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Essays on the measurement of waste and project evaluation Tsuneki, Atsushi 1987

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ESSAYS ON THE MEASUREMENT OF WASTE AND PROJECT EVALUATION By ATSUSHI TSUNEKI B.A., The U n i v e r s i t y o f Tokyo, Tokyo, 1981  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES THE DEPARTMENT OF ECONOMICS  We a c c e p t t h i s t h e s i s as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA June 1987 CcT) ^Atsushi  Tsuneki,  1987  In  presenting  requirements of  British  it  freely  agree for  this for  an  available  that  I  understood  that  financial  by  his  or  June  reference  and  study.  I  extensive  her or  shall  f l ,  copying  granted  by  the  of  publication  not  be  allowed  Columbia  /9£?  of  make  further this  head  representatives.  of  30  University shall  The U n i v e r s i t y o f B r i t i s h 1956 Main M a l l V a n c o u v e r , Canada V6T 1Y3  Date  the  the  Library  permission.  Department  at  of  the  may b e  copying  gain  degree  fulfilment  that  for  purposes  or  partial  agree  for  permission  scholarly  in  advanced  Columbia,  department  for  thesis  It  this  without  thesis  of  my  is  thesis my  written  ABSTRACT  Harberger's methodology for the measurement of deadweight loss i s reformulated in a general equilibrium context with adopting the A l l a i s Debreu-Diewert approach and i s applied to various problems with imperfect markets. class of  We also develop second best project evaluation rules for the same economies.  Chapter 1 i s devoted to the survey of various welfare i n d i c a t o r s . especially discuss the two welfare indicators due to A l l a i s ,  We  Debreu, Diewert  and Hicks, Boiteux i n r e l a t i o n to Bergson-Samuelsonian s o c i a l welfare function.  We f i r s t show that these two measures generate a Pareto inclusive  ordering across various s o c i a l states, but they are rarely w e l f a r i s t ,  so that  both are unsatisfactory as Bergson-Samuelsonian s o c i a l welfare functions.  We  next show that second order approximations to the Allais-Debreu-Diewert measure of waste can be computed from l o c a l information observable at the equilibrium, whereas second order approximations to the Hicks-Boiteux measure of welfare or to the Bergson -Samuelsonian s o c i a l welfare function require information on the marginal u t i l i t i e s  of income of households,  unavailable with ordinal u t i l i t y theory.  which i s  F i n a l l y , we give a diagrammatic  exposition of the two measures and t h e i r approximations to give an i n t u i t i v e insight into the economic implications of the two measures. Chapter 2 and Chapter 3 study an economy with public goods.  In Chapter  2, we compute an approximate deadweight loss measure for the whole economy when the endogenous choice of public goods by the government i s nonoptimal  and the government revenue i s raised by distortionary taxation by extending the Allais-Debreu-Diewert approach discussed i n Chapter 1. measure of waste i s related to i n d i r e c t tax rates, public goods,  The r e s u l t i n g  net marginal benefits of  and the derivatives of aggregate demand and supply functions  evaluated at an equilibrium.  In Chapter 3, cost-benefit  rules for the p r o v i -  sion of a public good are derived when there exist tax d i s t o r t i o n s . derive the rules as giving s u f f i c i e n t  We  conditions for Pareto improvement, but  we also discuss when these rules are necessary conditions for an i n t e r i o r s o c i a l optimum.  