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

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OPTIMAL PUBLIC POLICIES IN SMALL OPEN ECONOMIES By ARJA HELENA TURUNEN B.Sc, The Un i v e r s i t y of H e l s i n k i , 1976 M.Sc, The Un i v e r s i t y of H e l s i n k i , 1978 L i e . 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 i n THE FACULTY OF GRADUATE STUDIES (The Department of Economics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 1985 ® Arja Helena Turunen, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of ^ C o v ^ 1 The University of B r i t i s h Columbia 1956 Main M a l l Vancouver, Canada V6T 1Y3 Date DE-6 (3/81) - i i -ABSTRACT U n t i l r e c e n t l y , p r o o f s e s t a b l i s h i n g the e x i s t e n c e of g a i n s from t r a d e have used the a s s u m p t i o n t h a t the government can >alter the d i s t r i b u t i o n o f income by a s e t of lump sum t r a n s f er-s , i .e ., t h e government has a t i t s d i s p o s a l a s e t o f h o u s e h o l d s p e c i f i c t r a n s f e r i n s t r u m e n t s . However, r e c e n t work has-been d e v o t e d to s i t u a t i o n s . w h e r e t h e s e t r a n s f e r i n s t r u m e n t s a r e i n a d m i s s i b l e . D i x i t and Norman (1980: 79-80) d e m o n s t r a t e t h a t a government t h a t can a l t e r a l l d o m e s t i c commodity t a x e s can e n s u r e t h a t no i n d i v i d u a l i s made worse o f by moving from a u t a r k y to f r e e t r a d e . I t t u r n s o u t , however, t h a t t h i s D i x i t and Norman p r o o f o f the g a i n s from t r a d e shows o n l y t h a t the a u t a r k y e q u i l i b r i u m can be r e p l i c a t e d under f r e e t r a d e and not t h a t p o s i t i v e g a i n s w i l l o c c u r . One o f the purposes o f t h i s t h e s i s i s to i n v e s t i g a t e the problem o f t h e g a i n s from t r a d e when a v a r i e t y o f tax and t r a n s f e r i n s t r u m e n t s are a v a i l a b l e . I t i s f r u i t f u l to r e g a r d the p r o b l e m o f the g a i n s from t r a d e as a p o l i c y r e f o r m q u e s t i o n : can the government i n the home c o u n t r y 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 c o u n t r y ' s i n i t i a l i n t e r n a t i o n a l t r a d e p r o h i b i t i v e t a r i f f s w h i c h , accompanied w i t h a s u i t a b l e ( 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 c o u n t r y ' s commodity t a x 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 P a r e t o improvement? I n o r d e r to answer the 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 p r e s e n t e d i n C h a p t e r 2 . I t i s e s t a b l i s h e d t h a t , under some v e r y weak c o n d i t i o n s , t h e r e a r e ( d i f f e r e n t i a l ) t a r i f f p e r t u r b a t i o n s t h a t improve the c o u n t r y ' s i n i t i a l net b a l a n c e o f t r a d e . In C h a p t e r 4, i t i s shown t h a t t h e s e - i i i -pr o d u c t i v i t y gains can be d i s t r i b u t e d to the consumers i n the economy i n a s t r i c t Pareto improving way by s u i t a b l y adjusting the country's i n i t i a l commodity tax rates. The p r i n c i p a l tool for esta b l i s h i n g these r e s u l t s i s a d u a l i t y theorem: Motzkin's Theorem. Chapter 3 develops two approximative formulae for measuring the productivity gain accruing from a change of t a r i f f s . Some examples of s t r i c t Pareto improving perturbations i n commodity taxes and t a r i f f s are given i n Chapter 7. These include proportional and uniform reductions of t a r i f f s as well as a change toward uniformity in the country's i n i t i a l t a r i f f s t r u c t u r e . Next, the government i s assumed to be able to adjust only the home country's i n i t i a l vectors of t a r i f f s and lump sum transfers but not the vector of commodity taxes. Conditions f o r s t r i c t Pareto improving t a r i f f and transfer perturbations to e x i s t are developed. In Chapter 9 i t i s shown that neither the existence of s t r i c t gains from trade under commodity taxation or under lump sura compensation n e c e s s a r i l y implies the other. Examples of s t r i c t Pareto improving changes in t a r i f f s , taxes and tra n s f e r s are given in Chapter 10. These include proportional reductions of t a r i f f s and/or taxes and movements toward uniformity i n the tax rates for domestic and tradeable commodities. The ro l e of normality of commodities i n consumption in po l i c y recommendation r e s u l t s i s also discussed. Chapter 11 develops s u f f i c i e n t conditions for a perturbation i n the home country's tax structure, which causes i n t e r n a t i o n a l trade, to be s t r i c t Pareto improving. - i v -In Chapter 12 the goal 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 associated with the i n i t i a l observed e q u i l i b r i u m i n the economy. I t i s shown that i n e q u a l i t y reducing perturbations i n commodity taxes and t a r i f f s e x i s t , i f the preferences and i n i t i a l commodity endowments of the consumers s a t i s f y c e r t a i n c o n d i t i o n s . - v -TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS v LIST OF FIGURES v i i ACKNOWLEDGEMENTS v i i i 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 2.3 Local Controllability of the Production Sector and Changes in the Home Country's Net Balance of Trade 16 2.4 Existence of Productivity Improving Changes in Tariffs 25 3. HOW LARGE IS THE PRODUCTIVITY GAIN? 34 4. STRICT PARETO IMPROVING CHANGES IN COMMODITY TAXES AND TARIFFS 46 4.1 A General Equilibrium Model..... 46 4.2 Existence of Strict Pareto Improving Changes in Commodity Taxes and Tariffs 53 4.3 Necessary Conditions for Pareto Optimality: Nonexistence of Strict Pareto and Productivity Improving Tax and Tariff Perturbations 62 4.4 Strict Pareto and Productivity Improving Changes in Commodity Taxes and Tariffs When no Domestic Goods Exist 68 5. EXISTENCE OF STRICT GAINS FROM TRADE WHEN LUMP SUM TRANSFERS ARE NOT A FEASIBLE GOVERNMENT POLICY INSTRUMENT 72 - v i -6. EXISTENCE OF STRICT PARETO AND PRODUCTIVITY IMPROVEMENTS WHEN ONLY A LIMITED SET OF COMMODITY TAXES AND TARIFFS CAN BE PERTURBED 78 '7. SOME PIECEMEAL POLICY RESULTS WHEN NO LUMP SUM TRANSFERS • ARE USED AS GOVERNMENT POLICY INSTRUMENTS 85 8. PARETO IMPROVING POLICY PERTURBATIONS WITH LUMP SUM TRANSFERS 98 8.1 A Second Model f or the Production Side of an Economy.... 98 8.2 Existence 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 8.3 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 Transfers 113 8.4 Necessary Conditions f o r Pareto Optimality: Nonexistence of S t r i c t Pareto and P r o d u c t i v i t y Improving T a r i f f and Transfer Perturbations 115 9. MORE ON GAINS FROM TRADE 124 10. PROPORTIONAL REDUCTIONS IN DISTORTIONS AND SOME PIECEMEAL POLICY RESULTS 132 11. COMMODITY TAXATION AS A CAUSE OF TRADE 160 12. ECONOMIC INEQUALITY AND PUBLIC POLICIES 164 12.1 Existence of Inequality Reducing P o l i c y Perturbations... 166 12.2 Existence of Inequality Reducing and Welfare Improving P o l i c y Perturbations 174 13. CONCLUSIONS. 180 FOOTNOTES 184 REFERENCES 202 APPENDIX 1 204 APPENDIX 2 208 - v i i -LIST OF FIGURES Figure Page 1 Rela t i v e Numbers of Domestic Commodities and Production Industries: Continuity of Total Industry Net Supplies 13 2 Rank of the Matrix Y and Continuity of T o t a l Industry Net Supplies 15 3 S t r i c t Pareto Improving Perturbations i n T a r i f f s and Commodity Taxes 60 4 Existence of S t r i c t Gains from Trade 75 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 122 6 Existence of S t r i c t Gains from Trade Under Commodity Taxation and Lump Sum Compensation 127 7 Normality of Commodities and the E f f e c t s of a P o l i c y Change 152 8 B-Optimality and the Hatta Normality Condition 154 9 Pareto Improving and Inequality Reducing P o l i c y Perturbations 165 - v i i i -ACKNOWLEDGEMENTS I thank the following persons and i n s t i t u t i o n s f o r t h e i r invaluable help and encouragement during the long process of wr i t i n g t h i s thesis: The Department of Economics/U.B.C. The Yrjo Jahnsson Foundation My Supervisory Committee: Professors Erwin Diewert, John Weymark, David Donaldson Professors John Cragg, D. Paterson, Keizo Nagatani, Jim Brander, W. Ziemba Professor Alan Woodland Professors Seppo Honkapohja, Aarni Nyberg Doctor Robert Hewko My fr i e n d s : A l e x i s Fundas, Jayne Stuebing, Scott Schatz, Dan and G l o r i a Gordon, Michel and Carole Patry, Andreas Phingsten, Paul and Diana P r i c e , Beryl and Jim Skinner, Roy Dahlstedt, Anders Baudin, Tapani Luukkainen, La u r i Jauhiainen, A l l a n Nurminen, Ken Krueger, David K i n a l , Tomaso and Wanda DiCarlo, Jason Foley, L e o t i s Watson My soul brothers: Senyo Adjibolosoo, C h r i s t i a n Jones My friend of many years: Pekka Turunen My s i s t e r , I r j a Holopainen, and her family My parents, Aune and Veikko Halttunen: My work, as always, i s dedicated to them. K i i t t a e n omistan taman tyon vanhemmilleni. My s k i l l f u l t y p i s t s : Teresa Patterson, Jeeva Jonahs. The Audio-Visual Services, U.B.C. - 1 -1. INTRODUCTION The question of,.,whether there are gains from trade i s an i n t e r e s t i n g one. U n t i l recently, proofs e s t a b l i s h i n g the existence of gains from trade have used the assumption that the government can a l t e r the d i s t r i b u t i o n of income by a set of lump sum transfers; i . e . , the government has at i t s disposal a set of household s p e c i f i c transfer instruments. However, recent work has been devoted to s i t u a t i o n s where these transfer intruments are inadmissible. D i x i t and Norman (1980: 79-80) demonstrate that a government that can a l t e r a l l domestic commodity taxes can ensure that no i n d i v i d u a l i s made worse of by moving from autarky to free trade. However, Kemp and Wan (1983) provide an example of an economy which shows that the a v a i l a b i l i t y of commodity tax instruments alone (without the use of lump sum transfers) i s not s u f f i c i e n t to ensure that a Pareto improvement w i l l occur moving from autarky to free trade. Thus the D i x i t and Norman proof of the gains from trade shows only that the autarky equilibrium may be r e p l i c a t e d under free trade and not that p o s i t i v e gains w i l l occur. One of the purposes of this thesis i s to investigate the problem of the gains from trade when a v a r i e t y of tax and transfer instruments are a v a i l a b l e . It i s f r u i t f u l to regard the problem of the gains from trade as a p o l i c y reform question. Suppose that the i n i t i a l autarky equilibrium i n the home country i s a consequence of the government t a r i f f p o l i c y ; i . e . , the i n i t i a l t a r i f f s on i n t e r n a t i o n a l l y tradeable commodities, in a - 2 -country open for i n t e r n a t i o n a l trade, are chosen to be such that the res u l t i n g producer prices faced by the domestic production sector coincide with the autarky equilibrium prices for tradeables. Can the government find a small ( d i f f e r e n t i a l ) perturbation in the i n i t i a l 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 t a r i f f s which, accompanied with a sui t a b l e ( d i f f e r e n t i a l ) perturbation in the home country's commodity tax structure, r e s u l t s i n a s t r i c t Pareto improvement, i . e . , in a s t r i c t welfare improvement for a l l households in the economy? If this i s pos s i b l e , the government should adopt the p o l i c y of ( i n f i n i t e s i m a l l y ) changing the country's i n i t i a l tax and t a r i f f s t r u c t u r e — a policy which also g a i n f u l l y opens the country for i n t e r n a t i o n a l trade. The problem of the gains from trade thus becomes the following: under what conditions do s t r i c t Pareto improving perturbations of commodity taxes and t a r i f f s exist? In order to answer this question, a model for the production side of an economy i s presented in Chapter 2. The model i s written i n d u a l i t y terms, assuming that there are K . constant returns to scale production i n d u s t r i e s i n the home country. It i s established in Theorem 2.1 that, under some very weak conditions on the production sectors' technologies, there exist perturbations i n the home country's i n i t i a l t a r i f f structure that improve the country's i n i t i a l net balance of trade, i . e . , the net amount of tradeables revenue generated by the K i n d u s t r i e s . In Chapter 4, i t i s shown that these p r o d u c t i v i t y gains can be d i s t r i b u t e d to the consumers i n the economy i n a s t r i c t Pareto improving way by s u i t a b l y adjusting the country's i n i t i a l commodity tax rates. The p r i n c i p a l tool for esta b l i s h i n g these - 3 -results is 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 t a r i f f s . It is 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 de t a i l . In Chapter 6 i t is assumed that only a subset of the country's i n i t i a l 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 in ta r i f f s and commodity taxes. These include proportional and uniform reductions of the home country's i n i t i a l t a r i f f s . It is 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 is 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 specific transfers. The commodity tax rates in the economy are assumed to be fixed. The problem considered i s : under what conditions are there strict Pareto improving (differential) perturbations in the country's i n i t i a l t a r i f f s and transfers? As in Chapter 2, the conditions for strict 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 from trade i s readressed in Chapter 9 : the s u f f i c i e n t conditions for the existence of s t r i c t gains from trade under commodity taxation and under lump sum compensation are compared. It i s shown that neither the existence of s t r i c t gains under commodity taxation or under lump sum compensation n e c e s s a r i l y implies the other. In Chapter 10 some examples of s t r i c t Pareto improving changes i n t a r i f f s , taxes and lump sum transfers are given. These include proportional reductions of t a r i f f s and taxes, proportional reductions of e i t h e r domestic or tradeables commodity taxes, and movements toward uniformity i n the tax rates for domestic and tradeable commodities. The role of normality of commodities i n consumption i n p o l i c y recommendation results i s also discussed. Chapter 11 considers the welfare e f f e c t s of a change in the home country's commodity tax structure (without a change in t a r i f f s or lump sum t r a n s f e r s ) . I f 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 autarky equilibrium, the conditions given i n Theorem 11.1 can be interpreted to be s u f f i c i e n t f o r a perturbation of taxes, which causes i n t e r n a t i o n a l trade, to be s t r i c t Pareto (hence, welfare) improving. In Chapter 12 the main po l i c y goal of the government i s assumed to be the reduction of economic i n e q u a l i t y in the s o c i e t y . The goal of the analysis i n this chapter i s to o p e r a t i o n a l i z e the concept of economic.inequality i n such a way that p r a c t i c a l p o l i c y questions can be answered. I t i s established that economic i n e q u a l i t y reducing commodity tax and t a r i f f perturbations exist i f the consumers' preferences and i n i t i a l commodity endowments s a t i s f y c e r t a i n conditions. For example, - 5 -there must e x i s t a commodity with respect to which the preferences 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 . In t h i s case, a p r o p o r t i o n a l reduction of t a r i f f s can be made i n e q u a l i t y reducing by s u i t a b l y perturbing the home country's commodity tax s t r u c t u r e . - 6 -PRODUCTIVITY IMPROVING CHANGES IN TARIFFS 2 . 1 E q u i l i b r i u m f o r the Production Side of an Economy The production side of an economy i s assumed to consist of K constant returns to scale s e c t o r s 1 , indexed k=l,...,K. There are N + M commodities, N of which are domestic (not i n t e r n a t i o n a l l y tradeable). N The prices of 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 i s assumed to be small. The technology of the kth industry (or producer) i s represented by i t s 2 k k k k u n i t production 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 vector y = (y^,«..,y^) of domestic (net) supplies and the vector k / k k T f = ( f , . . . , f ) of (net) exports are producible by sector k when i t operates at unit scale. If y^ < 0 , n e [1,...,N], the nth domestic good i s used as an input in sector k and i f f < 0 , m e [1,...,M], the mth tradeable good i s an import for sector k, k=l,...,K. The unit production sets C are defined with respect to some always used input (or with respect to some always produced output). For example, i f for sector k an always used input exists (e.g., land), some amount of this input i s chosen as the unit l e v e l , and the set C then consists of a l l k k vectors (y , f ) that the sector can produce using one unit of land. Each set C , k=l,...,K, i s assumed to be nonempty, closed and bounded from above. Define z e R + as the amount of the always used input (or the always produced output) in sector k, k=l,...,K. This v a r i a b l e gives the - 7 -scale of sector k. Using the scale z , the t o t a l production p o s s i b i l i t y set of sector k i s defined as T k = {z kC k: z k >_0}. Each T k, k=l,...,K, i s a nonempty, closed cone. According to d u a l i t y theory,: since the K production sectors are assumed to behave competitively, ;the unit production p o s s i b i l i t y sets C , k=l,...,K, can equivalently be represented using the production k 5 sectors' unit p r o f i t functions T defined for k=l,...,K by ( 2 . 1 ) T : K (p, w + T ) = max { p T y k + (w + x ) T f k : ( y k , f k ) e C k}, y \ f k M where T e R i s a vector of trade taxes and/or subsidies. The vector (w + x) i n ( 2 . 1 ) i s thus the vector of prices for tradeable commodities faced by domestic producers. If 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 used as an input by a l l sectors or produced as an output by a l l sectors, the components of the vector x = ( T , . . . , T ^ ) may be interpreted as follows: i f f > 0 and x^ > 0 (< 0) , net exports of the mth i n t e r n a t i o n a l l y traded good by sector k are subsidized (taxed); i f f < 0 and x > 0 « 0 ) , m m ' net imports of good m into sector k are taxed ( s u b s i d i z e d ) 6 . In what i f follows, x i s c a l l e d the t a r i f f vector. The unit p r o f i t functions TT , k=l,...,K, are well-defined since, by assumption, each set C i s closed and bounded from above. Furthermore, each unit p r o f i t function i s convex and l i n e a r l y homogenous in the prices (p, w + x). Assuming that the unit p r o f i t functions n , k=l,...,K, are twice continuously d i f f e r e n t i a b l e , the s e c t o r a l price-dependent input output k k c o e f f i c i e n t s y and f , k=l,...,K, can be determined using H o t e l l i n g ' s Lemma: - 8 -(2.2) y k = V P TT k(p, w + x), k = 1.....K; Y = [ y 1 , . . . , y K ] (2.3) f k = V T r k ( p , w + x), k=l,...,K; F = [ f 1 , . . . , f K ] . w k k The vectors V IT (p, w + x) and V TT (p, w + x) denote the f i r s t order p w p a r t i a l d e r i v a t i v e s of the functions ir , k=l,...,K, with respect to the vectors p and w, r e s p e c t i v e l y . Linear homogeneity of the unit p r o f i t functions and Euler's Theorem imply (2.4) TT k(p, W + X ) = p T V TTk(p, W + X) + ( W + X ) T V TTk(p, W + x) p w = p T y k - (w + x ) T f k , k=l,...,K. k / k % k k k k If the scale z i s p o s i t i v e (z > 0), the vectors y z and f z give sector k's t o t a l domestic and net export supplies. Hence, the t o t a l p r o f i t earned by sector k i s (2.5) i r k z k = p T y k z k + (w + x ) T f k z k , k=l,...,K. Then, using the matrices Y and F defined in (2.2)-(2.3), the (row) vector T T of the industry t o t a l p r o f i t s can be written as p Yz + (w + x) Fz. The producers' aggregate symmetric s u b s t i t u t i o n matrix S i s defined by (2.6) S S S PP pw S S wp WW K 2 , K 2 I ' M P . w + T > z I V * <P. w + T ) z k=l PP k=l k=l P w K 2 7 WW I V T f k ( p , W + X ) z k I V T T k(p, W + X ) Z K k=l - 9 -In (2.6), the matrix block gives the responses of the domestic t o t a l net supplies to changes in domestic prices p, S gives the responses of p w the domestic t o t a l net supplies to changes i n tradeables prices (w + x), and gives the responses of the t o t a l net export supplies to changes i n prices (w,_+ x) . The matrix S i s p o s i t i v e semidef i n i t e , since the unit p r o f i t functions tr , k=l,...,K, are convex in prices (p, w + x). Linear homogeneity of the unit p r o f i t functions and Euler's Theorem imply This means that the producer 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 which i s the vector of producer prices (p, w + x). The equilibrium conditions for the production side of the economy, assuming that each sector k, k=l,...,K, i s operating at a p o s i t i v e s c a l e , are: (2.7) [ p T , (w + T ) T ] S = 0. N+M* 7 (2.8) * *k w + x )z = y » A (2.9) T r k ( p * , w + x*) = 0, k = 1 , . . • , K, (2.10) I w V it (p , w + x )z = b . k=l According to (2.8)-(2.9), at an equilibrium (indexed with an a s t e r i s k ) , net A A supply of domestic commodities equals an exogenously given y (y can be, for example, the consumers' net demand vector or a vector of domestic goods - 10 -endowments) and a l l industries make zero pure p r o f i t s . Equation * (2.10) defines b , the net amount of foreign exchange earned by the domestic producers. The N + K + 1 equations (2.8)-(2.10) endogenously determine the * equilibrium;vector of domestic prices p , the equilibrium industry scales * * z , and the equilibrium net balance of payments b . The exogenous variables * i n the model are the net output of domestic commodities, y , the constant * i n t e r n a t i o n a l prices w and the t a r i f f s t . It i s assumed that there e x i s t s an i n i t i a l equilibrium where (2.8)-(2.10) are s a t i s f i e d , and the vectors of domestic prices and industry scales 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 , given y and the prices of tradeable commodities T" r TT (w + X ) . For the subsequent analysis i t i s required that the endogenous p , * * z and b be regarded as (once continuously d i f f e r e n t i a b l e ) i m p l i c i t * functions of the prices (w + x ). The conditions that guarantee the existence of these i m p l i c i t functions can be derived by t o t a l l y 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 quilibrium: 10 (2.11) B Ap* + B Az* + B, Ab* = B Ax*, p z b x ' where B = P PP T Y T w S wp B = Y °KxK w F N °K -1 and B = -S pw - F T -wTS WW ft * According to the I m p l i c i t Function Theorem, the functions p (w + x ), A * * * z (w + x ) and b (w + x ) around the i n i t i a l e q u ilibrium e x i s t i f the - 11 -matrix [Bp, B z, B^] i s i n v e r t i b l e . Under this supposition, the d i r e c t i o n a l "k Jc * "k * A d e r i v a t i v e s of the functions p ( w + T ) , z ( w + r ) and b (w + T ) evaluated at the i n i t i a l e q u ilibrium are determined by the matrix [ v v v " 1 v Diewert and Woodland (1977: Appendix) show that necessary and s u f f i c i e n t conditions for the matrix [B , B , B, ] to be i n v e r t i b l e are: p z b (2.12) rank Y = K (< N) (2.13) rank (S + YY T) = N. PP I t i s assumed henceforth that (2.12)-(2.13) are s a t i s f i e d at the i n i t i a l e q uilibrium. Economic i n t e r p r e t a t i o n s f o r assumptions (2.12) and (2.13) are discussed i n Sections 2.2 and 2.3. 2.2 Continuity of the Producers' T o t a l Net Supply Functions In order to develop an i n t e r p r e t a t i o n for assumption (2.12), the 11 / *\ economy's GNP function G(w + T, y ) must be f i r s t defined: (2.14) G(w + x, y*) = max { I (w + T ) T f k Z k : ( y k , f k ) e C k, z k > 0 k = 1 • I y k z k >. y*; k = i f . . . ,K}. k=l In (2.14), the producers' net revenues from the sales of tradeables are maximized with respect to the constraint that the production sectors, i n - 12 -the aggregate, supply a predetermined amount y of domestic commodities. ( I f domestic good n, n e [1,...,N], i s an input, the producers' t o t a l * demand for t h i s factor must not exceed the given endowment - y^ >_ 0.) Using the K a r l i n (1959: p. 201) — Uzawa (1958: p. 34) Saddle Point Theorem,^ the concaveprogramming problem (2.14) can be written in an equivalent dual form: (2.15) G(w + T, y*) = min {- p Ty*: - i r k ( p , w + T ) >_ 0, k = 1,...,K}. P>°N In (2.15), the producers' costs from using the domestic commodity vector y as a net input are minimized with respect to the constraint that no production sector earns p o s i t i v e (pure) p r o f i t s . In order to i l l u s t r a t e problem (2.15), consider an economy where N = 2 and K = 3. In this p a r t i c u l a r case, N i s le s s than K and assumption (2.12) i s v i o l a t e d . Figure 1 i s drawn by s l i g h t l y modifying the f a c t o r p r i c e diagram presented in Woodland (1982: p. 48). The shaded area in F i g . la) i s the f e a s i b l e s o l u t i o n set for problem (2.15). For the fixed y depicted in the f i g u r e , 1 3 the cost minimizing vector of domestic commodity prices i s represented by the point p^, where * the unit p r o f i t l e v e l curves of the production sectors (at fixed (w + T ) ) i n t e r s e c t . 1 4 C l e a r l y , there exist industry scales z 1 > 0, z 2 > 0, z 3 > 0 such that y* = - V T ^ Z 1 - V TT 2 Z 2 - V i r 3z 3 i . e . , in F i g . la) a l l P P P three industries operate 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 perturbed, the producers' zero unit p r o f i t curves s h i f t , as depicted In F i g . 2.1b). As a r e s u l t , one of the - 1 3 -F i g u r e 1 - R e l a t i v e Numbers of Domestic Commodities and Pr 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 . - 14 -three i n d u s t r i e s i s l i k e l y to cease production at a p o s i t i v e scale: i n F i g . l b ) , sector 1 does not operate at the new cost minimizing prices p^, since i t s unit p r o f i t s at these prices are negative. Hence, i t can be seen that i f the number of production sectors K exceeds the number of domestic commodities N, as in the example above, the producers' t o t a l net supply k k k k functions y z and f z , k=l,...,K, are l i k e l y to be discontinuous — an outcome to be avoided i f d i f f e r e n t i a l analysis i s to be a p p l i e d . 1 5 The assumption that K <^  N can thus be regarded as a c o n t i n u i t y constraint on the s e c t o r a l t o t a l net supply functions. Continuity of the sectoral t o t a l net supply functions also depends on the rank of the matrix Y. To see t h i s , consider Figure 2 . Figure 2 i s drawn assuming that there are two production industries and two domestic commodities with prices p = (p^, P 2 ) in the economy. In F i g . 2 a ) the gradients of the sectoral unit p r o f i t functions IT , k = 1 , 2 , with respect to the domestic goods prices are l i n e a r l y dependent, i . e . , the rank 1 2 0 of (y , y ) i s 1 (< K = 2 ) . At the prices p both sectors are operating at a p o s i t i v e s c a l e . Suppose now that the prices (w + x ) change. The change in the tradeables producer prices causes a s h i f t in the sectoral unit p r o f i t l e v e l curves; a possible outcome i s depicted in F i g . 2 b ) . After the change in (w + x ) only sector 2 can earn zero pure p r o f i t s and hence, only sector 2 w i l l stay operative. Sector 1 w i l l close down, causing a d i s c o n t i n u i t y in i t s t o t a l net supplies. 1 2 I f , however, the gradients of the unit p r o f i t functions ir and ir with respect to the domestic commodity prices are l i n e a r l y independent, i . e . , - 15 -- 16 -1 2 the rank of (y , y ) i s 2 (= K), the s e c t o r a l zero unit p r o f i t curves w i l l i n t e r s e c t both before and a f t e r a small perturbation i n the tradeables ft producer prices (w + x ). In this case, both sectors w i l l stay operative despite the tradeables price change. Assuming that the unit p r o f i t functions TT , k = 1,2, are twice continuously d i f f e r e n t i a b l e , the k k k k s e c t o r a l t o t a l net supply functions y z 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 of the Production Sector and Changes In the  Home Country's Net Balance of Trade The goal of this section i s to provide i n t e r p r e t a t i o n s f o r assumption (2.13) and for a r e s t r i c t i o n imposed l a t e r i n the analysis on the net * * balance of trade function b (w + x ). I t w i l l be required that (2.16) V b(w + x*) * o3-x M It w i l l be seen that both assumptions (2.13) and (2.16) are related to l o c a l c o n t r o l l a b i l i t y of production i n the home country, where the concept of l o c a l c o n t r o l l a b i l i t y i s that defined by Guesnerie (1977) and Weymark (1979). T Let us s t a r t by considering the assumption that the matrix S + YY PP i s p o s i t i v e d e f i n i t e , 1 6 i . e . , assumption (2.13). As shown i n Section 2.3, this supposition i s s u f f i c i e n t (together with assumption (2.12)) f o r * * * * * * the i m p l i c i t functions p ( w + x ) , z ( w + x ) and b (w + x ) to be w e l l defined. Hence, from the mathematical point of view, assumption - 17 -(2.13) i s needed to guarantee the exi s t e n c e of the inverse demand 1 7 * A * * f u n c t i o n s 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 balance of trade f u n c t i o n b (w + x ) around the 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 assumption (2'.13), i t i s necessary to b r i e f l y e x p l a i n the agenda for the f o l l o w i n g s e c t i o n s . The production s i d e of an economy, described by the model (2.8)-(2.10) , i s analyzed. * The a u t h o r i t y choosing the exogenous v e c t o r of t a r i f f s x i s c a l l e d the government of the home country. The government i s assumed to have a p o l i c y g o a l : to improve the country's i n i t i a l net balance of trade 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 maintaining the aggregate domestic net supply at i t s i n i t i a l l e v e l y . In other words, the government i s assumed to search f o r a p e r t u r b a t i o n of the i n i t i a l t a r i f f s x such t h a t , a f t e r the country's production sector has * adjusted to the change i n the r e l a t i v e producer p r i c e s (w + x ), a higher l e v e l of net export revenue i s a t t a i n e d without s a c r i f i c i n g any of the * i n i t i a l domestic net supply y . It i s evident that 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 producers' domestic and tradeables net s u p p l i e s . Hence, to achieve i t s p o l i c y g o a l , the government must be able to i n f l u e n c e domestic goods production i n the home country i n such a way t h a t , i n the aggregate, the change i n the domestic net supply i s zero even though the s e c t o r a l net su p p l i e s do not g e n e r a l l y s t a y at t h e i r i n i t i a l l e v e l s . Consider the Guesnerie-Weymark d e f i n i t i o n of 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: (Guesnerie (1977), Weymark (1979)) The government i s said to have l o c a l control of production in an economy characterized by the equations (2.8)-(2.10) , i f the rank of the producer s u b s t i t u t i o n matrix S i s maximal (= N + M - 1) so that i t i s possible to induce a d i f f e r e n t i a l change in supplies i n any d i r e c t i o n on the economy's production p o s s i b i l i t y 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 of producer p r i c e s . Returning to the government's p o l i c y problem, i t seems cl e a r that i f the government has l o c a l control of production at the i n i t i a l e quilibrium, i t can induce the p a r t i c u l a r kind of change in the industry net exports and in the domestic net supplies ( i f such a change exists) that w i l l leave the * aggregate y constant while, at the same time, the i n i t i a l net balance of * trade b i s being improved. It can also be seen that when the producer s u b s t i t u t i o n matrix S i s of maximal rank (= N + M - 1), assumption (2.13) 18 i s s a t i s f i e d . There seems thus to be a connection between assumption (2.13) and l o c a l c o n t r o l l a b i l i t y of production in the country. Inspection of assumption (2.13) shows, however, that the producer s u b s t i t u t i o n matrix S need not be of maximal rank for assumption (2.13) to be s a t i s f i e d , i . e . , l o c a l c o n t r o l l a b i l i t y of production, in the sense of Guesnerie and Weymark, i s not necessary for (2.13) to hold. A weaker concept of c o n t r o l l a b i l i t y of production i s i n order: D e f i n i t i o n 2.2: Domestic goods production in an economy described by the model (2.8)-(2.10) i s 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 e quilibrium (which - 19 -s a t i s f i e s (2.8)-(2.10)) , i f there exist once continuously d i f f e r e n t i a b l e * * * * * * functions p (y , w + x ) and z (y , w + x ) such that (2.8)-(2.10) i s s a t i s f i e d when the i n i t i a l vector of t a r i f f s i s x . Here, l o c a l c o n t r o l l a b i l i t y of domestic goods production i s defined to mean ^ ^ & "ft "fi & that there exist functions " p (y , w + x ) and z (y , w + x ) (around the i n i t i a l equilibrium) which can be used to solve the appropriate * * perturbations i n the equilibrium domestic prices p and industry scales z , once a.suitable (net balance of trade improving) change in the t a r i f f s x * has been established. (The change in the vector y i s zero i f the i n i t i a l e quilibrium net supply of domestic commodities i s maintained). Lemma 2.1: Suppose that assumptions (2.12)—(2.13) are s a t i s f i e d . Then, the government has l o c a l control of domestic goods production i n the home country i n the sense of D e f i n i t i o n 2.2. Proof: Consider equations (2.8)-(2.9) which endogenously determine domestic prices * * * p and industry scales z , given t a r i f f s x and a vector of domestic net A supplies y . D i f f e r e n t i a t i n g (2.8)-(2.9) around the i n i t i a l values of p, z, x, and y: A A A (2.17) B Ap + B Az = B Ay , P z y where B = P PP > B z " KxK and B = y KxN - 20 -* * It follows that the domestic prices p and industry scales z can be regarded as (once continuously d i f f e r e n t i a b l e ) i m p l i c i t functions of the A A —J domestic net supplies y and t a r i f f s T , i f the inverse matrix [B , B 1 p z e x i s t s . According to Diewert and Woodland (1977: Appendix), the matrix [ B o ) B 1 i s i n v e r t i b l e i f and only i f (2.12)-(2.13) are s a t i s f i e d , which P z A A A has been assumed. Hence, the i m p l i c i t functions p (y , w + x ) and A A A z (y , w + x ) e x i s t . QED Using Lemma 2.L, assumption (2.13) can be given an i n t e r p r e t a t i o n as a s u f f i c i e n t condition 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 i . . 