); Fukushiraa (1979: p. 431); D i x i t (1975: p. 107); Smith (1980: pp. 8 - 9 ) ; H a t t a (1977: pp. 1865-66).  4.  Note that t h i s corresponds t o the e a r l i e r i d e a of a c o n s t a n t u t i l i t y p r o d u c t i v i t y improvement which can be converted to a P a r e t o improvement through a p e r t u r b a t i o n i n the i n i t i a l v e c t o r of lump sum t r a n s f e r s .  -  198  -  5.  i s an u n s t a b l e e q u i l i b r i u m . H a t t a shows that the e q u i l i b r i u m i n the economy d e s c r i b e d by h i s model i s s t a b l e , i f the H a t t a normality condition is s a t i s f i e d .  6.  Diewert  7.  t  ^T  (1983b):  Theorem  2 [ V _ B] < 0 i s one tt *n  6.  of the s u f f i c i e n t  conditions for a  * r e d u c t i o n i n a p o s i t i v e domestic tax r a t e t , n e [1,...,NJ, to be s t r i c t P a r e t o improving (assuming that the t r a n s f e r s g are adjusted a p p r o p r i a t e l y ) . 8.  See  formula (12) i n H a t t a  9.  Lemma 2 i n H a t t a (1977a).  (1977a). Note t h a t , u s i n g formula (12) i n H a t t a  (1977a),  i f o n l y one  tax r a t e  t i s reduced, n  for  change to be w e l f a r e i m p r o v i n g  a sufficient JUrrt  this  which i s (10.31).  i s that p  condition JL  (£^[p  Note that i n H a t t a ' s n o t a t i o n E  ]). > n  0,  = F. qq  Chapter  11:  1.  M e l v i n (1970: p.  2.  See  3.  The proof of Theorem 11.1 c o u l d a l s o be used to e s t a b l i s h that there e x i s t some s t r i c t Pareto and p r o d u c t i v i t y improving i n c r e a s e s i n the p o s i t i v e commodity tax r a t e s ( t * , s * ) , o r s t r i c t P a r e t o and p r o d u c t i v i t y improving r e d u c t i o n s i n the n e g a t i v e ( t * , s ). In the f i r s t case, the i n c r e a s e d taxes on commodities bought by the consumers are balanced with i n c r e a s e d s u b s i d i e s on t h e i r work effort. In the l a t t e r case, taxes on f a c t o r s s o l d are i n c r e a s e d s i m u l t a n e o u s l y w i t h i n c r e a s e s i n s u b s i d i e s on commodites demanded by the consumers.  4.  Suppose W(u)  problem  68).  (4.26).  T  ft  ft  = l u. I f the v e c t o r s t = 0. and s = 0., are H N M . o p t i m a l when T = 0^, there must e x i s t a v e c t o r X e H+N+K+1 I T  T  R  that  XA - ij. - x \  (See  (4.28)).  T  A  > 0 and  T  X [B ,  vector satisfying  B  p  B ,  z >  f  B] = o j ^ ^ . g  T  X [B , B , B 1 = p  Z  s u c  t  oF.^^.. N+K.+N  and  |  1  -  A  -X B  b  proof  > 0 i s of the form A of Theorem 2.1.  (2.27).  Choose A. = 0 . I n U  T T A A = 1 . R  (Note that T  that P J : + wr = qu vu T  If x  199 -  1  = [ A , (p =  + 6) , y , 1] using the  ° » then 6 = 0 M  Then, A  T  N  and y = 0^,  using  T  = [ 0 ^ , p * , 0^, 1] and U K .  the money m e t r i c  s c a l i n g of u t i l i t i e s  implies  ih. H  C h a p t e r 12: 1.  