When i n d i r e c t taxes are f u l l y f l e x i b l e but lump-sum trans-  fers are r e s t r i c t e d , we recommend a rule which generalized the  cost-benefit  rule due to Atkinson and Stern (1974) to a many-consumer economy. i n d i r e c t taxes and lump-sum transfers are f l e x i b l e , i s based on Diamond and Mirrlees'  When both  we suggest a rule which  (1971) productive efficiency p r i n c i p l e .  When only lump-sum transfers are variable, we obtain a version of the Harberger (1971)-Bruce-Harris (1982) cost-benefit  rules.  Chapters 4 and 5 study an economy with increasing returns to scale i n production and imperfect competition.  In Chapter 4, we discuss a methodology  for computing an approximate deadweight loss due to imperfect regulation of monopolistic industries by extending the Allais-Debreu-Diewert approach to incorporate the nonconvex technology.  With the assumption of the quasi-con-  cavity of production functions and fixed number of firms, we can derive an approximate deadweight loss formula which i s related to markup rates of firms, and the derivatives of aggregate demand functions,  factor supply and  demand functions and the derivatives of marginal cost functions.  We also  iv  d i s c u s s v a r i o u s l i m i t a t i o n s o f our approach and t h e r e l a t i o n between our work and  t h a t o f H o t e l l i n g (1938).  I n Chapter 5, we c o n s i d e r c o s t - b e n e f i t r u l e s  of a l a r g e p r o j e c t a p p l i c a b l e i n t h e presence of i m p e r f e c t c o m p e t i t i o n .  We  show t h a t t h e i n d e x number approach due t o N e g i s h i (1962) and H a r r i s (1978) can be extended t o h a n d l e s i t u a t i o n s w i t h i m p e r f e c t  competition.  V  TABLE OF CONTENTS Page Abstract  i i  L i s t of Figures  v i i  Acknowledgements  viii  Chapter 1  2  3  APPLIED WELFARE ECONOMICS  1  1-1.  The Mesures of Deadweight Loss  2  1-2.  The Approximation Approach  1- 3.  A Diagrammatic  t o the Measurement of Waste  Exposition  11 26  Footnotes  35  Appendices  38  THE MEASUREMENT OF WASTE IN A PUBLIC GOODS ECONOMY  50  2- 1.  Introduction  50  2-2.  The Model  51  2-3.  An A l l a i s - D e b r e u - D i e w e r t Measure o f Waste  55  2-4.  Second  57  2- 5.  Conclusion  Order Approximations  63  Footnotes  66  PROJECT EVALUATION RULES FOR THE PROVISION OF PUBLIC GOODS  70  3- 1.  Introduction  70  3-2.  The Model  72  3-3.  P i g o v i a n Rules R e c o n s i d e r e d  76  3-4.  Cases where Lump-Sum T r a n s f e r s a r e A v a i l a b l e  85  vi  3- 5.  4  5  Conclusion  92  Footnotes  94  Appendices  95  INCREASING RETURNS, IMPERFECT COMPETITION AND THE MEASUREMENT OF WASTE  99  4- 1.  Introduction  99  4-2.  The Model  4-3.  The A l l a i s - D e b r e u - D i e w e r t  4-4.  Second Order Approximations  108  4- 5.  Conclusion  115  Footnotes  117  101 Measure o f Waste  104  PROJECT EVALUATION RULES FOR IMPERFECTLY COMPETITIVE ECONOMIES  121  5- 1.  Introduction  121  5-2.  Compensation C r i t e r i a  f o r Cost-Benefit  A n a l y s i s Reconsidered  123  5-3.  The Model  127  5-4.  P r o j e c t E v a l u a t i o n Rules  129  5-5.  Conclusion  132  Footnotes  133  REFERENCES  135  vii  LIST OF FIGURES  Page Figure  1.  The ADD Measure of Waste  44  F i g u r e 2.  The HB Measure of Welfare  45  F i g u r e 3.  The ADD Measure:  46  F i g u r e 4.  The ADD Measure and i t s Approximations:  A One-Consumer Two-Goods Economy  A One-Consumer Two-Goods Economy  47  F i g u r e 5.  The HB Measure:  48  F i g u r e 6.  The HB Measure and i t s Approximations:  A One-Consumer Two-Goods Economy  A One-Consumer Two-Goods Economy  49  F i g u r e 7.  The ADD Measure i n a P u b l i c Goods Economy  67  F i g u r e 8.  