19 production. Let us now turn to consider tradeables production and the net balance A A of trade function b (w + x ). As shown in Section 2.1, assumptions (2.12)-(2.13) are s u f f i c i e n t for this function to e x i s t . Diewert (1983a) shows that assumptions (2.12)-(2.13) are also s u f f i c i e n t for the economy's A GNP function G(w + x, y ) to be twice continuously d i f f e r e n t i a b l e . Woodland (1982: p. 59) proves that the producers' aggregate net supply of A A tradeable commodities, f(w + x , y ), can be obtained as a vector of 20 p a r t i a l d e r i v a t i v e s of the GNP f u n c t i o n : (2.18) f(w + x , y ) = T V TTK (p , w + x )z = V G(w + x , y ) . k=l W W Then, applying the r e s u l t s of Diewert (1983a: pp. 189-190), under assumptions (2.12)-(2.13) , - 21 -(2.19) V ^ G ( w + x*. y*) = [- S w p D n - F D ^ , 1^] 11 wp 12 hi where the. matrices and a r e blocks i n the symmetric inverse matrix (2.20) D D l l °12 D21 °22 S Y PP KxK -1 2 * * and I„ i s an (M x M) i d e n t i t y matrix. The matrix V G(w + x , y ) gives M J ww y J o * the (net) export supply responses to changes i n prices (w + x ) holding A 9 A A domestic net outputs constant at y . The matrix v G(w + x , y ) i s ww * * p o s i t i v e semidefinite since the GNP function G(w + x , y ) i s convex i n * prices (w + x ). Because the GNP function i s also l i n e a r l y homogenous i n ft 2 ft ft (w + x ), the matrix V G(w + x , y ) has at le a s t one zero eigenvector: ww ft the vector of tradeables producer prices (w + x ). The following lemma connects the net balance of trade function * A 2 A A b (w + x ) and the matrix V G(w + x , y ) . ww Lemma 2.2: ft ft The gradient of the net balance of trade function b (w + x ) with respect A to the t a r i f f vector x i s : (2.21) V b*(w + x*) = - x* T [S - (S , F)D(S , F ) T ] x ww wp wp' AT ? , * *x = - x G(w + x , y ), WW ' J where the matrix D i s defined in (2.20). - 22 -Proof: Aft e r d i f f e r e n t i a t i n g (2.8)-(2.10) at the i n i t i a l e q u i l i b r i u m , the gradient * * V b (w + x ) can be solved from: t (2.22) PP KxK 0 0 N * T A T -T -T XF wp K -1 * V p -s pw * V z a -F T * *T„ V b T S T WW Since the inverse matrix D e x i s t s , (2.2) can be solved for V b (w + x ): T (2.23) V b*(w + T * ) = - x* TS + x* T(S D f 1 + FD 0 1)S + x* T(S D.0 T ww wp 11 21 pw w p l 2 + F D 2 2 ) F ww wp wp T T Consider now the quadratic form [- S D,. - FD._, I.,] S [- S D., - FD.-., n L wp 11 12 M wp 11 12 - I 1T - 23 -ww (2.24) [- S w p D u - F D j 2 , I m ] S [- S w p D u - FD* 2, ! „ ] T = S D..S D..S + S D..S D 1 0 F T + FD^.S D,, S + FD^.S D F^ wp 11 pp 11 pw wp 11 pp 12 12 pp 11 pw 12 pp 12 - S D..S - S D 1 0 F T - S D..S - FD?„S + S wp 11 pw wp 12 wp 11 pw 12 pw ww S D..S + FD? 0S D. 0F T - S D..S - S D . „ F T - S D..S 1 wp 11 pw 12 pp 12 wp 11 pw wp 12 wp 11 pw - Fuf 0S + S 12 pw ww - F D 0 0 F T - S D. 0F T - S -D...S - FD^.S + S 22 wp 12 wp 11 pw 12 pw S - (S , F)D(S , F ) T . ww wp wp Hence, V Tb (w + x ) = - x G(w + x , y ) using (2.19). QED Using Lemma 2.2, assumption (2.16) can be given two i n t e r p r e t a t i o n s . On one hand, i f the government wishes to induce an improvement in the home * country's i n i t i a l net balance of trade by changing the t a r i f f s x , i t seems * natural to require that the gradient V^b i s nonzero. In f a c t , in Section 2.4 i t w i l l be shown that (2.16) i s a necessary condition for s t r i c t improvements i n b to e x i s t . 2 1 On the other hand, assumption (2.16) i s related to l o c a l c o n t r o l l a b i l i t y of the tradeables production i n the * country (keeping domestic goods production fixed at y ): i f the rank of the matrix G(w + x , y ) i s maximal (= M - 1), then, according to D e f i n i t i o n 2.1, net export production in the home country i s l o c a l l y 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 change in net export supplies on the * economy's production p o s s i b i l i t y f r o n t i e r for tradeables (at fixed y ) can be induced by a su i t a b l e change in t a r i f f s x*. This means - 24 -that i f s t r i c t p r o d u c t i v i t y improving d i r e c t i o n s of net export production * change e x i s t , they can be attained by perturbing the i n i t i a l t a r i f f s T appropriately. Are there such d i r e c t i o n s of change i n net tradeables production? It w i l l be shown in section 2.4 that a necessary condition for s t r i c t p r o d u c t i v i t y improvements to exist i s that the " i n i t i a l vector of * t a r i f f s x i s nonzero and nonproportional to the i n t e r n a t i o n a l prices w, i . e . , the r e l a t i v e prices for tradeables at home and abroad do not coincide. But i f the i n i t i a l t a r i f f s x are nonzero and nonproportional to 9 ie ic the i n t e r n a t i o n a l tradeables prices w, and i f the matrix T" G(w + x , y ) ww ft ft i s of maximal rank, the gradient V^b (w + x ) must be nonzero, i . e . , (2.16) i s s a t i s f i e d : (2.16) can be regarded as a combined assumption concerning the existence of 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 home country's net export supply and the economy's a b i l i t y to 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, however, that l o c a l c o n t r o l l a b i l i t y of tradeables production, in the sense of Guesnerie and Weymark, i s not necessary for * * (2.16) to be s a t i s f i e d : the gradient V b (w + x ) can be nonzero even x 9 ie ic though the rank of the matrix 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 i s not a zero eigenvector of the matrix 2 * * V G(w•+ x , y ) ) . In this case, the tradeables production p o s s i b i l i t y ww ft f r o n t i e r in the home country (given a fixed y ) i s ridged and/or kinked, but, around the i n i t i a l equilibrium, at least some d i r e c t i o n s of change i n the economy's net export supply (that can be attained by a d i f f e r e n t i a l it 22 perturbation of the t a r i f f s x ) e x i s t . For the condition (2.16) to be - 25 -s a t i s f i e d , these d i r e c t i o n s of net export supply change must also be such that 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 function i s nonzero in the corresponding d i r e c t i o n of change i n the t a r i f f s x • 2.4 Existence of Productivity-Improving Changes i n T a r i f f s In the previous section, the government's p o l i c y goal was defined as * follows: f i n d a ( d i f f e r e n t i a l ) change in the t a r i f f s x such that the home country's i n i t i a l net balance of.trade b i s improved, while keeping the * domestic goods net supply at i t s i n i t i a l l e v e l y . In th i s section, the goal of the analysis i s to derive s u f f i c i e n t conditions that make this p o l i c y goal f e a s i b l e , i . e . , the aim i s to develop minimal s u f f i c i e n t conditions f o r : "fi ife ife .fe (2.25) there exist Ap , Az , Ab , Ax such that (2.11) i s s a t i s f i e d and Ab > 0. ' * A change of x for which (2.25) holds i s c a l l e d a pr o d u c t i v i t y improving * change of t a r i f f s x . Theorem 2.1 Suppose i) rank Y = K < N, i i ) rank (S + YY T) = N, i i i ) x* TV 2 *^ — ' PP w w . G(w + x*, y*) * 0^. Then, there e x i s t s a pr o d u c t i v i t y improving change i n * t a r i f f s x . The proof of Theorem 2.1 makes use of two lemmas, which are established i n Appendix 1. - 26 -Lemma 2 .3: N+K+l T T Any vector X e R s a t i s f y i n g the equations X [B , B ] = 0., „ must be p z N+K of the form (2.26) X T = k[(p* + 6 ) T , y T, 1], k e R, where (2.27) « T = x * T [ S w p D u + P D j 2 ] , Y T - t * T [ S w p D 1 2 4- F D ^ ] . Lemma 2.4: For the vector X solved in Lemma 2.3, (2.28) X TB = kx* TV 2 G(w + T* , y * ) , k z R. T WW Proof of Theorem 2.1: * A s u f f i c i e n t condition for a pr o d u c t i v i t y improving change in t a r i f f s x to exist i s (2.25). An equivalent form for this condition can be derived 25 using a theorem of a l t e r n a t i v e , Motzkin's Theorem: (2.29) there does not exist a vector X e R N + K + 1 s u c h that X T[B , B , -B ] p' z' x J - °N+K+M> X \ < °' On the contrary, suppose such a X e x i s t s . By Lemma 2.3, a vector X that T T T solves the equations X [B , B ] = 0„,.,, must be of the form X = p z N+K k[(p* + 6 ) T , y T , 1], k e R. For such a X, X TB f e = -k. Thus, for X TB f c to be negative, k > 0 (and k may be chosen to be one). By Lemma 2.4, - 27 -X TB = T * T V 2 G(w + x*, y * ) . By assumption i i i ) , X TB * 0^, a c o n t r a d i c t i o n . Q E D The following two propositions give examples of p r o d u c t i v i t y improving t a r i f f changes. Proposition 2.1: If the assumptions of Theorem 2.1 are s a t i s f i e d , a proportional reduction  of t a r i f f s x w i l l increase the amount of f o r e i g n exchange produced by the domestic production sector without diminishing the net supply of domestic commodities. Proof ; A s u f f i c i e n t 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 k k k (2.30) there exist Ap , Az , Ab , Ax such that (2.11) holds, * k k Ab > 0 and Ax = -rx , r >_ 0. 26 Applying Motzkin's Theorem, an equivalent condition i s : N+K+l T (2.31) there does not e x i s t a vector X e R such that X [B , B 1 = p z T Proceeding as in the proof of Theorem 2.1, the equations X [B^, B z] = 0^1T, are solved for X. For this X, X TB x* = k x ^ V 2 G(w + x*, y*)x*, N+K x ww ' 3 T where the p r o p o r t i o n a l i t y factor k must be p o s i t i v e since X B^ < 0. Thus, - 28 -* k may be set equal to one. By assumption, the t a r i f f vector T i s not a 2 * * zero eigenvector of the positive semidefinite matrix V G(w + T , y ) . ww Hence, X TB T* = x* TV 2 G(w + x*, y*)x* > 0. QED X WW ' J P r o p o s i t i o n 2.2: * Suppose that x >^  0^ (imports are taxed and exports are subsidized) and the assumptions of Theorem 2.1 are s a t i s f i e d . Then, there exists at least one tradeable such that lowering the t a r i f f on that good leads to a p r o d u c t i v i t y improvement. Proof : A A A*£ 2 A A According to Lemma 2.2, V b (w + x ) = -x V G(w + x , y ). Hence, •o x ww ' A l 1 0 A A A A A 9 A A x NT G(w + x , y ) x = - V b ( w + x ) x > 0 (the matrix T G(w"+ x , y ) ww J x ww * J ' * i s p o s i t i v e semidefinite and, by assumption, x i s not a zero eigenvector of the matrix G(w + x , y ) ) . If the vector x i s nonnegative, at least one of the components in the vector x G(w + x , y ) must be ft p o s i t i v e . Pick any t a r i f f x , m e [1,...,M], corresponding to the p o s i t i v e * * * number ( = - 3b /3x ). Then, lowering the t a r i f f x w i l l increase the m m amount of foreign exchange earned by the production sectors, given that a ft fixed y i s being supplied. QED A p o l i c y implication of Proposition 2.1 i s that small, competitive countries could 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 and exports are taxed), a r e s u l t p a r a l l e l to Proposition 2.2 holds: there e x i s t s at l e a s t one tradeable commodity m, m £ [1,...,M], such that an increase i n i s p r o d u c t i v i t y improving. Assumption i i i ) in Theorem 2.1 can be written in terms of the aggregate producer s u b s t i t u t i o n matrix S . Using (2.19), the following expression i s found: (2.32) T*V G(w + x*, y*) = T * T [ - S D,, - FD?_ , ww J wp 11 12' S The equations (2.32) can also be written as (2.33) x* TV 2 G(w + T*, y*) = [-<5T, x* T] S -D..S 11 pw T where the vector 5 i s defined by (2.34) * T = T * T [ S w p D l l + F D 1 2 ] « Then, since the matrix S i s p o s i t i v e semidefinite, * T 9 * * * T *T (2.35) T V G(w + x , y ) x = [- 6 , T ] S > 0. - 30 -Assumption i i i ) can now be replaced by: (2.36) the vector [- 6 , x ] i s not p r o p o r t i o n a l to any zero eigenvector of the producer s u b s t i t u t i o n matrix S. I f the only zero eigenvector of the matrix 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 to e i t h e r (2.37) t a r i f f s x (* 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 p r i c e s w or (2.38) the vector of producer p r i c e s f o r domestic commodities p i s not p r o p o r t i o n a l to the vector 5 defined i n (2.34). Lemma 2.5 i n Appendix 1 e s t a b l i s h e s the equivalence of (2.37) and (2.38). Diewert (1983b: p. 273) shows that the vec t o r (p + 6) i s the appropriate p r o d u c t i v e e f f i c i e n c y vector of shadow p r i c e s ( f o r domestic commodities) f o r choosing 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 taxes are present i n the economy. Using (2.37) and (2.38), i t can be added that * the domestic producer p r i c e s p should be used as the shadow p r i c e s * f o r cost b e n e f i t a n a l y s i s only i f i ) 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 zero, i . e . x = 0^, or i f i i ) 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 p r i c e s w. Under the rather strong 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 complicated assumption i i i ) i n Theorem - 31 -2.1 s i m p l i f i e s to an e a s i l y understood nonproportionality condition. Furthermore, i f the producer s u b s t i t u t i o n matrix S i s of maximal rank, the domestic goods producer s u b s t i t u t i o n matrix i s of f u l l rank (= N), which means that assumption i i ) of Theorem 2.1 i s s a t i s f i e d . P roposition 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 vector ft of t a r i f f s T 0^) i s not proportional to the inte r n a t i o n a l prices w. ft Then, there exists a produ c t i v i t y improving change in t ; the change in x may be chosen to be a proportional reduction. If there i s only one aggregate production sector in the home country, even the rank assumption on the domestic goods net supply matrix Y i n Proposition 2.3 may be erased. The r e s u l t i n g extremely simple version of Theorem 2.1 reveals the basic economic conditions which are s u f f i c i e n t for pro d u c t i v i t y improving t a r i f f changes to e x i s t : there must be s u b s t i t u t i o n i n production, and the r e l a t i v e producer prices for tradeable commodities in the home country must d i f f e r from the i n t e r n a t i o n a l prices w. The add i t i o n a l suppositions i n Theorem 2.1 are needed to cover the more general cases where the economy's t o t a l production p o s s i b i l i t y set may be kinked and the number of production sectors exceeds one. The ce n t r a l role of the s u b s t i t u t a b i l i t y and nonproportionality assumptions i s emphasized in Theorem 2.2 and in i t s c o r o l l a r y below: they e s t a b l i s h that 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, i . e . , t a r i f f changes that cause a s t r i c t increase in the home country's i n i t i a l net - 32 -balance of trade b*, cannot exist i f assumption i i i ) of Theorem 2.1 i s v i o l a t e d . Let us f i r s t consider the most general necessary conditions for nonexistence of 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 perturbations. These conditions can be regarded equivalently as 'necessary conditions for  p r o d u c t i v i t y optimality of the i n i t i a l e quilibrium. Theorem 2.2: A necessary condition for a 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 change to not exist i s : (2.39) there i s a vector X e R n + k + 1 s u c h that X T[B , B , - B ] = 0T, , , , p z' x N+K+M X T B U < 0. b Proof: A necessary condition for 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 librium i s : 1c 1c 1c ~k (2.40) there do not ex i s t Ap , Az , Ab , Ax such that (2.11) i s * s a t i s f i e d and Ab > 0. Using Motzkin's Theorem and the proof of Theorem 2.1, i t can be seen that (2.39) and (2.40) are equivalent. QED C o r o l l a r y 2.1: Suppose i) rank Y = K < N, i i ) rank (S + YY T) = N, i i i ) x* T V 2 — pp ww - 33 -it k X G(w + T , y ) = 0^. Then, the i n i t i a l e quilibrium s a t i s f i e s the necessary condition for pr o d u c t i v i t y optimality given in Theorem 2.2, and no s t r i c t 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 in t a r i f f s T e x i s t . Proof: Assumptions i ) - i i ) of Corollary 2.1 and Lemma 2.3 imply that a vector A T T which solves the equations A [B , B ] = 0„ T, must be of the form p z N+K * fj T A = k[(p + 6) , y , 1], where k e R. Choose k = 1 so that ATB, = - 1 < 0. By Lemma 2.4, A TB = x* TV 2 G(w + x* y * ) . b 1 x ww J T T Hence, by assumption i i i ) , A B = 0^ and (2.39) i s s a t i s f i e d . QED C o r o l l a r y 2.1 implies that (2.16) i s a necessary condition for s t r i c t  p r o d u c t i v i t y improvements i n t a r i f f s T to e x i s t . In other words, i f •k -k -k the gradient of the net balance of trade function b , b (w + T ), i s a zero M-vector (and (2.12) and (2.13) are s a t i s f i e d ) , the i n i t i a l e q u i l i b r i u m s a t i s f i e s the necessary condition (2.39) for productivity o p t i m a l i t y . However, (2.39) i s not s u f f i c i e n t for the i n i t i a l equilibrium to be a pr o d u c t i v i t y maximum, i . e . , such that the maximum for the net & ft balance of trade function b (w + x ) i s attained. - 34 -3. HOW LARGE IS THE PRODUCTIVITY GAIN? It was shown in the previous chapter that, under c e r t a i n rather weak conditions, p r o d u c t i v i t y improving changes in t a r i f f s e x i s t . How large are these.productivity gains, i . e . , the improvements in the home country's i n i t i a l net balance of trade b ? Approximating formulae for the increase in b can be developed using the Allais-Diewert method of measuring the deadweight loss due to i n e f f i c i e n t schemes of taxation. Before considering the approximations for the p r o d u c t i v i t y gain, i t i s useful to give a b r i e f review of the d e r i v a t i o n of the Allais-Diewert deadweight loss measure. 1 Diewert (1983c) defines the maximal amount of foreign exchange a v a i l a b l e to the production sector, under the condition that the production sector in the aggregate supplies at l e a s t the i n i t i a l e q u i librium amount y* of domestic commodities, as the s o l u t i o n b° to o r ^ T k k ^ k k * (3.1) b = max { £ w f z : E y z ^ y , k=l,...,K|. (y , f ) e C z > 0 K The value b° that solves (3.1) i s at least as large as the i n i t i a l v * T *k *k 2 observed net balance of trade b ( = £ w f • z )• The A l l a i s -k=l Diewert production loss pertaining to the i n i t i a l equilibrium i s then defined as - 35 -(3.2) ^ = b - b > 0. Using the K a r l i n (1959: p. 201) - Uzawa (1958: p. 34) Saddle Point Theorem, (3.1) can be written in an equivalent dual form: o r ^ k k T * i (3.3) b = max min { E TT (p,w)z - p y : k=l,...,Kj, z >_ 0 K p > 0 N k=l where the vector of Lagrange m u l t i p l i e r s p i s the (producer) shadow p r i c e vector for domestic commodities when no t a r i f f d i s t o r t i o n s are present in the economy. The f i r s t order necessary conditions for (3.3) are: (3.4) Tr k(p,w) = 0, k = l , . . . , K , (3.5) Z V Tr k(p,w)z k = y*. k=l P It i s assumed that a unique s o l u t i o n z e R , p e R f o r (3.4) -++ ++ (3.5) e x i s t s , and that the industry scales z and domestic prices p can be regarded as i m p l i c i t functions of the exogenous (w, y*) around the s o l u t i o n for (3.4) - (3.5). 3 Let us now compare the optimal reference equilibrium s a t i s f y i n g (3.4) - (3.5) to the i n i t i a l d i s t o r t e d equilibrium characterized by the e q u a t i o n s (2.8) - ( 2 . 9 ) . The o p t i m a l e q u i l i b r i u m can be mapped to the d i s t o r t e d e q u i l i b r i u m by d e f i n i n g a g - e q u i l i b r i u m f o r w h i c h (3 .6) i r k ( p ( 0 , w + x * 0 = 0 , k=l ,. .. , K, (3.7) E V 7 r k ( p ( 5 ) , w + T * ) z k ( 5 ) = y*. k=l P I f 5 = 0, (3.6) - (3.7) d e f i n e s the same o p t i m a l e q u i l i b r i u m as the e q u a t i o n s (3.4) - ( 3 . 5 ) , and i f £ = 1, (3.6) - (3.7) d e f i n e s the i n i t i a l o b s e r v e d e q u i l i b r i u m t h a t s o l v e s ( 2.8) - ( 2 . 9 ) . F o r each e q u i l i b r i u m o f the economy i n d e x e d by 5 ( 0 <_ 5 <_ 1 ) , D i e w e r t d e f i n e s the A l l a i s o b j e c t i v e f u n c t i o n A(Q f o r the economy as the net amount of f o r e i g n exchange t h a t the p r o d u c t i o n s e c t o r ( i n the a g g r e g a t e ) p r oduces a t the e q u i l i b r i u m i n d e x e d by (3.8) A(C) = Z w T V k ( p ( 5 ) , W + T * ? ) ^ ? ) . 1 * k=l U s i n g ( 3 . 8 ) , the A l l a i s - D i e w e r t l o s s measure A^ d e f i n e d i n (3.2) can be w r i t t e n as (3.9 ) ^ = A ( 0 ) - A ( l ) . - 37 -Next, the value A( 1 ) i s approximated using the second order Taylor Series expansion of A around 5 = 0 , A ( 0 ) + A ' ( 0 ) ( 1 - 0 ) + 1/2 A " ( 0 ) ( 1 - 0 ) 2 . Using this formula, the Allais-Diewert loss AL becomes approximately - A ' ( 0 ) - 1/2 A " ( 0 ) . It can be shown that the d e r i v a t i v e of the A l l a i s objective function at 5 = 0 , A ' ( 0 ) , i s zero and hence, ( 3 . 1 0 ) A L = -1/2 A " ( 0 ) , 5 where ( 3 . 1 1 ) - A " ( 0 ) = [ p ' ( 0 ) T , T* T] Sc P ' ( 0 ) * T > 0. In ( 3 . 1 1 ) , p ' ( 0 ) denotes the d e r i v a t i v e s of the domestic commodity prices p with respect to 5, evaluated at 5 = 0 . These d e r i v a t i v e s can be calculated by d i f f e r e n t i a t i n g the equations ( 3 . 4 ) - ( 3 . 5 ) at the solution of ( 3 . 4 ) - ( 3 . 5 ) and by s e t t i n g 5 = 0 . The producer s u b s t i t u t i o n matrix S° in ( 3 . 1 1 ) i s evaluated at the optimal undistorted equilibrium; p o s i t i v e semidefiniteness of S° implies that - A " ( 0 ) >_ 0. Let us now return to the problem of measuring the p r o d u c t i v i t y gain accruing from a p r o d u c t i v i t y improving perturbation of t a r i f f s * x . Suppose that the i n i t i a l t a r i f f d i s t o r t e d equilibrium i s indexed by 1, the new ( a f t e r the t a r i f f change) equilibrium by 2, and the optimal undistorted equilibrium by 0. It seems natural to measure the producti-v i t y gain, AQ, between the e q u i l i b r i a 1 and 2 by the d i f f e r e n c e - 38 -(3.12) A_ = b 2 - b 1 > 0, LJ — where bl i s the observed net balance of trade in equilibrium 1 (formerly denoted by b*) and b^ i s the net balance of trade after the change in t a r i f f s (formerly denoted by T*). If the perturbation in the i n i t i a l t a r i f f s T 1 i s s t r i c t p r o d u c t i v i t y improving, the gain AQ i s nonnegative. 6 Adding and subtracting the maximal net balance of trade b° on the r i g h t hand side of (3.12), the gain AQ can be written as (3.13) A G - (b° - b 1) - (b° - b 2) = A ° + 1 - A ° * 2 . In (3.13), A^ ^ denotes the A l l a i s - Diewert p r o d u c t i v i t y loss between the optimal and actual observed e q u i l i b r i a , and A^^ 2 i s the A l l a i s -Diewert loss between the optimal and new ( a f t e r the change i n t a r i f f s ) e q u i l i b r i a . The p r o d u c t i v i t y gain AQ has 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 losses between the three e q u i l i b r i a 0, 1 and 2. Applying Diewert's second order approximation rule (3.10) -(3.11) to (3.13), the approximate gain accruing from a s t r i c t p r o d u c t i v i t y improving change in t a r i f f s i s : (3.14) A G » I [ p ' ( 0 ) T , x 1 T ] S° [ P ' ( 0 ) T , T 1 T ] T -J [ P ' ( 0 ) T , T 2 T ] S° [ P ' ( 0 ) T , T 2 T ] T , - 39 -where x2 denotes the vector of t a r i f f s at the equilibrium 2 . In order to calculate the gain approximation (3.14), the government must have information about the aggregate producer s u b s t i t u t i o n matrix S^, the domestic price d e r i v a t i v e s p'(0) (which depend on the net supply matrices and at the optimal equilibrium), and the po l i c y v a r i a b l e s and x2. In p r a c t i c e , unfortunately, i t may be Impossible to form estimates for the unobserved matrices S^, Y°, and F^. Further approximations to (3.14) are thus c a l l e d f o r . One p o s s i b i l i t y i s , i f the optimal and observed e q u i l i b r i a are not too far from each other, to replace the matrices S°, F° and Y° i n (3.14) by the observed matrices S 1, F 1 and Y^. This approximation, however, involves an error, the size of which i s not known. Is there then any way of measuring the p r o d u c t i v i t y gain as accurately as i n (3.14), using only the observed information at equilibrium 1? It turns out that t h i s i s indeed possible, i f the following version of the Quadratic Approximation Lemma i s employed. Lemma 3.1: (The Quadratic Approximation Lemma; Diewert (1976: p. 118)). For a twice continuously d i f f e r e n t i a b l e function f ( z ) , z e R^ , (3.15) f U 1 ) - f(z°) = Vf(z°) T(z 1 - z°) + 1 ( z 1 - z ° ) T V 2 f(z°) ( z 1 - z°) I 7.2. i f and only i f (3.16) f(z) = a Q + a Tz + i z TAz, - 40 -where A^ = A, or i f and only i f (3.17) f ( z X ) - f(:z°) = 1 [Vf(z°) + V f ( z 1 ) ] T ( z 1 - z°). Lemma 3.1 establishes two exact expressions f o r the change i n the value of a quadratic function f ( z ) : (3.15) uses the second order Taylor Series expression, while (3.17) uses the inner product of the average gradient l/2[Vf(z°) + V f t z 1 ) ] with the differe n c e ( z 1 - z°). I f the function f ( z ) i s not quadratic, i . e . , of the form (3.16), (3.15) gives a second order approximation to the change f ( z ^ ) - f ( z ^ ) . The formula (3.17) also provides an approximation to f ( z ^ ) - f ( z ^ ) which, by rewriting (3.17) i n the form (3.18) fCz1) - f(z°) « I [Vf(z°)T(z1 - z0)] + I [Vf(z 1) T(z 1 - z 0)], can be given a new i n t e r p r e t a t i o n : the term Vf(z^)T (z* - z^) i n (3.18) y i e l d s a f i r s t order approximation to the change f ( z * ) -f ( z ^ ) around the point z^, whereas the term V f ( z ^ ) ^ ( z ^ - z^) provides a f i r s t order approximation to f ( z * ) - f ( z ^ ) around the point TS . Thus, i n (3.18), the change i n the value of the function f i s approximated using the average of two f i r s t order approximations. Lemma 3.1 shows that, for a quadratic function, (3.18) provides exactly as good an approximation to the change f ( z ^ ) - f ( z ^ ) as the second order expression (3.15); both are in fact exact. If the function f i s not quadratic, the approximation (3.18) i s approximately as accurate as the second order formula (3.15). - 41 -The above r e s u l t can be used to derive another approximation formula for the p r o d u c t i v i t y gain measure AQ defined in (3.12). Using (3.13), AQ can be regarded as a d i f f e r e n c e of two loss measures A^*^ and A^ 2 which themselves are defined as changes in the A l l a i s objective function A(£). Let us approximate the l o s s A^*^, not to the second order around the optimal equilibrium, but using the average of the two f i r s t order losses around the e q u i l i b r i a 0 and 1. To the f i r s t order, around the equilibrium 1, (3.19) A ^ 1 = A(0) - A ( l ) * A ' ( l ) and to the f i r s t order, around the equilibrium 0, (3.20) A ^ 1 = A(0) - A ( l ) - -A'(0) Hence, by taking the average of (3.19) and (3.20), (3.21) A ^ 1 - I [A'(l) - A'(0)] Diewert (1983c: p. 170) shows that A*(0) = 0, which implies (3.22) * ™ - ^ A . ( l ) . - 42 -S i m i l a r l y , f or the measure (3.23) A ^ Z . I A ' ( 2 ) . 7 The average l o s s measures (3.22) and (3.23) are approximately as accurate as the corresponding second order measures derived using the formulae '('3.10) - (3.11). Applying (3.22) - (3.23), the p r o d u c t i v i t y g ain measure AQ d e f i n e d i n (3.13) can be approximated by Using the r e s u l t s i n Diewert (1983c) , the d e r i v a t i v e s of the A l l a i s o b j e c t i v e f u n c t i o n , A ' ( l ) and A 1 ( 2 ) , can be shown to equal (3.24) A G « I [ A ' ( l ) - A ' ( 2 ) ] . (3.25) A ' ( l ) = [ p ' ( l ) T , x 1 T ] S 1 [ p * ( l ) T , T 1 T ] T and ( 3 . 2 6 ) A'(2) = [ p ' ( 2 ) T , x 2 T ] S 2 [ p ' ( 2 ) T , T 2 T ] T . 9 Formulae ( 3 . 2 4 ) - ( 3 . 2 6 ) provide the second approximation f o r the p r o d u c t i v i t y gain AQ i n t h i s s e c t i o n . The measure ( 3 . 2 4 ) depends on the observed producer 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 matrices Y and F . It also depends on the corresponding-unobserved matrices i n the new ( a f t e r the t a r i f f change) equilibrium. I f , however, the perturbation i n t a r i f f s X* i s small, i t i s reasonable to assume that the changes in the producer s u b s t i t u t i o n matrix S^- are also small. Hence, the matrix S 2 can be approximated by the matrix S 1. After this adjustment, the p r o d u c t i v i t y gain measure (3.24) The approximation error involved i n s u b s t i t u t i n g the matrix Sl for the matrix S 2 i n the second term of (3.27) i s l i k e l y to be considerably smaller than the error i n (3.14), i f , i n (3.14), the matrix i s used instead of the matrix S^. 2 2 If the net output matrices a f t e r the t a r i f f change, Y and F , can be assumed to be close to the i n i t i a l observed matrices Y* and F*, the d e r i v a t i v e s p'(2) in (3.27) can be replaced by the i n i t i a l e quilibrium price d e r i v a t i v e s p ' ( l ) . In t h i s case, the gain measure (3.27) s i m p l i f i e s to becomes (3.27) A G - 1 , ,,..T IT. c l r ,,.*T 1T.T 2 [P (1) , x ] S [ p ' ( D , x ] - ^ [ P ' ( 2 ) T , x 2 T ] s1 [ P ' ( 2 ) T , x 2 T ] V ° (3.28) A G « wp p ' ( l ) + p ' ( l )T ( x 1 - x 2 ) + X IT S WW 1 2T c l 2, x - x S x J. ww - 44 -This approximation depends only on observable v a r i a b l e s . 1 1 I t can be seen from (3.28) that the p r o d u c t i v i t y gain AQ increases p r o p o r t i o n a l l y with the production s u b s t i t u t i o n terms S^, S 1 and S* , i . e . , i f a l l the matrices ,. S1 and are m u l t i p l i e d wp ww pw' wp ww r by a s c a l a r a > 0, the gain AQ i s replaced by OCAQ: the more  s u b s t i t u t i o n there i s i n the domestic production s e c t o r , the l a r g e r the  gains from p r o d u c t i v i t y improving t a r i f f p o l i c i e s are l i k e l y to be. In the case of 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 of the approximate g a i n (3.28) w i t h respect to a p r o p o r t i o n a l i t y f a c t o r . Assume, for example, that T 2 = k x 1 , k e ( 0 , 1 ) . Using (3.28), (3.29) A - i [(1 - k) x 1 T S 1 p ' ( l ) + Cr Z Wp (1 - k) p» ( 1 ) T S 1 T 1 + T 1 T S 1 T 1 - k 2 T 1 T S1 T 1 ] . pw WW WW Hence, (3.30) ^ - = T [ - t 1 T s 1 P'(D " P ' ( 1 ) T S 1 T 1 - 2k T 1 T S 1 T 1 ] , dk 2 wp pw ww I f the producer s u b s t i t u t i o n matrix f o r domestic and tradeable commodities i s zero (S = 0„ . , ) , the d e r i v a t i v e (3.30) becomes pw NxM ' (3.31) -JT-^ = -2k T A SL T < 0. dk ww — - 45 -The weak i n e q u a l i t y i n (3.31) follows from the p o s i t i v e semidefiniteness of the matrix . From (3.31) i t can be seen that as the ww 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 . , as the proportional reduction of t a r i f f s x^, which i s s t r i c t p r o d u c t i v i t y improving by Proposition 2.1, becomes l a r g e r ) , the p r o d u c t i v i t y gain AQ i n c r e a s e s . 1 2 F i n a l l y , i t should be noted that the p r o d u c t i v i t y gain measure AQ defined in (3.12) i s only a p a r t i a l , production side, measure of gain as opposed to a general equilibrium gain formula. Nonetheless, i t i s valuable as a lower bound for the t o t a l general equilibrium gain accruing from a p r o d u c t i v i t y improving change of t a r i f f s . If a more accurate approximation of the general equilibrium gain i s needed, the method of measuring the p r o d u c t i v i t y gain presented in t h i s section could be adapted to the general equilibrium context by employing the Debreu-Diewert measure of deadweight los s defined in Diewert (1984). - 46 -4- STRICT PARETO IMPROVING CHANGES IN COMMODITY TAXES AND TARIFFS 4.1 A General E q u i l i b r i u m Model There are H consumers (households), indexed by h=l,...,H, in the economy. The preferences of the consumers are represented by t h e i r expenditure functions ran(un, q, v ) , which are defined as the minimum net cost ( f a c t o r supplies are indexed negatively) of achieving a given u t i l i t y l e v e l u n . In addition to u^1 the expenditure functions mn, h=l,...,H, depend on the N-dimensional vector of domestic consumer prices q = (p + t) e R^ , where p i s the vector of domestic producer prices and t i s a vector of taxes or subsidies on domestic commodities. The expenditure functions m*1, h=l,...,H, also depend on a vector of consumer prices for i n t e r n a t i o n a l l y tradeable goods equal to M v = (w + x + s) e R , where w i s the world price vector, T i s the t a r i f f vector and s i s a vector of taxes or subsidies on i n t e r n a t i o n a l l y tradeable commodities. The expenditure functions mh, h=l,...,H, are assumed to be twice continuously d i f f e r e n t i a b l e . They are also concave and l i n e a r l y homogenous in prices (q, v ) . Using Shephard's Lemma, the consumers' Hicksian (compensated) net demand functions for domestic and tradeable commodities can be derived as f i r s t order p a r t i a l d e r i v a t i v e s of the expenditure fuctions mnt (4.1) x h(u h,q,v) = V qm h(u h,q,v), h=l H, - 47 -and (4.2) e h ( u h , q, v) = V vm h(u h, q, v ) , h=l,...,H. The consumer net demand matrices for domestic and tradeable goods are defined using (4.1) and (4.2): (4.3) X = [ x * , . . . , x H ] ; E = [ e ^ , . . . , e H ] . The aggregate consumer s u b s t i t u t i o n matrix E i s defined by (4.4) E = E E qq qv E E vq vv H E h=l u 2 h V m qq „2 h V m vq V m qv „2 h V m vv In (4.4), the matrix block E q v , for example, gives the aggregate net demand responses to changes i n tradeables consumer prices for domestic commodities. The other blocks of the s u b s t i t u t i o n matrix E have analogous i n t e r p r e t a t i o n s . The matrix E i s symmetric, negative semidefinite (since the expenditure functions mh, h=l,...,H, are concave in prices (q, v)) and s a t i s f i e s (4.5) [ q T , v T ] E = 0^ + M - 48 -This follows because the expenditure functions m , h=l,...,H, are l i n e a r l y homogenous i n prices (q, v ) . Equations (4.5) imply that the rank, of the matrix E can be at most N + M - 1. It may be that some of the N domestic commodities are producer durables or supplied independently of prices by the consumers. Suppose the n t n domestic good i s such a commodity. Then, the n t n row and column of the matrix Eqq consist of zeros. 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 enter into the preferences of any consumer, or i f i t passes through the domestic production sector before being a v a i l a b l e to consumers, the column m of the matrix Eq V i s a column of zeroes, and the m1-^  row of the matrix E i s a row of zeroes. Since the expenditure functions m^u*1, q, v) , h=l,...,H, are nondecreasing in u n , h=l,...,H, the u t i l i t y of each household can be measured i n terms of income, holding the consumer prices (q, v) fixed at some given l e v e l . If the consumer prices are fixed at the i r i n i t i a l f^e & h h h ^ ^ equilibrium l e v e l s (q , v ), the equations u = m (u , q , v ), h=l,...,H, define the consumers' u t i l i t y l e v e l s u*1 i n the neighbourhood of the i n i t i a l e q uilibrium. This money metric s c a l i n g of u t i l i t i e s implies the following r e s t r i c t i o n s : (4.6) V h m h(u* h, q*, v*) =1, h=l,...,H, u and (4.7) V 2 h h m h(u* h, q*. v*) = 0, h=l,...,H. u u In addition, Diewert (1978: p. 146) shows that, i f the money metric s c a l i n g of u t i l i t i e s i s applied, the matrices E and E defined below ° r ' qu vu can be interpreted as the income d e r i v a t i v e matrices of the consumers' ordinary demand functions: (4.8) E = [m1 ,...,mH ], qu qu . qu where mh = V 2 m h(u* h, q*, v * ) , h=l,...,H, qu qu and (4.9) E '= [m1 ,...,mH ], vu vu vu ' h 2 h, *h * * where m = V m ( u , q , v ) , h=l,...,H. vu vu The government in the economy imposes t a r i f f s on tradeables and taxes both domestic and tradeable commodities. Furthermore, the government may give the households ( p o s i t i v e or negative) lump sura t r a n s f e r s . The vector of lump sum transfers i s denoted by g e R^. With i t s income, the government buys domestic and tradeable commodities 0 N 0 M , . J i n amounts x e R + and e e R +. These commodities are used to produce public goods and services for the private sector. The vectors x^ and - 50 -e u are assumed to stay constant throughout the analysis so that no p u b l i c goods e x p l i c i t l y appear i n the model. There are H + N + K + 1 equations that characterize an e q u i l i b r i u m (indexed with an a s t e r i s k ) when the demand side of the economy i s taken into account: h *h * '* *h (4.10) m (u , q , v ) => g , h=l,...,H, (4.11) T f k ( p * , w + x*) = 0, k=l,...,K, ^ h *h * * 0 ^ k * * * k (4.12) Z V m (u , q , v ) + x = E V ir (p , w + x ) z , h=l q k=l P ii i i \ T ? n W * h * *^ T 0 (4.13) w £ V m ( u , q , v ) + w e = h=l V L w V 7 T ( p , W + T ) Z - b . w k=l According to (4.10) - (4.13), the consumers (households) equate t h e i r expenditures on domestic and tradeable commodities minus the i r factor incomes to t h e i r lump sum revenues (which may be z e r o ) 1 , the K constant returns to scale production sectors earn zero (pure) p r o f i t s , consumer aggregate net demand for domestic commodities equals t h e i r aggregate net supply, and the balance of payments net surplus equals b . - 51 -If (4.10) - (4.13) are s a t i s f i e d , by Walras' Law, the government budget constraint (4.14) t * T x° + ( t * + s * ) T e° + t * T E x* h + s* T ? e* h h=l h=l * T H A h K * k + T [ E e - E f z j h=l k=l •*T o . *T o . J *h * = q x + v e + E g + b h-1 also holds. With a simple manipulation (4.14) can be rewritten as: ,. ... *T ? *h . *T ? * t i *T r J *h * *k *k, (4.15) t E x + s E e + x [ E e - E f z ] h-1 h-1 h=l k=l * T ° x T ° j. ? * h u. K * = p x + w e + E g + b h=l This form of the budget constraint implies that the government expenditures on domestic and tradeable commodities plus the lump sum transfers forwarded to the consumers plus the balance of payments net surplus must equal the government tax and t a r i f f income. The exogenous variables in (4.10) - (4.13) are the i n t e r n a t i o n a l prices w which also provide a price normalization in the model, and the 1c 1c 1c 1c government p o l i c y instruments g , t , s and x • The endogenous - 52 -v a r i a b l e s , which are determined i n ( 4 . 