I t w i l l be assumed i n t h i s c h a p t e r that a l l consumers u t i l i t i e s , which are measured using a money m e t r i c , are f u l l y comparable. I n the p r e v i o u s c h a p t e r s , when o n l y P a r e t o improvements were c o n s i d e r e d , no c o m p a r a b i l i t y assumption was needed. Y e t , when the s o c i a l w e l f a r e f u n c t i o n W(u) = 8 u , 8 > 0^, was i n t r o d u c e d , i t was i m p l i c i t l y assumed that the household u t i l i t i e s a r e f u l l y comparable. T  2.  The n o t i o n that only the households' n o n - l a b o r incomes are s h i f t e d toward e q u a l i t y u s i n g government tax and t r a n s f e r p o l i c i e s corresponds to Diewert's (1984) C a p i t a l Income F a i r e q u i l i b r i u m concept. T h i s i n t u r n i s based on V a r i a n ' s (1976) i d e a of Opportunity Fairness.  3.  I t turns out to be convenient to denote the consumer u t i l i t i e s by uft> h = 1,...,H, i n s t e a d of u , h=l,...,H. Note that the d e f i n i t i o n of consumer e x p e n d i t u r e s i n (12.1) d i f f e r s from the d e f i n i t i o n of m i n the e q u a t i o n ( 4 . 1 0 ) . I n (12.1), the consumer e x p e n d i t u r e s are d e f i n e d as the d i f f e r e n c e between the value of ( g r o s s ) purchases of domestic and t r a d e a b l e commodities minus the v a l u e of any l a b o r s e r v i c e s s u p p l i e d . I n (4.10), the consumer e x p e n d i t u r e s were c a l c u l a t e d as the d i f f e r e n c e between the v a l u e of the consumers' net consumption and the v a l u e of t h e i r labor supply. n  n  4.  Note that the observed i n i t i a l e q u i l i b r i u m consumer p r i c e s (q , v ) a r e chosen to serve as the r e f e r e n c e p r i c e s f o r the money m e t r i c u t i l i t y f u n c t i o n s . Hence, a l l ^ i n t e _ r p e r s o n a l u t i l i t y comparisons In t h i s s e c t i o n are based on (q , v ) . The choice of the s o c i a l w e l f a r e f u n c t i o n W(u) i s a v a l u e judgement. An economist can only suggest p r o p e r t i e s of the f u n c t i o n W(u) that the s o c i e t y might t h i n k as d e s i r a b l e . The f u n c t i o n W(u) may, f o r example, be r e q u i r e d to be q u a s i c o n c a v e , symmetric, i n c r e a s i n g , and c a r d i n a l l y scaled. The l a s t p r o p e r t y means that s o c i a l w e l f a r e i s i n c r e a s i n g along the e q u a l l y d i s t r i b u t e d r e a l income l i n e . (This a l s o means that W(Aljj) = A f o r a l l A e R.) An example o f a  - 200 -  s o c i a l w e l f a r e f u n c t i o n t h a t s a t i s f i e s the above mentioned c o n d i t i o n s i s the mean o f o r d e r r f u n c t i o n H j 1/r W(u) = [ E i uJ] , r < 1, r * 0, u _> 0 , h=l 1  ? II  /  H  , r = n0. h=l F o r an i n c r e a s i n g W(u), the w e l f a r e w e i g h t s S a r e p o s i t i v e . I f the government wants to a s s i g n n e g a t i v e w e i g h t s to those households whose incomes exceed some " a c c e p t a b l e " upper l i m i t , the government u  can use the mean-variance s o c i a l w e l f a r e f u n c t i o n W(u) = u ,T 1 T H y [ - ( U - u l ) ( u - u l ) I , u = - j j — , y > 0. U  R  R  5.  