The ADD Measure and i t s Approximations  i n a Public  Goods Economy F i g u r e 9.  68  An Example where Approximations of the ADD Measure are I n a c c u r a t e  69  Figure  10.  The ADD Measure w i t h I n c r e a s i n g Returns t o S c a l e  Figure  11.  The ADD Measure and i t s Approximations with Returns t o S c a l e  119  Increasing 120  v i i i  ACKNOWLEDGEMENTS F i r s t and  of  chairman  a l l , of  encouragement several  other  I  two  valuable q u a l i t y  i n  should  t h e s i s  c l a s s e s  should  a l s o of  of  my  t h e s i s  comments,  and  i n my  thesis,  to  thank  t h e s i s  not  comments  only  my  on  E n g l i s h  l i f e .  I  graduate  students  of  the  Department  Columbia  who  my  study  gave  w r i t i n g  the  Ishikawa, Komiya, from  I  a l s o  Itoh,  Keimei  and  John  of  must and  express  Kaizuka,  f o r my  p a t i e n t l y  of  Canada  who  on  deep  my  t h e s i s  me  the  t h i s  other  at  the  both to  I  as  the  teaching  e x c e l l e n t l y .  f i r s t  parents work  of  i n  and outside  while  I  Ryutaro  of  support  F i n a l l y ,  by  E x t e r n a l  a s s i s t a n t .  my  was  Tsuneo  administered  who  and  B r i t i s h  f i n a n c i a l  Department  my  adjust  members  Hatta,  Nationals  my  to  professors  Kanemoto,  of  Keizo  comfortable  Tatsuo  the  g i v i n g  U n i v e r s i t y  appreciate  of a  f a c u l t y  encouragement  Yoshitsugu  a p p r e c i a t i o n to  complete  help  f o r  the  the  expression  kind  Foreign  me  f o r  f o r  Hamada,  behalf  employed  t y p i n g  to  to  improve  to  warm  Okano.  the  the  repeated  t h a n k f u l  thanks  and  Weymark  on  Weymark,  a l s o  1983-1987  K o i c h i  Awards  John  on  complete  f o r  to  kind  comments  am  to  deep  John  helped  a l s o  Economics  my  Canada  Weymark  P a i s l e y  of  Yukihide  Service  and  and  to  a d v i s e r  and  d e t a i l e d  Donaldson  which  I  t h a n k f u l  and  David  but  p r i n c i p a l  i t p o s s i b l e  appreciate  comments  Motoshige  Negishi,  and  content  express  important  my  suggestions  Blackorby  t h e s i s  during  i n c l u d e :  Government  generously  there  which  Jeanette  least,  be  t h e s i s ,  U n i v e r s i t y  A f f a i r s , Ms.  many  Takashi  the  World  me  must  made  c a r e f u l l y .  American  who  my I  the  North  UBC  which  s t i m u l a t i n g d i s c u s s i o n s  should  Diewert,  important  Charles  on  to  I  many  committee,  f o r  challenging.  f o r  Erwin  p r i v a t e d i s c u s s i o n s , and  Nagatani  made  thank  conspicuously.  c o r r e c t i n g many  of  and  to  committee  l i k e  my  d i s c u s s i o n s  l i k e  and  v e r s i o n s  members  extensive thesis  my  e a r l i e r  thesis.  I  but  always academic  I  thank not  helped career.  me  1.  CHAPTER 1 APPLIED WELFARE ECONOMICS  The  purpose of t h i s c h a p t e r i s t o compare a l t e r n a t i v e c r i t e r i a f o r  s o c i a l waste o r w e l f a r e from s e v e r a l v i e w p o i n t s and choose one which s u i t s our purpose b e s t .  criterion  I n d o i n g so, we p r e s e n t our b a s i c s t r a t e g i e s  f o r the measurement of deadweight l o s s and d i s c u s s t h e i r pros and  cons  compared w i t h o t h e r approaches t o a p p l i e d w e l f a r e economics. In s e c t i o n 1,  we i n t r o d u c e two c r i t e r i a f o r measuring deadweight l o s s ;  t h a t i s , the A l l a i s - D e b r e u - D i e w e r t measure of s o c i a l waste and the H i c k s B o i t e u x measure o f s o c i a l w e l f a r e .  