1 0 ) - ( 4 . 1 3 ) as i m p l i c i t functions of the exogenous v a r i a b l e s , are u* (household u t i l i t y l e v e l s ) , p* (domestic producer p r i c e s ) , z*(industry scales) and b* (balance of trade net surplus). It i s assumed that an i n i t i a l equilibrium s a t i s f y i n g ( 4 . 1 0 ) -ie ie ( 4 . 1 3 ) and (p , z ) » 0 N + £ e x i s t s . T o t a l d i f f e r e n t i a t i o n of ( 4 . 1 0 ) - ( 4 . 1 3 ) at the i n i t i a l values of the variables of the model y i e l d s : ie ie ie ie ie ie ie • ie ( 4 . 1 6 ) AAu = B Ap + B Az + B, Ab + B AT + B At + B As + B Ag , p z b x t s g where the matrices A, Bp,...,Bg are defined as follows: 1* A = qu KxH T w E vu -X -E + S qq P P T T -w E + w S vq wp HxK KxH T w F N °K -1 B = T -E -E + S qv pw T T -w E + w S VV WW - 53 --X -E qq °KxN -w E vq -E -E qv 0, KxN -w E vv H NxH KxH T i H Using the I m p l i c i t Funct ion Theorem, the d e r i v a t i v e s of the endogenous * * * * * * * * u , p , z and b with respect to the exogenous x , t , s , and g are determined by the matrix [A, -B , -B , -B, ] _ 1 [B , B_, B , B ]. J ' p' z' b x t s 2 4.2 Existence of S t r i c t Pareto Improving Changes i n Commodity Taxes and T a r i f f s A s t r i c t Pareto improving change i n the commodity tax rates ( t , s ) and t a r i f f s x i s one which leads to a u t i l i t y increase for each consumer (household) i n the economy. The goal of t h i s section i s to develop s u f f i c i e n t conditions for such a tax and t a r i f f perturbation to exist when, in addition, the t a x / t a r i f f change i s required to lead to an increase i n the home country's i n i t i a l net balance of trade. More p r e c i s e l y , the problem Is the following: under what conditions, s t a r t i n g from an i n i t i a l equilibrium which s a t i s f i e s (4.10) - (4.13), * * * * * * * * do there e x i s t Au , Ap , Az , Ab , At , As , Ax and Ag such that (4.16) holds and A u » 0 H, Ab > 0, Ag = 0 R ? 5 One of the s u f f i c i e n t conditions given i n Theorem 4.1 below i s a r e s t r i c t i o n on the consumers' preferences and on t h e i r i n i t i a l commodity - 54 -endowments. It i s required that (4.17) there i s no solution a > 0 U to a T [ X T , E T] = 0^tx.. H N+M This supposition i s s a t i s f i e d i f there i s some domestic good n, n e [1,...,N], which i s i n net demand or i n net supply by every household, i . e . , > 0 or < 0 for a l l h=l,...,H. A l t e r n a t i v e l y , i t i s s u f f i c i e n t 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], i n net demand or i n net supply by every consumer; i n t h i s case, e S 0 or < 0 for a l l h=l,...,H. Goods that are demanded or supplied by a l l consumers i n the economy are often c a l l e d Diamond-Mirrlees goods. Weymark (1979: pp. 176-177) shows that the existence of a Diamond-Mirrlees good (or a composite Diamond-Mirrlees commodity) i n the economy guarantees that some Pareto improving d i r e c t i o n s of consumer pr i c e changes ( i . e . , commodity tax changes) e x i s t . Without t h i s assumption, the household preferences might be c o n f l i c t i n g to such a degree that no Pareto improvement through a perturbation of the economy's i n i t i a l commodity tax structure i s possible. Assumption (4.17) thus implies s u f f i c i e n t homogeneity of consumer preferences without which Theorem 4.1 cannot be e s t a b l i s h e d . 6 Theorem 4.1 Suppose ( i ) rank Y = K •<_ N, ( i i ) rank [ S p p + YY T] = N, ( i i i ) x* T V 2 G(w + T*, y*) * 0^ and ( i v ) there i s no solution a > 0„ WW M ri - 55 -to a T [ X T , E T ] = 0 ^ + M . Then, a ( d i f f e r e n t i a l ) s t r i c t Pareto and pro d u c t i v i t y improving change in the economy's i n i t i a l t a r i f f and commodity tax structures e x i s t s , holding the i n i t i a l vector of lump sum transfers constant. This change i n taxes and t a r i f f s also improves the home country's net balance of trade. Proof: Applying Motzkin's Theorem, a s u f f i c i e n t condition for a s t r i c t 7 Pareto and pr o d u c t i v i t y improving tax and t a r i f f change to exi s t i s : ( 4 . 1 8 ) there i s no vector A T = [ A 1 T , A 2 T, X 3 T, X 4] e R h + n + k + 1 s u c h that X T [ B p , B z, B x, B t, B g] = 0 ^ ^ , X T[A, -Bfa] > 0 ^ . T T Consider the equations A [B , B , B ] = 0 „ , „. Subtract the N p z t N+K+N equations corresponding to the matrix B t from the f i r s t N equations corresponding to the matrix Bp. This implies (4 .19) [ X 2 T , A 3 T, A 4] PP ,T T w S wp KxK T w F = 0 N+K Equations (4 .19) have already been solved in Lemma 2 . 3 . Hence, - 56 -i f X 4 = k e R, X 2 T = k(p* + 6 ) T and X 3 T = k y T , where the v e c t o r s 5 and y T T are those defined i n (2.27). The i n e q u a l i t i e s X [A, -B,] > 0„,. imply b H+l 4 that X = k ^ 0. In order to determine i f (4.18) can be s a t i s f i e d , two cases need to be considered. ( i ) X 4 = k = Q: 2T T 3T T I f k = 0, then also X = 0„ and X = 0T,. In order to s a t i s f y N K (4.18), the f o l l o w i n g must h o l d : (4.20) X TA = X 1 T > ol, X TB = - X 1 T X T = oJ, X TB = - X 1 T E T = 0^. H t N s M By assumption, there 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). ( i i ) X 4 = k > 0: T T Set k = 1. Let us consider the equations X B^ = 0^. The goal here i s to show that these equations cannot be s a t i s i f e d i f T 1T A T T 1 X = [ X , (p + 6 ) , y , 1] (the v e c t o r X may be solved from the T T 1 equations X [B^, B g] =» ^N+M' ^ U t knowledge °f ^ * s n o t required f o r the proof of the theorem). Using the above defined v e c t o r X, (4.21) X TB = - X 1 T E T - (p* + 6 ) T E + (p* + 6 ) T S + y T F T - wT E T K / q v ^ p w w T + w S ww = (p* + 6 ) T S + Y T F T + w TS , pw WW - 57 -since X B g = 0^. Applying Lemma 2.4, i t can be seen that (4.22) X TB = -x* TV 2 G(w + x*, y*) * 0T,. QED T WW M ; . Theorem 4.1 shows that a s t r i c t p r o d u c t i v i t y improving change in A the i n i t i a l equilibrium t a r i f f s x can be converted to a s t r i c t Pareto * A * improving change of the t a r i f f s x and commodity tax rates (t , s ) , without a change i n the consumers' i n i t i a l lump sum incomes. The f i r s t three assumptions needed to e s t a b l i s h the r e s u l t have been encountered i n Theorem 2.1. They imply that a 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 change, s t a r t i n g from the i n i t i a l e q uilibrium, e x i s t s . In order to d i s t r i b u t e these pro d u c t i v i t y gains to the households i n a s t r i c t Pareto improving way, the fourth assumption which involves the consumer preferences and endowments, must be s a t i s f i e d . The role of assumptions ( i ) - ( i v ) i s further c l a r i f i e d , i f Theorem 4.1 i s compared to some r e s u l t s of Diamond and Mirrleess (1971). In their c l a s s i c paper, Diamond and Mirrlees consider, among other things, the existence of Pareto improving changes i n commodity taxes in a closed economy. They show that i f ( i ) a l l production in the 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 equilibrium production choice l i e s inside the home country's production p o s s i b i l i t y set , and ( i i i ) a Diamond-Mirrlees good e x i s t s , then a Pareto Improving change in the economy's i n i t i a l commodity tax rates i s possible. They argue further that i f , a l t e r n a t i v e l y , production in the economy takes - 58 -place i n a private production sector, Pareto-improving tax changes s t a r t i n g from the i n i t i a l equilibrium s t i l l e x i s t , i f the producer and consumer prices i n the economy can be perturbed independently from each < 8 other and the above mentioned assumptions ( i i ) - ( i i i ) are s a t i s f i e d . Consider now the existence r e s u l t i n Theorem 4.1. Let us define i n t e r n a t i o n a l trade as an ad d i t i o n a l production technology made av a i l a b l e for the home country. This a r t i f i c i a l technology can be expressed as the set — r K k K k K k T K k (4.23) T = {( E y , E f ) : E y < 0 , w E f < o}. k=l k=l k=l k=l The economy's t o t a l production p o s s i b i l i t y set i s then generated by the sum of the domestic production technologies T^, k=l,...,K and the set T. In Figure 3, which i s drawn assuming that there are two i n t e r n a t i o n a l l y tradeable commodities (M = 2) and one consumer (H = 1) in the home country, the curve PP' gives the domestic production p o s s i b i l i t y f r o n t i e r f o r tradeables keeping domestic goods net supply constant. The f r o n t i e r PP' i s generated by the domestic technology sets T^, k=l,...,K. The l i n e denoted by w defines the production k — p o s s i b i l i t y f r o n t i e r for the t o t a l technology E T + T. k Suppose f i r s t that a l l production i n the country i s under d i r e c t p u b l i c c o n t r o l , and that the l i n e denoted by w i n F i g . 3 i s the relevant production p o s s i b i l i t y f r o n t i e r for tradeables. Suppose further, that - 59 -the i n i t i a l equilibrium i n the economy corresponds to the points A and B i n Figure 3. Since the i n i t i a l production choice A l i e s inside the f e a s i b l e production p o s s i b i l i t y set, using the Diamond-Mirrlees argument, the only condition needed ;for a Pareto improving change in the X X X commodity tax rates (t , s ) and t a r i f f s x to e x i s t , i s that there i s a Diamond-Mirrlees commodity i n the economy. Assumption (iv) in Theorem 4.1 i s the weakest s u f f i c i e n t version of this assumption. Suppose then that the domestic production sector i n the home country consists of K constant returns to scale i n d u s t r i e s . In t h i s case, the production p o s s i b i l i t y f r o n t i e r PP' generated by the sum of the s e c t o r a l technologies T k, k=l,...,K, becomes a constraint: the government can choose a s t r i c t Pareto improving change in the i n i t i a l commodity tax rates and t a r i f f s only i f the change i s such that the new production choice, established a f t e r the perturbation i n the taxes and t a r i f f s , l i e s on the f r o n t i e r PP'. Assumptions ( i ) - ( i i i ) i n Theorem 4.1 guarantee that a net balance of trade improving t a r i f f change, that s a t i s f i e s this condition, e x i s t s . This change moves the economy's production choice from A, i n F i g . 3, toward the point C along the curve PP'. Since the consumer preferences s a t i s f y Assumption (iv) in Theorem 4.1, and since, i n Theorem 4.1, the consumer and producer prices of 9 commodities can be perturbed independently of each other, the government can adjust the i n i t i a l commodity tax rates (t , s ) ft simultaneously with a change i n t a r i f f s x so that a s t r i c t Pareto improvement i s attained. Theorem 4.1 can thus be regarded as a generalized open economy version of the r e s u l t s of Diamond and M i r r l e e s . - 60 -GOOD 2 F i g u r e 3 - 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 i n T a r i f f s and Commodity Taxes. - 61 -F i n a l l y , one more comment should be made about what was not assumed in Theorem 4.1: nothing was said about the i n i t i a l values of the commodity tax rates t* and s*. It was only assumed that the government i s able to adjust a l l of these tax rates i f need be. Would Theorem 4.1 s t i l l hold, i f the i n i t i a l tax rates ( t * , s*) happened to be Diamond-Mirrlees optimal, i . e . , they maximize some s o c i a l welfare function W(u) with respect to the constraints of the general equilibrium model (4.10) - (4.13)? It turns out that only the properties of the i n i t i a l t a r i f f vector T* matter. I f , at the i n i t i a l equilibrium, the t a r i f f s x* and the commodity tax rates (x*, s*) are Diamond-Mirrlees optimal, then no s t r i c t Pareto and prod u c t i v i t y •k improving changes i n them e x i s t . But i f the vector of t a r i f f s x i s a r b i t r a r y , Diamond-Mirrlees o p t i m a l i t y of the commodity tax rates does not change the conclusion of Theorem 4.1. Rather, i f the commodity taxes are not optimal at the i n i t i a l e q u i l i b r i u m , there exist s t r i c t Pareto improving tax and t a r i f f changes with Ax = 0M i . e . , the i n i t i a l t a r i f f s x need not be perturbed at a l l . Considering p r a c t i c a l p o l i c i e s , t h i s means that as long as the i n i t i a l commodity tax  rates ( t , s ) i n the home country are not Diamond-Mirrlees optimal, the  government can a t t a i n s t r i c t Pareto (hence, welfare) improvements by  changing only the commodity taxes. A f t e r these 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 involving  changes i n t a r i f f s are needed. - 62 -A.3 Necessary Conditions f o r Pareto Optimality; Nonexistence of S t r i c t  Pareto and P r o d u c t i v i t y Improving Tax and T a r i f f Changes Having established s u f f i c i e n t conditions implying the existence of s t r i c t , P a r e t o and pr o d u c t i v i t y improving t a r i f f and commodity tax perturbations, i t i s natural to enquire when these p o l i c y changes do not e x i s t . A r e s u l t in this vein was already established in Section 2.4 as Theorem 2.2, where i t was shown that s t r i c t p r o d u c t i v i t y improving ft changes i n the i n i t i a l equilibrium t a r i f f s x are not possible i f the gradient of the net balance of trade function b (w + T ) with respect to the t a r i f f s T i s zero. Theorem 2.2 can also be interpreted to give a set of necessary conditions that the i n i t i a l e q uilibrium of the economy must s a t i s f y , i f i t i s a l o c a l p r o d u c t i v i t y optimum. The p r a c t i c a l s i g n i f i c a n c e of t h i s r e s u l t l i e s i n preventing a search for d i f f e r e n t i a l improvements in the government t a r i f f p o l i c y when none e x i s t . In this section, the most general necessary conditions for Pareto and p r o d u c t i v i t y optimality of the i n i t i a l equilibrium are established. Thereafter, some spe c i a l cases are considered. Theorem 4.2: A necessary condition for s t r i c t Pareto and pr o d u c t i v i t y improving commodity tax and t a r i f f changes to not e x i s t , i . e . , a necessary condition f o r Pareto and p r o d u c t i v i t y optimality of the i n i t i a l equilibrium, i s : - 63 -(4.24) there i s a vector X e R .H+N+K+l such that A [A, - B ] > 0 H+l ' N+K+N+M+M* Proof: .. A necessary condition for the nonexistence of s t r i c t Pareto and produc t i v i t y improvements i s : 9c 9c 9c 9c 9c 9c 9c (4.25) there do not exist Au , Ap , Az , Ab , At , As , Ax such Using Motzkin's Theorem and the proof of Theorem 4.1, i t can be shown that (4.25) and (4.24) are equivalent. QED Although Theorem 4.2 advises the government not to search for s t r i c t Pareto and pr 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 i s rather d i f f i c u l t to see from the statement of the r e s u l t , how i t i s related to Theorems 2.1 and 4.1. In order to in t e r p r e t Theorem 4.2, a s p e c i a l kind of i n i t i a l equilibrium for the economy i s introduced. tax rates ( t , s ) and t a r i f f s x, i f i t solves the nonlinear programming problem that (4.16) holds and Au » 0 H' Ab > 0. An equilibrium i s c a l l e d B-optimal with respect to the commodity (4.26) max 5 {3 Tu: (4.10) - (4.13) hold, g - constant}. u,p,z,b ,t ,s ,x - 64 -In ( 4 . 2 6 ) , the government i s assumed to choose the tax and t a r i f f rates ( t , s , T ) to maximize a s o c i a l welfare function of the form W(u) = 8 T u , 3 > 0H» The welfare weights 3 can be regarded as the gradient vector V^W(u) of some general s o c i a l welfare function W(u). (The function T — S u i s thus a l o c a l l i n e a r i z a t i o n of W(u)). The constraints i n ( 4 . 2 6 ) r e s t r i c t the 3-optimum to be a competitive e q u i l i b r i u m 1 0 with a fixed ( p o s s i b l y zero) vector of lump sum t r a n s f e r s . E s s e n t i a l l y , ( 4 . 2 6 ) corresponds to the (closed economy) Diamond, Mi r r l e e s ( 1 9 7 1 ) - Diewert ( 1 9 7 8 ) optimal tax problem; the only d i f f e r e n c e i s that, i n ( 4 . 2 6 ) , the trade t a r i f f s T are also assumed to be set so as to maximize s o c i a l welfare. If the i n i t i a l equilibrium of the economy i s a 8-optimum with respect to the commodity tax rates ( t * , s*) and t a r i f f s T*, then i t must n e c e s s a r i l y s a t i s f y : * * * * * * * ( 4 . 2 7 ) there do not exist Au , Ap , Az , Ab , At , As , AT such that ( 4 . 1 6 ) holds, 3 TAu* > 0 , Ab* > 0 and Ag* = 0 „ . rt I f ( 4 . 2 7 ) 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 welfare maximum. Applying Motzkin's Theorem, ( 4 . 2 7 ) can be written as (4.28) there exists a vector X e R H + N + K + 1 s u c h that X TA = 3 T (> oJ), - X \ > 0 and X T[B p, B,, B t, B ^ = o j ^ ^ . - 65 -Comparing (4.28) and (4.24), i t can be seen that, i f the i n i t i a l e q u i librium i s a Pareto and p r o d u c t i v i t y optimum, i t i s also a welfare maximum for the s o c i a l welfare function (X^A)u, where the vector X i s defined by (4.28). Proposition 4.1: I f , at the i n i t i a l equilibrium of the economy, no s t r i c t Pareto and p r o d u c t i v i t y improving commodity tax and t a r i f f changes e x i s t , the T e q u i l i b r i u m i s a welfare maximum for the s o c i a l welfare function (X A)u, where the vector X solves X T[B , B , B , B , B ] = (£ T., „, „ , - X T B , > 0. p z t s x N+K+N+M+M b Proposition 4.1 i s a Negishi (1960) type r e s u l t showing that a Pareto and p r o d u c t i v i t y optimum i s also a welfare maximum with respect to some welfare function of the form W(u) = B Tu. In p a r t i c u l a r , i f , i n i t i a l l y , the government i s not assumed to choose the tax and t a r i f f rates i n the country so as to maximize s o c i a l welfare, and i f (4.28) i s s a t i s f i e d , the observed equilibrium i s revealed to be a welfare maximum with respect to a welfare function which may or may not have s o c i a l l y acceptable welfare weights. It was assumed i n the optimization problem (4.26) that the government can adjust the t a r i f f s T i n any way deemed optimal. But i f a l l or some of the t a r i f f s x are f i x e d , i t would be useful to know, i f any p a r t i c u l a r values of the i n i t i a l vector of t a r i f f s x s a t i s f y (4.24) under the supposition that only the domestic commodity tax rates - 66 -t* and s* can be chosen optimally. P r o p o s i t i o n 4.2: Let the i n i t i a l equilibrium be S-optimal with respect to the commodity tax rates ( t * , s * ) , and suppose that the assumptions ( i ) -*T 2 * ( i i ) and ( i v ) of Theorem 4.1 are s a t i s f i e d . Then, i f T V G(w + x , ww ft T y ) = 0^, the i n i t i a l equilibrium s a t i s f i e s the necessary condition for Pareto and p r o d u c t i v i t y optimality given i n Theorem 4.2, and no s t r i c t Pareto and pr o d u c t i v i t y improving d i r e c t i o n s of change in the t a r i f f s T* and commodity taxes ( t * , s*) e x i s t . Proof: B-optimality of the i n i t i a l e q u i l i b r i u m implies that there exists a vector X e R H + N + K + 1 s u c h t h a t x T[A, - B j > 0* , X T[B , B , b H+1 P z T B_, B ] = 0 . T | T , A p p l y i n g the proof of Theorem 2.1, for a vector X t s N+K+N+M J ° T T T that solves the equations X [B , B 1 = 0.Tlt,, the vector X B equals ^ p' z J N+K' x n T V G(w + x , y ). By assumption, T V G(w + x , y ) = 0,.. WW WW M Then, i n addition to the optimality conditions given above, also (4.24) i s s a t i s f i e d . QED Proposition 4.2 implies that the gradient vector of the net balance of ft ft ft>j> ^ ie ie trade function, V^b (w + x ), which equals -x G(w + x , y ), must be nonzero for s t r i c t Pareto and p r o d u c t i v i t y improvements in * * * taxes (t , s ) and t a r i f f s x to e x i s t . - 67 -The assumption i n Proposition 4.2 that the vector *T 2 * * T G(w + T , y ) equals zero can be replaced by a condition i n v o l v i n g the aggregate producer s u b s t i t u t i o n matrix S: (4.29) [-S T, T * T ] S = 0^ + M, 12 where the vector 5 i s defined i n (2.27). Then, i f the matrix S i s of maximal rank ( = N + M - 1 ) , a s u f f i c i e n t condition implying (4.29) i s that the r e l a t i v e producer prices for tradeables i n the home country 13 * * and abroad coincide. It follows that i f , i n i t i a l l y , x = 0^, or i f x i s some multiple of the i n t e r n a t i o n a l prices w, no s t r i c t Pareto and pr o d u c t i v i t y improving commodity tax and t a r i f f perturbations, s t a r t i n g from the i n i t i a l equilibrium, are possible; i . e . , the i n i t i a l e q uilibrium s a t i s f i e s the necessary condition (4.24) for Pareto and p r o d u c t i v i t y optimality. Furthermore, i t can be argued, using the programming problem ( 3 . 3 ) , that when x = % or when x i s some multi p l e of the i n t e r n a t i o n a l prices w, the i n i t i a l equilibrium must i n fact be a pro d u c t i v i t y maximum. Hence, f o r a small country that cannot influence 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 are Pareto and p r o d u c t i v i t y optimal even though lump sum tr a n s f e r s are not a f e a s i b l e government p o l i c y instrument as long as the domestic commodity tax rates can be chosen optimally, the producer s u b s t i t u t i o n matrix S i s of maximal rank, and assumptions ( i ) and ( i v ) of Theorem 4.1 are s a t i s f i e d . The above statement can also be interpreted as an e f f i c i e n c y of t o t a l production r e s u l t . Assuming that the producer s u b s t i t u t i o n matrix - 68 -S i s o f maximal r a n k , and a s s u m p t i o n s ( i ) and ( i v ) of Theorem 4.1 h o l d , s t a r t i n g from an i n i t i a l e q u i l i b r i u m t h a t s a t i s f i e s (4.10) - (4.13) w i t h * * *x 2 * * T V b (w + x ) = x V G(w + x , y ) * 0.,, t h e r e e x i s t s a p a t h o f x ww y M 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 l e a d i n g to a P a r e t o and p r o d u c t i v i t y optimum, where the r e l a t i v e p r o d u c e r p r i c e s w and (w + x ) have been e q u a l i z e d . T h i s P a r e t o and p r o d u c t i v i t y optimum i s e f f i c i e n t k — w i t h r e s p e c t to the t o t a l p r o d u c t i o n t e c h n o l o g y E T + T d e f i n e d i n ( 4 . 2 3 ) . ( I n F i g . 3, t h e P a r e t o 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 to the f i r s t b e s t e q u i l i b r i u m (C,D).) U s i n g D i a m o n d - M i r r l e e s (1971) t e r m i n o l o g y , i t can be s a i d t h a t 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 i s k d e s i r a b l e ( w i t h r e s p e c t to the t e c h n o l o g y Z T + T) i n a s m a l l c o u n t r y , k i f t he 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 o f maximal r a n k , a s s u m p t i o n s ( i ) and ( i v ) of Theorem 4.1 are s a t i s f i e d , and the commodity t ax r a t e s * * ( t , s ) can be chosen o p t i m a l l y to maximize s o c i a l w e l f a r e . 4.4 S t r i c t Pareto and P r o d u c t i v i t y Improving Changes i n Commodity Taxes  and T a r i f f s when no Domestic Goods E x i s t In the f o r m u l a t i o n of Theorem 4.1, i t was i m p l i c i t l y assumed t h a t 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 , i . e . , N > 0. I t i s t e m p t i n g to s i m p l y note t h a t i f a l l goods i n the home c o u n t r y 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 the theorem can be e r a s e d and the - 69 -r e s u l t i s restored. A closer inspection of the theorem shows, however, that although the s u f f i c i e n t conditions for s t r i c t Pareto and p r o d u c t i v i t y improvements to e x i s t , when N equals zero, are very s i m i l a r to those given in. Theorem 4.1, the i n t e r p r e t a t i o n of the general equilibrium model (4.10) - (4.13) changes i f N = 0. If the number of production sectors i n the model (4.10) - (4.13) equals the number of domestic commodities i . e . , N = K (> 0), the ft s e c t o r a l production technologies and the tradeables prices (w + T ) * 15 determine the equilibrium prices p for domestic goods i n (4.11). ( I f the number of the production i n d u s t r i e s i s less than the number of domestic goods (K < N), the domestic market equilibrium conditions (4.12) also a f f e c t the prices p •) I f there are no domestic commodities i n the economy, the tradeables prices (w + T*) determining . the industry (pure) p r o f i t s 7r k , k=l,...,K, instead of the prices p* i n (4.11), i . e . , the production i n d u s t r i e s do not generally earn zero p r o f i t s when N = 0. The existence of these possibly p o s i t i v e p r o f i t s creates K a r t i f i c i a l domestic f a c t o r s , to which the p r o f i t s are imputed. Hence, i n the end, N, the number of domestic commodities, must be p o s i t i v e . If the only domestic commodities i n the economy are the K factors to which the s e c t o r a l p o s i t i v e p r o f i t s are being imputed, the domestic net supply matrix Y becomes an (K x K ) - i n d e n t i t y matrix: each sector supplies one unit of i t s "ownership", as the newly created domestic commodities might be c a l l e d . The aggregate producers' zero p r o f i t A T A ^ ^ A condition can then be written as p + (w + T ) 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 profits. The producer substitution matrix S is of the form (4.30) S = KxK 3MxK KxM ww The consumers hold endowments of industry ownership shares, denoted by h H h x , h=l,...,H ( Z x, = 1 , for a l l k=l,...,K). h=l fc Theorem 4.3: Let the only domestic commodities in the home country be the ownership shares in the production sectors k, k=l,...,K. Then, i f (i) T T T T *T T there is no solution a > 0.7 to a [X , E ] = a,,.,, and ( i i ) T S * 0 W. H K+M ww M there exists a st 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 is K (= N), and the matrix T YY is positive definite. Hence, assumptions (i) - ( i i ) of Theorem 4.1 *X T are satisfied. Since x S * 0 W, assumption (iv) of Theorem 4.1 is ww M r 9 1c 1c also satisfied. (Note that V G(w + x , y ) = S , i f S is of the form ww ww (4.30).) QED - 71 -Assumption ( i i ) in Theorem 4.3 has a similar i n t e r p r e t a t i o n as assumption ( i v ) has in Theorem (4.1); i f only the production side of the economy were considered, then proceeding as in Theorem 2.1, i t would be possible to show that t h i s supposition implies the existence of s t r i c t p r o d u c t i v i t y improving changes i n t a r i f f s x*. These p r o d u c t i v i t y gains can be d i s t r i b u t e d to the households in a s t r i c t Pareto improving way, i f the consumer preferences are s u f f i c i e n t l y homogenous, i . e . , assumption ( i ) of Theorem 4.3 i s s a t i s f i e d . If the only domestic commodities are the ownership shares in the production i n d u s t r i e s , t h i s condition i s e a s i l y met: i t i s enough to have a production sector k, k £ [1,...,K], such that x^ > 0 f o r a l l h=l,...,H. Then, lowering the i n i t i a l domestic tax rate t ^ i s a s t r i c t Pareto improving tax change d i r e c t i o n . - 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 rather weak assumptions about the i n i t i a l e q u i l i b r i u m , the government can cause a s t r i c t p r o d u c t i v i t y and Pareto improvement by ad j u s t i n g the i n i t i a l commodity tax rates ( t * , s*) and t a r i f f s x* a p p r o p r i a t e l y . 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 easy.to prove the existence of p o s i t i v e gains 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 autarky  e q u i l i b r i u m , i . e . , there i s no i n t e r n a t i o n a l trade. In order to descr i b e autarky using the open economy model (4.10) - (4.13), i t i s assumed that the l a c k of i n t e r n a t i o n a l trade i s caused by the government t a r i f f p o l i c y . To t h i s end, 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 are defin e d as * a (5.1) x =w -w, where w3 i s the autarky e q u i l i b r i u m v e c t o r of tradeables p r i c e s 1 and w i s 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 . I t follows from (5.1) that (w + x*) = w3. This means that i f the t a r i f f s x*, defined by (5 . 1 ) , are used i n the open economy model (4.10) - (4.13), the model 2 c h a r a c t e r i z e s an autarky e q u i l i b r i u m . - 73 -In order to apply Theorem 4.1, i t i s assumed that the conditions ( i ) - ( i v ) of the theorem are s a t i s f i e d . Then, there exists a s t r i c t ft Pareto and produ c t i v i t y improving change in the i n i t i a l t a r i f f s T and commodity tax rates (t , s ) . It follows that, i f the government allows i n t e r n a t i o n a l trade by perturbing the 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 t a r i f f s x*, i t can also change the i n i t i a l commodity tax rates ( t * , s*) in such a way that 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 gains from trade e x i s t . It should be noted, however, that a perturbation of the t a r i f f s ft x i s not always necessary f o r s t r i c t gains from trade to e x i s t : i f ft ft Lf the autarky commodity tax rates ( t , s ) are not 8-optimal, s t r i c t ft Pareto improvements can be found by changing only the tax rates ( t , ft s ). At the new (perturbed) l e v e l s of the commodity tax rates the ft i n i t i a l t a r i f f s 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 ; trade w i l l thus be opened up, causing a p r o d u c t i v i t y and welfare improvement. 5 Proposition 4.2 gives necessary conditions for 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 equilibrium, when only commodity tax and t a r i f f rates are used as government p o l i c y instruments. In the present context, Proposition 4.2 may be interpreted to give necessary conditions for the nonexistence of s t r i c t gains from trade. For ft ft example, i f the i n i t i a l commodity tax rates ( t , s ) are g-optimal, and i f the i n i t i a l trade p r o h i b i t i v e t a r i f f s x* equal zero (or i f they are proportional to the i n t e r n a t i o n a l prices w) so that the gradient of the net balance of trade function b*(w + x*) i s a zero M-vector,the p o t e n t i a l p r o d u c t i v i t y gains accruing from the p a r t i c i p a t i o n in - 74 -i n t e r n a t i o n a l trade have already been exhausted in autarky, and no further s t r i c t gains from trade are p o s s i b l e . The i n i t i a l autarky equilibrium can also s a t i s f y the condition V b (w + T ) = 0 W when T * 0 M, and x i s not proportional to the x M M f t-i n t e r n a t i o n a l prices w. In this case, since the home country's production p o s s i b i l i t y f r o n t i e r i s ridged and/or kinked, no d i f f e r e n t i a l s t r i c t Pareto and p r o d u c t i v i t y improving perturbations i n the t a r i f f s x* and commodity tax rates ( t * , s*) e x i s t . Consider Figure 4. There are two tradeable commodities in the economies depicted in Figures 4a) and 4b). The production p o s s i b i l i t y f r o n t i e r for these commodities, given a fixed net supply of domestic goods y*, i s represented by the curve PP'. The i n i t i a l autarky equilibrium in F i g . 4a) i s denoted by A: the producers face the tradeables prices w3 (= w + x*), while the consumer prices for tradeables are v a(= wa + s ). At these prices the consumer at t a i n s the u t i l i t y l e v e l u^. The point B i n F i g . 4a) shows the economy's production choice under free undistorted trade (x = 0 W, s = 0 W ) . The l i n e w denotes M M the exogenous i n t e r n a t i o n a l p r i c e s . Let us denote the consumer i n d i f f e r e n c e curve tangent to the l i n e w (not shown in Fig. 4a)) by u^. (The point C denotes the tangency point between the l i n e w and the curve u^.) The f i r s t best autarky equilibrium in F i g . 4a) l i e s at A' where the consumer and producer prices for tradeables coincide (w a = v a ) . - 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 Gains from Trade - 76 -Because of a government tax revenue requirement and the lack of nondistortionary tax instruments, the actual i n i t i a l autarky equilibrium at A i s only a second best equilibrium, i . e . , u^ < u^ . But i f the conditions of Theorem 4.1 are s a t i s f i e d , the government can move the equilibrium from A, i . e . , from autarky, toward the f i r s t best (free trade) equilibrium at B and C i n such a way that the consumer i n the economy s t r i c t l y b e n e f i t s . The s h i f t i s accomplished by s u i t a b l y perturbing the i n i t i a l commodity tax rates ( t * , s*) and t a r i f f s ft ft fi x . I f , f o r example, the t a r i f f s x are reduced to zero, production w i l l s h i f t to B, while the consumer w i l l reach the in d i f f e r e n c e curve tangent to the consumer price l i n e v(= w + s a ) . The consumer's u t i l i t y l e v e l on this i n d i f f e r e n c e curve can be higher or lower ( i n the Fig. 4a) i t i s lower) than the f i r s t best autarky u t i l i t y l e v e l u^', but i t i s not lower than u^ and not higher than u^. I f , s t a r t i n g from the equilibrium with zero t a r i f f s , the commodity taxes can be changed f u r t h e r , the gains from trade (the d i f f e r e n c e between u^ and the in d i f f e r e n c e curve tangent to the l i n e v) can be made even l a r g e r . It i s easy to see in Fi g . 