ft U s i n g F i g u r e 9, a r e d u c t i o n i n I a l s o c o r r e s p o n d s to a s h i f t toward the u\ = U2 l i n e s t a r t i n g from the i n i t i a l e q u i l i b r i u m a t S ( i f the s o c i a l w e l f a r e f u n c t i o n W i s s t r i c t l y q u a s i - c o n c a v e ) . I f W i s u t i l i t a r i a n , the measure I i s i d e n t i c a l l y z e r o r e f l e c t i n g the governments l a c k o f i n t e r e s t i n the d i s t r i b u t i o n of r e a l income i n the economy.  6.  See Theorem 2.1.  7.  At l e a s t one household must have nonzero ( n e t ) demand o r s u p p l y o f the good i n q u e s t i o n .  8.  Note t h a t i f 8 = 0 , i . e . , 8 U  n.  =  f o r a l l h, h=l,...,H,  strict  H.  i n e q u a l i t y r e d u c t i o n s u s i n g commodity t a x and t a r i f f p e r t u r b a t i o n s do n o t e x i s t . I n t h i s case, t h e s o c i a l w e l f a r e f u n c t i o n W(u) i s u t i l i t a r i a n , o r t h e r e a l incomes u , h=l,...,H, have been e q u a l i z e d a t the i n i t i a l e q u i l i b r i u m . n  9.  I t i ssufficient that  X [B , T  T  4  B ,  Z  X B  equations implies X  B ,  P  G  RH N K+1  to show t h a t t h e r e i s no v e c t o r X e G  = 0  B ] = 0 ^ ^ ,  XA = 8 , X  imply X  The i n e q u a l i t y  R  = k >_ 0.  T  t  1  = 0 . R  T  T  I f k=0, then X A = 0  T  T  R  * 8 . T  B  FA  +  +  < 0. The X B T  B  _< 0  ( F o r k :> 0, t h e T  proof of Theorem 8.3 shows t h a t X^ = k ( p * + e ) , X^ = k 0 ) . I f T k = 1, then X B T  = w  T  2 ~ T 7 B(w) * 0,, by a s s u m p t i o n . WW M •* r  201 -  10.  I t i s n e c e s s a r y f o r a s t r i c t P a r e t o improving change i n commodity taxes and t a r i f f s to e x i s t that the g e n e r a l i z e d Diamond-Mirrlees condition (4.17) i s s a t i s f i e d . But i t can be seen t h a t i n s p i t e of the c o n d i t i o n ( 4 . 1 7 ) being s a t i s f i e d , the v e c t o r 8 [X  - C ,E  third  asumption i n Theorem 1 2 . 1 . ( T h i s i s because 8 i OJJ.) ( I f  B^fX^  - C ^ , E^ - D^l = 0?L.„, . N+M  - D ] may s t i l l  i n e q u a l i t y reductions equilibrium.)  11.  be equal to zero, v i o l a t i n g the  i t can be shown that no s t r i c t  can e x i s t , s t a r t i n g from the i n i t i a l  There may e x i s t an a > 0 „ such that a [X . H the v e c t o r  0 [X  - C ,E  - c" , E  I f equation (12.16)  T  - D ] i s nonzero (8 i OJJ)..  rri  12.  - if ] = 0 . . „ b u t N+M  rri  i n the form 8 [X  i s written  rri  T T T T = - r 8 [X - C , E - D ] , i t can be seen that there to ( 1 2 . 1 6 ) X E  i f the v e c t o r s  8 [X T  T  X - D  ] a r e of the same s i g n .  - C , E T  T  rti  - D~ ]  i s no s o l u t i o n  - D ] and 3 [ X T  rii  - C" , E  T  T  - CT , T  - 202  -  REFERENCES  B l a c k o r b y , C. and D. Donaldson [1977], " U t i l i t y vs E q u i t y , " J o u r n a l of P u b l i c Economics 7, 365-381. Diamond, P.A. and J.A. M i r r l e e s [1971], "Optimal T a x a t i o n and P u b l i c P r o d u c t i o n , I - I I , " American Economic Review 61, 8-27 and 261-278. D i e w e r t , W.E. [1976], "Exact and S u p e r l a t i v e Index Numbers," J o u r n a l of Econometrics 4, 115-145. Diewert, W.E. 