A f t e r e x p l a i n i n g t h e i r i n t u i t i v e meanings  by i l l u s t r a t i o n s we c o n s i d e r whether t h e y can s e r v e as P a r e t o - i n c l u s i v e and individualistic  (or w e l f a r i s t ) s o c i a l w e l f a r e f u n c t i o n s .  We  show t h a t t h e s e  measures a r e P a r e t o - i n c l u s i v e , but not i n d i v i d u a l i s t i c except when e i t h e r Gorman's p r e f e r e n c e r e s t r i c t i o n i s s a t i s f i e d o r the p r o d u c t i o n set i s l i n e a r .  possibilities  The A l l a i s - D e b r e u - D i e w e r t measure of waste i s a f f e c t e d by  the  c h o i c e of t h e r e f e r e n c e bundle of goods i n terms o f which t h e s c a l e of e f f i c i e n c y l o s s i s d e t e r m i n e d whereas the H i c k s - B o i t e u x measure of w e l f a r e i s a f f e c t e d by t h e c h o i c e o f t h e o p t i m a l a l l o c a t i o n o f r e a l income.  T h i s means  t h a t the former measure i s a f f e c t e d by the v a l u a t i o n of each good f o r s o c i a l e f f i c i e n c y w h i l e the l a t t e r i s a f f e c t e d by t h e v a l u a t i o n o f each i n d i v i d u a l i n the measure o f s o c i a l w e l f a r e .  Thus, the A l l a i s - D e b r e u - D i e w e r t measure i s  a pure e f f i c i e n c y waste measure whereas the H i c k s - B o i t e u x measure shows a change o f s o c i a l w e l f a r e i n c l u d i n g b o t h e f f i c i e n c y and e q u i t y a s p e c t s . I n t h e second s e c t i o n , t h i s p o i n t i s f u r t h e r e l a b o r a t e d by t a k i n g a second o r d e r a p p r o x i m a t i o n  t o the two measures when t a x d i s t o r t i o n s  prevail.  We show that the Allais-Debreu-Diewert measure i s computable from the second order derivatives of expenditure functions and p r o f i t functions evaluated at the observed equilibrium while the Hicks-Boiteux measure or the BergsonSamuelsonian s o c i a l welfare function requires information on the difference between the inverse of the marginal u t i l i t y of income and the marginal s o c i a l importance to evaluate the equity l o s s .  Since t h i s information i s not  available with ordinal u t i l i t y theory, i t i s d i f f i c u l t to use the Hicks-Boiteux measure i n applied welfare economics.  This provides the main  reason why we use the Allais-Debreu-Diewert measure in this essay. and  The pros  cons of the approximation approach we adopt i n this essay are next  compared with an a l t e r n a t i v e i n f l u e n t i a l approach, applied (or numerical) general equilibrium models.  A numerical general equilibrium model computes  the exact value of s o c i a l welfare indicators by r e s t r i c t i n g the functional forms of production and u t i l i t y to overly simple forms.  Our approach, on the  other hand, computes approximate values of s o c i a l welfare indicators from more general functional forms and observable information. of  Finally, in light  the measurement of waste approach for welfare economics, we reconsider the  theory of second best.  Our conclusion here i s that this theory i s not a  replacement for the measurement of deadweight loss, even though several positive r e s u l t s derived i n second best theory are useful. Finally,  i n section 3, in order to give insight into the economic  implications of our approach, we give diagrammatic expositions of the two measures and t h e i r approximations for a one-consumer two-goods economy.  1-1.  