4a) that, i f the i n t e r n a t i o n a l prices w and the autarky producer prices (w + x*) coincide, s t r i c t gains from trade can be found only by changing the commodity tax rates (t , s * ) , i f they are not B-optimal at the i n i t i a l equilibrium ( s t a r t i n g from an equilibrium, where production takes place at B, and the consumer attain s the u t i l i t y l e v e l corresponding to the consumer prices v, the consumer's welfare can be improved, i f the tax rates s* (and t*) are perturbed optimally; i n the Figure 4a) the consumer price l i n e v could - 77 -be rotated toward the l i n e w as far as the government revenue constraint allows). Figure 4b) i l l u s t r a t e s the case where the autarky equilibrium at A s a t i s f i e s the necessary conditions for Pareto and p r o d u c t i v i t y o p t i m a l i t y given i n Proposition 4.2 even though the trade p r o h i b i t i v e t a r i f f s T are nonzero and nonproportional to the i n t e r n a t i o n a l prices w: the gradient of the net balance of trade function b*(w + T*) at A 2 * * i s a zero 2-vector since, at A, the matrix V G(w + x , y ) i s a zero ww J (2 x 2)-matrix. It follows that no s t r i c t ( d i f f e r e n t i a l ) Pareto and p r o d u c t i v i t y improving tax and t a r i f f changes s t a r t i n g from A are possible: s t r i c t gains from i n t e r n a t i o n a l trade do not e x i s t . The existence of the gains from trade i s further discussed in Chapter 9. In that chapter, lump sum transfers are assumed to be admissible, and the conditions under which s t r i c t gains from trade exist when either commodity taxation or lump sum compensation i s used to d i s t r i b u t e the gains to the consumers, are compared. - 78 -6. EXISTENCE OF STRICT PRODUCTIVITY AND PARETO IMPROVEMENTS WHEN ONLY A LIMITED SET OF COMMODITY TAXES AND TARIFFS CAN BE PERTURBED Suppose that, instead of being able to adjust a l l the N + M i n i t i a l commdity tax rates t* and s* at w i l l , the government i s constrained to perturb only the N domestic tax rates t * . Are s t r i c t Pareto and pr o d u c t i v i t y improving changes i n t a r i f f s and taxes s t i l l possible? In p a r t i c u l a r , are the N domestic tax rates s u f f i c i e n t to d i s t r i b u t e the gains from i n t e r n a t i o n a l trade tb the households i n the economy in a s t r i c t Pareto improving way? Theorem 6.1: Suppose that ( i ) rank Y = K £ N, ( i i ) S p p + YY T i s p o s i t i v e d e f i n i t e , ( i i i ) there i s no so l u t i o n to a > 0 to a^X^ = 0^, H N ftT"" ry ft ft r p ft ( i v ) T X V G(w + x , y ) * Of., and v) g = 0„. Then, there exists a WW M H s t r i c t Pareto and productivity improving change in t a r i f f s x* and commodity tax rates t * . Proof: It i s s u f f i c i e n t to show that there does not exist a vector X e R H + N + K + 1 such that X T[A, - 1^ ] > 0 ^ , X T [ B p , B^, B j = 0 ^ , T T X B^ = 0^. Following the proof of Theorem 4.1, the vector X that T T solves the N + K equations X [B , B 1 = 0„T must be of the form p z N+K k[(p +6) , y , 1], k e R, in i t s three l a s t components. Assumption - 79 -( i i i ) implies that i f k = 0, no vector X s a t i s f y i n g the equations and i n e q u a l i t i e s given above e x i s t s . Suppose k = 1: T T Consider the equations A = 0^. Using the d e f i n i t i o n of i n (4.16), these equations can be written as (6.1) X TB = - X?E T - (p* + 6 ) T E - w TE + (p* + 5 ) T S x 1 ^ qv w y. pw + y T F T + wT S . WW T T By assumption, X B t = 0N> This implies, using (4.16), that rri rrt ^ ^ r p ^ m ^ (6.2) - A,X q - (p + 6) E q - w E q = 0, l qq v q where q* = (p* + t * ) . Using the homogeneity of the expenditure functions, (6.2) becomes (6.3) [ X T E T + (p* + 6 ) T E + wT E ] v* - X? g* = 0, 1 qv vv 1 where v* = (w + s* + x*). Since g* = 0 U, (6.3) y i e l d s X TB v* = 0. It follows, using (6.1), that H s X AB v = [(p + 6 ) 1 S + Y F + wxS ] v = -x 1 G(w + x , y ) v . x pw WW WW - 80 -By assumption, T V G(w + x , y ) * 0..; hence, since v » 0 W, J ww M ' M T * T X B^v * 0 and the vector X B^ cannot be zero. QED Theorem 6.1 shows that s t r i c t Pareto and p r o d u c t i v i t y improving t a r i f f and tax perturbations are s t i l l possible even thought the government i s constrained to adjust only the N domestic commodity tax rates t in addition to the t a r i f f s x . Theorem 6.1 also implies that the separation of the consumer and producer sectors i s not  necessary f or s t r i c t Pareto 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.1 (In Theorem 4.1, the assumption that a l l commodity tax rates (t , s ) can be f r e e l y adjusted means that that the consumer and producer prices can be perturbed independently from each other.) The assumptions of Theorem 6.1 are not much s t r i c t e r than the assumptions of Theorem 4.1. The only a d d i t i o n a l r e s t r i c t i v e supposition i s that the (po s s i b l y composite) Diamond-Mirrlees good, that i s supposed to e x i s t , must be a domestic good. The assumption that there are no lump sum transfers at the i n i t i a l equilibrium i s not necessary for the r e s u l t to hold. Inspection of the proof for Theorem 6.1 shows that the * T * *7J1 • * t r a n s f e r vector g must only be such that X g * -x v G(w + x , 1 ww * * T T / * vT y ) v , where the vector X^  solves the equations X^X + (p + <5) T T T T E + w E = 0„T, i . e . , the equations X B = 0 X T. Nonexistence of qq vq N t N trans f e r s at the i n i t i a l equilibrium i s s u f f i c i e n t for the inequa l i t y X^ g* * -x* TV 2 G(w + x*, y*) v* to be s a t i s f i e d . 1 ww - 81 -Theorem 6.1 proves that p r o d u c t i v i t y gains from i n t e r n a t i o n a l trade can be d i s t r i b u t e d to the households in a s t r i c t Pareto improving way also when the set of c o n t r o l l a b l e tax instruments i s r e s t r i c t e d to t* , the set of domestic commodity tax rates. Yet, existence of fixed tax d i s t o r t i o n s s* in the .home country gives r i s e to some complications. F i r s t , i n contrast to the r e s u l t s of the previous sections, zero t a r i f f s (free trade) are not generally optimal for a small country, i f nonoptimal commodity taxes that cannot be adjusted,are  present i n the system. This can be seen as follows: write equation (6.1) in the form (6.4) X TB = - X T E T + t * T I + (x* + S * ) T £ - x* T[S D,, + FD ? J I x qv vv wp 11 12 qv *T ? , * *. + x V G(w + x , y ) • ww If the i n i t i a l t a r i f f s x* are zero, but ( t * , s*) * 0"N+M> equation (6.4) does not reduce to a zero i d e n t i t y ; i n other words, even though the t a r i f f s x* at the i n i t i a l equilibrium were zero, some s t r i c t Pareto and p r o d u c t i v i t y improving changes i n x and commodity tax rates t* would s t i l l exist (assuming that the other conditions of Theorem 6.1 a r e s a t i s f i e d . ) Yet, there i s a case where free trade, i . e . , zero t a r i f f s , i s Pareto and p r o d u c t i v i t y optimal f o r a small country, i r r e s p e c t i v e of the fi x e d i n i t i a l commodity tax rates s*. This occurs when there i s no 3 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 )• It i s well known that taxes on goods in fixed demand are nondistortionary; hence, - 82 -free trade under these circumstances i s optimal for a small country. The above re s u l t that free trade i s not generally Pareto and p r o d u c t i v i t y optimal for a small country i f some d i s t o r t i o n a r y taxes i n 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 of t o t a l  production e f f i c i e n c y under the presence of market d i s t o r t i o n s . In t h i s form, the statement i s an a p p l i c a t i o n of Guesnerie's (1977) e a r l i e r propositions to an open economy. Consider then the existence of s t r i c t Pareto and produc t i v i t y improvements i n t a r i f f s and commodity taxes when only one i n i t i a l t a r i f f * x , m e [1,...,M], can be varied. In th i s case, Theorems 2.1 and 4.1 m can s t i l l be applied, i f a l l commodity tax rates ( t * , s*) or at le a s t the rates t are adjustable: a s u f f i c i e n t condition for a k k k s t r i c t Pareto and pr o d u c t i v i t y improvement i n (t , s ) and x to e x i s t , m supposing that assumptions ( i ) , ( i i ) and (iv) of Theorem 4.1 are x x xT *? x s a t i s f i e d , i s that the d e r i v a t i v e V b ( w + x ) ( = x V G(w + x , X WW m k 2 & k rh y ) ) i s nonzero. (The notation V G(w + x , y ) refer s to the m •m ww .m 2 * * column of the matrix V G(w + x , y ).) ww • ' J * If only one i n i t i a l domestic commodity tax rate t , n e [1,...,N], : n in addition to the t a r i f f s x* i s v a r i a b l e , s t r i c t Pareto improvements * * in x and t n are not possible, but the government can s t i l l generate a s t r i c t welfare (and productivity) improvement in an exogenously given s o c i a l welfare function W(u) = 8*^11. - 83 -Theorem 6.2: T Suppose that ( i ) rank Y = K < N, ( i i ) S + YY i s po s i t i v e — pp j c . • , / j * *\ i_ , ' „H+N+K+1 , ,. , , d e f i n i t e , and ( i n ) the vector A e R defined by (6.5) A T =VT, 0 j + K + 1 ] [ A , - B - B z, - (B^.J" 1 T T does not solve the equations A B^ = 0^. Then, there exists a s t r i c t welfare and productivity improving change in the t a r i f f s x* and the * tax rate t , n e [1,...,N]. Proof: A s u f f i c i e n t condition for a s t r i c t welfare and produ c t i v i t y * * improving change in the t a r i f f s x and the tax rate t , n e [1,...,N] , to exist i s : * * * * * * (6.6) there exist Au , Ap , Az , Ab , At^, Ax such that (4.16) i s ^ T 9c "J1 9c s a t i s f i e d and 3 Au » 0 , Ab > 0. ri E q u i v a l e n t l y , by Motzkin's Theorem: (6.7) there must not exist a vector A e R such that A A = B , x T [ v v <V.J - X \ = ° M ' - X \ > ° -- 8 4 -By the Implicit Function Theorem, the inverse matrix [A, - B , - B z > - B ] * e x i s t s , and hence (6.5) defines a unique X P Z n JL for the vector of welfare weights 3 . By assumption, f o r this A, the vector A TB T i s not zero; thus (6.7) holds. QED Why i s i t that only a s t r i c t welfare (not a s t r i c t Pareto) improving change in t a r i f f s T and the tax rate t , n £ [1,...,N], can can be found? The i n t u i t i v e explanation i s that, to produce Pareto improvements, the government needs a s u f f i c i e n t l y large number of free u tax instruments; for welfare improvements, only one adjustable tax rate can be s u f f i c i e n t . The technical reason for the nonexistence of s t r i c t Pareto improvements above i s that the vector A, which can be solved T from (6.5), cannot be determined from the equations A [B p, B z > ( B > n ) ] T = C v , , T r . i that would have to be used in order to show the existence N+K+l * * s of s t r i c t Pareto improvements in T and t ^ , n e [1,...,N]. * If a tradeable commodity tax rate s^, m £ [1,...,M], were the free tax instrument, a theorem analogous to Theorem 6.2 could be established. In this case, in (6.5), the column (-B ) > n would be replaced by the column (-B ) , but the other assumptions of the theorem s «m would stay the same. - 85 -7. SOME PIECEMEAL POLICY RESULTS WHEN NO LUMP SUM TRANSFERS ARE USED AS GOVERNMENT POLICY INSTRUMENTS The previous sections have been e n t i r e l y concerned with the existence of s t r i c t Pareto and p r o d u c t i v i t y improving government p o l i c y perturbations. Thus f a r , very l i t t l e has been said about the s p e c i f i c nature of the Pareto and pr o d u c t i v i t y improving p o l i c y changes. It was established e a r l i e r that, for example, a proportional reduction of the i n i t i a l equilibrium t a r i f f s x* could be s t r i c t p r o d u c t i v i t y improving. In this s e c t i o n , i t w i l l be shown that such a reduction i s also a s t r i c t Pareto improvement, i f the i n i t i a l commodity tax rates ( t , s ) are adjusted accordingly. Some other examples of s t r i c t Pareto improving t a r i f f and tax perturbation p o l i c i e s are also given. ft ft Consider f i r s t a reduction of the i n i t i a l t a r i f f s x when x _> ft ft i Cty, and an increase of x when x <^  0^. Theorem 7.1; Let assumptions ( i ) - (iv) of Theorem 4.1 be s a t i s f i e d so that a s t r i c t Pareto and pr o d u c t i v i t y improving change in the i n i t i a l t a r i f f s x* and commodity taxes ( t * , s*) e x i s t s . Then, ( i ) i f the t a r i f f s ft ft ft x are nonnegative, i . e . , x >_ 0^, the change in x may be taken to be a reduction (Ax* _< 0^) , and ( i i ) i f the t a r i f f s x* are nonpositive, i . e . , x <_ 0^, the change in x may be taken to be an increase (Ax* >^ 0^) . - 86 -Proof: 9c Consider the case ( i ) where x >^  0^. A s u f f i c i e n t condition f o r a s t r i c t Pareto and p r o d u c t i v i t y improvement in t a r i f f s and commodity taxes to exist i s : (7.1) there i s no vector X e R h + n + k + 1 s u c h that X T [ A , - B j > b x T [V V V V = ° N + K + N + M ' A \ > ° M V The condition (7.1) i s derived using Motzkin's Theorem; the constraint * T T Ax < 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). Consider the — M x — M equations T * T (7.2) A B (w + x ) = X x -E -E + S qv pw F T -wT E + w TS vv WW ( W + T ) T * T * * = X X q + E s - g * * * E q + E s - S p qq qv pp r T * - Y p T ' * T * T * w E q - w S p + w E s vq wp vv - 87 -The homogeneity of the expenditure and unit p r o f i t functions, and the A T A T A T A T A T T equations q X + v E = g and p Y + ( w + x ) F = 0 y i e l d the l a t t e r K. X A x T form of X B (w + T ). Since X [B , B ] = 0,_ by assumption, x t' s N+M 3 v ' (7.3) X TB (w + T*) = - X V - X* S p * - X M p * - wTS p * T 1 L p p J Wp ,T * • ,T „T * ,T_ * T r * = - X 1 § - X1 X p - X 2 E q q p - w E V Q P , using X TB = G\T. Further, because X TB = oJT, p N t N (7.4) X T B t (w + T*) = - X ^ g * = - x [ [ X T q * + E Tv*] = X * X T q * - A*ETv*. T T The equations X [B . B ] = 0 „ , w imply t s N+M (7.5) X TB (w + x ) = (p + 6) E q + w TE q + (p + 6 ) T E v X r qq n vq n r qv T * + w E v = 0, w using the homogeneity of the expenditure functions mn, h=l,...,H. In X * T T (7.5) the fact that the vector X must be of the form [X^, (p + 6 ), y , 1] has also been employed. By assumption, the vector (w + x*) i s T ^T _2 s t r i c t l y p o s i t i v e (» 0^) and the vector X B^ i s nonzero (since x 7 ^ * * X G(w + x , y ) * 0 W; see the proof of Theorem 4 .1) . Thus, the vector - 88 -T X must contain at l e a s t one negative (and positive) element, i . e . , (7.1) i s v i o l a t e d . ie * Consider now the case where x <^  Oj^. The vector x i s such that (w + x ) » Oj^. A s u f f i c i e n t c o n d i t i o n for a s t r i c t Pareto and p r o d u c t i v i t y improving increase in x* to exist i s : (7.6) there i s no vector X e R h + n + k + 1 s u c h that * T t A , " V > 0 j + 1 , X T [ B P , B Z , B T , B G ] - 0 j + K + N + M , X T B t < Oj. Using the same reasoning as above, i t can be seen that for any X T T s a t i s f y i n g (7.6) (except the i n e q u a l i t i e s X B ^ <^  0^) , the i n e q u a l i t i e s T T X B < must be v i o l a t e d . Hence (7.6) holds, and a s t r i c t Pareto and x — M * p r o d u c t i v i t y improving perturbation in the t a r i f f s x and tax rates ie ie ( t , s ) e x i s t s . QED Theorem 7.1 shows that a movement toward free trade, i . e . , zero t a r i f f s , i s s t r i c t Pareto and p r o d u c t i v i t y improving for a small country, i f a l l the i n i t i a l commodity tax rates ( t , s ) can be adjusted s u i t a b l y and the other conditions of the theorem are s a t i s f i e d . However, a closer inspection of the proof of Theorem 7.1 shows that also an increase of p o s i t i v e t a r i f f s and a reduction of 2 negative t a r i f f s can be s t r i c t Pareto and p r o d u c t i v i t y improving. How can t h i s seemingly c o u n t e r i n t u i t i v e r e s u l t be explained? The economic - 89 -j u s t i f i c a t i o n seems to be that a movement toward equalized r e l a t i v e producer prices for tradeables at home and abroad, i . e . , toward equalized r e l a t i v e prices (w + T ) and w, i s s t r i c t Pareto and pr o d u c t i v i t y improving under the conditions of "Theorem 4.1, whether the * change i s accomplished "by reducing (increasing) the t a r i f f s T >_ 0^ ( T _< 0^) or by increasing (reducing) them. It may sometimes be desirable to choose a p a r t i c u l a r kind of reduction in the home country's trade b a r r i e r . For example, a proportional or uniform reduction of t a r i f f s may be considered p o l i t i c a l l y f a i r . It turns out that a proportional reduction of t a r i f f s i s s t r i c t Pareto improving under the conditions of Theorem 4.1 but that a uniform reduction of t a r i f f s requires somewhat stronger assumptions; the responses of the domestic production sectors' net export supply functions to changes in tradeables producer prices must be such that a ft decrease i n each t a r i f f x , m e [1,...,M], leads to an increase in the home country's net balance of trade. Theorem 7.2: Let assumptions ( i ) - ( i i ) and ( i v ) of Theorem 4.1 be s a t i s f i e d . A A A T 0 A A T Assume further that V b (w + x ) = -x T~ G(w + x , y ) < 0 W. Then, x ww M ' ft a s t r i c t Pareto improving reduction of t a r i f f s x , accompanied by a * * change in the i n i t i a l commodity tax rates (t , s ), e x i s t s , and the * change in the t a r i f f s x can be chosen to be a uniform reduction, i . e . , Ax m = -h, h > 0, m e [1,..,M].H - 90 -Proof: T • cc- • u , H+N+K+l It i s s u f f i c i e n t to snow that there i s no vector A e R such that A T[A, - B, ] > 0*-, A T[B , B , B , B ] = O*^.^., X TB h M > 0, b ..-H+l p z t s N+K+N+M T M — where h^ i s an M-vector consisting of numbers h(> 0 ) . As shown in the T proof of Theorem 4.1, for a vector X s a t i s f y i n g X [B , B , B , B ] = J ° L p' z t' s T T ^T 9 A 1c 0 X T,„.„,„, the following must hold: X B = - T V G(w + x , y ). Then, N+K+N+M T ww rj* 9crV 2 " A the i n e q u a l i t i e s 1 8 ^ ) 0 can be written as -x G(w + x , y ) b^ = b (w + x ) h^ >_ 0. But, by assumption, V^b (w + x ) < 0^ and h » 0 . Hence, V b*(w + x*) h„ < 0. QED M x M The assumptions i n Theorem 7.2 are also s u f f i c i e n t for a thir d kind of reduction in t a r i f f s x* to be s t r i c t Pareto and pr o d u c t i v i t y improving. Suppose that x* » i . e . , net exports are subsidized and net imports are taxed. In this case, the government may be interested i n bringing the t a r i f f rates x* closer to each other; i n other words, a change toward uniformity i n t a r i f f rates may be de s i r a b l e . P r o p o s i t i o n 7.1: 1c 1c Let the i n i t i a l vector of t a r i f f s x be p o s i t i v e , i . e . , x » 0^ Suppose that the assumptions of Theorem 7.2 are s a t i s f i e d . Then, there e x i s t s a s t r i c t Pareto and pro d u c t i v i t y improving change in the i n i t i a l ie ie 1c t a r i f f s x and commodity tax rates ( t , s ), and the change of t a r i f f s x* can be chosen to be a reduction toward a (nonnegative) uniform t a r i f f structure. - 91 -Proof: Define x^ = ( x , . . . , x ) T (>_ 0 ) as Che uniform set of t a r i f f s , toward which the i n i t i a l t a r i f f s x* are perturbed. The d i r e c t i o n of * * * change in x i s Ax = -(x - x) < 0^. Using the proof of Theorem 7.2, for a s t r i c t Pareto and p r o d u c t i v i t y improvement to exist 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 r. „ , . „T ,T r„ „ „ « T vector X e R such that X [A, - B , ] > C- , X [ B , B , B . B ] = b H+1 p' z' t s °N+K+N+M' x T b t ( t * " T ) > °- B y assumption, X T B t = ? x b*(w + x*) < 0^ and thus, X B (x - x) < 0„ when (x - x) > 0 W. QED x M M It should be noted that, i f the i n i t i a l vector of t a r i f f s x i s negative, i . e . , net export are taxed and net imports are subsidized, under the conditions of Proposition 7.1, there exists a s t r i c t Pareto and p r o d u c t i v i t y improving decrease in t a r i f f s x* toward a uniform t a r i f f s t r u c t u r e . 5 Proposition 7.1 i s c l o s e l y related to a r e s u l t established by Hatta (1977b). Hatta showed that, under c e r t a i n conditions, in a one consumer one producer economy, a reduction of the highest (ad valorem) t a r i f f rate to the l e v e l of the next highest t a r i f f rate improves the welfare of the consumer. To derive his r e s u l t , Hatta assumed that lump sum transfers are admissible, and that no d i s t o r t i o n a r y commodity taxes exist in the home country. In Proposition 7.1, in contrast, no lump sum transfers are assumed to be a v a i l a b l e , d i s t o r t i o n a r y ( s p e c i f i c ) commodity taxes are present, and the numbers of consumers and - 92 -production i n d u s t r i e s are not r e s t r i c t e d to one. The assumption that the gradient of the net balance of trade function b*(w + x*) with respect to t a r i f f s x* i s negative at the i n i t i a l equilibrium appears to be a g e n e r a l i z a t i o n of a supposition used by Hatta: the good with the highest t a r i f f rate must be a substitute in production to the other * *T 2 tradeable commodities. (If x » 0 W , for the vector x V M ww G(w + x , y ) to be p o s i t i v e , p o s i t i v e terms in the matrix V ww * * G(w + x , y ), i . e . , s u b s t i t u t i o n in production, must dominate.) Is there anything more to be said about the changes in the * ie i n i t i a l tax rates t and s ? For example, i s i s possible to lower also them, when t a r i f f s x* (>_ % ) are being reduced? Let us suppose that a l l i n i t i a l commodity tax rates in the economy are p o s i t i v e , i . e . , t * > 0 N » S * > 0(4« (This means that the government i s taxing commodities bought by the consumers, whereas factors of production sold by the households are being subsidized.) Then, assuming that the i n i t i a l equilibrium i s not a B-optimum with respect to the tax and t a r i f f rates ( t * , s*, x*) ( i n which case, no s t r i c t Pareto and p r o d u c t i v i t y improvement in them could e x i s t ) , a s t r i c t Pareto and p r o d u c t i v i t y improvement can be attained by simultaneously reducing t , * * s and x . Theorem 7.2: Suppose that the i n i t i a l commodity tax rates ( t * , s*) are ie ie ie p o s i t i v e , i . e . , t > 0 N , s > C^, and the t a r i f f s x s a t i s f y - 93 -x x x x T _> 0^. Assume in addition that ( t , s , x ) do not solve the problem (7.7) max {g Tu: (4.10) - (4.13) hold, g = 0 R}, u,p,z,b ,t,s,x f o r any S > 0g. Then, there exists a s t r i c t Pareto and produc t i v i t y improving simultaneous reduction in t , s and x ( i . e . , At <_ 0^, As* < 0 M, Ax* < 0 M ) . Proof: A s u f f i c i e n t condition for a s t r i c t Pareto p r o d u c t i v i t y improving simultaneous reduction in t*, s* and T* to exist i s : (7.8) there i s no vector X e R H + N + K + ^ s u c h that X T[A, - ] > °^ +^, X T [V V = °N +K> X T [V V B x ] > °N+M+M' Since the i n i t i a l equilibrium i s not a B-optimum, f or any vector X s a t i s f y i n g X T[A, - B ] > O^ ., and X T[B , B ] = o3,„ , the vector b H+1 p z N+K T X [B , B , B ] i s not a zero (N+M+M)-vector. It follows from the t s x i n e q u a l i t i e s X TA > 0^ that X * O^ ..,,,,.. Then, for any H H+N+K+l X e RH+N+K+l ( , Q ) i . - 94 -(7.9) X 1[B t, B s > B x] W + T = xJ T -X - E T - E T -Z qq -E qv -E + S qv pw °KxN °KxM F T -w. E -w E T T -w E + w S vq vv vv WW W + X T * * X P - g * * E p - S p qq pp T * -Y P X * x * w E p - w S p v q r wp^ T * x x * using the equations X B p = 0^ when g = 0^. * * * Since, by assumption, t > 0 N, s > 0^ and (w + x ) » 0^, the vector X X [B . B , B ] must contain negative elements ( i t i s not zero because t s x & the i n i t i a l equilibrium i s not a B-optimum) . Thus, (7.8) i s s a t i s f i e d . QED C o r o l l a r y 7 .2 .1: * If t > 0 , s > 0 M, x > 0 M and rank [B , B , B , B , B ] = N M — M p z t s x H+N+K+l, a s t r i c t simultaneous Pareto and p r o d u c t i v i t y improving ii it it reduction In t , s and x e x i s t s . - 95 -Proof: If rank [ B , B , B , B , B ] = H+N+K+l, the only solution to the p z' t S T equations X T [ B P , \, \, \, \] = 0 N + K + N + M + M i s X = 0 R m + r Then, T T T X A = OJJ and X B^ = 0, which contradict the conditions X A > 0^ and X T B U < 0 in (7.8). QED b If the i n i t i a l commodity tax rates ( t * , s*) are. negative (t < 0^, s < 0^) so that commodities bought by the consumers are subsidized, and factors of production sold by the households are taxed, i t can be shown, using the same analysis as in the proof of Theorem 7.2, that a s t r i c t Pareto and p r o d u c t i v i t y improving increase i n t , s and T (<_ 0 M) i s possible. This p o l i c y change translates to reduced subsidies on consumer goods, reduced taxes on factors of production (e.g., l a b o r ) , smaller import subsidies and lower export taxes, i f such e x i s t . The conclusion of Theorem 7.2 may not seem so sur p r i s i n g ; i t i s only claimed that a simultaneous reduction of (p o s i t i v e ) d i s t o r t i o n a r y taxes and t a r i f f s i s de s i r a b l e . Yet, i t i s not self-evident that a s t r i c t Pareto and productivity improvement can be achieved through a change in ( t * , s*) and x*, without a change in the i n i t i a l e quilibrium vector of lump sum transfers ( i n Theorem 7.2, g* = and Ag* = 0^). 7 A c t u a l l y , according to Theorem 7.2, i f the assumptions of Theorem 7.2 are s a t i s f i e d , there e x i s t s an entire path of - 96 -simultaneous s t r i c t Pareto and p r o d u c t i v i t y improving reductions in the p o s i t i v e ( t , s ) and T which, in the l i m i t , lead to a nondistortionary government which behaves l i k e an a d d i t i o n a l private 3 production sector in the economy. One should, however, be cautious i n i n t e r p r e t i n g this r e s u l t as a p r a c t i c a l p o l i c y recommendation: the general equilibrium model (4.10) - (4.13), which i s used to derive Theorem 7.2, characterizes a s t a t i c , p e r f e c t l y competitive economy, where no e x t e r n a l i t i e s i n consumption or production are present. Furthermore, the outcome of a s t r i c t Pareto and p r o d u c t i v i t y improvement g may be unacceptable from an economic e q u a l i t y point of view. If the government i s r e s t r i c t e d to adjust a subset of the JL JL commodity tax rates (t , s ), r e s u l t s analogous to Theorem 7.2 can s t i l l be established. For example, the following statements can be proved using si m i l a r techniques as in the proof of Theorem 7.2: A * ie ( i ) I f t > 0l7, s = 0.. and the i n i t i a l tax rates t and the t a r i f f s N M T are not 3-optimal, there e x i s t s a simultaneous s t r i c t Pareto A ie and p r o d u c t i v i t y improving reduction of t and T . ie ic ie ( i i ) I f t = 0.7, s > 0„. and the i n i t i a l tax rates s and the t a r i f f s N M ie T are not 3-optimal, there e x i s t s a simultaneous s t r i c t Pareto ie * and p r o d u c t i v i t y improving reduction of s and T . ( i i i ) I f t* > 0, n e [1,...,N], t * n = O ^ , 1 0 s* = 0^, g* = 0 H and the - 97 -* * i n i t i a l tax ra t e t and the t a r i f f s x are not 3-optimal, there n e x i s t s a simultaneous s t r i c t Pareto and p r o d u c t i v i t y improving r e d u c t i o n of t and x . n - 98 -8. PARETO IMPROVING POLICY PERTURBATIONS WITH LUMP SUM TRANSFERS 8.1 A Second Model f o r the Production Side of An Economy Consider a model of an economy's production side consisting of the domestic supply equals demand equations (4.12), the zero p r o f i t r e l a t i o n s (2.9), and the net balance of trade equation (4.13). These N+K+l re l a t i o n s h i p s endogenously determine the equilibrium vector of domestic prices p*, the equilibrium vector of industry scales z*, and the home country's equilibrium net balance of trade b , given the exogenous t a r i f f s x , the vector of i n t e r n a t i o n a l prices w, a fixed vector of domestic commodity tax rates ( t * , s * ) , and a fixed vector ft of household u t i l i t i e s u . The new model of the economy's domestic production sector d i f f e r s from the model (2.8) - (2.10) in that the production sector i s not required to supply a fixed vector y* of domestic commodities but, instead, the consumers are assumed to be kept ft L at fixed u t i l i t y l e v e l s u , h=l,...,H, which equal the observed i n i t i a l equilibrium l e v e l s of consumer welfare. It i s assumed that a solution to the equations (4.12), (2.9) and (4.13) e x i s t s . It i s also assumed that the domestic p and the A industry scales z that solve (4.12), (2.9) and (4.13) are s t r i c t l y p o s i t i v e . D i f f e r e n t i a t i n g the equations (4.12), (2.9) and (4.13) around the i n i t i a l solution p* » 0 N, z* » 0 K, b* e R: (8.1) B Ap* + B Az* + B, Ab* = B A T * , p z b T ' - 99 -where B = P S - E pp qq T Y w T(S - E ) wp vq KxK T w F K -1 B = T -S + E pw qv - F T -wT(S - E ) WW vv A ie Applying the Implici t Function Theorem, the endogenous p , z and b can be regarded as i m p l i c i t functions of the exogenous tradeables prices (w + T*) (at fixed u*, t* and s * ) , i f the matrix [Bp, B z, B^] i s i n v e r t i b l e . Under t h i s supposition, the d i r e c t i o n a l A A A A A A d e r i v a t i v e s of the functions p ( w + x ) , z ( w + T ) and b (w + T ) , evaluated at the i n i t i a l s o l u t i o n to (4.12), (2.9) and (4.13), are given by the matrix [B , B , B , ] - 1 B . p z b x Using the r e s u l t s of Diewert and Woodland (1977: Appendix), i t can be seen that the necessary and s u f f i c i e n t conditions for the matrix [Bp, B z, B b] to be i n v e r t i b l e are (2.12) and (8.2) rank (S - E + YY ) = N. pp qq Henceforth, i t i s assumed that (2.12) and (8.2) are s a t i s f i e d at the i n i t i a l s olution to (4.12), (2.9) and (4.13). 1 Assumption (8.2) has a s i m i l a r economic i n t e r p r e t a t i o n as assumption (2.13). Consider the following version of D e f i n i t i o n 2.2: - 100 -D e f i n i t i o n 8.1: Domestic goods production i s 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 s o l u t i o n to (4.12), (2.9) and (4.13), i f there exist * A A A A A A continuously d i f f e r e n t i a b l e functions p (u , w + xs , t , s ) and z (u , A A A w + x , t , s ) around the i n i t i a l s o l u t i o n to (4.12), (2.9) and (4.13) which s a t i s f y (4.12), (2.9), and (4.13) when the exogenous variables u, x, t and s assume the i r i n i t i a l equilibrium values. The idea behind D e f i n i t i o n 8.1 i s simple. It i s assumed that the p o l i c y goal of the government i s to find such a perturbation of the i n i t i a l t a r i f f s T that the amount of net foreign exchange earned by the production i n d u s t r i e s , i n the aggregate, i s increased, while the consumers in the economy are kept at t h e i r i n i t i a l equilibrium u t i l i t y l e v e l s u*. In order to achieve i t s target, the government must be able to influence domestic goods production i n the country i n such a way that as the t a r i f f s x (and hence, the consumer prices f o r tradeables A A (w + x + s )) are perturbed from th e i r i n i t i a l l e v e l s , the induced change in the consumers' welfare i s zero. Local c o n t r o l l a b i l i t y of domestic goods production in the sense of D e f i n i t i o n 8.1 i s s u f f i c i e n t A A A A A A A to t h i s end: the functions p ( u , w + x , t , s ) and z (u , A A A w + x , t , s ) define the appropriate changes in domestic goods prices A A A and industry scales corresponding to given changes in u , 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 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  production i n the sense of D e f i n i t i o n 8.!.3'1* If the consumer - 101 -s u b s t i t u t i o n matrix for domestic goods Eqq i s a zero (N x N)-matrix, cond i t i o n (8.2) coincides with condition (2.13), and c o n t r o l l a b i l i t y of domestic goods production in the sense of D e f i n i t i o n 8.1 coincides with the c o n t r o l l a b i l i t y concept of D e f i n i t i o n 2.2. 5 If the matrix Eqq i s nonzero, however, condition (8.2) i s l e s s r e s t r i c t i v e than condition (2.13). 6 As before, the gradient vector of the net balance of trade function b (w + T ) w i l l play an important r o l e i n the subsequent a n a l y s i s . Since the model consisting of the equations (4.12), (2.9) and (4.13) d i f f e r s from the model (2.8) - (2.10), a new expression for the vector V^b (w + x ) must be found. The gradient V^b (w + x ) i s given by the l a s t row of the matrix [B , B , B , ] - 1 B : 7 p z b x (8.3) V b*(w + x*) = w T{[S - E ] - [S - E , F] D x ww vv J 1 wp vq* 1 [S - E , F ] T } , wp qv where the symmetric inverse matrix D i s defined by - 1 — — (8.4) D = °11 °22_ S - E Y pp qq T Y 0 KxK - 102 -Assumptions (2.12) and (8.2) guarantee that the inverse matrix D e x i s t s . In order to develop an i n t e r p r e t a t i o n for the matrix on the right hand side of (8.3), a new f u n c t i o n , c a l l e d the home country's constant  u t i l i t y tax adjusted balance of trade function B, i s defined: (8.5) B(w, w + x, w + T + s, t) = max {-w Te° + E (w + T ) T f k Z k z,Y,F,X,E k=l 5 ' / J . ^ > T H ? . T H v k k - E . ( w + T + s ) e - E t x : E y z -h=l h=l k=l ^ h 0 . . , k ..k. _k , h h N Jx. *h, E x - x >_ 0 , (y , f ) e C , (x , e ) e M (u ), h=l IN h=l,... ,H, z >_ 0„}. The function B gives the maximal net amount of foreign exchange that the private production i n d u s t r i e s can earn, net of government tax revenue, when the consumes are kept at their i n i t i a l e q u ilibrium u t i l i t y l e v e l s ^ h h ^ h u ; the sets M (u ) in (8.5) represent the net consumption vectors (x^, e* 1), h=l,...,H, than can achieve u t i l i t y l e v e l u*'1, h=l,...,H. The function B i s convex and l i n e a r l y homogenous i n i t s arguments. Using the K a r l i n (1959: p. 201) - Uzawa (1958: p. 34) Saddle Q Point Theorem, the convex programming problem (8.5) can be written in a dual form which gives an a l t e r n a t i v e expression for the function B: - 103 -r T O (8.6) B(w, w + x, w + x + s, t) = max min {-w e + z 2. °K» p 2. °N £ * / . N ^ ^ h *h . Z ir (p, w + x) z - Z m (u , p + t , w + x + s ) k=l h=l T 0, - p x ] . In (8.6), the functions n^, k=i,...,K, and mn, h=l,...,H, are the previously defined sectoral unit p r o f i t and household expenditure functions, r e s p e c t i v e l y . Problem (8.6) shows that B i s equal to the net value of private production valued at the prices (p, w + x) minus the net value of public production valued at the prices (p, w) minus the net household expenditures valued at the prices (p + t, w + x + s ) . * * * If x = x , s = s and t = t , the f i r s t order conditions for problem (8.6) become the zero p r o f i t and domestic supply equals demand equations (2.9) and (4.12) which are assumed to be s a t i s f i e d at the economy's i n i t i a l equilibrium. Hence, i f conditions (2.12) and (8.2) are s a t i s f i e d , the tax and t a r i f f v a r i a b l e s are fixed at t h e i r i n i t i a l <fe «fe values (t , s , x ), and the household u t i l i t i e s are fixed at u , the equations (2.9) and (4.12) can be used to determine the industry scales z and domestic prices p as i m p l i c i t functions of the i n t e r n a t i o n a l prices w. This means that B becomes a function of w: (8.7) B(w) = B(w, w + x*, w + x* + s*, t*) . - 104 -The gradient of the function B can be calculated by d i f f e r e n t i a t i n g the objective function in (8.6): 0 ^ k * k (8.8) V B(w) = -e + E V TT (p(w) , w + T ) z (w) k=l H h ^ h ^ A A E V m ( u ,p(w) + t , w + x + s ) . h=l V Thus, VwB(w) i s the net excess supply vector of i n t e r n a t i o n a l l y traded commodities produced by the constant u t i l i t y economy. In order to develop the Hessian matrix of the function B the gradient V w B(w) i s d i f f e r e n t i a t e d with respect to the i n t e r n a t i o n a l prices w. Using the formulae for the d e r i v a t i v e s Vwp(w) and V wz(w) obtained by d i f f e r e n t i a t i n g the equations (4.12) and (2.9) with respect Q to w, p and z: (8.9) V 2 B(w) = S - E - [S - E , F] D[S - E , F ] T , ww ww w wp vq' wp vq Hence, using (8.3), (8.10) V b* (w + T*) = wT V 2 B(w) T WW - 105 -Lemma 8.1 The matrix V B(w) can be written in the form WW (8.11) V^BCw) = [ - ( S W P - E V Q ) D U - P D j 2 f I M ] [S - Z] -D,. (S - Z ) - D._ F' 11 pw qv 12 M where the matrices and D^2 are blocks i n the inverse matrix D defined i n (8.4). The proof of Lemma 8.1 i s given in Appendix 2. Lemma 8.1 implies 2 ** 10 that the matrix V B(w) i s p o s i t i v e semidefinite, but the zero ww eigenvectors of the matrix are generally unknown. It can be shown, however, that i f the i n i t i a l equilibrium commodity tax rates are zero, * * T O i . e . , t = 0^ and s = 0^, the function B + w e , defined using (8.5), * must be l i n e a r l y homogenous i n the tradeables prices (w + x ). This 2 " means the matrix V B(w) must s a t i s f y the constraint ww (8.12) v i T B ( w ) ( w + x * ) = 0 M, WW i f t* = 0 N and s* = Oj^. - 106 -8.2 Existence of Constant U t i l i t y Productivity Improving Changes i n  Ta r i f f s The government's policy goal i s defined as follows: find such a ( d i f f e r e n t i a l ) change in the home country's i n i t i a l equilibrium vector JU JU of t a r i f f s T that the country's i n i t i a l net balance of trade b i s improved, while the consumers i n the economy are kept at their i n i t i a l e q u ilibrium l e v e l s of welfare u* n, h=l,...,H. 