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Allocation,  - 204  -  APPENDIX 1  P r o o f f o r Lemma  2.3:  T T C o n s i d e r the e q u a t i o n s X [B , B ] = 0„, ,: p z N+K T  n  (Al)  T  T  T  [Xj, Xj,  X^]  = 0, N+K  pp T KxK T w S  wp  T w F  * Set X. = k £ R. 3 T  p* Y + (w + x * )  Using T  F = Cv  the e q u a t i o n s ((2.7) and  S  pp  p + S  (2.5)),  (w + T ) = 0„ and pw N  ( A l ) can be w r i t t e n as  * T  (A2)  Y  S  * T  X  = k[x S  PP  wp  + p  * T  S  , x pp'  * T  F + p  Y]  T  Y  Assumptions  0KxK  (i) - (iii)  i n Theorem 2.1  hand s i d e of (A2) can be i n v e r t e d and defined i n (2.20). and  The  T T (X. , X„ ) :  that the matrix on  the  left  the i n v e r s e i s the matrix D  p r o p e r t i e s of t h i s m a t r i x are g i v e n i n Diewert  Woodland (1977: Appendix).  vector  imply  Equations  (A2)  can be s o l v e d f o r the  - 205 -  [x[, XJ]  (A3)  = k[(p* +  T  T  S), ] , Y  whe re  (A4)  5  T  T  = x* [S  D.. + FD* ] , y wp 11  Equations  D  :  D  12  Proof  = D  S  pp  Y  ]  D  =  [ I  wp 12  0 0  ].  22  the f o l l o w i n g p r o p e r t i e s of the matrix  ]  N'°NxK -  Q  E D  X  T  (2.11),  B = -[(p*+ 5 ) S T pw T  T  + Y F  T  Applying  the d e f i n i t i o n s  T  X B  T  t  = -k[p* S  T  T  + wS  + x* (S  ].  WW  of the v e c t o r s  T  pw  T  + wS  r  (A6)  D._ + F D  of Lemma 2.4: Using  (A5)  [  T  = x* [S  12  (A3) are d e r i v e d u s i n g  21>  T  5 and y i n ( A 4 ) ,  D.. + FDj„)S + T* (S D._ + F D „ ) F wp 11 12 pw wp 12 22 T  T  0  ] WW  = -k[T* (S T  T  D..S + FD^„S + S D F + wp 11 pw 12 pw wp 12 1 0  FD F ) - T* S ] 22 ww T  0 0  T  - 206 -  s i n c e , by the homogeneity of the u n i t  -T  profit  functions, p  S + w S pw ww  *T  S . ww  Using  the e q u a t i o n s D..S D, l l p p l l  &  M  = D,., D,,S D,„ = 0 and 11' 11 pp 12  T D  D  12^pp 12  may  =  D  ~ 22 §  i  v  e  n  i  n  D  i  e  w  e  r  t  and Woodland (1977:  Appendix), (A6)  be w r i t t e n as  (A7)  T  T  X B  = kt* [-S  D.. - FD 12 wp 11  x  -D..S - D.„F 11 pw 12  :  hi 9  * T  kx  V  *  G(w +  T  *  , y ),  WW  u s i n g Lemma 2.2.  QED  Lemma 2.5: T  k  k  = k(w + T ) ,  k  k * 0,1, i f and o n l y i f -5 = kp .  Proof: k  T  Suppose  k  k  = k(w + T ) , k * 0,1.  [ I f k = 1, then w = 0^ which v i o l a t e s  Then, T  = aw with a = k / ( l - k ) .  the assumption that w »  k  k  k = 0, then T Using  (A8)  = 0^, which v i o l a t e s  the d e f i n i t i o n  6  T  0^. I f  T  = x* [S  the assumption that  x  of the v e c t o r 6 i n ( A 4 ) , i f x* = aw,  D.. + FTVJ wp 11  12  T  = aw [S  D. . + FDT,]. wp 11  12  * ^M*^  - 207 -  The homogeneity of the u n i t p r o f i t f u n c t i o n s and the  [  S  Y  P  ]  D  (A9)  6  [I  =  P  T  <W  M>  ^y  ±m  T  = a[(- * S P  T  - x* S )  p p  w p  = -ap  S *T  r  D  W FD{ T  N  +  *T  AT  = -ap  at  D.,-  ppll  - ap  *T  T  T YD:,,  12  - ax  *T  S  = -a[p  L  + x  D..  