The Measure of Deadweight Loss In a long series of papers on the measurement of deadweight loss (or  'welfare cost' or 'waste,' terms which are used interchangeably i n this  thesis) which includes H o t e l l i n g (1938), Hicks (1941-2), A l l a i s (1943, 1977), Boiteux (1951), Debreu (1951, 1954), Harberger (1964, 1971), and DiamondMcFaddon (1974), two types of welfare c r i t e r i a are c h i e f l y used:  the  Allais-Debreu-Diewert measure of waste (the ADD measure hereafter) and the Hicks-Boiteux measure of surplus (the HB measure h e r e a f t e r ) . 1 Let us set up the model of our economy to discuss these two measures. There are H consumers having quasi-concave u t i l i t y functions f (x ), h = 1, . . . , H defined over a translated orthant P.* where x* = (x^ , . . . ,x^) 1  1  1  consumption vector of goods 1 , . . . , N by the hth consumer.  T  is a  The i n i t i a l  endowment vector of the hth consumer i s given by x* , h = 1 , . . . , H .  There are  1  k k K firms and firm k produces y using the production p o s s i b i l i t i e s set S , k = 1,...,K. problem?  We can define the ADD measure i n terms of a primal programming :  ADD  ( 1 )  s  ~=  r  m a X r  , h x  r y  k  = h=1  { r  E  f (x ) h  +  P  '  r  1  E  k=l  y  > U j J , h = 1, . . . , H ; y  +  K  E  h=1  X  ;  e S , k = 1,...,K}, k  > 0^ i s an a r b i t r a r i l y chosen reference bundle of  where p = ( p ^ , . . . , p ^ ) commodities.  h  X  To i n t e r p r e t t h i s problem we rewrite (1) i n an a l t e r n a t i v e  N . The following notation i s used. R i s the N-dimensional nonnegative K k k orthant. E i d i r e c t sum of the production p o s s i b i l i t i e s sets S . manner.  +  S  s  t  n  e  K = 1  S(u ) 1  = {x : E f . , x 1 x; f ( x ) h  h  h  h  > UjJ, h = 1,...,H} i s the Scitovsky set  corresponding to a u t i l i t y a l l o c a t i o n u rewritten as  1  1 1 = (u.,...,u„). I  H  Now (1) can be  4.  (2)  max  {r  r  : 6 r  e Q = Z^x  + E ^S  h  k  (2) has a s t r a i g h t f o r w a r d i t e r p r e t a t i o n : set  k  - S(u )}. 1  maximize the s c a l e of the  reference  of goods i n Q where Q i s the s e t of goods p r o d u c i b l e from the a g g r e g a t e  p r o d u c t i o n p o s s i b i l i t i e s p l u s endowments which g i v e consumers a t l e a s t the 1 u t i l i t y vector u  when the goods are a p p r o p r i a t e l y d i s t r i b u t e d .  1 we d e p i c t the ADD  measure, i n a two goods economy where p  p r i c e of the programming problem ( 1 ) .  is a  In F i g . support  p^ i s d e t e r m i n e d up t o a m u l t i p l i c a -  t i o n by a p o s i t i v e number so t h a t we can choose p^ = 1; i . e . , the o p t i m a l p r i c e of the f i r s t good i s u n i t y w i t h o u t l o s s of g e n e r a l i t y . F u r t h e r m o r e , OT OT can choose the s c a l e of 8 so t h a t p 8 = 1 . Then, r = (p 8 ) r e q u a l s AB OT since p  pr i s the d i f f e r e n c e between the v a l u e of p r o d u c t i o n minus consump-  t i o n e v a l u a t e d a t p^.  Note t h a t the c h o i c e of the r e f e r e n c e bundle 8 i s  c r u c i a l i n the e v a l u a t i o n of the ADD L e t us now  measure (see Diewert  t u r n t o the HB measure. 0  ( 1985a;50)) .  3  We b e g i n from an a t t a i n a b l e and 0 0 T (u.,...,u ) . We a l s o assume t h a t t n 0 0 0 T t h e r e e x i s t s a p r i c e v e c t o r p = (p^,...,P ) which s u p p o r t s the s o c i a l l y o p t i m a l a l l o c a t i o n of r e s o u r c e s . Then we can d e f i n e the HB measure L..,. as s o c i a l l y optimal u t i l i t y a l l o c a t i o n u  s  N  HD  follows:  (3)  = E,  - E,  where we d e f i n e the e x p e n d i t u r e f u n c t i o n : ^  (4)  we  ra (p,u ) h  h  = min  {p x T  h  h  : f ( x ) > u,}, h  h  where p > 0  N  and  The measure L  H B  e Range f  defined by (3) can be interpreted as the sum of the  negative  of the equivalent variations obtained i n moving from a s o c i a l l y optimal u t i l i t y vector u^ to the observed distorted u t i l i t y vector u . 