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 conditions f o r : k k k k (8.12) there exist Ap , Az , Ab , Ax such that (8.1) holds and Ab* > 0. k A perturbation 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 * constant u t i l i t y p r o d u c t i v i t y improving change of t a r i f f s T . Theorem 8.1: Suppose ( i ) rank Y = K < N, ( i i ) rank (S - E + YY T) = N and • - P P qq T 2 ~ T ( i i i ) w V w wB(w) * 0 M. Then, there e x i s t s a constant u t i l i t y p r o d u c t i v i t y improving change in t a r i f f s x*. Proof: The proof makes use of two preliminary lemmas, the proofs of which are given in Appendix 2. 1 0 7 -Lemma 8 . 2 : Any vector X e R N+K+l s a t i s f y i n g the equations X [B , B ] N+K must be of the form ( 8 . 1 3 ) X T = k[(p* + e ) T , 8 T, 1] , k e R, where ( 8 . 1 4 ) wp' *T x F] D. Lemma 8 . 3 : For the vector X solved in Lemma 8 . 2 , the following holds: ( 8 . 1 5 ) X TB = -k w T V 2 B(w), k e R. x ww ' Proof of Theorem 8 . 1 : A s u f f i c i e n t condition for a constant u t i l i t y p r o d u c t i v i t y improving change i n t a r i f f s x* to exist i s ( 8 . 1 2 ) . Using Motzkin's Theorem as in the proof of Theorem 2 . 1 , an equivalent condition can be derived : ( 8 . 1 6 ) there must not exi s t a vector X e R N + K + 1 such that - 108 -X T [V V -V = °N+K+M' * \ < <>. On the contrary, suppose such a A e x i s t s . By Lemma 8.2, a vector A that T T solves the equations A [B , B ] = 0„,T, must be of the form • p z N+K f * T T T A = k[(p + e) , 8 , 1], k e R. For such a A, A Bfe = -k. Thus, f o r T A B^ to be negative, k > 0 (and k may be chosen to be one). Using Lemma .8.3, A TB = -w TV2 B(w). By assumption, ATB * 0^, a co n t r a d i c t i o n . T WW X M QED An example of constant u t i l i t y p r o d u c t i v i t y improving t a r i f f changes i s provided by Proposition 8.1. Pro p o s i t i o n 8.1: If the assumptions of Theorem 8.1 are s a t i s f i e d , a change of the * t a r i f f s x i n the d i r e c t i o n of the world prices w w i l l be p r o d u c t i v i t y  improving (keeping 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 Ax = rw, r > 0. Using the proof of Proposition 2.1, i t i s s u f f i c i e n t to show that there i s no vector A e R N + K +'' - such that A T[B , B ] = oJT,„, ATB,_ < 0, ATB w = 0. Using Lemmas 8.2 and 8.3, i t p' z N+K' b ' x & ' N^*K.~H 1 T can be seen that f o r the vector A e R that s a t i s f i e s A [B , B ] = P z - 109 -T T T 2 ~ 0»,.„» the scalar X B w must be equal w V B(w)w. The matrix N+K x ww 2 T 2 ~ T V B(w) i s p o s i t i v e semidefinite and, by assumption, w V B(w) * 0 M. Hence, .wTV^B(w) w > 0. QED Proposition 8.1 implies that small, competitive countries can  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 t h e i r t a r i f f  structure toward the i n t e r n a t i o n a l p r i c e s w, without s a c r i f i c i n g the welfare of t h e i r consumers. Assumption ( i i i ) in Theorem 8.1 may be written in an a l t e r n a t i v e form which involves the producer and consumer s u b s t i t u t i o n matrices S and E. Using the proof of Lemma 8.3, i n Appendix 2, the following formula can be derived: (8.17) w T7 2 wB(w)w = w T[-(S - E ) D n - FD^, ^ [ S - E] -D. . (S - E ) - D. _F' 11 pw qv 12 M w [-(p* + £ ) T , wT] (S - Z) "(p + e) w > o, 11 where the vector E i s defined i n (8.14). Employing (8.17), assumption ( i i i ) in Theorem 8.1 can be replaced by: - n o -(8.18) the ve c t o r [-(p + e) , w ] i s not p r o p o r t i o n a l to any zero eigenvector of the s u b s t i t u t i o n matrix (S - I ) . I f there are no d i s t o r t i o n a r y commodity taxes at 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 M > and both matrices S and E are of 12 maximal rank (= 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) the ve c t o r of domestic producer p r i c e s p* i s not p r o p o r t i o n a l to the vect o r £ defined i n (8.14). The equivalence of (8.19) and (8.20) can be derived using 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. Diewert (1983: p. 289) shows that the ve c t o r (p + e) i s the ap p r o p r i a t e productive e f f i c i e n c y v e c t o r of shadow p r i c e s ( f o r domestic 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 are a v a i l a b l e f o r the p o l i c y choosing government. The c o n d i t i o n s (8.19) and (8.20) imply that the domestic ft producer p r i c e v e c t o r p should be used as the shadow p r i c e v e c t o r f o r domestic commodities only i f no commodity tax d i s t o r t i o n s are present i n the home country, and the i n i t i a l t a r i f f s x* are zero or p r o p o r t i o n a l - I l l -to the in t e r n a t i o n a l prices w. If the i n i t i a l commodity tax rates it it ( t , s ) d i f f e r from zero, the appropriate shadow prices for ie domestic commodities are (p + e) . Using (8.19), Theorem 8.1 may be written in a s i m p l i f i e d form which i s p a r a l l e l to Proposition 2.3. Pro p o s i t i o n 8.2: Suppose ( i ) rank Y = K_< N, ( i i ) rank S = rank Z = N+M-l, ( i i i ) * * * ( t , s ) = 0.TI.,, ( i v ) the i n i t i a l vector of t a r i f f s T i s nonzero and N+M not proportional to the i n t e r n a t i o n a l prices w. Then, there e x i s t s a constant u t i l i t y p r o d u c t i v i t y improving change in t a r i f f s x . If the consumer s u b s t i t u t i o n matrix £ i s a zero (N + M) x (N + M) - matrix, Theorems 8.1 and 2.1 coin c i d e : a constant u t i l i t y p r o d u c t i v i t y improving change of t a r i f f s i s also a pr o d u c t i v i t y improving change of t a r i f f s , when domestic goods net supply i s kept at 13 i t s i n i t i a l equilibrium l e v e l . Hence, applying Proposition 2.1, the JU constant u t i l i t y p r o d u c t i v i t y improving change in t a r i f f s x can be taken to be a proportional reduction of x*, i r r e s p e c t i v e of the JU JU i n i t i a l domestic commodity tax rates ( t , s ) . Consider then the necessary conditions for constant u t i l i t y  p r o d u c t i v i t y optimality of the i n i t i a l e q u i l i b r i u m . Under these circumstances, no s t r i c t 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 x* e x i s t . - 1 1 2 -Theorem 8 . 2 : A necessary condition for 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 equilibrium i s : ( 8 . 2 1 ) there exists a vector X e R N + k + 1 s u c h that X T [ B p > B z, -B T] = 0 ^ + K + M , X \ < 0. Proof: A necessary condition for a s t r i c t constant u t i l i t y p r o d u c t i v i t y improving change in T* to not e x i s t i s : ft ft ft ft ( 8 . 2 2 ) there do not exist Ap , Az , Ab , AT such that ( 8 . 1 ) i s ft s a t i s f i e d and Ab > 0. Motzkin's Theorem gives the equivalent form ( 8 . 2 1 ) for ( 8 . 2 2 ) . QED C o r o l l a r y 8 . 2 . 1 : Suppose ( i ) rank Y = K < N, ( i i ) rank (S - I + YY T) = N, — pp qq T 2 ~ T ( i i i ) w V B(w) = 0 M. Then, the i n i t i a l equilibrium s a t i s f i e s the WW w necessary condition for 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 given i n Theorem 8 . 2 , and no s t r i c t constant u t i l i t y 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 a r i f f s T e x i s t . The proof of Corollary 8 . 2 . 1 i s analogous to the proof of C o r o l l a r y 2 . 1 . 1 . - 113 -8 . 3 S t r i c t Pareto and Productivity Improving Changes In Tariffs and  Lump Sum Transfers In this section, the problem to be considered i s that of d i s t r i b u t i n g the gains accruing from a constant u t i l i t y p r o d u c t i v i t y improving change of t a r i f f s to the consumers. The government in the home country i s assumed to have lump sum transfer instruments i n i t s d i s p o s a l , but the i n i t i a l equilibrium commodity tax rates ( t , s ) are assumed to be kept unchanged. More p r e c i s e l y , the government's p o l i c y problem i s : * * * * * * * * (8.23) f i n d Au , Ap , Az , Ab , AT , At , As , Ag , such that (4.16) * * * * i s s a t i s f i e d and Au » 0TT, Ab > 0, At = 0„T, As = 0,,. L i * * M M Theorem 8.3: Suppose ( i ) rank Y = K < N, ( i i ) rank (S - E + YY T) = N and - P P qq T 2 T ( i i i ) w V B(w) * 0„. Then, there e x i s t s a s t r i c t Pareto and ww M ' JU JU p r o d u c t i v i t y improving change in t a r i f f s T and transfers g , without a change i n the home country's i n i t i a l commodity tax structure. Moreover, the change in t a r i f f s and transfers s t r i c t l y improves the country's i n i t i a l net balance of trade b . - 114 -Proof: Applying Motzkin's Theorem, a s u f f i c i e n t condition for a s t r i c t Pareto and pr o d u c t i v i t y improving transfer and t a r i f f change to exist i s : ( 8 . 2 7 ) there i s no vector X e R H + n + k + 1 s u c h that X T [ B , B , B ] P z' g J T T T T T = °AI L^TT> * [A> ~ Bt.] > °TT,I > ^ B = 0,., where the matrics A N+K+H b H+1 x M and [B , B , B , B,] are those defined i n (4.16). P z g o T T Consider the equations X [B , B , B ] = 0,Tlt, , r T. The equations p z g N+K+H ^ T T X Bg = 0^ imply that the f i r s t H components of the vector X must be zero, i . e . , X. = 0 . Then, using Lemma 8.2, the other components of the vector X can be solved. It follows that X T = k[0„, (p* + e ) T , 9 T, 1], k e R, where the vectors £ and 9 are defined i n (8.14). T The i n e q u a l i t y X Bfe > 0 implies k >_ 0. But i f k = 0, then T T T T X A = 0 , which contradicts the assumption that X A > 0„. Hence, k > 0 H H T T and i t may be set to one. Consider now the equations X B^ = 0^. Using T Lemma 8.3 and the fact that X, = 0TT, i t can be seen that X B = 1 H' x T 2 ~ w V B(w). (The change of the sign in the previous formula i s caused ww by the d i f f e r i n g d e f i n i t i o n s of the matrices B T in Lemma 8.3 and T 2 ~ T (4.16).) By assumption w V^^w) * 0 M- QED - 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 improving change of t a r i f f s can be converted to a s t r i c t Pareto improving change of t a r i f f s and lump sum t r a n s f e r s , without changing the home country's i n i t i a l commodity tax s t r u c t u r e . The assumptions needed for the r e s u l t to be e s t a b l i s h e d are e x a c t l y the same as the c o n d i t i o n s f o r a constant u t i l i t y p r o d u c t i v i t y improving t a r i f f change to e x i s t . In p a r t i c u l a r , no homogeneity assumption on consumer preferences i s present i n Theorem 8.1. This i s because a Diamond-Mirrlees commodity always e x i s t s i n an economy where household s p e c i f i c t r a n s f e r s are a d m i s s i b l e . l t f 8.4 Necessary Conditions f o r Pareto Optimality: Nonexistence of S t r i c t  Pareto and P r o d u c t i v i t y Improving T a r i f f and Transfer Changes Consider f i r s t the most general necessary 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 the government i s assumed to use lump sum t r a n s f e r s and trade t a r i f f s as i t s p o l i c y instruments. Thorem 8.4: A necessary c o n d i t i o n for s t r i c t Pareto 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 changes to not e x i s t , i . e . , a necessary c o n d i t i o n 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 i s the f o l l o w i n g : - 116 -(8.25) there exists a vector X e R such that X [A, -B. ] > O.V. , D n+1 x T [ B p V B Z » B g ] = °N+K+H and X TB = 0*. x M Proof: If the i n i t i a l equilibrium i s a Pareto and p r o d u c t i v i t y optimum, * * * * * * there must not exist Au , Ap , Az , Ab , Ax , Ag such that (4.16) i s * * s a t i s f i e d and Au » 0 and Ab > 0. Then (8.25) follows by using ti Motzkin's Theorem as in the proof of Theorem 4.2. QED Although Theorem 8.4 establishes conditions under which s t r i c t d i f f e r e n t i a l Pareto and pr o d u c t i v i t y improvements are not possible, the res u l t i s rather d i f f i c u l t to i n t e r p r e t . It i s helpful to develop some examples of 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 with respect to t a r i f f s and t r a n s f e r s . An equilibrium i s said to be 8-optimal with respect to t a r i f f s x and  transfers g, i f the vectors x and g solve the nonlinear programming problem (8.26) max [8 Tu: (4.10) - (4.13) hold, t = constant, u,p,z,b,g,x s = constant] . In (8.26), the vectors g and x are chosen so as to maximize the s o c i a l T welfare function W(u) = B u, B > 0 U, with respect to the constraints n - 117 -of the general equilibrium model (4.10) - (4.13). The commodity tax rates t and s are assumed to be kept at their previously set l e v e l s . Suppose now that the i n i t i a l equilibrium of the economy i s a 8-optimum with respect to the transfers g and t a r i f f s x . Then ne c e s s a r i l y , * * * * * * (8.27) there must not exist Au , Ap , Az , Ab , Ag , Ax such that T * A * (4.16) i s s a t i s f i e d and g Au > 0, Ab > 0, At = 0>T and • N As* - 0 M. Motzkin's Theorem y i e l d s an equivalent form for (8.27): (8.28) there must exist a vector X e R such that X TA = g T, - X \ > 0, X T [ B p , B z, B g, BT] - 0 ^ ^ . Comparing (8.28) and (8.25), i t can be inferred that i f the i n i t i a l e q u i librium of the economy i s a Pareto and p r o d u c t i v i t y optimum, i t must also be a welfare maximum with respect to the s o c i a l welfare function T T W(u) = B u, where the welfare weights B are equal to (X A) and the vector X i s defined i n (8.28). This r e s u l t corresponds to Proposition 4.1. ft Suppose then that only the domestic lump sum transfers g , but not the t a r i f f s x*, can be chosen to maximize s o c i a l welfare. - 118 -Theorem 8.5: Let the i n i t i a l equilibrium be a S-optimum with respect to the transf e r s g*, and suppose that assumptions (2.12) and (8.2) are 9c T 0 " T s a t i s f i e d . Then, i f V b (w + x ) = w V B(w) = 0,., no s t r i c t Pareto x ww M 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 in t a r i f f s x and ie transfers g e x i s t . Proof: 8-optimality of the i n i t i a l equilibrium implies that there i s a vector X e R H + N + K + 1 s u c h t h a t X TA = B T(> 0 T T), -XTB,_ > 0, X T[B , B , B ] H b p z g T = 0_TJ^IIT. It was shown in the proof of Theorem 8.3 that for a vector N+K+H T T T X, which solves X [B , B , B ] = 0 . . , , , , „ , the vector X B equals • p z g N+K+H' x H T 2 T 2 ~ T w V B(w). Since, by assumption, w V B(w) = 0.,, i t follows that ww ' J r ' ww M T T X B = 0,, and (8.25) i s s a t i s f i e d . QED x M When i s the gradient of the net balance of trade function * * * * X b (w + x ) zero? Using (8.17), the equation V^b (w + T ) = ° M c a n b e replaced by the condition (8.29) [-(p* + e ) T , wT] (S - Z) = ojj. It follows that V Tb*(w + x*) equals zero at lea s t i f both aggregate s u b s t i t u t i o n matrices S and E are zero (N + M) x (N + M) -matrices, i . e . , there i s no subsitution i n consumption or production - 119 -at the i n i t i a l equilibrium. Under these circumstances, no d i f f e r e n t i a l change of r e l a t i v e prices in the home country can change the economy's equilibrium vectors of net supply and net demand, and hence no s t r i c t Pareto and p r o d u c t i v i t y improvements through ( d i f f e r e n t i a l ) changes in the i n i t i a l t a r i f f s are p o s s i b l e . 1 6 Using (8.29), i t can be seen that VTb*(w + x*) i s zero also i f the vector [-(p* + e)^, w*-] i s a zero eigenvector of the matrix (S - E). In general, i t i s hard to i n f e r when th i s condition might be s a t i s f i e d because the zero eigenvectors of the matrix (S - E) are unknown, unless the i n i t i a l tax rates ( t * , s*) are zero. Yet, at l e a s t one i n t e r e s t i n g p o l i c y r e l a t e d r e s u l t can be derived. Suppose that, at the i n i t i a l e quilibrium, i n t e r n a t i o n a l trade i s f r e e , i . e . , x* = 0^* Using (8.14), i t can be i n f e r r e d that, i f either one of the consumer s u b s t i t u t i o n matrices E and E d i f f e r s from zero and qq qv *T; *T T ( t , s ) * 0„, w, the vector e must be nonzero. This implies that, N+M r ' at l east when the matrices S and E are of maximal rank, the vector [-(p + e ) T , w j cannot be a zero eigenvector of the matrix (S - E). * * Thus, at the i n i t i a l e q uilibrium, V b (w + x ) * 0 W, and some s t r i c t x M Pareto and p r o d u c t i v i t y improving t a r i f f changes exist (assuming that the other conditions of Theorem 8.3 are s a t i s f i e d ) . In other words, fo r a small country, zero t a r i f f s ( f ree trade) are not Pareto and  p r o d u c t i v i t y optimal i f there are d i s t o r t i o n a r y (nonoptimal) commodity  taxes i n the economy and the producer and consumer s u b s t i t u t i o n matrices  are of maximal rank (= N + M - 1) , even i f lump sum transfers could be - 120 -employed as government p o l i c y instruments. This r e s u l t can also be regarded as an e f f i c i e n c y of production r e s u l t : t o t a l production e f f i c i e n c y with respect to the technology T + Z T defined in (4.23) k is.-nolt d e s i r a b l e i n a small country, i f there are nonoptimal commodity  taxes i n the system and the matrices S and Z are of maximal rank, even  i f lump sum transfers could be chosen optimally to maximize s o c i a l welfare. I f , however, the i n i t i a l commodity tax rates (t , s ) are zero, using (8.19), free trade can be said to be Pareto and p r o d u c t i v i t y optimal for a small country, i f the lump sum transfers g are chosen to maximize s o c i a l welfare and the s u b s t i t u t i o n matrices S and Z are of 18 maximal rank. Under these circumstances, t o t a l production e f f i c i e n c y _ k with respect to the technology T + Z T i s d e s i r a b l e , and the k appropriate shadow prices for tradeables for cost benefit analysis are the i n t e r n a t i o n a l prices w. There i s also a case, where t o t a l production e f f i c i e n c y i s d e s i r a b l e in a small country, even though the commodity tax structure in the home country a r b i t r a r y . Consider an i n i t i a l equilibrium where Z = X (N+M) i , e * > there i s no s u b s t i t u t i o n in consumption. Then, 2 ? * * the matrix V B(w) coincides with the matrix v G(w + T , y ) defined WW WW * * i n (2.19). It follows that the vector of producer prices (p , w + x ) 2 ~ * must be a zero eigenvector of the matrix V B(w). Hence, i f x = 0.. , ww 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 p r i c e s w, the g r a d i e n t * * T 2 ~ V b (w + T ) ( w h i c h e q u a l s w V B(w)) must be a zero M - v e c t o r , x ww ie ie i r r e s p e c t i v e o f the i n i t i a l commodity t a x r a t e s ( t , s ) . The d i s c u s s i o n above i s I l l u s t r a t e d i n F i g u r e 5. F i g u r e 5 i s drawn assuming t h a t t h e r e are two t r a d e a b l e c o m m o d i t i e s i n the economy; the c u r v e PP' denot e s the p r o d u c t i o n p o s s i b i l i t y f r o n t i e r f o r these goods k e e p i n g net s u p p l i e s o f o t h e r c o m m o d i t i e s c o n s t a n t . C o n s i d e r f i r s t the F i g . 5a). At the i n i t i a l e q u i l i b r i u m , p r o d u c t i o n t a k e s p l a c e a t A and the one consumer i n the economy a t t a i n s u t i l i t y l e v e l u^- a t * * consumer p r i c e s (w + x + s ) . 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 i n g change o f t a r i f f s and t r a n s f e r s moves the economy's p r o d u c t i o n c h o i c e toward the p o i n t B, w h i c h i s the p r o f i t m a x i m i z i n g o u t p u t c h o i c e i f i n t e r n a t i o n a l t r a d e i s f r e e . The consumer i s s h i f t e d toward the p o i n t C, w h i c h c o r r e s p o n d s to the consumer's f i r s t b e s t optimum. Y e t , s i n c e the commodity t a x r a t e s s cannot be p e r t u r b e d , the f i r s t b e st e q u i l i b r i u m a t B and C ca n n o t be r e a c h e d . T h i s i s because the consumer p r i c e s (w + s*) t h a t would be o b s e r v e d i f t a r i f f s were reduced to z e r o , do not g e n e r a l l y s u p p o r t the 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. Hence, 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 o p t i m a l f o r the c o u n t r y . 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 d e n o t e d by x i n F i g . 5a), w h i c h I s D i a m o n d - M i r r l e e s o p t i m a l g i v e n t h e JL e x i s t i n g t a x d i s t o r t i o n s s . The c o r r e s p o n d i n g s e c o n d - b e s t e q u i l i b r i u m i s denoted by D i n F i g . 5a). The consumer f a c e s the p r i c e s * ° (w + s + x) at D. The u t i l i t y l e v e l a t t a i n e d by the consumer a t t h e s e - 122 -• GOOD 1 Figure 5 - S t r i c t Pareto and P r o d u c t i v i t y Improvements in T a r i f f s and Transfers. - 123 -p r i c e s i s at l e a s t as high as but g e n e r a l l y l e s s than the f i r s t p best l e v e l u . * * In F i g . 5b), the case where t = 0\T and s = O w i s d e p i c t e d . N M . 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 that both matrices S and E are of maximal rank, i t i s p o s s i b l e to adjust the i n i t i a l t a r i f f s and t r a n s f e r s so that the f i r s t best e q u i l i b r i u m at B and C ( t o t a l production e f f i c i e n c y ) i s a t t a i n e d . In F i g . 5 c ) , the consumer has L-shaped i n d i f f e r e n c e curves, i . e . , ^ = <"'(N+2)x(N+2) * ^- s o under these circumstances, the f i r s t e q u i l i b r i u m at B and C can be reached. This i s because, at C, the consumer p r i c e s (w + s*) are a support vector of the i n d i f f e r e n c e curve u^. - 124 -9. MORE ON GAINS FROM TRADE Assuming that the i n i t i a l vector of t a r i f f s x* equals the 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 t a r i f f vector (w 3'- w) defined in (5.1), Theorem 8.3 may be applied to show the existence of s t r i c t gains from trade when the government uses lump sum transfers as income r e d i s t r i b u t i o n instruments. This r e s u l t should be compared to the discussion in Chapter 4, where i t was established that also commodity taxes alone (without lump sum compensation) can be used to d i s t r i b u t e the pro d u c t i v i t y gains accruing from the p a r t i c i p a t i o n in i n t e r n a t i o n a l trade. What i s the connection between these two propositions? Do s t r i c t gains from trade always exist when only commodity taxation i s admissible, i f the conditions implying the existence of s t r i c t gains under lump sum compensation are s a t i s f i e d ? Or v i c e versa: do s t r i c t gains from trade under lump sum compensation always e x i s t , i f the s u f f i c i e n t conditions implying the existence of s t r i c t gains under commodity taxation are s a t i s f i e d ? Kemp and Wan in t h e i r mimeo1 analyze these questions. They claim that s u f f i c i e n t conditions for s t r i c t gains from trade to exist whether or not household s p e c i f i c transfers are admissible are: (9.1) i . each tradeable good i s produced in the home country, i i . the production p o s s i b i l i t y surface of the home country i s smooth and s t r i c t l y concave, i i i . a l l consumers in the economy have l o c a l l y unsatiated preferences in autarky. - 125 -The goal of t h i s chapter i s to generalize and strengthen this Kemp and Wan r e s u l t . Before the generalized proposition can be established, i t i s necessary to understand, why s t r i c t gains from trade may not exi s t i f lump sum compensation.is not possible (Kemp and Wan provide an example of a s i t u a t i o n where this i s the case). It can also be shown that, under c e r t a i n circumstances, s t r i c t gains from trade do not e x i s t , unless commodity taxation i s a f e a s i b l e government p o l i c y option (an example i s offered l a t e r in this s e c t i o n ) . Let us f i r s t consider the Kemp-Wan example i n more d e t a i l . Kemp and Wan assume that there are two consumers i n the home country. The consumers i n e l a s t i c a l l y supply two primary f a c t o r s . There are also two producers manufacturing two tradeable commodities using fixed c o e f f i c i e n t s technologies. In the present notation: H = 2, N = K = 2, M=2, £ = 0- _ , S = 0. , . qq 2x2 4x4 Suppose f i r s t that lump sum transfers are admissible, but that the commodity tax rates cannot be changed from their i n i t i a l values ( t , s ). Do s t r i c t gains from trade exist i n these circumstances? To answer the question, Theorem 8.3 i s applied. Assumption ( i ) of the theorem i s s a t i s f i e d , since the producers in the economy supply separate commodities and N = K = 2. Also assumption ( i i ) i s s a t i s f i e d , since the matrix YY T i s p o s i t i v e d e f i n i t e when N = K and the rank of the matrix Y equals K. In order to confirm assumption ( i i i ) , the matrix V B(w) ww must be ca l c u l a t e d . By assumption, E = O;,^ * *^^ s implies that 2 2 ~ T £ - On i> and i t can thus be seen that 7 B(w) = E and w E * qv 2x2 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 and s t r i c t gains from i n t e r n a t i o n a l trade e x i s t . Consider then the case where only commodity tax rates, but not lump sum tra n s f e r s , can be perturbed from th e i r i n i t i a l autarky l e v e l s . It i s immediately obvious that assumption ( i i i ) of Theorem 4.1 i s v i o l a t e d . This i s because the producer s u b s t i t u t i o n matrix S i s assumed to a zero (4 x 4)-matrix, which implies that v G(w + T , y ) = CL „. ww Zxz Hence, according to Proposition 4.2, i n t e r n a t i o n a l trade, caused by a ( d i f f e r e n t i a l ) perturbation of the t a r i f f s x , i s not s t r i c t l y g a i n f u l . How can these d i f f e r i n g conclusions be explained? Consider the Figure 6. Figure 6 represents a two tradeable commodities one consumer economy, where the production p o s s i b i l i t y f r o n t i e r for tradeables (keeping domestic goods net supply fixed) i s given by the curve PP'. At the autarky equilibrium A, the producers face the tradeables prices (w + x*), whereas the consumer prices for tradeable are (w + x* + s * ) . At these prices the consumer attains the i n d i f f e r e n c e curve u \ At the autarky equilibrium point the economy's production p o s s i b l i t y f r o n t i e r for tradeables i s kinked. This means that no ft d i f f e r e n t i a l change in the autarky t a r i f f s x can s h i f t the producers' net supply of tradeables from A.1* Then, according to Proposition 4.2, no s t r i c t increase in the net amount of foreign exchange earned by the production industries i s possible. It follows, that the welfare of the - 127 -GOOD 2 ure 6 - Existence of S t r i c t Gains from Trade Under Commodity Taxation and Lump Sum Compensation. - 128 -consumer cannot be s t r i c t l y improved by changing the commodity tax rates s (and t ). In other words, s t r i c t gains from trade under commodity taxation do not e x i s t . ., . I f , however, the i n i t i a l e q u i l i b r i u m lump sum transfer g for •the consumer can be al t e r e d , i . e . , lump sum transfers are a f e a s i b l e government p o l i c y instrument, the s i t u a t i o n of the consumer i n F i g . 6 can be improved. Suppose, for example, that the i n i t i a l t a r i f f s T are perturbed toward the t a r i f f s x depicted i n Fi g . 6. As a consequence, the producers aggregate net supply of tradeables does not change from i t s i n i t i a l l e v e l but the consumer i s moved along his ind i f f e r e n c e curve u^-.7 I f , simultaneously with the t a r i f f change, the consumer i s given a transfer i n the d i r e c t i o n of Ag i n F i g . 6, the consumer i s made s t r i c t l y better o f f . Hence, i n t e r n a t i o n a l trade, caused by a perturbation i n the i n i t i a l t a r i f f s x* i s s t r i c t l y g a i n f u l i f 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 existence of a "s u b s t i t u t i o n gap" between the Theorems 4.1 and 8.3. According to the l a t t e r r e s u l t , s u b s t i t u t i o n i n consumption can make a s t r i c t Pareto improvement possible even though a s t r i c t Pareto and pro d u c t i v i t y improvement i n the sense of Theorem 4.1 does not e x i s t . But the " s u b s t i t u t i o n gap" between Theorems 4.1 and 8.3 can be employed also i n another fashion. Suppose that a l l the conditions of Theorem 4.1 are s a t i s f i e d ; i n p a r t i c u l a r , the producer s u b s t i t u t i o n matrix S i s not a zero (N + M) x (N + M) - matrix (S * °(N+M)x(N+Mp• b u t s u P P o s e » i n addition, that the consumer s u b s t i t u t i o n matrix E i s such that - 129 -(9.2) wT V 2 B(w) = oJ. ww M Then, assumption ( i i i ) of Theorem 8.3 i s v i o l a t e d and, • according to Theorem 8.5, no s t r i c t Pareto improving ( 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 ft ft t a r i f f s x and transfers g e x i s t . In other words, s t r i c t gains from trade under lump sum compensation are not possible although i n t e r n a t i o n a l trade under commodity taxation would be s t r i c t l y g a i n f u l . The 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 existence 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 that neither the existence of s t r i c t gains from trade under commodity taxation or the existence of s t r i c t gains from trade under lump sum compensation n e c e s s a r i l y implies the other. Yet, with the help of Theorems 4.1 and 8.3, a general result can be established. Theorem 9.1: I. Suppose that the conditions of Theorem 8.3 are s a t i s f i e d so that s t r i c t gains from trade under lump sum compensation e x i s t . Then, T i f i n autarky, ( i ) rank ( Spp + YY ) = N, ( i i ) there Is no so l u t i o n rp rp rp rp AT 9 A A T a > 0 H to aL[XL, E ] = 0 ^ , and ( i i i ) T G(w + T , y ) * 0*, s t r i c t gains from trade under commodity taxation e x i s t . I I . Suppose that the conditions of Theorem 4.1 are s a t i s f i e d so that s t r i c t gains from trade under commodity taxation e x i s t . Then, i f - 130 -T 2 ~ T w V B(w) t 0 W at the i n i t i a l autarky equilibrium, s t r i c t gains from ww M trade under lump sura compensation e x i s t . The conditions ( i ) - ( i i i ) in the f i r s t part of Theorem 9.1 are the generalizations of the three Kemp-Wan assumptions (9.1). Kemp and Wan suppose that the consumer preferences are unsatiated around the autarky equilibrium and th i s assumption i s i m p l i c i t l y present also in Theorem 9.1. But the Kemp-Wan requirement that each tradeable good must be produced in autarky i n the home country seems to translate to the condition that the same K production sectors that operate In autarky, o operate also after i n t e r n a t i o n a l trade has become possi b l e . The Kemp-Wan assumption that the country's production p o s s i b i l i t y f r o n t i e r i s g l o b a l l y smooth and s t r i c t l y concave generalizes to assumption ( i i i ) in Theorem 9.1: kinks and ridges in the production p o s s i b i l i t y f r o n t i e r are allowed but, at the autarky e q u i l i b r i u m , the gradient of the net * * balance of trade function b (w+ T ) with respect to the trade p r o h i b i t i v e t a r i f f s T must be nonzero. Assumption ( i ) in the part I of Theorem 9.1 i s needed when the numbers of production sectors and domestic commodities do not coincide as i n the Kemp-Wan case. The r o l e of t h i s assumption i s to guarantee l o c a l c o n t r o l l a b i l i t y bf domestic goods production in the home country i n the sense of D e f i n i t i o n 2.2. Assumption ( i i ) in the Part I of Theorem 9.1 ensures that some s t r i c t Pareto improving d i r e c t i o n s of consumer price changes ( s t a r t i n g from the autarky equilibrium) e x i s t . This condition i s s a t i s f i e d in 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 necessary c o n d i t i o n f o r s t r i c t gains from trade when commodity taxations i s used to r e d i s t r i b u t e consumer income. The second part of Theorem 9.1 emphasizes the f a c t that s t r i c t gains from trade under lump sum compensation need not e x i s t whenever trade under commodity t a x a t i o n i s s t r i c t l y g a i n f u l . I t seems, however, that the l i k e l i h o o d of t h i s abnormality i s s m a l l : f o r example, the T 2 v e c t o r w V B(w) cannot be a zero M-vector i f both the s u b s t i t u t i o n ww matrices S and Z are of maximal rank, and the i n i t i a l e q u i l i b r i u m * / 9 t a r i f f s x 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 p r i c e s w. - 132 -10. PROPORTIONAL REDUCTIONS IN DISTORTIONS AND SOME PIECEMEAL POLICY RESULTS )• The goal of the analysis in this chapter is to develop some examples of s t r i c t Pareto and productivity improving policy changes whe household specific lump sum transfers are an admissible policy instrument. The results found are often similar in nature to those presented in Chapter 7, where lump sum transfers were not allowed, but some differences do occur. F i r s t , the existence of arbitrary tax distortions in the economy makes i t harder to establish explicit policy recommendations—often to derive a result, i t is necessary to * * , assume either that t = 0„, s = 0 W (tax distortions at the i n i t i a l N M equilibrium do not exist), or that there is no substitution in consumption between some or a l l commodities. On the other hand, i f lum sum transfer changes are possible, more results, where the changes in the policy instruments are either proportional or shift the policy variables toward uniformity, can be proved. Let us start by considering a s h i f t toward the international * prices w in the i n i t i a l equilibrium t a r i f f s x—a policy that was shown to be constant u t i l i t y productivity improving in Proposition 8.1. If lump sum transfers may 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 are satisfied. Then there exists a s t r i c t Pareto and productivity improving change in the - 133 -i n i t i a l equilibrium t a r i f f s and t r a n s f e r s , and the change in t a r i f f s x can be chosen to be a movement toward the world price vector w. Proof: * Let Ax = rw, where r > 0. A s u f f i c i e n t condition for a s t r i c t ft Pareto and pro 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 to exist i s : (10.1) there does not exist a vector X e ^ H+N+K+l ^at X T[A, -B b] > og + 1, X T [ B p , B z, Bg] = O £ + k + h , - XTB.w > 0. T T A vector X s a t i s f y i n g the equations X [B , B , B ] = 0„.T, 1 T T must be of p z g N+K+H rp rjt >£. rp rp the form X = k[0 , ( p + e ) , 0 , 1 ] , k e R. The i n e q u a l i t i e s H T T T T X [A, -B ] > 0 imply k > 0. Since X A > 0 , k must be p o s i t i v e , b H+1 — H T T 2 Hence, choose k=l. Using Lemma 8.3, X B^ = w B(w) . (The change T 2 i n the sign of the vector w V B(w) occurs because the matrices B ° ww x defined in (8.1) and (4.16) have opposite signs.) By assumption, T 2 ~ T T T 2 ~ w V B(w) * 0 M. It follows that, -X B w = -w V B(w) w < 0 since matrix ww M ' x ww 9 V^wB(w) i s pos i t i v e semidefinite. QED Theorem 10.1 i s a counterpart of Theorem 7.1. According to Theorem 7.1, i f the commodity tax rates ( t , s ) can be adjusted, a s t r i c t Pareto and pro 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 - 134 -ft ft (nonnegative) t a r i f f s x . In Theorem 10.1, the change in t a r i f f s x i s not a reduction as such, but i t i s a reduction in the d i s t o r t i o n a r y gap between the domestic and i n t e r n a t i o n a l producer prices for tradeables. Hence, also i t r e s u l t s i n a reduction of the home country's trade b a r r i e r . * A reduction of 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 , accompanied by ft a perturbation of the transfers g , may be shown to be a s t r i c t Pareto and p r o d u c t i v i t y improvement, i f eit h e r there are no d i s t o r t i o n a r y commodity taxes i n the economy, i . e . , t = 0^ and s = 0^, or i f the consumer s u b s t i t u t i o n matrix 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 exists a ft s t r i c t Pareto and pro d u c t i v i t y improving change in t a r i f f s x and ft tra n s f e r s g . The change in t a r i f f s may be taken to be a proportional reduction. Proof: A s u f f i c i e n t condition for the existence of a s t r i c t Pareto and p r o d u c t i v i t y improvement in t a r i f f s x and transfers g i s (10.1), X X * where the Ineq u a l i t i e s -X B^ w > 0 are replaced by X B^x > 0. * * (a) I f t = 0„, s = 0 W, and the vector X i s of the form N M rp rp £ rp ^ = [0 , (p + e), 8 , 1 ] , using (4.16) and the homogeneity of the unit p r o f i t andexpenditure functions, i t can be seen that - 135 -T T T T *T * A T (10.2) X B = 9 F + e (S - E ) - x S + (x + s ) E . x pw qv ww vv The vectors e and 8 are defined in (8.14). Using these d e f i n i t i o n s * * when t = 0., and s = 0.., N M (10.3) A B_ = x [(-E V + S D 1 0 + FD 0 0) F x 1 vq 12 wp 12 22 + <-EvqDl  + Swp DH + F ° 1 2 ) ( Swp - V " ( Sww~ W = - T * T [ - ( S w p " V D l l " F D 1 2 ' ^  <S " V [-(S - E ) D. - FD* T ] T wp vq 11 12 M = -x* TV 2 B(w) . WW The l a s t equation above i s derived using Lemma 8.1. In order to derive the quadratic form in (10.3), the following properties of the inverse matrix D have been used: D,.