L  S  wp  Then, 6  T  3  = --p- — p* 1 + a  T  + x  2  ]  T  S D . . + aw wpll  FD,„ 12  S D , , + aw wp 11  w h i c h , u s i n g the z e r o p r o f i t c o n d i t i o n p  (A10)  equations  *T  T  F D , , 0  12'  ,  T F = 0 , yields  *NT  Y + (w + x )  = -a(p  FD,„]  J. J.  T  1  + S) .  1Z  T  = kp* .  S i n c e a l l s t e p s of the proof are i f and o n l y i f s t a t e m e n t s ,  equivalence  has been shown. Q E D  * Note that p r o p o r t i o n a l i t y of the v e c t o r s * a l i t y of x  x  and w i m p l i e s p r o p o r t i o n -  * and (w + x ) , and hence (2.37) and (2.38) are e q u i v a l e n t .  - 208 -  APPENDIX 2  P r o o f o f Lemma 8.1 Let  us develop  the matrix  the q u a d r a t i c form  (S - £) i s denoted  ^  (Bl)  by  rp  V B(w) = AD,.BD,,A ww 11 11  rp  T  - AD  rp  = -AD A U  since D ^ B D ^ Woodland  (B2)  = D , u  1 2  F  T  DjjBD^  1 2  = 0,  - AD A rp  ro  -  D  B 1  D  2  i2  T  i2  +  AD,.BD F 11 12 10  - FD^A  rp A  rn rp  rp  - AD,,A 11  U  rp  - AD F  (1977 : A p p e n d i x ) ) .  ww  rp  0  1 2  To s i m p l i f y n o t a t i o n  A C  + FD7 BD. , A 12 11  + FD^ BD F 2  B A  (8.11).  1  + C  rp  "  F D  22  -D  22  F  +  C  '  ( s e e Diewert and  Then,  B(w) - C - [A, F] D  which i s ( 8 . 9 ) .  QED  P r o o f o f Lemma 8.2: Using *T  p  ft Y + (w +  (8.1) and the equations .T T  )  T  F = 0^,  S  p p  p  ft ft + S (w + T ) = 0 , p w  N  - 209 -  r (B5)  A  X [B P  , B ] = X z  S  T  - E pp  qq KxK * T  S  - T  -p  S  wp  Set  X  3  = k e R.  - T  * T  F -p  Y  vq  pp  Then,  S  (B6)  ~W E  - E pp  qq KxK  = k [ x LS  Assumptions  wp  + p  ( i ) - ( i i ) i n Theorem 8.1  s i d e of (B6) can be i n v e r t e d .  Call  S  pp  + w E  vq  imply that  , x  F + p  Y]  c  •  the m a t r i x on l e f t  the i n v e r s e D =  D  ll  D  12  D  21  D  22  Then,  rp  (B7)  rp  Jt.  [ X f , \\] = k [ ( p  rp  rp  T  T  + e) , 9 ]  where  (B8)  T  T  [e , 9 ] =  T  + T* S  T  [-t* E  - (x* + s * ) E qq  T  vq  T  , x* F] wp'  D.  hand  - 210 -  Equations  (B8) are d e r i v e d u s i n g the e q u a t i o n s  (Spp  _  D  ^qq^  l l  ^21  +  I , q E + v* Z = oL and (S - E )D + YD ~ = 0„ „. N qq qv N PP qq 12 22 NxK T  T  K T  1 0  0  P r o o f of Lemma 8.3: Using  (B9)  (8.1) and ( B 7 ) ,  = k[-(p  l  \Z  \^  T  k  Using  T  + e) S  + (p pw  -  Then,  T  - w (S  qV  - E )], WW  (B5), the homogeneity of the zero p r o f i t condition p  £  rp  Y + ( W + T ) F =  rji  W  '  T  [ ( p * + e ) , e ] = -w [S - E , F] ' wp vq' T  r  T  f u n c t i o n s and the zero Jj.  0 , the v e c t o r [(p K  be w r i t t e n as  (BIO)  eV  R.  e  Jjfrp  profit  r  + e) E  K  D.  rjt  rjt  + e) , 8 ] can  - 211 -  (BID  T  T  kw [(Swp - Evq)D,11 , +'FD"F ] (Swp - Eqv ) + kw 12  T  XB  [(S  wp  - E ) D? + F D „ ] F vq 12 22 0  T  = kw [(S  wp  T  T  x  P r o o f o f Theorem  10.4:  (a)  X [B  T  + FD F 22 0 0  T  T  - E ) vv  + Fuf-CS - E ) 12 wp vq  - (S - E )] ww w  T  QED  T  T  such that X [ A , - B ] > 0 ^ , X [ B b  B ](t  , s  g  T1T  J  T  B ] = 0*  p )  g  ) _> 0. I t has been e s t a b l i s h e d e a r l i e r  T T X s a t i s f y i n g X [B , B , B ] = 0. ,,„ must ° p z' g N+K+H T  (p* + e ) , 9 , 1 ] , k e R.  t h a t k > 0.  \ [B , t  g  that a v e c t o r  TI  T  H+1  (4.16), the homogeneity o f the u n i t p r o f i t and  B ](t \  ,  The I n e q u a l i t i e s X [ A , - B. ] > 0 ^ . imply  T f u n c t i o n s , and X B  expenditure  L  Using  + K + H  T T be o f the form X = k [ 0 , H'  b  (B12)  T  l c i s s u f f i c i e n C Co show ChaC Chere i s no v e c t o r  H + N + K + 1  t >  ww  - E ) + (S - E , F) D(S - E , F) ] vv wp vq wp qv  ww  = - k w ^ ^ B t w ) , k e Ri  Using- (8.10), X B  T  - kw (S  - E )D..(S - E ) vq 11 wp vq  1 0  = kw [-(S  T  0  + (S - E ) D F wp vq 12  X e R  T  0  s V  p  T = 0^,  = [A*. X^, X* ]  -Spw w +  E w qv  -F w -w  T  Swww  T + w Z w vv  - 212 -  T  T  T  T  Thus, X [B ., B ] [ t * , s * ] t' s  T  rj,  By assumption, w  2  w = -w  V B(w)w ww  T  x  2  V  T  = -X B  1  using Lemma 8.3, &  X *T *T T Hence, X [B , B ] [ t , s ] < 0.  B(w) * 0 .  (b) I t must be shown that there i s no X e R T  T  X [A, - B J For  X  T  > Oj, X [ B , B , B ] = 0 ^ p  T  T  g  T  + K + H  , X B * > 0. gS  T  = [ 0 , (p* + e ) , 6 , 1],  JU  rp  (B13)  z  such that  X B s s  JU  = -(p ^ = -p  JU  rp  + e) E s vq  *T ~ * x + s E D,,E s - wE vq 11 qv w  *  *X  E s vq  c  JU  rp  - w E s vv  ft ft using (B8) with t = 0,, T N  *  s, '  = 0... M  T  Then,  (B14)  T  XB  T  s  s* = s * [ E  = -s  + E D., E ]s* vq 11 qv  vv  1  V  z  Bs ss  *X  since, by assumption, s  V  < 0,  Bs SS  * * 0 x and the matrix V2 B i s positive SS  LL  semidef i n i t e . If E  qv  X  = 0. NxM  then X B s s  T  yLrjri  assumption, s  X  E * 0. vv M  thus E w = -E s*). vv vv  W  *  x  =-wE  w  s  *  = w w  T  E  w<0,  since, by J  ^  (If E = 0,, „, then E (w + s ) = 0., and vq MxN w M  - 213 -  / (c) Now  T * the v e c t o r X B t must  be shown to be n e g a t i v e .  t  Using  (4.15),  T  (B14)  T  XB  T  t* = -(p* + e ) E  t* - w E  qq  t = -p  AT  E  qq  *  vq  T  t  *  - w E t vq  + t  XT'  = t  + E  If  E  qu  *T 2 V _B  9  = 0„ „, NxM  T  (B16)  XB  V  tt  2  B  t  <  A  E  t  °»  V  2  B i s positive  ,  (P  * 0.  A T  + t  semidefinite.  qq  A  E  D,, E  + E  *  D..E ]t qq 11 qq T  + t ) = 0^. QED  t  qq n qq  qq  A  *T  B  r  = t [E  t  *  t * = -<p* + e ) E t * qq  ^  q q  ]t  T  t  AT  „  *  then  = -p  because £  D., E  T * 0^ and the m a t r i x  t(  ~  E D, . E t qq 11 qq  qq n qq  qq = "t  * T  ?V  [E AT  since t  t*  A  Hence, X B t t  AT  = -t  _9  v^Bt  A  < 0,  since  - 214 -  Proof  o f P r o p o s i t i o n 10.1;  * Suppose t vector T = 0  H+N+K+l A e R  H+ix+ti  the  t  0^.  to prove that there  i s no  T T T such that A [A, - B j > 0 , A [B , B , B ] ' b H+l' p z' g T T L l  t  the v e c t o r h i s d e f i n e d as the d i r e c t i o n of  h = ( t - t ) > 0^, where  — — t = (t,...,t)  N e R . +  *T 2 T V B < 0 „ and h > 0 . tt N N  QED  It is sufficient  T , A B h > 0, where  change i n t :  taxes,  »  lT  t i s the v e c t o r o f uniform  T *T 2 Then, A B h = t h v  t  The o t h e r  cases  B  t  t  < 0 i f , as assumed,  a r e proved i n a s i m i l a r  way.  

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