(S - E ) D , „ = -D„„, D,, 12 pp qq 12 22 11 (S - £ ) D. = D,., D.,(S - E ) D. = 0. pp qq 11 11 11 pp qq 12 Since, using (10.3), XTB = -x*V 2 B(w), i t follows that X TB x* = » * & > • ' » T w w ' X * T _ 9 * -x A\T (w) x < 0. WW 2 ~ (b) If E = 0 / V T,„, N ,„ w N , the matrix V B(w) equals the matrix (N+M)x(N+M) ww O J^ J^ J^ l^T JU w rri JU v G(w + x , y ) for any t e R and s e-R . In th i s case, X B x = WW . X 9c*T 0 9c 9c 9c — x A V ^ G(w + x , y ) x < 0. QED Let us now assume that although the commodity tax rates i n the home country are a r b i t r a r y at the i n i t i a l e quilibrium, the government i s - 136 -* it able to adjust them in addition to the t a r i f f s x and transfers g . It was shown in Theorem 7.2 that, under c e r t a i n conditions, commodity taxes and t a r i f f s can be simultaneously reduced to produce a s t r i c t Pareto and pr o d u c t i v i t y improvement. This r e s u l t did not require lump sura transfers to be admissible, but i f they are, the simultaneous  reduction i n taxes and t a r i f f s may be chosen to be a proportional  reduction. Theorem 10.3: Suppose that assumptions ( i ) - ( i i i ) of Theorem 8.3 are s a t i s f i e d . Then, there exists a s t r i c t Pareto and p r o d u c t i v i t y * * * * improving change in t , s , x and g , and the change in the commodity tax rates and t a r i f f s can be chosen to be a proportional reduction. Proof: A condition s u f f i c i e n t to imply the existence of a s t r i c t Pareto and p r o d u c t i v i t y improving proportional reduction in commodity taxes and t a r i f f s i s : * * * * * * * * (10.4) there e x i s t s Au , Ap , Az , Ab , Ax , At , As and Ag * * * such that (4.16) i s s a t i s f i e d Au » 0 U, Ab > 0, Ax = H ic •& x x x - r t , At = - r t , As = - r s , r > 0. The Motzkin dual equivalent to (10.4) i s : - 1 3 7 -( 1 0 . 5 ) there i s no vector X e R h + n + k + 1 s u c h that X T[B , B , B ] P z g = 0 N+K+H , X [A, - B b] > 0 R + 1 , X [ B t , B B T] > 0 . As shown before, the vector s a t i s f y i n g the equations X [ B p , , B^] = T T T T T ^ T 0 N + R + H and X [A, - B b] > 0 H must be of the form X = [ 0 , (p + E ) , T 0 , 1]. For this X, using (4.16), (10.6) A [B , B , B ] i^2' ^3» T * F T E p + 1 w + S T qq qv pw X * T X * w I p + w E w + w S T vq vv ww T T The equations X B^ = 0 N and the homogeneity of the unit p r o f i t and expenditure functions have been used to derive ( 1 0 . 6 ) . Applying now the d e f i n i t i o n of the vector X, - 138 -(10.7) A [B t, B G , B T ] , * T * A T = (p + e) S p + (p + e) Z w PP qv , * * T * T T . * + (p + E ) S x + BLYl (p + e) pw T * T T * + w S p + w Z w + w S T , wp W WW where also the equation x F = e Y derived from (B5) i n Appendix 2 has been employed. Applying the equations (B5) in Appendix 2, (10.8) A i [ B t , B , B ] , * T * * T = (p + e) S p + (p + e) Z w PP qv , * . T * . * T + (p + e) iS x + (p + E) (Z - S ) pw ^ qq pp 7 9c 9crV 9c 9crV 9c (p + e) + x 1S (p + e) + p LS (p + e) wp pp + wTZ (p* + e) + wTS p* + w TZ w vq wpr vv T * + wiS x WW [(p* + s ) T , wT] (Z - S) * p + e w -wTV2 B(w)w, ww - 139 -T 2 T using Lemma 8.1. Since, by assumption w V B(w) * 0 W, the scalar ww M T 2 -w V B(w)w must be negative. QED ww Theorem 10.3 implies that, i f the assumptions of the theorem are s a t i s f i e d , s t a r t i n g from an i n i t i a l equilibrium with a r b i t r a r y commodity tax and t a r i f f structures, there e x i s t s a path of tax and t a r i f f  reductions which, i n the l i m i t , leads to a nondistortionary government financed by lump sum taxation. If the i n i t i a l equilibrium i s an autarky * * * equilibrium, the f i n a l equilibrium with t = 0.T, s = 0 W and x = 0 W N M M coincides with the f i r s t best equilibrium depicted in F i g . 3 by the points C and D. In other words, i f the assumptions of Theorem 10.3 are s a t i s f i e d , the domestic economy can be s h i f t e d to t o t a l production e f f i c i e n c y with respect to the technology T + ET defined in (4.23) by k * * p r o p o r t i o n a l l y reducing the commodity tax and t a r i f f rates t , s * and x . * Proportional reductions i n the i n i t i a l commodity tax rates t ft ft and/or s without changes i n the t a r i f f s x w i l l also u s u a l l y be s t r i c t Pareto and p r o d u c t i v i t y improving. To develop precise r e s u l t s , a new Hessian matrix has to be defined. Consider problem (8.6) which defines the constant u t i l i t y tax adjusted balance of trade function ft ft ft ft B(w, w + x , w + x + s , t ) . The f i r s t order conditions for (8.6) are the zero p r o f i t and supply equals demand equations (2.9) and (4.12). These equations are assumed to be s a t i s f i e d at the i n i t i a l equilibrium - 140 -of the economy and they can thus be used to define the equilibrium / ft ft domestic price and industry scale vectors p and z as i m p l i c i t ft ft ft functions of the variables w, x , t and s . D i f f e r e n t i a t i n g the equations (2.9) and (4.12) at the i n i t i a l equilibrium: (10.9) S - £ Y pp qq T Y 0 KxK * Ap * Az E - S qv pw AT + qq KxN At + qv KxM As Equations (10.9) determine the d e r i v a t i v e s of the i m p l i c i t functions ft ftftft ft p (w + x , t , s ) and z ( w + T , t , s ) with respect to the exogenous ft ft ft commodity tax rates ( t , s ) and the t a r i f f s x at the i n i t i a l e q u i l i b r i u m . ft ft ft ft D i f f e r e n t i a t e the function B(w, w + x , w + x + s , t ) with * * * respect to x , t and s . Using (8.6), K k * * * k ^ h * h * * (10.10) V B = E V TT (p , w + x ) z - E V m (u , q , v ), T k=l T h=l T (10.11) V xB = * n h *h * - E V m (u , q , v ) , h=l - 141 -(10.12) V B = - E V m h(u* h, q* , v*) . S h=l S The second order p a r t i a l d e r i v a t i v e s of the function B with respect to ie ie ie the t a r i f f s T and tax rates ( t , s ) at the i n i t i a l equilibrium can be calculated by d i f f e r e n t i a t i n g equations (10.10) - (10.12) and by using (10.9) : (10.13) V B = 2 2 2 V B, V B, V B TT Tt ' TS 2 2 2 V" B, B, B tT ' tt ' tS 2 2 2 V B, V B, V B ST St SS S - E , -E , -E ww vv vq vv qv' w -E , -E qq qv -E , -E vq w S - E , F wp vq -E qq -E vq , 0 , 0. NxK MxK S - E , F wp vq -E qq -E vq , 0. , 0. NxK MxK The inverse matrix D i n (10.13) i s d e f i n i t e d in (8.5). Since B i s a o convex function of i t s arguments, the matrix V B i s p o s i t i v e semidefinite. Consider now, for example, a proportional reduction i n the * tradeables taxes s . One of the s u f f i c i e n t conditions for this p o l i c y perturbation to be s t r i c t Pareto and p r o d u c t i v i t y improving i s that the *T 2 * T 7 T vector s V B i s nonzero, i . e . , s v B * 0 W: the i n i t i a l vector of ss ss M - 142 -tradeables taxes s must not be a zero eigenvector of the matrix v B. 0 ss Theorem 10.4: Suppose that ( i ) rank Y = K < N and ( i i ) rank (S - £ + YY T) — pp qq = N at the i n i t i a l e quilibrium. Then, * T 2 T (I) I f T = C-T and w V B(w) * 0... there 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 in the i n i t i a l commodity taxes ft ft ft ft ft* ( t , s ) and transfers g . The change in ( t , s ) may be chosen to be a proportional reduction. (II) If t* = 0„, T* = 0 M and s * T V 2 B * oJ or i f £ = 0„ M N' M ss M — qv NxM ftrp 2 J and s V B * 0 W, there exists a s t r i c t Pareto and p r o d u c t i v i t y ss M' J improving change in the i n i t i a l tax rates s* and transfers g*. The change in the tradeables taxes s* may be taken to be a proportional reduction. (II I ) I f s* = 0„, T* = 0^ and t * T V 2 B * C-3 or £ = 0„ „ and M M tt N qv NxM ftrj, 2 Y t ^ t t B * there exists a s t r i c t Pareto and p r o d u c t i v i t y improving ft ft change in the i n i t i a l tax rates t and transfers g . The change in ft t may be chosen to be a proportional reduction. The proof of Theorem 10.4 i s rather tedious and hence deferred to Appendix 2. The s u f f i c i e n t conditions for proportional reductions i n the commodity tax rates t and/or s to be s t r i c t Pareto and pr o d u c t i v i t y improving given in Theorem 10.4 are r e s t r i c t i v e : although not much i s required concerning the s u b s t i t u t i o n matrices S and £ (the - 143 -r p 2 ~ * T 2 *T 2 vectors w 7 B(w), s V B and t V B depend on the matrices S and ww ss ss * E) , i n t e r n a t i o n a l trade i s assumed to be free, i . e . , x = 0^, and some * * of the tax rates (t , s ) are also assumed to be i n i t i a l l y zero. I f these conditions are not s a t i s f i e d , a proportional reduction of t* and/or s*, even i f accompanied with a change in household s p e c i f i c t r a n s f e r s , may cause a welfare and p r o d u c t i v i t y l o s s . Theorem 10.5: Suppose ( i ) rank Y = K < N, ( i i ) rank (S - £ + YY T) = N, and — pp qq ( i i i ) the i n i t i a l equilibrium t a r i f f s x* are such that (10.14) V b*(w + x*) (w + x*) < 0. x Then, there exists a s t r i c t Pareto and pr o d u c t i v i t y deproving change of commodity tax rates ( t * , s*) and lump sum transfers g*, where the change in ( t * , s*) i s a proportional reduction. Proof: A s u f f i c i e n t condition for a s t r i c t Pareto and pro d u c t i v i t y A A A A A deprovement in (t , s ) and g to e x i s t , when the change in ( t , s ) i s a proportional reduction, i s : (10.15) there does not exist a vector X e RH+N+K+1 g u c h t h a t - 144 -A T [ A , - B B ] < 0 ; + 1 > X T [ B P , B Z , B G ] = 0 T + K + H , X [B t, B s] > 0. T T A vector s a t i s f y i n g the equations X [B , B , B ] = 0„ ' , 'must be of p z g N+K+H *TI Hp JL rrt frt the form X = k[0„, (p + e) , 9 , 1], k e R. The i n e q u a l i t i e s bl T T X [A, - B^] < 0 H imply k < 0; choose k = -1. Then, using the proof of Theorem 10.4, (10.16) X i[B ( ;, B ] T O * = w V B(w) (w + T ), WW Using (8.10) and (10.14), Je Jc & = V^b (w + T )(w + x ) < 0. QED Theorem 10.5 shows that, under very reasonable conditions, i f the i n i t i a l t a r i f f vector x i s a r b i t r a r y and nonzero, a proportional reduction of the commodity tax rates (t , s ) may lead to a s t r i c t Pareto and p r o d u c t i v i t y deprovement i n s p i t e of the fact that lump sum transfers can be used to r e d i s t r i b u t e consumer income: i t i s only (10.17) X A [ B T , B G ] - 145 -required that the producer price weighted net balance of trade A A d e r i v a t i v e s b (w + x ), m=l,...,M, sum to a negative number, ra If the strong conditions of Theorem 10.4 are s l i g h t l y modified, i t can be shown that movements toward uniformity (at a lower l e v e l of  taxation) i n p o s i t i v e commodity taxes are s t r i c t Pareto and p r o d u c t i v i t y  improving. Proposition 10.1: Suppose that Assumptions of Theorem 10.4 are s a t i s f i e d , except 2 ~ 2 2 that the conditions on the matrices V B(w), V B and V B are replaced ww ss t t T 9 T A T 9 T A T 9 T by: w V B(w) < Of., s v B < 0„, t V ^ B < 0... Then, there exist 3 ww M ss M' t t N ' s t r i c t Pareto and p r o d u c t i v i t y improving changes in (I) (t , s ) » A A A A A C-W™ a n d g , (II) s » 0 U and g , and (III) t » 0.T and g , where the perturbations in the commodity tax v a r i a b l e s are chosen to be movements toward a lower uniform l e v e l of taxation. The proof of Proposition 10.1 i s given in Appendix 2. The s u f f i c i e n t conditions for s t r i c t Pareto and p r o d u c t i v i t y improvements to exist given in Proposition 10.1 are c l o s e l y related to a well known r e s u l t of Hatta (1977a). Hatta showed that, under c e r t a i n conditions, the reduction of the highest tax rate (on a domestic good) In a one consumer closed economy i s welfare improving. In Proposition 10.1, a s i m i l a r kind of p o l i c y recommendation r e s u l t i s achieved in an open economy without r e s t r i c t i n g the numbers of consumers, producers and - 146 -commodities (except that K_< N). The assumptions about the matrices 2 2 V B and V B in Proposition 10.1 are g e n e r a l i z a t i o n s of the Hatta t t ss ° assumption that the good with the highest (domestic) tax rate must be substitutable for a l l other (domestic) commodities in the economy. This JL can be seen as follows: suppose, for example, that t » 0 N. Then, *T 2 T using (10.13), the condition t ^ t t B ^ * s equivalent to *T " T t (E + E D, , £ ) > 0 M.For the l a t t e r vector to be p o s i t i v e , the qq qq 11 qq N ~ ' off-diagonal terms i n the matrix E q q must be s u f f i c i e n t l y p o s i t i v e , i . e . , i n a sense, s u b s t i t u t i o n in consumption of the domestic commodities must dominate. Yet, the connection between Proposition 10.1 and the above mentioned Hatta r e s u l t i s not straightforward since Hatta used ad valorem (not s p e c i f i c ) commodity tax rates i n h i s model. Furthermore, one of the assumptions used in his theorem was the Hatta normality condition: the sum of the consumer demand income d e r i v a t i v e s weighted 3 by the corresponding producer prices must be p o s i t i v e . No such supposition was needed in Proposition 10.1 or in the p o l i c y recommendation re s u l t s of the previous chapters. In order to resolve the di f f e r e n c e s between Proposition 10.1 and the r e s u l t s of Hatta, the model employed by Hatta i s examined in d e t a i l . Let us f i r s t write the Hatta model in the form (4.10) - (4.13). The home country i s assumed to be closed with N+l domestic commodities. There are N producers, each one of whom supplies one of the N commodities, indexed by n e [1,...,N], The [N + l ] t h domestic good i s a fixed resource which i s used as an input to produce the other domestic commodities. The fixed resource serves as the numeraire commodity i n - 147 -the model with a price denoted by w = 1 (hence, the fixed factor i s regarded as a "tradeable" good). The government imposes ad valorem taxes t £ on the N domestic commodities and gives lump sum transfers g e R to the single consumer in the country. The i n i t i a l e quilibrium of the Hatta economy i s characterized by the equations: (10.18) m(u , 1, q ) = g , (10.19) i r n ( l , p ) =' P ~ a n = 0. n=l,...,N, (10.20) Vqm(u*, 1, q*) = y* , N N (10.21) Z V TT (1, p ) y = Z a y = r = V m( u , 1, q ) , n=l n=l (10.22) q* = [1 + t*] p*. ft In (10.22), the matrix [1 + t ] i s an (N x N)-matrix with diagonal * elements equal to (1 + t n ) , n=l,...,N. The other elements of the ft matrix [1 + t ] are zero. According to (10.18), the consumer's expenditures on the N domestic commodities minus his revenue from s e l l i n g the fixed factor - 148 -equal his lump sum income g . The producers' unit p r o f i t functions are of the form (10.19), where the number i s the n t n sector's input c o e f f i c i e n t for producing one unit of the n t n output using the fixed factor as an input. The equations (10.20) - (10.21) are market clearing conditions: consumer demand for the N commodities equals their supply y , and the producer demand for the factor equals i t s fixed k supply r . The industry output l e v e l s y serve as the scale variables for the N producers. This choice of the production scale i s permissible because each production sector supplies only one output. The 1+N+N+l equations (10.18) - (10.21) determine endogenously k the equilibrium u t i l i t y l e v e l of the consumer u , the equilibrium k k prices p , the equilibrium output l e v e l s y and one of the p o l i c y instruments, given the exogenous v a r i a b l e s a = (a ,...,a^), r , t and g*. D i f f e r e n t i a t i o n of (10.18) - (10.21) at the i n i t i a l equilibrium (which solves (10.18) - (10.21)), y i e l d s : (10.23) A Au* = B Ap* + B Ay* + B At* + B Ag* P y • ' t g 6 where A = — — 1 CL N E qu 0 -x A[i + t ] -E [1 + t ] qq T , B = ' y N °NxN I., N T -a - 149 -B = g 1 The normality condition given in Hatta (1977a) i s a E > 0. qu Using the equation (10.19), t h i s condition can be written in the form It i s easy to see that (10.24) i s s a t i s f i e d , i f a l l commodities n, n=l,...,N, are normal, i . e . , i f a l l elements of the vector Eq U are p o s i t i v e . The following theorem shows the connection between the Hatta normality assumption and the r e s u l t s of the previous chapters. Theorem 10.6; If the i n i t i a l equilibrium in the economy that s a t i s f i e s the equations (10.18) - (10.22) i s a g-optimum with respect to the i n i t i a l t ransfer g , the Hatta normality condition (10.24) i s s a t i s f i e d . (10.24) P E qu > 0. - 150 -Proof: Suppose the i n i t i a l equilibrium i s a 8-optimum with respect to * X T T T T the transfer g . Then, there exists a vector X = [X^, X^, X^, X^] e I + M + M + I T T T R such that X A = B = 1 > 0 , X [B , B , B ] = 0„ L X T L.. (Since P y S N+N+l T H = 1 , B can be set to equal 1.) The equations X B =0 imply that X^  = 0. Hence, (10.25) [X*, X*, xj] N -E qq [i + t ] o. N NxN -a = 0 N+N It follows that X^ = X. a T and X^ - \li E [1 + t*] = 0^. Set X. = k, 3 4 2 3 qq N 4 X T T T * T k £ R. Then, X = ka , X = ka E [1 + t ]. The equality X A = 1 3 2 qq J T becomes ka E = 1 . Hence, k * 0. In addition, i t can be inferred qu ' that, i n f a c t , k > 0. (This can be seen by replacing the equality (10.21) with a weak ine q u a l i t y (<). Although, at the i n i t i a l e quilibrium, (10.21) i s s a t i s f i e d as an equali t y , allowing the p o s s i b i l i t y of free disposal for the fixed factor r imposes a nonnegativity constraint on the l a s t component of the vector X T T when the Kuhn-Tucker conditions X A = 1, X [B , B , B ] = 0„ „ , , p z' g N+N+l' X £ R 1 + N + N + 1 > a r e derived for the problem maxfu: (10.18) - (10.21) rn rrt hold}). Setting k = 1, X A = 1 (> 0) i s equivalent to a E q u = 1 > 0. - 151 -Thus, using a (10.19), the Hatta normality condition (10.24) i s s a t i s f i e d . QED Why i s i t that normality of goods in demand and S-optimality of the i n i t i a l equilibrium are connected? In order to answer the question, consider Figure 7 presented by Hatta (1977a). There are two goods, and X £, in the economy. The l i n e PP' defines the country's production p o s s i b i l i t y f r o n t i e r for x^ and X2 given the amount of the fixed factor r . The i n i t i a l equilibrium for the economy i s at x^ where the consumer attai n s the u t i l i t y l e v e l xP. At x^, the good X£ i s taxed at a higher l e v e l than the good x^. Suppose the tax rate for X2 i s reduced and as a consequence, the equilibrium s h i f t s to x^. The change from x^ to x^ may be decomposed to two parts: f i r s t , the consumer i s moved along his * i o r i g i n a l i n d i f f e r e n c e curve to x and then, up to x 1 along his Engel (income consumption) curve. 4 If the consumer's Engel curve cuts the production p o s s i b i l i t y f r o n t i e r PP' from the inside to the outside, the move from x^ to x^ must be a welfare improvement. When does the Engel curve have the form required? Consider the * T Hatta normality'condition. If p E > 0, an increase in the qu consumer's transfer income ra i s e s the value of h i s t o t a l consumption, i . e . , his Engel curve must cut the production p o s s i b i l i t y f r o n t i e r from the inside to the outside. Hence, the normality condition in the p o l i c y recommendation r e s u l t of Hatta (1977a) i s needed to guarantee that a p o l i c y change (here, a reduction of the highest commodity tax rate) i s welfare improving. - 152 -x2 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 of a P o l i c y Change. - 153 -I f , on the other hand, the i n i t i a l e q u i l i b r i u m i n the economy i s a B-optimum with respect to the t r a n s f e r g*, i t can be seen that the consumer's Engel curve must cut the production p o s s i b i l i t y f r o n t i e r from the i n s i d e to the o u t s i d e . I f t h i s were not the case, the i n i t i a l e q u i l i b r i u m could not be a welfare-maximum: i t would be p o s s i b l e to reduce the consumer's lump sum income and move along h i s Engel curve to a new e q u i l i b r i u m w i t h a higher l e v e l of consumer w e l f a r e . The s h i f t would also 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 lead to an e q u i l i b r i u m i n s i d e the economy's production p o s s i b i l i t y s e t . The s i t u a t i o n i s depicted i n Figure 8. The point x^ i n F i g . 8 cannot be a * 5 B-optimum with respect to the t r a n s f e r g . I t can thus be concluded that the Hatta n o r m a l i t y c o n d i t i o n which i s present 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 e q u i l i b r i u m of the Hatta economy i s a welfare maximum with respect to the lump sum t r a n s f e r g . But, as can be r e c a l l e d , the 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 were not c o n d i t i o n a l on the i n i t i a l e q u i l i b r i u m of the economy being a 8-optimum with respect to the v e c t o r of t r a n s f e r s g*. How are the e a r l i e r r e s u l t s then to be understood? How can they imply that some welfare improvements do e x i s t without any normality c o n d i t i o n imposed on the domestic or tradeable commodities? The answer i s rather simple. I f the normality c o n d i t i o n i n the Hatta 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 a 8-optimum w i t h respect to the lump sum t r a n s f e r g . This means that some s t r i c t Pareto Improving p e r t u r b a t i o n only i n the t r a n s f e r g* e x i s t s , without a change i n the commodity tax rates t . However, once - 154 -x 2 i F i g u r e 8 - B - O p t i m a l i t y and the H a t t a N o r m a l i t y C o n d i t i o n . - 155 -a l l such Pareto Improvement p o s s i b i l i t i e s have been exhausted, the r e s u l t i n g equilibrium i s a B-optimura with respect to g , the Hatta normality condition i s s a t i s f i e d , and then, further Pareto improvements can be attained by perturbing some of the other p o l i c y instruments (commodity taxes) together with the lump sum transfer g , as the e a r l i e r theorems demonstrate. The Hatta model (10.18) - (10.22) i s a very much s i m p l i f i e d v ersion of the model (4.10) - (4.13). Hence, the normality condition appropriate for the general model i s not the simple Hatta assumption (10.24). It was shown in the proof of Theorem 8.1 that the vector X s R^ +N +K+l a s s o c x a t e d with a B-optimum with respect to the lump sum transfers g i n the economy described by (4.10) - (4.13) must be T T T proportional to the vector X = [0 , (p + e) , 9 , 1], where the H vectors e and 9 are defined in (8.14). For this vector X, the T T i n e q u a l i t i e s X A > 0 tra n s l a t e to H (10.29) (p* + e ) T £ + wT I > 0* qu . vu H which i s the generalized Hatta normality, condition for the model (4.10) - (4.13). I f (10.29) i s observed to be v i o l a t e d , the i n i t i a l equilibrium of the economy i s not a B-optimum with respect to the transfers g and s t r i c t Pareto and p r o d u c t i v i t y improving changes i n * only transfers g e x i s t . The condition (10.29) d i f f e r s from the Hatta normality condition (10.24) in that the tradeables income d e r i v a t i v e s E v u (weighted by the - 1 5 6 -i n t e r n a t i o n a l prices w) are added to the formula. Also the domestic income d e r i v a t i v e s Eq u are weighted by the shadow prices (p + e) ft ft instead of the producer prices p . The prices (p + e) can be shown to be nonnegative i f the i n i t i a l equilibrium i s a 8-optimum with respect to the i n i t i a l transfers g*. 6 Hence, at a B-optimura, (10.29) i s s a t i s f i e d i f a l l commodities in the economy are normal. If lump sum transfers are not a f e a s i b l e government p o l i c y instrument, the (H+N+K+l) vector of Lagrange m u l t i p l i e r s associated with a B-optimum with respect to the commodity tax rates ( t * , s*) i s rn rp rp rp proportional to the vector X = [X^, (p + 6) , y , 1 ] , where X solves T T the equations X [ B , B , B . B ] = 0.T ,T. In this case, the p z t ' s N+K+N+M ' T T generalized normality condition, implied by the i n e q u a l i t i e s X A > 0 , H i s (10.30) X? + (p* + 5 ) T E + w T E > oL 1 qu vu H I f (10.30) i s v i o l a t e d , s t r i c t Pareto and p r o d u c t i v i t y improving * * perturbations i n only (t , s ) e x i s t . Let us now return to consider P r o p o s i t i o n 10.1 and the Hatta (1977a) r e s u l t according to which a reduction of the highest domestic tax rate in a closed one consumer economy i s welfare improving. - 157 -Theorem 10.7: (Hatta (1977): Theorem 1) Suppose that the model (10.18) - (10.22) i s used to characterize an economy where the tax rate t ( » 0), n e [1,...,N], on the n * t domestic commodity i s the highest of a l l tax rates t . Then, i f the n domestic commodity i s substitutable for a l l other domestic goods in the sense that (10.31) p* T(£ [p*]) > 0, ^ th ^ where (E [p ]) denotes the n column of the (N x N)-matrix E [p ] qq «n qq * there exists a s t r i c t welfare improving change of the tax rate t ^ , * * n e [1,...,N], and transfer g . The change In t ^ may be taken to be a reduction toward the l e v e l of the next highest domestic tax rate * t , ( ^ 0 ) , n' e [ l , . . . , n - l , n+l,...,N]. Proof: A s u f f i c i e n t condition for a s t r i c t welfare improvement of the above form to exist in the Hatta economy i s : (10.32) there i s no vector X e R 1 + N + N + 1 s u c h t h a t X T A > 0, AT[V V V = °5«+l» ^V'n h > °' W h e r e - 158 -T Using the proof of Theorem 10.6, the vector A s a t i s f y i n g A [B , B , B ] p y © = O X T . , , . 1 and A A > 0 must be of the form A = [ 0 , a E [1 + t ] , N+K+l qq T T a , 1]... Hence, i t s u f f i c e s to show that A ( B t ) . n < 0. Using (10.23), (10.19) and (10.31) , (10.33) A T ( B J = -a T(Z [p*]) < 0. t • n q q ^ « n QED In his version of Theorem 10.7, Hatta assumes that the normality condition (10.24) i s s a t i s f i e d . This condition holds, i f the i n i t i a l ft equilibrium i s a 3-optimura with respect to the transfer g . If this i s not the case, then there are some welfare improving changes i n only g* (At* = 0, n=l,... ,N) . n Theorem 10.7 can be regarded as a special case of Proposition 10.1, since a movement toward uniform domestic taxation means reducing the highest domestic tax rate, i f a l l other domestic commodity tax rates equal some common rate t (>_ 0). The apparent d i s s i m i l a r i t y of the *T 2 * 7 condition t (^ttB^«n ^ ^ * n P r o p o s i t i 0 1 1 10.1 and the condition *X * p ( Z q q [ p ] ) . n > 0 in Theorem 10.7 can be explained by taking account of the di f f e r e n c e between s p e c i f i c and ad valorem taxation: Hatta derives an expression for the consumer welfare change caused by a perturbation in the economy's commodity tax structure, and the sign of the welfare change depends on the signs of the vector p*^ E q q [ p * ] ; 8 - 159 -Chen, using the homogeneity condition EqqU + t ] p = 0 N, Hatta a r r i v e s at an a l t e r n a t i v e form for the vector p E which can be interpreted qq to imply the s u b s t i t u t a b i l i t y of the n t n domestic commodity for a l l other domestic goods, as assumed by Hatta in his theorem. 9 - 160 -11. COMMODITY TAXATION AS A CAUSE OF INTERNATIONAL TRADE I n a paper on the r e l a t i o n s h i p between commodity t a x a t i o n and i n t e r n a t i o n a l t r a d e M e l v i n 1 gives- an i n t e r e s t i n g s t a t e m e n t : "When t r a d e i s caused by a consumption t a x , the c o u n t r y i m p o s i n g the t a x may be made worse o f f by t r a d e so t h a t a p r o h i b i t i v e t a r i f f would be a p p r o p r i a t e . " Two q u e s t i o n s i m m e d i a t e l y a r i s e : f i r s t , how does a commodity t a x "c a u s e " t r a d e , and s e c o n d l y when i s i n t e r n a t i o n a l t r a d e , i n d u c e d by a change i n the c o u n t r y ' s commodity tax s t r u c t u r e , w e l f a r e r e d u c i n g ? In o r d e r to answer the f i r s t p r o b l e m , c o n s i d e r an economy a t an a u t a r k y e q u i l i b r i u m . The a u t a r k y ( i n t e r n a t i o n a l t r a d e p r o h i b i t i v e ) t a r i f f s T* a r e those d e f i n e d i n (5.1). The t a r i f f s x* a r e d e f i n e d w i t h r e s p e c t to the i n i t i a l e q u i l i b r i u m commodity t a x r a t e s and the i n i t i a l t r a n s f e r s i n the home c o u n t r y . Hence, i f any o f these p o l i c y v a r i a b l e s a r e p e r t u r b e d from t h e i r a u t a r k y l e v e l s , w i t h o u t s i m u l t a n e o u s l y c h a n g i n g the t a r i f f s x*, t h e r e l a t i v e p r o d u c e r and consumer p r i c e s i n the economy change, c r e a t i n g a p o s s i b i l i t y f o r i n t e r n a t i o n a l t r a d e ; a d j u s t m e n t s i n t h e home c o u n t r y ' s a u t a r k y commodity  t a x 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 t r a d e by i n d i r e c t l y c h a n g i n g  t h e a u t a r k y p r e s e r v i n g t r a d e b a r r i e r x . Whether the t r a d e i n d u c i n g change i n the a u t a r k y commodity t a x e s o r t r a n s f e r s i s 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 i n g depends on the p r o p e r t i e s o f the i n i t i a l commodity t a x r a t e s ( t , s ) and t r a n s f e r s g*• I f , f o r example, ( t * , s*) a r e B - 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 : (4.10) - (4.13) h o l d , x = x and g = g ) , - 161 -no perturbation of only (t , s ) (which would cause i n t e r n a t i o n a l trade) can be s t r i c t Pareto or welfare improving. S i m i l a r l y , i f the i n i t i a l autarky equilibrium i s a S-optimum with respect to the transfers ft g (assuming that the home country's commodity tax structure i s ft f i x e d ) , no change of the transfers g alone can be s t r i c t Pareto and welfare improving. On the other hand, i f the i n i t i a l commodity tax * * rates ( t , s ) are adjustable but a r b i t r a r y i n autarky, s t r i c t Pareto  and p r o d u c t i v i t y improving perturbations i n them e x i s t ; trade caused by  these changes i n the home country's commodity tax structure i s welfare  improving. Theorem 11.1: Suppose that i n autarky ( i ) rank Y = K _< N, ( i i ) rank ( S p p + YY T) = N, ( i i i ) t* > 0 N, s* > 0 M, g* = 0 R, (iv) ( t * , s*) are not T 8-optimal with respect to any s o c i a l welfare function W(u) = B u, B > 0„, i . e . , any vector X e R H + N + K + 1 t h a t s o l v e s X T[A, - B j > 0^ , tl b H+1 and X T [ B p , B z] = 0^ + R does not s a t i s f y X T[B t, B g] = 0^ + M- Then, there ft ft exists a s t r i c t Pareto and pr o d u c t i v i t y improving reduction of (t , s ) , i . e . , At <^  0 N, As _< 0^,without a change in the autarky t a r i f f s T . I f the autarky tax r a t e s ( t , s ) are nonpositive, there exists a s t r i c t ft ft Pareto and pr o d u c t i v i t y improving increase of ( t , s ), without a change ft in the i n i t i a l t a r i f f s x . - 162 -Proof: Suppose that t > 0.., s > 0,,. It s u f f i c e s to show that there i s — N — M „H+N+K+1 > T r . „ . v „T . T r „ . T no vector X e R such that X [A, - B, ] > 0 T T 1 1 , X [B , B ] = 0„ „, b H+l p z N+K T T X [ B ^ , B g ] >^  OJJ+M* Using ( 4 . 1 6 ) , the homogeneity of the expenditures functions, and assumption ( i i i ) , T T T T (11.1) XL[Bt, Bs] (q , V 1 ) ' = 0. T By assumption, (q,v) » 0 and the vector X [ B t > B ] i s nonzero. It T follows that the vector X [B , B ] must contain negative elements, i . e . , t s T T the 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 * * If t < 0„ and s < 0,,, i t s u f f i c e s to show that there i s no — N — M vector X e R H + N + K + 1 s u c h t h a t X T [ A , - B, ] > 0* , X T[B , B ] = 0 ^ „ , D rl+1 p Z N+tv T T X [B^, Bg ] _< 0ji+^« Using (11.1) and a s i m i l a r reasoning as above, T T the 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 . QED t s — N+M Theorem 11.1 implies that the government i n a small autarkic country, where lump sum transfers are not admissible, can find such a reduction of the i n i t i a l nonoptimal taxes (subsidies) on the commodities in net demand (supply) by the consumers that the country's i n i t i a l net balance of trade i s s t r i c t l y improved and every household in the economy i s made s t r i c t l y better of. Equally w e l l , i f the country's commodity - 163 -tax structure in autarky i s such that goods in net demand by the consumers are subsidized and the factors sold by the households are taxed, there exists some s t r i c t Pareto and 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 subsidies and taxes. The example, on which Melvin based his statement c i t e d in the beginning of this chapter, i s in fact c l o s e l y related to Theorem 11.1. Melvin considered a world consisting of two countries with i d e n t i c a l production p o s s i b i l i t y sets. In both countries, government expenditures are financed using lump sum transfers without commodity taxation ( i . e . , * * t = 0„, s = 0 W ) . Because the demand sides of the two economies are N M assumed to be s i m i l a r , the autarky e q u i l i b r i u m prices i n the two countries c o i n c i d e . It follows that the autarky t a r i f f s T* in each country equal zero. However, i t can be shown that the zero commodity tax rates ( t * , s*) in the two countries must be 8-optimal for the T ^ IL s o c i a l welfare function W(u) = 1 u i f x = 0 . Hence, assumption ( i v ) H M of Theorem 11.1 i s v i o l a t e d . Furthermore, using Proposition 4.2, i t can * * * be seen that when T = 0^ and the tax rates (t , s ) are B-optimal ( f o r 8 = l j j ) , no s t r i c t Pareto 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 commodity tax rates ( t , s ) and the t a r i f f s T e x i s t ; in Melvin's example, the consumers in the country which imposes a nonzero tax on a tradeable good are a c t u a l l y made worse of. - 164 -12. ECONOMIC INEQUALITY AND PUBLIC POLICIES The analysis i n the previous chapters has been e n t i r e l y concerned with the existence of s t r i c t Pareto improvements. A s t r i c t Pareto improvement i s , of course, also a s t r i c t welfare Improvement for any increasing s o c i a l welfare function, but i t could be argued, following Blackorby and Donaldson (1977: p. 374), that accepting Pareto improvements as s o c i a l welfare improvements may "allow too much inequali t y to creep into a s o c i a l arrangement." Consider Figure 9. Suppose that at the I n i t i a l equilibrium 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^) depicted by the point S i n the f i g u r e , and that the u t i l i t i e s u^ and are f u l l y comparable. 1 The point S' gives the symmetric d i s t r i b u t i o n of (u^, u 2) with respect to the equal u t i l i t y l i n e u^ = u 2» Sta r t i n g from S, a s t r i c t Pareto improving change i n the government p o l i c y instruments s h i f t s the consumers to a new u t i l i t y ( r e a l income) d i s t r i b u t i o n point i n the cone ASB. This p o l i c y change i s also ( r e l a t i v e ) i n e q u a l i t y reducing, i f the perturbation causes the consumers 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 increasing, i f the households are being s h i f t e d i n the cone CSB. A p o l i c y change that causes a s h i f t i n the consumer u t i l i t i e s i n the cone S'SA i s ( r e l a t i v e ) i n e q u a l i t y reducing, but not Pareto improving. The primary p o l i c y goal of the government, i n th i s chapter, i s assumed to be to reduce 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 i n the home country. The aim of th i s chapter i s to operationalize the concept - 165 -F i g u r e 9 - P a r e t o Improving and I n e q u a l i t y R e ducing P o l i c y P e r t u r b a t i o n s . - 166 -of economic I n e q u a l i t y , i . e . , to d e f i n e i t i n a way that allows the a n a l y s i s of p r a c t i c a l p o l i c y questions: .when do i n e q u a l i t y reducing commodity tax and 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 ever, are simple p o l i c i e s (e.g., p r o p o r t i o n a l reductions 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 that each household h, h=l,...,H, i n the economy possesses nonnegative endowments of domestic —h —h and i n t e r n a t i o n a l l y tradeable commodities, denoted by c 0> 0 N) and d (>_ 0^) r e s p e c t i v e l y . I t i s a l s o assumed that i f , f o r example, the nth domestic 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*1 i n h i s endowment v e c t o r c*1 i s equal to zero. The reasons f o r the n l a t t e r assumption are twofold: 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 that i t needs i n order to tax t h e i r a b i l i t i e s and secondly, i f taxes on the consumers' labor endowments are allowed, i t i s p o s s i b l e that the i n d i v i d u a l s i n the s o c i e t y are forced to donate t h e i r time to the government—an outcome that corresponds to s l a v e r y and i s obviously o i n c e n t i v e i n compatible. The 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 : h * * * *h *T —h *T —h (12.1) m (u^, < 3 » v ) = g + q c + v d , h=l,...,H, - 167 -(12.2) E V m h(u* q*, v*) + x° = h=l q h ^ k * * * k ^ _ r ) E V TT (p , W + X ) Z + E C , k=l P h=l (12.3) Tr k(p*, w + T*) = 0, k=l,...,K, T • h * * * T n (12.4) w E V m (u , q , v ) + w e = h=l V T V * * * k T —h * E w V Tr (p , w + x ) z + w E d - b . k=l W h=l Let us denote the matrices of consumer endowments by (12.5) C = [ c 1 , . . . , ^ ] ; D E [ d 1 , . . . , ^ ] . D i f f e r e n t i a t i o n of the model (12.1) - (12.4) at the i n i t i a l e quilibrium y i e l d s : (12.6) AAu* = B Ap* + B Az* + B,Ab* + B At* + B As* + B Ax*, p ^ z b t s x • where - 168 -A = — — I H , B = I p qu °KxH W L vu T —T -X + C -Z + S qq P P T Y T T -w Z + w S vq wp B = t T —T -X + C -Z qq KxN T r -w vq B = T —T -E + D -Z qv °KxN T „ -w vv > B T —T -E + D -Z + S qv pw T T -w Z + w S vv WW The matrices B and B, i n (12.6) are those defined in (4.16). z b It i s assumed that the government agrees on the form of a s o c i a l welfare function W(u); the gradient of this function at the i n i t i a l e q uilibrium i s denoted by (12.7) 8 = V u W ( u ). Define the household average r e a l income u ,at the i n i t i a l e q u ilibrium as _* i h * * * (12.8) u = rr Z m (u, , q , v ). H h=l ^ - 169 -Next, define the Kolm (1969) equal equivalent of the r e a l income vector * * * * \i_ as the s c a l a r K(u ) which s a t i s f i e s W(K(u )1^) = w ^ u ^ ^ ' ^he s c a l a r K(u ) gives the amount of real income which, i f attained by each i n d i v i d u a l i n the economy, would give r i s e to the same l e v e l of s o c i a l ft welfare as the actual observed vector of r e a l income u . Assuming that the s o c i a l welfare function W(u) i s c a r d i n a l l y scaled, the equation (12.9) W(u*) = K(u*) i s s a t i s f i e d at the i n i t i a l e quilibrium. Hence, using the Kolm (1969) measure of i n j u s t i c e , a monetary measure of economic i n e q u a l i t y at the ft i n i t i a l e q u i l i b r i u m , I , can be defined: (12.10) I* = u* - W(u*). A reduction of economic i n e q u a l i t y i s defined to mean a reduction * * 5 i n the measure I ( A l < 0). When can such reductions be reached using the i n i t i a l commodity tax rates ( t * , s*) and the t a r i f f s x* as the government p o l i c y instruments? Let us rewrite the equation (12.10) i n * * _* an equivalent form -I = W(u ) - u . Then, i t can be seen that a reduction of economic i n e q u a l i t y i s equivalent to an increase i n the - 170 -i n i t i a l value of the s o c i a l welfare function W(u) = W(u) - u. What are _ * the s u f f i c i e n t conditions for an improvement in W(u ) to exist? The problem i s to determine the minimal s u f f i c i e n t conditions f o r : k k k k k k k (12.11) there e x i s t Au , Ap , Az , Ab , At , As , Ax such that (12.6) holds, 3 T Au > 0 and Ab >_ 0, where 8 T = JU cn rri np rx% 7 u W(u ) = (8 - 1 R/H), (B * Og). Theorem 12.1: Suppose that ( i ) rank Y = K <_ N, ( i i ) rank ( S p p + YY T) = N, ( i i i ) B T [ ( X T - C T ) , ( E T - D T)] * oJ. M, ( i v ) x* T V 2 G(w + x*, y*) * 0^ at the INTiM WW M i n i t i a l equilibrium. Then, there e x i s t s a s t r i c t welfare and economic equ a l i t y improving change i n the t a r i f f s x and commodity tax rates ( t * , s*) (without a change i n the lump sum transfers g*) which does not reduce the country's i n i t i a l net balance of trade. The change in t a r i f f s can be chosen to be a proportional reduction. Proof: Motzkin's Theorem y i e l d s an equivalent condition to (12.11): - 171 -(12.12) there must not exist a vector A e R h + n + k + 1 s u c h that X T [V V V B s ] = °N+K+N+M' X T B x " °M> X T A - ^> XTB,_ < 0. D T The i n e q u a l i t y X Bfa < 0 implies X^  _> 0. Set ^ = k > 0. As i n the proof T T of Theorem 2.1, the equations X [B , B , B . B ] = ()„,.,,„,„ can be used p z t s N+K+N+M T to solve the components X^ and X^  of the vector X when X^  = k: X^  = k(p + 6) , X^ => k y » where the vectors 6 and y a r e defined i n (2.27). If k = 0, X 2 = 0 N and X 3 = 0 R. Then, the equations i n (12.12) s i m p l i f y to (12.13) X 1 = B 1, - Xj [X 1 - C 1] = Oj, -Xj [E - D ] = 0*. By assumption, there i s no s o l u t i o n X^ = g T to (12.13). T x *T 2 Suppose k > 0. Then, the equations X B^ = 0^ are equivalent to x G(w + x , y ) = 0 ' w h e r e the matrix V G(w + x , y ) i s defined i n M ww J (2.12). By assumption, T * T V 2 G(w + x*, y*) * 0^, i . e . , (12.12) i s ww M s a t i s f i e d . Proposition 2.1 shows that Ax may be chosen to be a proportional reduction. QED - 172 -C o r o l l a r y 12.1.1: If rank [X T —T T - C , E D T] = H (_< N + M), condition ( i i i ) i n Theorem 12.1 i s s a t i s f i e d . Proof: If rank [X T —T T - C , E T N+M i s A., = 0. H* Hence, since 8 0 T H' assumption ( i i i ) i n Theorem 12.1 must be s a t i s f i e d . QED Theorem 12.1 shows that the s u f f i c i e n t conditions that guarantee the existence of improvements i n economic eq u a l i t y are very s i m i l a r to those implying the existence of s t r i c t Pareto improvements (when commodity taxes and t a r i f f s are the admissible government p o l i c y instruments): assumptions ( i ) , ( i i ) and ( i v ) i n Theorem 12.1 are s u f f i c i e n t f o r a s t r i c t p r o d u c t i v i t y improving perturbation i n the t a r i f f s x* to e x i s t , 5 and assumption ( i i i ) ensures that the gains accruing from a produ c t i v i t y improving perturbation of x can be d i s t r i b u t e d to the households i n a way that s h i f t s the d i s t r i b u t i o n of rea l income toward equality. Assumption ( i i i ) can also be given an i n t e r e s t i n g i n t e r p r e t a t i o n . Let us p a r t i t i o n the households i n the economy into three classes: those with a p o s i t i v e welfare weight 8 , ^h those with a zero welfare weight 8 , and those with a negative welfare A h / weight 8 . The f i r s t class of consumers i s c a l l e d "the poor" and the l a s t "the r i c h . " For assumption ( i i i ) i n Theorem 12.1 to be s a t i s f i e d , - 173 -i t i s s u f f i c i e n t to assume that there exists some good (domestic or i n t e r n a t i o n a l l y tradeable) i n (net) demand (or supply) by a l l "the poor" and i n (net) supply (or demand) by a l l "the r i c h . " 7 In other words, the preferences of "the r i c h " and "the poor" must s i g n i f i c a n t l y d i f f e r from each other at l e a s t i n the case of a commodity. Under this ft supposition, a proportional reduction of the t a r i f f s x can be made 8 economic i n e q u a l i t y reducing. Using Proposition 4.2, i t can be seen that s t r i c t welfare and ft economic equality improving d i r e c t i o n s of change i n the t a r i f f s x and X X commodity taxes (t , s ) cannot e x i s t , i f the i n i t i a l e q u ilibrium i s a 8-optimum with respect to the commodity tax rates ( t * , s*) and the * * gradient of the net balance of trade function, V b (w + x ), equals zero. One can thus conclude that zero t a r i f f s (free trade) are not only  Pareto but also " e q u a l i t y optimal" f o r a small country, i f the domestic  commodity tax rates are f r e e l y adjustable and the producer s u b s t i t u t i o n  matrix S i s of maximal rank ( = N + M - 1). If the government in the home country can adjust the i n i t i a l X vector of lump sum transfers g , i t can be shown that the assumptions of Theorem 8.3, which imply the existence of s t r i c t Pareto improving ft ft perturbations i n the t a r i f f s x and transfers g , are also s u f f i c i e n t f o r a s t r i c t welfare and equality Improving t a r i f f and 9 transfer change to e x i s t . - 174 -12.2 Existence of Inequality Reducing and Welfare Improving P o l i c y  Perturbations By comparing the assumptions of theorem 12.1 to the conditions of Theorem 8.3, which guarantee the existence of s t r i c t Pareto improving perturbations i n the commodity tax rates (t , s ) and the t a r i f f s x*, the conclusions drawn from Figure 9 can be confirmed: ( i ) a s t r i c t Pareto improvement ( i f i t e x i s t s ) does not n e c e s s a r i l y imply a s t r i c t reduction in economic i n e q u a l i t y , 1 0 and ( i i ) a s t r i c t i n e q u a l i t y reduction can e x i s t even i f a s t r i c t Pareto improvement does n o t . 1 1 A A A When can the government f i n d changes i n (t , s , x ) that are both  Pareto and e q u a l i t y improving? And when are commodity tax and t a r i f f perturbations welfare improving f o r both s o c i a l welfare functions W(u) and W(u)? Theorem 12.2: Suppose that ( i ) rank Y = K _< N, ( i i ) rank ( S p p + YY ) = N, ( i i i ) x* T V 2 G(w + T* y*) * Cv at the i n i t i a l e q uilibrium, ww J M I. Then, i f B T = V W(u*) = (B T - lJ/H) and u H (12.14) there i s no solution a > C>H to a T [X T - C T, E T - D T] = ojJ+M> and - 175 -(12.15) there i s no solution r > 0^ to ( r T + ^ T ) [ X T - C T, E T - D T ] = 0 ^ + M , there e x i s t s a s t r i c t Pareto and eq u a l i t y improving change i n the ft ft ft i n i t i a l commodity tax rates (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 vector of lump sum transfers g ). The po l i c y perturbation does not reduce the l e v e l of the economy's i n i t i a l net ft balance of trade b . II . I f (12.16) there i s no solu t i o n r > 0 to (r g T + 6 ) [X T - C T , E - D T] = 0 ^ + M , and (12.17) sHx 1 - c \ E 1 - V ] * 0TmM, 6V- C T , E T - D T 1 *0l+M, there e x i s t s a s t r i c t welfare and eq u a l i t y improving perturbation of the A A A i n i t i a l commodity tax rates (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 vector of lump sum transfers g )• The change does not reduce the l e v e l of the economy's i n i t i a l net balance of trade - 176 -Proof: (I) A s u f f i c i e n t condition for a Pareto and equality improving change in ( t * , s*) and x* to e x i s t i s : JU JL JU JU JL yu ju (12.18) there exist Au , Ap , Az , Ab , At , As , Ax such that (4.16) Jf JL *Srp Jf i s s a t i s f i e d , Ab > 0, Au » 0„ and tr Au > 0. — H Using Motzkin's Theorem, an equivalent condition for (12.18) can be derived: 1 T (12.19) there i s no vector X e R such that X [B , B , B„, B , p z ' t ' s ' V - ° N + K + N + M + M ' X T B b < °« A ? A - V l + 7 2 ? ' < V V > <W T Using X B^ _< 0, X^ = k _> 0. Then, as shown i n the proof of Theorem 2.1, X^ = k(p* + 6 ) T , X^ = k Y T« I f k = 0, X 2 = 0 N and X 3 = 0 R. I t T T follows that X A = X^. For (12.19) to be s a t i s f i e d , there must not e x i s t a s o l u t i o n to rp rp rp ^rp rp rri rp rri (12.20) x{ = 7f + V ' g 1 , - xj[x - c \ E - TJ ] = o N + M , ( ^ , 7 2) > o R + 1 . Since (v , v ) > 0 , (12.20) i s equivalent to 1 2. H*r1 - 177 -(12.21) X^ = v* (> 0 R) or X* = v* + 8 T, V l > 0 R, and Using assumptions (12.14) - (12.15) and (12.21), i t can be seen that there is no sol u t i o n to (12.20). If k > 0, choose k - 1. Then, XTB = x* T V 2 G(w + T * , y*) * o3 by assumption. Hence (12.19) i s s a t i s f i e d . (II) Now i t is required that * * * * * * * (12.22) there e x i s t Au , Ap , Az , Ab , At , As , A T such that yu ^ rp! JU rfl yu (4.16) i s s a t i s f i e d and Ab > 0, 3 Au > 0, 3 Au > O. Equi v a l e n t l y , (12.23) there must not e x i s t a vector X e R H + N + K + 1 such that XT[V V. V V V = ° N + K + N + M + M > X \ < ° - ^ = v x 8 + v 2 8 , ( v x , v 2 ) > 0 2. Proceeding as above, X <^  0 implies ^ = k > 0. T T If k = 0, X 2 = 0 N and X 3 = 0K» Then, X A = ^ . For (12.23) to be s a t i s f i e d , there must not e x i s t a s o l u t i o n to - 178 -"TP ^rp rp rp rp rp rp rp rp (12.24) A| = v ^ 1 + v ^ 1 , - A j t X 1 - C 1, E - D j = 0 * + M > ( v j , > 0,,. Since (v^, v 2 ) > 0 2, (12.24) i s equivalent to (12.25) • Aj = 6 T + v 2 g T , v 2 _> 0 or A^ = V j B T + 3 T, > 0, and rp rp rp rp rp rp -A* [X T - C T, E - T) ] = 0 j + M . Using assumptions (12.16) - (12.17) and (12.25), i t can be seen that there i s no s o l u t i o n to (12.24). X T *T J2 I f k > 0. choose k = 1. The equations A B = 0 W s i m p l i f y to x v x M ww * A X G(w + x , y ) = 0^. Using assumption ( i i i ) , these equations must be v i o l a t e d . QED C o r o l l a r y 12.2.1: I f rank [ X T - C T, E T - D T ] = HC< N+M), c o n d i t i o n s (12.21) -(12.22) of Theorem 12.2 are s a t i s f i e d . I f rank [ X T - f J T , E T - D T ] = H (<_ N+M) and 8 > 0 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: I f rank [ X T - C T, E T - D j = H (<_ N+M), the only s o l u t i o n f o r the equations A T [ X T - E T - D 1 ] = i s A = 0 . Hence, (12.14) i s - 179 -s a t i s f i e d . Then, since 3 * - r , r > 0„, the equations ( r + 3 ) H [ X T - C T, E T - D j = 0^ + M have no s o l u t i o n r > 0 . ( I f E B h = 1, h= 1 not a l l 0 h (= B h - 1/H) can be negative.) I f rank [ X T - C T, E T - D T] = 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 rp Furthermore, i f rank [X - C , E - D ] = H and 6 > 0Tjr, c o n d i t i o n (12.16) must be s a t i s f i e d ( s i n c e the v e c t o r ( r B + B) cannot be zer o ) . QED Assumption (12.14) i n P a r t I of Theorem 12.2 i s the g e n e r a l i z e d Diamond-Mirrlees c o n d i t i o n that i s needed f o r a Pareto improving d i r e c t i o n of change i n the i n i t i a l commodity tax and t a r i f f to e x i s t . C o n d i t i o n (12.15) guarantees that a Pareto Improving d i r e c t i o n of change i n the i n i t i a l commodity tax and t a r i f f rates can be made economic i n e q u a l i t y reducing. C o r o l l a r y 12.2.1 i m p l i e s that the rank c o n d i t i o n t h a t , according 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 f o r e q u a l i t y improvements to e x i s t , i s a l s o s u f f i c i e n t to ensure the existence of a ft ft ft s t r i c t Pareto and e q u a l i t y improving p e r t u r b a t i o n i n ( t , s , T ) . Condition (12.17) i n Part I I of Theorem 12.2 corresponds to assumption ( i i i ) i n Theorem 12.1. C o n d i t i o n (12.16) imposes a r e s t r i c t i o n on the welfare weights B (which are determined by the * gradient V^VKu ) ) . C o r o l l a r y 12.2.1 shows that (12.16) i s s a t i s f i e d at l e a s t i f 3 > 0„ and the matrix [ X T - CT, E T - IF 1] i s of f u l l (row) rank. - 180 -13. CONCLUSIONS One of the famous problems in the area of i n t e r n a t i o n a l trade theory has been the question of the gains from trade: i s i t possible for a l l the consumers i n an autarkic country to benefit i f the country i s opened up for i n t e r n a t i o n a l trade and i f so, under what conditions does this occur? It i s well-known that free i n t e r n a t i o n a l trade enlargens (or at l e a s t , does not reduce) the f e a s i b l e consumption p o s s i b i l i t y set for the consumers i n a previously autarkic country. Hence, i f the government i n the home country has lump sum transfer instruments at i t s d i s p o s a l , i t can ensure that every household i n the economy w i l l be better of (or at l e a s t , not worse of) under free i n t e r n a t i o n a l trade than i n autarky. But what i f lump sum r e d i s t r i b u t i o n of Income i s inadmissible? Is i t s t i l l possible to show that a l l consumers i n an autarkic country can benefit from free i n t e r n a t i o n a l trade? D i x i t and Norman (1980) i n t h e i r textbook provided a r e s u l t according to which the autarky welfare 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 under free trade i f the government can f r e e l y adjust the country's commodity tax structure. However, D i x i t and Norman did not show that s t r i c t gains from trade would occur, i . e . , that the s h i f t from autarky to free trade would produce a s t r i c t Pareto improvement. This thesis started as an attempt to find the s u f f i c i e n t conditions for s t r i c t gains from trade to e x i s t when only commodity taxation i s used to influence the d i s t r i b u t i o n of income. The problem i s approached as a p o l i c y reform question: when can the government find - 181 -s t r i c t Pareto improving ( 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 country'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 autarky t a r i f f s are defined to be such that i n t e r n a t i o n a l trade i s j u s t being p r o h i b i t e d ? It, turns out that under some weak conditions ( e s t a b l i s h e d i n Chapter 2 of the t h e s i s ) on the production technologies of the domestic production 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 there e x i s t such changes i n the economy's t a r i f f v e c t o r that the amount of f o r e i g n exchange earned by the domestic producers i s increased. 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 to be a p r o p o r t i o n a l r e d u c t i o n . In Chapter 4 of the t h e s i s the question of r e d i s t r i b u t i n g the increase i n the amount of f o r e i g n exchange earned by the domestic producers to the consumers i s considered. I t turns out that a c o n d i t i o n on the preferences and i n i t i a l endowments of the households has to be s a t i s f i e d ; 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 Diamond-Mirrleess assumptions that a good i n net demand or supply by a l l households e x i s t s . I t can be shown that i f the above mentioned c o n d i t i o n on the consumer preferences 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 domestic production technologies are s a t i s f i e d , then s t r i c t Pareto improving p e r t u r b a t i o n s i n the country'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 rates e x i s t . In 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 autarky 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 gains from trade to e x i s t . In Chapter 3 of the t h e s i s the problem of approximating the s i z e of the p r o d u c t i v i t y gain accruing from a p r o d u c t i v i t y improving t a r i f f - 182 -change i s analyzed. The measures developed i n t h i s chapter a l s o provide production side approximations f o r the s i z e of the gain from trade when the i n i t i a l l y trade p r o h i b i t i v e t a r i f f s are being perturbed. Having e s t a b l i s h e d the exi s t e n c e of s t r i c t Pareto improving t a r i f f and commodity tax p e r t u r b a t i o n s , the next question i s : what kind of examples of these p o l i c y changes can be found? In Chapter 7 of the t h e s i s i t i s shown t h a t , f o r example, reductions of p o s i t i v e t a r i f f s and increases of negative t a r i f f s , uniform reductions of t a r i f f s and changes toward u n i f o r m i t y i n the country's t a r i f f s t r u c t u r e can be s t r i c t Pareto improving i f the country's i n i t i a l commodity tax rates can be f r e e l y v a r i e d . In Chapter 8, lump sura t r a n s f e r s are 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 Pareto 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 to e x i s t are developed. These production side c o n d i t i o n s are a l s o s u f f i c i e n t f o r s t r i c t gains from trade to e x i s t i f lump sum t r a n s f e r s can be used to r e d i s t r i b u t e consumer income. In Chapter 9, the existence 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 instruments are compared. An i n t e r e s t i n g p r o p o s i t i o n derived i n t h i s s e c t i o n i s that s t r i c t gains from trade need not e x i s t under lump sura compensation even i f s t r i c t gains under commodity t a x a t i o n would be p o s s i b l e . Chapter 10 contains some examples of s t r i c t Pareto improving commodity tax, 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 : these include changes toward i n t e r n a t i o n a l p r i c e s i n t a r i f f s , simultaneous p r o p o r t i o n a l reductions 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 taxes. The r e s u l t s concerning 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 Hatta (1977a) r e s u l t according to which a reduction i n the highest commodity tax rate i n a closed one consumer economy (where lump sum t r a n s f e r s are admissible) i s welfare improving. In the l a s t chapter of the t h e s i s the government i s assumed to search f o r p o l i c y reforms that improve economic e q u a l i t y i n the country. In order to consider 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 reduction i n economic i n e q u a l i t y i s defined to correspond to an improvement i n an e s p e c i a l l y defined s o c i a l " w e l f a r e f u n c t i o n . One of the c o n d i t i o n s f o r i n e q u a l i t y reducing commodity tax and t a r i f f changes to e x i s t turns out to be a c o n d i t i o n on the consumer preferences: there must be a good with respect to which "the r i c h " and "the poor" are on the opposite sides of the (net) market. In the f u t u r e , the same methods that, i n t h i s t h e s i s , have been used to analyze p o l i c y reforms i n a s m a l l open economy could be a p p l i e d to study m u l t i c o u n t r y t a r i f f and other p o l i c y agreements. One might a l s o want to develop (production side) approximative measures f o r the  gains accruing to the c o u n t r i e s that agree to implement b e n e f i c i a l p o l i c y reforms. Another 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 of optimal tax reforms, imperfect i n f o r m a t i o n and  u n c e r t a i n t y . In t h i s case, the goal 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 (or a firm) a c t i n g under u n c e r t a i n t y about the other economic agents' goals and c h a r a c t e r i s t i c s . Of course, one could 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 gains from the trade to e x i s t under these circumstances. In a many country case, one could compare p o l i c i e s (and the gains accruing from them) under imperfect 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. If some of the production sectors exhibit diminishing returns to scale, new domestic commodities that correspond to the ownership shares of these sectors are added to the model. The new factors absorb the pure p r o f i t s earned by the diminishing returns to scale s e c t o r s . 2. The technologies of the production i n d u s t r i e s are described using t h e i r unit production p o s s i b i l i t y sets since the technologies of the sectors are assumed to exhibit constant returns to scale. I f the t o t a l production p o s s i b i l i t y sets of the industries were used, the s e c t o r a l t o t a l p r o f i t functions would assume only two values, zero and i n f i n i t y . Each p r o f i t function would thus be discontinuous and n o n d i f f e r e n t i a b l e . 3. A l l vectors i n this thesis are defined as column vectors; x^ denotes the transpose (row) vector of x. 4. There are several ways of d e f i n i n g the scale of a production sector. If the sector k, k=l,...,K, produces only one output, the scale of the sector k i s the amount of output produced i n that sector i n each time period. If j o i n t production i s present i n sector k, the scale of the sector k, k=l,...,K, can be defined as i n Ch. 2 by using an always needed input or a l t e r n a t i v e l y by employing units of value added (Woodland (1982 : p. 135)). 5. For unit p r o f i t functions and d u a l i t y , see Diewert and Woodland (1977: pp. 377-378). 6. If some sectors produce i n t e r n a t i o n a l l y traded good m, m e [1,...,M], while other sectors u t i l i z e good m as an input, i t may be necessary to redefine good m as two separate commodities: an input good and an output good. I t i s assumed that each technology set C k, k=l,...,K i s such that good m i s e i t h e r produced or used as an input, 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 (or not u t i l i z e d at a l l ) by each sector or used as an input by each sector. 7. Notation: O^ +M denotes an (N+M) - vector of zeroes. 8. If a l l production sectors have a common always used input (or an always produced output), the matrix S w i l l have a zero row and column corresponding to this commonly applied input. Then, the rank of the matrix S i s at most N+M-2. 9. The constant returns to scale assumption together with competitive p r o f i t maximization implies that a l l production sectors earn zero pure p r o f i t s i n equilibrium. I f the equilibrium 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 sector, the p r o f i t s of this sector could be driven to i n f i n i t e l y simply by increasing the industry's scale (since the sector's technology i s CRS). Hence, the zero p r o f i t conditions (2.9) constrain the equilibrium to the f i n i t e scale case. 10. Notation: Ap* denotes an i n f i n i t e s i m a l change i n p*, usually denoted by dp*. O ^ K i s a n (KxK)-matrix of zeroes. 11. The properties of the GNP function are given i n Woodland (1982: Ch. 3.7). 12. A s p e c i a l case of problem (2.15) i s presented i n Woodland (1982: Ch. 3). There, i t i s assumed that each production sector supplies only one tradeable good using domestic commodities as inputs. In this case, assuming that the sector k, k=l,...,K, produces the th k k tradeable, sector k's. unit p r o f i t function i s TT (p, w + x) = (w^ + x^) - c^(p), where c^(p) i s sector k's unit cost for th producing the k tradeable. The problem (2.15) can then be interpreted as minimizing the input cost under the constraint that th the producer price f o r the k tradeable, (w^ + x^), does not exceed the sector k's unit cost c^(p). 13. The net input endowment vector -y i s drawn i n the diagram taking p° as the o r i g i n . 14. Convexity of the unit p r o f i t functions implies convexity of the unit p r o f i t l e v e l curves -ir^(p, w + x ) = 0. 15. A l l the three production sectors can stay operative i n s p i t e of the change i n (w + x*) i f the change i s very s p e c i a l , i . e . , i t i s necessary that a l l the unit p r o f i t l e v e l curves corresponding to the new producer prices f o r tradeables i n t e r s e c t at p'. T 16. The matrix S p p + YY i s p o s i t i v e d e f i n i t e , for example, ( i ) i f S = 0.. „ and N = K ( i f rank Y = K(= N), the matrix YY T i s pp NxN * po 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 implies that rank S p p = N. Proof f o r ( i i ) : suppose on the contrary that rank S = N-l when rank S = N+M-l. Then, there e x i s t s x e RN, PP ' rri rn rri rti rri x * 0XT, such that x S = 0„. Choose y = (x , 0„). I t follows N pp N J M T that y Sy = 0; hence, y i s a zero eigenvector of the p o s i t i v e k k semidefinite S. Because both the vectors (p , w + x ) and (x, Oy) are zero eigenvectors of the matrix S (and the vectors are - 186 -x M not l i n e a r l y dependent since (w + T ) £ R ), rank S i s at most T T N+M-2, which contradicts the assumption that rank S = N+M-l. 17. Note that the vector y can be regarded as a net domestic demand vector. 18. I f rank S = N+M-l, then rank S p p = N and (2.13) i s s a t i s f i e d . 19. Note also that c o n t r o l l a b i l i t y of production i n the sense of Guesnerie and Weymark implies c o n t r o l l a b i l i t y of domestic goods production i n the sense of D e f i n i t i o n 2.2, but not conversely. 20. The vector of domestic goods prices p* appropriate i n (2.18) i s the one that solves problem (2.15). This p* i s the shadow price vector corresponding to the domestic goods constraint y • x x T x 21. I f V b (w + x ) = 0 W, one cannot determine i f the function b x M * (w + x ) is increasing, decreasing or stationary. Yet, i f x x rr* x b (w + x ) = 0 , b cannot be s t r i c t l y increasing i n i t s argument at (w + x ). 9 k k 22. If the matrix G(w + x , y ) i s a zero (MxM)-matrix, the condition (2.16) i s v i o l a t e d . Under these circumstances, no i n f i n i t e s i m a l change i n x can change the production sectors' aggregate production choice ( f o r tradeables); even i f pr 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 tradeables supply (given a fixed y ) would e x i s t , they could- not be attained through d i f f e r e n t i a l perturbations i n the prices (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^. When x e R, x » 0 i s equivalent to x > 0. 24. Note that the existence of a p r o d u c t i v i t y improving t a r i f f change is equivalent to the existence of an A l l a i s production gain. (Diewert (1983c) defines the A l l a i s production loss as the extra amount of foreign exchange the producers could earn by having an optimal i n t e r n a l rearrangement of production instead of one disturbed by the existence of d i s t o r t i o n a r y taxes and t a r i f f s . ) Another possible way of de f i n i n g a p r o d u c t i v i t y improving change i n t a r i f f s x - 187 -JU J ^ J ^ y^  yu i s to require that there e x i s t Ap , Az , Ab , Ax , Ay such that B Ap* + B Az* + B, Ab* + B Ay* = B Ax* and Ab* > 0, Ay* > 0„, p z b y J T ' ^ N T where B^ = [ - T - N > ^ ^ K ' * This approach corresponds to the Debreu measure of p r o d u c t i v i t y loss defined i n Diewert (1983c). 25. Motzkin's Theorem: Let , and Q 3 be given matrices with * 0. Then, e i t h e r ( i ) Q^p » 0, Q2P >^  0, Q 3p = 0 has a so l u t i o n or ( i i ) + a 2Q 2 + = 0 T, > 0, a.^ !> 0, has a solut i o n , but never both. Motzkin's Theorem and other theorems of al t e r n a t i v e are discussed i n Mangasarian (1969: pp. 17-37). In order to show the equivalence of (2.25) and (2.29), choose P T - [Ap* T, Az* T, Ab* T, Ax* T], Q l - [ 0 l x N , 0 l x K , 1, 0 ^ ] , Q 3 -[B , B , B, , -B ]. Then, there must not e x i s t a, e R , a, e R N + K + 1 p z ' b x ' 1 ' j T T such that CJQJ^ + "30.3 = °N+K+1+M' a l ^  ®' E q u i v a l e n t l y , there must not e x i s t , « 3 such that <^[B p, B z, - B T ] = oJj+K+1+M, c ^ B B = ~a^< 0. Choose X = a 3 and (2.29) follows. 26. Choose p T = [Ap* T, Az* T, Ab* T, Ax* T, r] , Q u = [ 0 l x ( N + R ) , 1, °lxN' ° ] ' Q12 - (0LK+1+M> 1 ] ' Q31 " [ B p ' B z ' V "V °(N+K+1) x l 1 ' Q32 = [°Mx(N+K+l)' V ^ U T h e n' t h e r e m u S t n 0 t e x i s t a 1 e R N + K + 1 , a 2 e R , 0 ^ e R M , e R such that oc2 > 0, a 4 > 0, and oc*[B p, B j = 0 ^ , a\ + <*2 = 0, - a ^ + a* = (£, X * a 3x + = 0. Equivalently, there must riot exist vectors a^, i=l,...,4, such that o£[B , B Z ] = 0^ + K, = - 0 ^ < 0, otj^ B^  = X X * N+K+l a„, a~x = -a. < 0. Hence, there must, not exi s t a, e R 3 3 4 — 1 such that aF[B , B ] = o3,„ , < 0, OTB X * < 0. Take X T = txT. 1 p z N+K ' l b ' I T — 1 - 188 -27. Note that an increase in a negative x , m e [1,...,M], i s a decrease in the magnitude of x . m ft ft m 28. V b (w + x ) * 0„ i s also one of the s u f f i c i e n t conditions for 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 3: 1. See Diewert (1983c) for more d e t a i l s . *k *k *k 2. The i n i t i a l e quilibrium s o l u t i o n (y , f , z ), k=l,..., K, i s f e a s i b l e for the problem (3.1) but not n e c e s s a r i l y optimal. 3. The i m p l i c i t functions e x i s t i f ( i ) rank Y = K _< N and ( i i ) S p p + YY^ i s p o s i t i v e d e f i n i t e . 4. A l t e r n a t i v e l y , A(O gives the net value of i n t e r n a t i o n a l l y traded goods produced by the entire production sector, when the goods are valued at the i n t e r n a t i o n a l prices w. 5. Diewert (1983c: pp. 169-170). 6. If 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 function 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 for a small f i n i t e change of t a r i f f s i n this d i r e c t i o n , the net balance of trade function must be increasing. 7. The formula (3.23) contains an abuse of notation. The term A'(2) refers to the derivative of the function A(£) at 5 = 1, where the value C = 1 corresponds to the new ( a f t e r the t a r i f f change) equilibrium, previously indexed as the equilibrium 2. 8. See Diewert (1983c): pp. 169-170. Set 5 = 1 . Using formulae (28) and (30) in Diewert (1983c), the expression (3.25) follows. Note that (28), i n Diewert (1983c), contains a typo: the next to k T k l a s t term i n (28) should be x y (?) z£ ( ? ) . 9. The same abuse of notation as encountered in formula (3.23) i s present here. See footnote 7. 10. The government might be able to give estimates f o r the net output matrices Y^ and that determine the price d e r i v a t i v e s p'(2). - 189 -11. Note that i f N = K, p ' ( l ) T = x 1 T F ^ Y 1 1 ) l . 12. Note that i f S = 0XT „, also A„ = (1 - k 2 ) T 1 T S 1 T 1 > 0. pw NxM G ww — Chapter 4: 1. Consumer preferences are assumed to be nonsatiated so that (4.10) holds as an e q u a l i t y . 2. If some public goods are inputs into the production process, the possible nonzero pure p r o f i t s generated by these factors can be imputed to domestic factors of production created for this purpose. & £ & & 3. The i m p l i c i t functions u , p , z , and b exist i f the matrix [A, - B , -B , - B, ] i s i n v e r t i b l e . p z b 4. Note that i n order to derive (4.16), the order of the equations (4.11) and (4.12) has been changed. ft 5. The Pareto improvement i s required to be s t r i c t (Au » 0 instead ti ft of Au > OJJ) i n order to avoid problems with actual ( f i n i t e ) changes i n u t i l i t i e s , as explained i n Diewert (1978: pp. 154-155). Assuming that ( i ) the i m p l i c i t functions that determine the endogenous variables as functions of the exogenous variables of the model ex i s t i n the neighborhood of the i n i t i a l equilibrium, and that ( i i ) the changes i n the exogenous variables have been normalized by requiring the sum of the changes squared to be one, ft ft ft ft the changes Au , Ap , Az and Ab may be interpreted 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 functions u , p , z and b given w, * * * * *h x , t , s and g . If for some consumer h, h e [1,...,H], 9u ft ft ft ft ft ft (w , T , t , s , g ) / 3 t n = 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 rate t i s perturbed from i t s from i t s i n i t i a l equilibrium value by a small f i n i t e amount. *h, * *h However, i f 3u /3t > 0, the function u i s increasing i n t at n n * t . n 6. A condition s u f f i c i e n t to imply that there i s no s o l u t i o n a > 0^ to a T [ X T , E T] = 0 ^ + M i s to require that rank [X T, E T] = H. This assumption means that the number of domestic commodities (N) plus the number of tradeable goods (M) must be equal or greater 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 Pareto improvements, the government must have a s u f f i c i e n t l y large number of free tax instruments (commodity taxes) i n i t s d i s p o s a l . 7. Choose p T = [Ap* T, Az* T, Ab*, A t * T , As* T, A T * T ] , Q . . = [ I u , T T °Hx (N+K+l+N+M+M) ] ' Q12 = [°H+ N + K ' ° N + M + M ] * Q3 = [ A ' "V ~ B z ' -B,, -B„, -B , -B 1. There must not ex i s t a, e R^+N+K+I ^H b t s x 1 2 ' a £ R such that of A + C L = 0?;, a?[-B , -B , -B , -B , -B ] = J 1 z H l p z t s x O-NVK+N+M+M' " a l B b + « 3 = 0» « 2 > 0 H, a 3 > 0. Choose X T = - c £ a n d (4.18) follows. .8. By changing the producer p r i c e s , the government can induce a change i n production that corresponds to the change i n consumer demand caused by a Pareto improving tax change. 9. This i s because a l l the commodity tax rates ( t * , s*) can be adjusted i n Theorem 4.1. If some producer price (w + x ) , m m * m e [ 1 , . . . , M ] i s changed, the tax rate can be perturbed so that the e f f e c t of the t a r i f f change on the consumers i s zero. This kind of separation of production and consumption sectors 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 (net) supplies by maximizing u t i l i t y and p r o f i t s , r e s p e c t i v e l y . The government can influence these (net) demands and (net) supplies i n d i r e c t l y through changes i n r e l a t i v e prices obtained 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' (net) demand vectors or the producers' (net) supplies. 11. The equilibrium that solves (4.26) i s usually a second best equilibrium, since the lump sum transfes g have not been chosen optimally. Only by a c c i d e n t — - i f g happened to be fixed at the optimal l e v e l — c o u l d the equilibrium be a f i r s t best optimum. 12. See Section 2.4. 13. See (2.37) in Section 2.4. 14. If N = 0, c o n t r o l l a b i l i t y of domestic goods production cannot cause problems and assumption ( i i ) i s thus not needed. If 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 supplies cannot occur i f the unit p r o f i t functions T T ^ ( W + T * ) , k=l,...,K, are twice continuously d i f f e r e n t i a b l e . Hence, assumption ( i ) i s not needed. - 191 -15. This i s a version of the McKenzie Factor P r i c e E q u a l i z a t i o n Theorem; McKenzie (1955). Chapter 5: 1. The prices w3 are obtained by so l v i n g the following autarky general equilibrium model: ( i ) mh ( u h , q, v) = g h , h=l,...,H ( i i ) 7rk (p, w + x) = 0, k=l,...,K ( i i i ) EV v m*1 (u^ 1, q, v) + e^ = EV w TTK (p, w + x) z k ( i v ) EV^ m h ( u h , q, v) + x° = IV p i r k (p, w + x) z k (v) x = 0 M ( v i ) Wj = 1 (pr i c e normalization). 2. Note that the t a r i f f s x* do not generate income to the government. Hence, the autarky government budget constraint i s s a t i s f i e d . 3. Note that i n order to reach p o s i t i v e gains from trade the t a r i f f s x need not be reduced to zero ( f r e e trade). ie ie 4. ( t , s ) are not g-optimal, i f they do not solve the problem max {B Tu: (4.10) - (4.13) hold, x* = (w a - w), g* = u,p,z,b,t,s constant}. 5. Welfare e f f e c t s of i n t e r n a t i o n a l trade that i s caused by changes i n the country's commodity tax structure are further discussed i n Chapter 11. 6. This change involves a series of i n f i n i t e s i m a l changes of x toward x* = Oj»i. 7. F i n i t e s t r i c t Pareto 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 exist at A. Chapter 6: 1. Note that the vector 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 producers. ft 2. A necessary condition for x to be Pareto and p r o d u c t i v i t y optimal i s that the vector X^B_ i s zero; see Proposition 4.2. - 192 -3 - I F 1 = ° ( N + M ) X ( N + M ) ' T H E E ^ A T L O N S * \ - °l i»piy -XIXT = <>£•. There I s no p o s i t i v e s o l u t i o n to these e q u a t i o n s , but choose rri , * T T * T T A, = 0„. Then, A A = (p + 5) E + w E = p £ + w E i f 1 H qu v u qu v u T =0... I t i s known t h a t p E + w E = 1„ (money m e t r i c M r qu vu H J s c a l i n g of u t i l i t i e s ) . Hence, f o r t h i s A, A TA > 0 R , X T [ B p , B z , B t ] rr% rpi rrt JL^rri JU JU r^i = ' ° M 4 . V 4 . M . ~X B K ^ 0 and A B = T V G(w + T , y ) = oA s i n c e N+K+N b x ww M * T = 0 M« T h i s i s s u f f i c i e n t to i m p l y t h a t no ( d i f f e r e n t i a l ) 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 e x i s t , s t a r t i n g f r o m the i n i t i a l e q u i l i b r i u m , i f T = C>M and E = ° ( N + M ) x ( N + M ) • T h e i n i t i a l 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 , the 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 , under f r e e t r a d e , the amount of f o r e i g n exchange earned by the p r o d u c t i o n s e c t o r i s maximal (see 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 ze r o m a t r i x , e x i s t e n c e o f non z e r o commodity t a x e s does not i n f l u e n c e the consumer net demand y i n ( 3 . 3 ) . 4. See Theorem 6.1. A s u f f i c i e n t c o n d i t i o n i m p l y i n g a s s u m p t i o n ( i i i ) i n Theorem 6.1 i s t h a t rank X T = H. T h i s means t h a t H _< N, i . e . , th e number of v a r i a b l e tax i n s t r u m e n t s must be at l e a s t as l a r g e as the number o f consumers ( o r consumer g r o u p s ) i n the economy. 5. 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 p r o d u c t i v i t y and it it improvement i n x and t , n e [ 1 , . . . , N ] , t o e x i s t i s : t h e r e i s no A e R H + N + K + 1 s u c h t h a t x T[A, - B. ] > oL, , A T [ B , B , (B ) ] ' b H+l ' p z' t *n = oJ.Vj.. , A TB = oj. N+K+l' T M C h a p t e r 7: 1. I f a good m, m e [1,...,M], i s a net e x p o r t f o r s e c t o r k, k=l,... , K , it it T > 0 ( T < 0) means t h a t p r o d u c t i o n of the good i s s u b s i d i z e d m — m — ( t a x e d ) , whereas i f the good i s a net 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 I s b e i n g t a x e d ( s u b s i d i z e d ) . JL JU 2. C o n s i d e r an 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 improvement t o e x i s t i s : - 193 -there does not exi s t a vector A e R such that A [A, - B b] > °H+1' X l t B p > V V V " °N+K+N+M' X \ < °M' Using the same reasoning as i n the proof of Theorem 7.1, this condition can be shown to be s a t i s f i e d . •k For a proportional reduction i n t a r i f f s x to be s t r i c t Pareto 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 vector A e R H + N + K + 1 such that A T[A, - B j > 0?; . , A T[B , B , B , b H+1 p z t B ] = 0M,„,.7,„, A B x > 0. For a vector A s a t i s f y i n g A [B , B , s N+K+N+M' x — J ° p' z' rp rp X T 0 A A B . B ] = 0.,^..^.., the scalar A B x equals -x V G(w + x , y ) t' s N+K+N+M' x H ww ' • J* 2 / * * v x . This number i s negative, since the matrix V G(w + x , y ) i s ww A *. T pos i t i v e seraidefinite and, 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 in the absolute value of x . The assumption that m ft ft V^b (w + x ) i s negative can be interpreted as follows: i f ft x >_ 0 (net exports are subsidized and net imports are taxed), s u b s t i t u t i o n i n production of tradeables must dominate i n the sense that x V G(w + x , y ) > CL,. In this case, the ww M ' * uniform reduction of x i s equivalent to a uniform reduction of ft subsidies for net expors and taxes f o r net imports. I f x 0^ (net exports are taxed and net imports subsidized), complementarity i n production must dominate i n the sense that A r P o at x T* A x V G(w + x , y ) < 0... Then, the uniform reduction of x ww ' J M ' amounts to a uniform increase i n the export taxes and to a uniform increase i n the Import subsidies. * Use the proof of Proposition 7.1 and note that when (x - x) > 0 , A A A A A 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 also be shown that a s t r i c t Pareto improving increase in p o s i t i v e t*, s and T * and a s t r i c t Pareto improving decrease i n negative t*, s* and x* e x i s t . 7. In Chapter 8 i t w i l l be shown that i f lump sum transfers are a v a i l a b l e , the simultaneous reduction i n taxes and t a r i f f s can be chosen to be a proportional reduction. A A A A 8. If t = C>N, s = 0^, T = ° M a n d g = 0 R, the government budget A T ' f\ TP (~\ constraint becomes p x + w e =0, which can only hold i f e i t h e r x^ = 0^ ,, e^ = 0^, or i f the government behaves l i k e a private producer with the government budget constraint serving the role of the zero p r o f i t constraint. This implies that some of the components of x^ and e^ must be negative, corresponding to inputs bought to produce other domestic or tradeable goods. 9. Government p o l i c y choices and economic i n e q u a l i t y are discussed i n Chapter 12. 10. Notation: t = [t, , ... ,t t , t 1 T ] T e R N 1 . -n 1 n-1 n+1 N Chapter 8: T 1. Assumption (8.2) i s s a t i s f i e d , i f f o r example, the matrix 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 matrix Z i s of f u l l rank N. This occurs when the matrix Z i s of maximal qq rank (= N+M-l). A A A 2. Note that the changes i n u , t and s are going to be zero. 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 values of the v a r i a b l e s . The assumptions (2.12) and (8.2) are necessary and A A A A A s u f f i c i e n t f o r the i m p l i c i t functions p (u , w + x , t , s ) and A A A A A z ( u , w + x , t , s ) t o e x i s t . Compare this to the proof of Lemma 2.1. 4. Note that l o c a l c o n t r o l l a b i l i t y of production in 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 production 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 matrix S p p must be N). However, l o c a l c o n t r o l l a b l i t y of production i n the sense of D e f i n i t i o n 2.1 i s not necessary for l o c a l c o n t r o l l a b i l i t y of domestic goods production i n the sense of D e f i n i t i o n 8.1. - 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 that 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 production coincide. 6. The matrix -£qq i s p o s i t i v e semidefinite. 7. See the proof of Lemma 2.2. 8. The sets C , k=l,...,K, and M (u ), h=l,...,H, are assumed to be closed and convex; i n addition, the S l a t e r constraint q u a l i f i c a t i o n condition i s assumed to be s a t i s f i e d . ( I f the u t i l i t y functions of the consumers are quasiconcave and continuous from above, the sets M , h=l,...,H, are convex and closed. If there i s a f e a s i b l e s o l u t i o n for (8.5) such that the i n e q u a l i t y constraints 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 condition i s s a t i s f i e d . ) 9. Note that the conditions (2.12) and (8.2) are s u f f i c i e n t for the function B(w) to be twice continuously d i f f e r e n t i a b l e . They are also s u f f i c i e n t for the function V B to be once continuously w J d i f f e r e n t i a b l e . 10. This could be seen also by noting that the function B(w) must be convex in prices w, as the function B i s convex in i t s arguments. 11. The weak i n e q u a l i t y follows from the p o s i t i v e semidefiniteness of the matrix (S - E). k k 12. If both matrices S and E are of maximal rank and t = 0„, s = 0 W, N M the matrices S and E have a common zero eigenvector, i . e . , the k k vector (p , w + T ). This vector i s also the zero eigenvector of the matrix (S - E). 13. The equivalence 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 matrix 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 could be i l l u s t r a t e d using F i g . 3. The Pareto improving change in t a r i f f s and lump sum transfers s h i f t s the economy's production choice toward the point 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. Yet, f i n i t e Pareto and pr o d u c t i v i t y improving changes i n t a r i f f s and transfers may e x i s t . 17. If rank E = N+M-l, then rank E = N. Assuming that N >_ 1, E must be nonzero. Hence, e * 0„ i f t * 0„. qq ' N N * * * 18. If t = 0.., s = 0.,, x = 0 W and lump sum transfers are chosen N M M optimally, the i n i t i a l e q u i l i b r i u m i s a pro d u c t i v i t y maximum, i . e . , the f i r s t best equilibrium the economy can a t t a i n under i n t e r n a t i o n a l trade. Chapter 9: 1. Kemp and Wan (1983). T T T 2. 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, since 2 v q l l q v 2 2 w 2 — ' the matrix E i s negative seraidefinite. It i s also known that T x„ E x„ < 0, because E i s negative semidefinite. If the matrix 2 vv 2 — vv £ q v were nonzero, the vector x could be chosen so as to v i o l a t e the i n e q u a l i t y x^ Ex _< 0. Hence, Eq V = % X M « * * 3. I f E = 0 o „, then E v = 0„. In the Kemp-Wan example, s = 0„ qv 2x2 w 2 r r ' 2 which implies E (w + x ) = 0„. The i n t e r n a t i o n a l trade vv 2 * p r o h i b i t i v e t a r i f f s x i n the Kemp-Wan economy are not proportional T to w. Then, w E must be nonzero because the matrix E i s of vv vv maximal rank (=1) in this example. 4. The matrix V w f(w + x , y ) = G(w + x , y ) i s a zero (2x2)-matrix at A. 5. I t i s assumed that the tax rates (t , s ) are chosen Diamond-Mirrlees optimally with respect to the constraints that * a * x = x (= w - w) and g = constant. Otherwise, a welfare improving change in only (t , s ) would e x i s t . - 197 -6. To be exact, one must assume that the i n i t i a l transfers g are not Diamond-Mirrlees optimal with respect to the constraints x = x (= wa - w) and t = t , s = s . 7. There must be su 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 production sector supplies only one good, as in the Kemp-Wan example, the requirements that each tradeable good i s produced i n autarky and that each sector operates at a pos i t i v e scale i n autarky are equivalent. T 2 " T T T 9. For w V B(w) = 0.. to be s a t i s f i e d the vector [-(p + e) , w ] WW M i \ r / > J must be a zero eigenvector of the matrix (S - E) (see 8 .18) . I t follows that, since the matrices S and -E are pos i t i v e semidefinite, T T the vector [-(p + e) , w ] must be a zero eigenvector of S and E. * * I f , however, the vectors w and x are not proportional, x t 0^, and the matrices S and E have only one zero eigenvector ( [ p T , (w + x*) T] S = 0jJ + M , [(p + t ) T , (w + x* + s ) T ] E = 0^ + M ) , then T T the vector [-(p + e) , w ] cannot be a zero eigenvector of S or E. Chapter 10: 1. I f x* = C^j, t h i s condition cannot be s a t i s f i e d since V b*(w + x*)w = w TV 2 B(w)w > 0 i f V b*(w + x*) * 0,T,. x ww x M 2. For example, i f t » 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 perturbation * _ of the i n i t i a l tax rates i n the d i r e c t i o n (t - t) i s s t r i c t Pareto and p r o d u c t i v i t y improving. 3. This condition also appears i n Hatta and Fukushima (1979: p. 509); Fukushiraa (1979: p. 431); D i x i t (1975: p. 107); Smith (1980: pp. 8-9) ; Hatta (1977: pp. 1865-66). 4. Note that this corresponds to the e a r l i e r idea of a constant 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 Pareto improvement through a perturbation i n the i n i t i a l vector of lump sum t r a n s f e r s . - 198 -5. i s an unstable equilibrium. Hatta shows that the equilibrium i n the economy described by his model i s stable, i f the Hatta normality condition is s a t i s f i e d . 6. Diewert (1983b): Theorem 6. ^T 2 7. t [ V _ B] < 0 i s one of the s u f f i c i e n t conditions for a tt *n * reduction in a p o s i t i v e domestic tax rate t , n e [1,...,NJ, to be s t r i c t Pareto improving (assuming that the transfers g are adjusted appropriately). 8. See formula (12) i n Hatta (1977a). 9. Lemma 2 i n Hatta (1977a). Note that, using formula (12) in Hatta (1977a), i f only one tax rate t i s reduced, a s u f f i c i e n t condition n JUrrt JL for this change to be welfare improving i s that p (£^[p ] ) . n > 0, which i s (10.31). Note that i n Hatta's notation E = F. qq Chapter 11: 1. Melvin (1970: p. 68). 2. See problem (4.26). 3. The proof of Theorem 11.1 could also 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 increases i n the p o s i t i v e commodity tax rates ( t * , s*), or s t r i c t Pareto and p r o d u c t i v i t y improving reductions in the negative ( t * , s ). In the f i r s t case, the increased taxes on commodities bought by the consumers are balanced with increased subsidies on t h e i r work e f f o r t . In the l a t t e r case, taxes on factors sold are increased simultaneously with increases i n subsidies on commodites demanded by the consumers. T ft ft 4. Suppose W(u) = l I T u . If the vectors t = 0.T and s = 0., are H N M . optimal when T = 0^, there must e x i s t a vector X e RH+N+K+1 s u c | 1 that XTA - i j . - x \ > 0 and X T[B p, B z > B f, Bg] = o j ^ ^ . (See (4.28)). A vector s a t i s f y i n g XT[B , B , B 1 = oF.^^.. and p Z t N + K . + N - 199 --X AB b > 0 i s of the form A = [A 1, (p + 6) , y , 1] using the proof of Theorem 2.1. If x = ° M» then 6 = 0 N and y = 0^, using (2.27). Choose A. = 0 U. Then, A T = [0^, p* T, 0^, 1] and I n U K . T T A A = 1 R. (Note that the money metric s c a l i n g of u t i l i t i e s implies that P T J : + wTr = i h . qu vu H Chapter 12: 1. I t w i l l be assumed in this chapter that a l l consumers u t i l i t i e s , which are measured using a money metric, are f u l l y comparable. In the previous chapters, when only Pareto improvements were considered, no comparability assumption was needed. Yet, when the s o c i a l welfare function W(u) = 8 Tu, 8 > 0^, was introduced, 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 are f u l l y comparable. 2. The notion that only the households' non-labor incomes are s h i f t e d toward equality using government tax and transfer 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 quilibrium concept. This i n turn i s based on Varian's (1976) idea 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, instead of u n , h=l,...,H. Note that the d e f i n i t i o n of consumer expenditures i n (12.1) d i f f e r s from the d e f i n i t i o n of mn i n the equation (4.10). In (12.1), the consumer expenditures are defined as the dif f e r e n c e between the value of (gross) purchases of domestic and tradeable commodities minus the value of any labor services supplied. In (4.10), the consumer expenditures were calculated as the difference between the value of the consumers' net consumption and the value of t h e i r labor supply. 4. Note that the observed i n i t i a l e q u i l i b r i u m consumer prices (q , v ) are chosen to serve as the reference prices for the money metric u t i l i t y functions. Hence, all^inte_rpersonal u t i l i t y comparisons In this section are based on (q , v ). The choice of the s o c i a l welfare function W(u) i s a value judgement. An economist can only suggest properties of the function W(u) that the society might think as d e s i r a b l e . The function W(u) may, for example, be required to be quasiconcave, symmetric, increasing, and c a r d i n a l l y scaled. The l a s t property means that s o c i a l welfare i s increasing along the equally d i s t r i b u t e d real income l i n e . (This also means that W(Aljj) = A f o r a l l A e R.) An example of a - 200 -s o c i a l welfare f u n c t i o n that s a t i s f i e s the above mentioned con d i t i o n s i s the mean of order 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 / H n II u , r = 0. h=l For an i n c r e a s i n g W(u), the welfare weights S are p o s i t i v e . I f the government wants to assign negative weights to those households whose incomes exceed some "acceptable" upper l i m i t , the government 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 U y [ - ( U - u l R ) (u - u l R ) I , u = - j j — , y > 0. ft 5. Using Figure 9, a reduction i n I a l s o corresponds 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 at S ( i f the s o c i a l welfare f u n c t i o n W i s s t r i c t l y quasi-concave). 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 zero r e f l e c t i n g the governments l a c k of 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 (net) demand or supply of the good i n question. 8. Note that i f 8 = 0 U, i . e . , 8 = f o r a l l h, h=l,...,H, s t r i c t n. H. i n e q u a l i t y reductions using commodity tax and t a r i f f p e r t u r b a t i o n s do not e x i s t . In t h i s case, the s o c i a l welfare f u n c t i o n W(u) i s u t i l i t a r i a n , or the r e a l incomes u n , h=l,...,H, have been equ a l i z e d at the i n i t i a l e q u i l i b r i u m . 9. 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 X e RH + N + K+1 that X T [ B P , B Z , B G , B t ] = 0 ^ ^ , X TA = 8 T, X T B F A < 0. The equations X T B G = 0 R imply X 1 = 0 R. The i n e q u a l i t y X T B B _< 0 im p l i e s X 4 = k >_ 0. I f k=0, then X TA = 0 R * 8 T. (For k :> 0, the proof of Theorem 8.3 shows that X^ = k(p * + e ) T , X^ = k 0 T ) . I f T T 2 ~ T k = 1, then X B = w 7 B(w) * 0,, by assumption. T WW M •* r - 201 -10. It i s necessary for a s t r i c t Pareto improving change i n commodity taxes and t a r i f f s to exist that the generalized Diamond-Mirrlees condition (4 .17) i s s a t i s f i e d . But i t can be seen that i n s p i t e of the condition (4 .17) being s a t i s f i e d , the vector 8 [X - C , E - D ] may s t i l l be equal to zero, v i o l a t i n g the th i r d asumption i n Theorem 1 2 . 1 . (This i s because 8 i OJJ.) (If B^fX^ - C ^ , E^ - D^l = 0?L.„, i t can be shown that no s t r i c t . N+M ine q u a l i t y reductions can e x i s t , s t a r t i n g from the i n i t i a l equilibrium.) 11. There may exi s t an a > 0„ such that a [X - c" , E - if ] = 0 . T . „ but . H N+M the vector 0 [X - C , E - D ] i s nonzero (8 i OJJ).. rri rri rri rii rti 12. If equation (12.16) i s written i n the form 8 [X - C" , E - D~ ] T T T T = - r 8 [X - C , E - D ], i t can be seen that there i s no so l u t i o n to (12.16) i f the vectors 8 T[X T - C T , E T - D T] and 3 T[X T - CTT, X X E - D ] are of the same sign. - 202 -REFERENCES Blackorby, C. and D. Donaldson [1977], " U t i l i t y vs Equity," Journal of  Public Economics 7, 365-381. Diamond, P.A. and J.A. Mirrlees [1971], "Optimal Taxation and Public Production, I - I I , " American Economic Review 61, 8-27 and 261-278. Diewert, W.E. [1976], "Exact and Superlative Index Numbers," Journal of  Econometrics 4, 115-145. Diewert, W.E. [1978], "Optimal Tax Perturbations," Journal of Public  Economics 10, 139-177. Diewert, W.E. [1983a], "A Fundamental Matrix Equation of Production Theory with Applications to the Theory of International Trade," pp. 181-194 i n Methods of Operations Research, V o l . 46, P. Staehly (ed.), Koenigstein, West Germany: Verlag Anton Hain. Diewert, W.E. [1983b], "Cost-Benefit Analysis and Project Evaluation: A Comparison of Al t e r n a t i v e Approaches," Journal of P u b l i c Economics 22, 265-302. Diewert, W.E. [1983c], "The Measurement of Waste Within the Production Sector of an Open Economy," Scandinavian Journal of Economics 85, 159-179. Diewert, W.E. [1984], "The Measurement of Waste and Welfare i n Applied General Equilibrium Models," Discussion Paper 1984-17, The Un i v e r s i t y of B r i t i s h Columbia, Department of Economics, May 1984. Diewert, W.E. and A.D. Woodland [1977], "Frank Knight's Theorem i n Linear Programming Rev i s i t e d , " Econometrica 45, 375-398. D i x i t , A.K. [1975], "Welfare E f f e c t s of Tax and Pric e Changes," Journal  of P u b l i c Economics 4, 103-123. D i x i t , A.K. [1979], "Price Changes and Optimum Taxation i n a Many-Consumer Economy," Journal of P u b l i c Economics 11, 143-157. D i x i t , A.K. and V. Norman [1980], Theory of International Trade, Welwyn, Herts, U.K.; James Nisbet. Fukushima, T. [1979], " T a r i f f Structure, Nontraded Goods and Theory of Piecemeal P o l i c y Recommendations," In t e r n a t i o n a l Economic Review 20-2, 361-369. Guesnerie, R. [1977], "On the D i r e c t i o n of Tax Reform," Journal of  Public Economics 7, 179-202. - 203 -Hatta, T. [1977a], "A Theory of Piecemeal P o l i c y Recommendations," The  Review of Economic Studies 44, 1-21. Hatta, T. [1977b], "A Recommendation for a Better T a r i f f Structure," Econometrica 45, 1859-1869. Hatta, T. and T. Fukushiraa [1979], "The Welfare E f f e c t s of T a r i f f Rate Reductions i n a many Country World," Journal of In t e r n a t i o n a l  Economics 9, 503-511. K a r l i n , S. [1959], Mathematical Methods and Theory i n Games,  Programming and Economics, Vol. 1, Palo A l t o , C a l i f o r n i a : Addison-Wesley. Kemp, M.C. and H.Y. Wan [1983], "Trade Gains Without Lump Sum Compensation?" mimeo, Un i v e r s i t y of New South Wales, A u s t r a l i a . Kolm, S.C. [1969], "The Optimal Production of S o c i a l J u s t i c e , " pp. 173-200 i n Public Economics, J . Margolis and H. Guitton (eds.), London: Macmillan. Mangasarian, O.L. [1969], Nonlinear Programming, New York: McGraw-Hill. McKenzie, L.W. [1955], "Equality of Factor Prices i n World Trade," Econometrica 23, 239-257. Melvin, J.R. [1970], "Commodity Taxation as a Determinant of Trade," Canadian Journal of Public Economics 3-1, 62-78. Negishi, T. [1960], "Welfare Economics and the Existence of an Equilibrium f or a Competitive Economy," Metroeconomica 12, 92-97. Smith, A. [1980], "Optimal Public P o l i c y i n Open Economies," Discussion Paper, London School of Economics, August 1980. Uzawa, H. [1958], "The Kuhn Tucker Theorem i n Concave Programming," pp. 32-37 i n Studies i n Linear and Nonlinear Programming, K.J. Arrow, L. Hurwicz and H. Uzawa (eds.), Stanford: Stanford U n i v e r s i t y Press. Varian, H.R. [1976], "Two Problems i n the Theory of Fairness," Journal  of Public Economics 5, 249-260. Weymark, J.A. [1979], "A R e c o n c i l i a t i o n of Recent Results i n Optimal Taxation Theory," Journal of P u b l i c Economics 12, 171-189. Woodland, A.D. [1982], International Trade and Resource A l l o c a t i o n , Amsterdam: North-Holland. - 204 -APPENDIX 1 Proof f o r Lemma 2.3: T T Consider the equations X [B , B ] = 0„, T,: n p z N+K T T T (Al) [Xj, Xj, X^] pp T KxK T T w S w F wp = 0, N+K * Set X. = k £ R. Using the equations S p + S (w + T ) = 0„ and 3 pp pw N p* TY + (w + x * ) T F = Cv ((2.7) and (2.5)), (Al) can be written as (A2) S Y PP T Y 0 KxK * T * T * T * T = k[x XS + p S , x F + p Y] wp pp' Assumptions ( i ) - ( i i i ) i n Theorem 2.1 imply that the matrix on the l e f t hand side of (A2) can be inverted and the inverse i s the matrix D defined in (2.20). The properties of this matrix are given i n Diewert and Woodland (1977: Appendix). Equations (A2) can be solved f o r the T T vector (X. , X„ ) : - 205 -(A3) [x[, XJ] = k[(p* + S)T, Y T], whe re (A4) 5 T = x* T[S D.. + FD* ], y T = x* T[S D._ + FD 0 0] . wp 11 12 wp 12 22 Equations (A3) are derived using the following properties of the matrix D : D 1 2 = D 2 1 > [ S p p Y ] D = [ I N ' ° N x K ] - Q E D Proof of Lemma 2.4: Using (2.11), (A5) X T B = - [ ( p * + 5 ) T S + Y T F T + wTS ]. T r pw WW Applying the d e f i n i t i o n s of the vectors 5 and y i n (A4), (A6) X T B t = -k[p* TS + x* T(S D.. + FDj„)S + T * T ( S D._ + FD„ 0)F T T pw wp 11 12 pw wp 12 22 + wTS ] WW = - k [ T * T ( S D..S + FD^„S + S D 1 0 F T + F D 0 0 F T ) - T * T S ] wp 11 pw 12 pw wp 12 22 ww - 206 -since, by the homogeneity of the unit p r o f i t functions, p S + w S pw ww * T - T S . Using the equations D..S D, = D,., D,,S D,„ = 0 and ww & M l l p p l l 11' 11 pp 12 T D12^pp D12 = ~ D22 § i v e n i n D i e w e r t and Woodland (1977: Appendix), (A6) may be written as (A7) X TB = k t * T [ - S D.. x wp 11 - FD 12 : -D..S - D.„F 11 pw 12 hi * T 9 * * kx V G(w + T , y ), WW using Lemma 2.2. QED Lemma 2.5: k k k T = k(w + T ) , k * 0,1, i f and only i f -5 = kp . Proof: k k k Suppose T = k(w + T ) , k * 0,1. Then, T = aw with a = k / ( l - k ) . [If k = 1, then w = 0^ which v i o l a t e s the assumption that w » 0^. If k k k = 0, then T = 0^, which v i o l a t e s the assumption that x * ^ M*^ Using the d e f i n i t i o n of the vector 6 i n (A4), i f x* = aw, (A8) 6 T = x* T[S D.. + FTVJ = aw T[S D. . + FDT,]. wp 11 12 wp 11 12 - 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 equations [ SP P Y ] D = [ IM> <W ±m^y (A9) 6 T = a [ ( - P * T S p p - x * T S w p ) D N + W T F D { 2 ] A T * T T T = -ap S D . , - at S D . . + aw FD,„ p p l l w p l l 12 * T * T T * T T T = -ap - ap Y D : , , - ax S D , , + aw F D , 0 , r 12 wp 11 12' *T , * N T T which, using the zero p r o f i t c o n d i t i o n p Y + (w + x ) F = 0 , y i e l d s (A10) SL = -a[p + x LS D . . + x F D , „ ] = -a(p + S ) 1 . wp J. J. 1Z Then, 6 T = --p- 3— p* T = kp* T. 1 + a Since a l l steps of the proof are i f and only i f statements, 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 x and w i m p l i e s p r o p o r t i o n -* * a l i t y of x and (w + x ), and hence (2.37) and (2.38) are e q u i v a l e n t . - 208 -APPENDIX 2 Proof of Lemma 8.1 Let us develop the quadratic form (8.11). To s i m p l i f y notation the matrix (S - £) i s denoted by B A A C ^ rp rp rp rp rn rp (Bl) V B(w) = AD,.BD,,A + FD70BD. , A - AD,,A + AD,.BD10F ww 11 11 12 11 11 11 12 + FD^ 2BD 1 2F T - A D 1 2 F T - A D U A T - F D ^ A 1 + C rp rp rp rp rp = -AD UA - AD 1 2F - roi2A " F D 2 2 F + C ' since D^BD^ = D u , D j j B D ^ = 0, D 1 2 B D i 2 Woodland (1977 : Appendix)). Then, -D 2 2 (see Diewert and (B2) B(w) - C - [A, F] D ww which i s (8.9). QED Proof of Lemma 8.2: ft ft Using (8.1) and the equations S p p p + S p w(w + T ) = 0 N, * T ft . T T p Y + (w + T ) F = 0^, - 209 -r (B5) X A[B , B ] = X P z S - E pp qq - T S - p S ~W E wp pp vq KxK * T * T - T F -p Y T Set X 3 = k e R. Then, (B6) S - E pp qq KxK = k[x LS + p S + w E , x F + p Y] wp pp vq • c Assumptions ( i ) - ( i i ) i n Theorem 8.1 imply that the matrix on l e f t hand side of (B6) can be inverted. C a l l the inverse D = D l l D12 D21 D22 Then, rp rp Jt. rp rp (B7) [Xf, \\] = k[(p + e ) T , 9 T] where (B8) [ e T , 9 T] = [-t* TE - (x* + s * ) T E + T * T S , x* TF] D. qq vq wp' - 210 -Equations (B8) are derived using the equations (Spp _ ^qq^ D l l + ^21 I K T, q T E + v* TZ = oL and (S - E ) D 1 0 + YD0~ = 0„ „. N qq qv N PP qq 12 22 NxK Proof of Lemma 8.3: Using (8.1) and (B7), (B9) \lZ = k[-(p + e) TS + (p + e ) r E - eV - w T(S - E ) ] , T \^ pw K qV WW W ' k e R. Using (B5), the homogeneity of the zero p r o f i t functions and the zero J j f r p £ r p r j i J j . rjt rjt p r o f i t condition p Y + ( W + T ) F = 0 , the vector [(p + e) , 8 ] can K be written as (BIO) [(p* + e ) T , e T] = -w T[S - E , F ] D. r ' wp vq' Then, - 211 -(BID X TB kwT[(S - E )D,, +'FD"F0] (S - E ) T + kwT wp vq 11 12 wp qv [(S - E ) D? 0 + FD„ 0]F T - kw T(S - E ) wp vq 12 22 ww vv = kw T[(S - E )D..(S - E ) T + Fuf-CS - E ) T wp vq 11 wp vq 12 wp vq + (S - E ) D 1 0 F T + F D 0 0 F T - (S - E )] wp vq 12 22 ww w = kw T[-(S - E ) + (S - E , F) D(S - E , F ) T ] ww vv wp vq wp qv Using- (8.10), X TB x = -kw^^Btw), k e Ri QED Proof of Theorem 10.4: (a) l c i s su f f i c i e n C Co show ChaC Chere i s no vector X e R H + N + K + 1 such that X T[A, - B b] > 0 ^ , X T [ B p ) B g] = 0* + K + H, X [ B t > B g ] ( t , s ) _> 0. It has been established e a r l i e r that a vector T T T T X s a t i s f y i n g X [B , B , B ] = 0.T1T,,„ must be of the form X = k[0 T I, J ° p z' g N+K+H H' (p* + e ) T , 9 T, 1], k e R. The I n e q u a l i t i e s X T[A, - B. ] > 0^. imply b H+1 that k > 0. Using (4.16), the homogeneity of the unit p r o f i t and T T expenditure functions, and X B p = 0^, (B12) \L[Bt, B g ] ( t \ s V = [A*. X^, X* ] -S w + E w pw qv -F w T T -w S w + w Z w ww vv - 212 -Thus, XT[B1., B ] [ t * T , s * T ] T = -XTB w = -wT V2 B(w)w using Lemma 8.3, t' s T ww & rj, 2 x X *T *T T By assumption, w V B(w) * 0 . Hence, X [B , B ][t , s ] < 0. (b) It must be shown that there is no X e R such that XT[A, - B J > Oj, X T[B p, B z, B g] = 0^ + K + H, X TB g S* > 0. For XT = [0 T, (p* + e ) T , 6T, 1], r p JU JU rp JU r p JU (B13) X B s = -(p + e) E s - w E s s ^ vq vv *X * *T ~ * x * = -p E s + s E D,,E s - wE s , c vq vq 11 qv w ' ft ft using (B8) with t = 0,T, T = 0... N M Then, (B14) XTB s* = s* T[E + E D., E ]s* s vv vq 11 qv = -s 1 Vz Bs < 0, ss * X * x 2 since, by assumption, s V Bs * 0 and the matrix V B is positive S S LL S S semidef in i t e . X * x * T If E = 0.T then X B s =-wE s = w E w<0, since, by qv NxM s w w J yLrjri X ^ assumption, s E * 0 W. (If E = 0,, „, then E (w + s ) = 0., and vv M vq MxN w M thus E w = -E s*). vv vv - 213 -/ T * (c) Now the vector X B t t must be shown to be negative. Using (4.15), (B14) X TB t* = -(p* + e ) T E t* - w TE t* t qq vq A T * T * * T ~ * = -p E t - w E t + t E D, . E t qq vq qq 11 qq XT' ?V = t [E + E D., E ]t qq qq n qq A T 9 * = " t V t t B t < °» *T 2 T 2 since t Vt(_B * 0^ and the matrix V B i s p o s i t i v e semidefinite. I f E = 0„ „, then qu NxM (B16) XTB t * = -<p* + e) TE t * t r qq A T A A T A = -p E t + t E D,, E t ^ qq qq n qq = t A [ E + E D..E ]t qq qq 11 qq „ , * T A A T _ 9 A because £ q q ( P + t ) = 0^. Hence, X B t t = - t v ^ B t < 0, since *T 2 t B * 0. QED - 214 -Proof of Proposition 10.1; * Suppose t » 0^. I t i s s u f f i c i e n t to prove that there i s no H+N+K+l T T T vector A e R such that A [A, - B j > 0 T T L l , A [B , B , B ] ' b H+l' p z' g T T = 0 , A B h > 0, where the vector h i s defined as the d i r e c t i o n of H+ix+ti t the change i n t : h = (t - t) > 0^, where t i s the vector of uniform — — N T *T 2 taxes, t = ( t , . . . , t ) e R +. Then, A B th = t v t t B h < 0 i f , as assumed, *T 2 T t V B < 0„ and h > 0lT. The other cases are proved i n a s i m i l a r way. tt N N QED 

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