1  The HB  measure evaluated i n units of the f i r s t good i n a two good economy by choosing  = 1 i s i l l u s t r a t e d i n F i g . 2.  Generally, the desirable properties of the ordering of s o c i a l states are summarized i n the Bergson-Samuelsonian s o c i a l welfare function (BSSWF hereafter) .  (See Samuelson (1956) for a discussion of the BSSWF and i t s proper-  t i e s l i s t e d below.)  We f i r s t assume that the underlying s o c i a l ordering i s  compatible with the Pareto p a r t i a l ordering ( i . e . , t i e s increase,  then so does s o c i a l welfare)  becomes Pareto-inclusive.  i f a l l individual  utili-  so that the r e s u l t i n g BSSWF  Suppose also that the evaluation of s o c i a l states  i s i n d i v i d u a l i s t i c (or w e l f a r i s t ) ;  i.e.,  the u t i l i t y vector u p r e v a i l i n g at  the state is the only information used i n the evaluation.  Also suppose that  the evaluation takes the form of a continuous ordering of u t i l i t y vectors. Then, Debreu's (1959;56) representation theorem i s applied to get the BSSWF, W(u).  Pareto-inclusiveness implies that W i s monotone increasing i n u. Recalling that the ADD measure and the HB measure evaluate the states of  the economy numerically, they generate orderings of the u t i l i t y vectors where the u t i l i t y vectors with smaller amounts of waste are ranked higher given the reference bundle p or the reference u t i l i t y vector u . 0  5  I t may, therefore,  be i n t e r e s t i n g to ask whether these measures are Pareto-inclusive^ and i n d i vidualistic; i . e . ,  whether they work as a kind of BSSWF.  The f i r s t question  6.  may be answered e a s i l y . ADD measure.  F i r s t , notice the d e f i n i t i o n (2) of the  Suppose that u  a  i s preferred to u  b  i n terms of the Paretian  p a r t i a l ordering, then S(u ) i s a subset of S ( u ) . a  Noting that production  b  p o s s i b i l i t i e s are fixed, Q(u ) i s a subset of Q(u ) and hence r ( u ) < r ( u ) . a  b  a  b  In the case of the HB measure, Pareto inclusiveness d i r e c t l y follows from the nondecreasingness of the expenditure function with respect to u (see Diewert (1982;541)) and i t s d e f i n i t i o n  (3).  The other question i s more d i f f i c u l t to solve.  The ADD measure r =  r(u,p) becomes a function of both u and p, so i t cannot be i n d i v i d u a l i s t i c ; i.e.,  i t i s always affected by the choice of B, which i s not related to i n d i -  viduals' welfare.  We extend the concept of an ' i n d i v i d u a l i s t i c ' evaluation  by saying that r i s o r d i n a l l y i n d i v i d u a l i s t i c i f and only i f the ordering of u t i l i t y vectors induced by r for given p i s not affected by the choice of p. This d e f i n i t i o n i s formalized as follows:  (5)  r(u ,p ) > r(u ,p ) iff r(u ,p ) > r(u ,p ) a  a  b  a  a  b  for a l l p  b  a  >0  N  b  and p  k  The p r o f i t function ir , k = 1 , . . . , K i s defined as  (6)  i r ( p ) = max {p y k  where p > 0 „ .  T  x  : y e S }, k  k = 1,...,K,  b  > 0 . N  7. The regularity properties of the p r o f i t function are summarized i n Diewert (1982;580-1). We assume below that the production p o s s i b i l i t i e s preferences are quasiconcave.  sets are convex and  Then, (1) i s equivalent to the following dual  max min problem:  (7)  L  A D D  ( u , p ) = r(u,p) = max m i n r  {r(1-p p)+E ^ p x +E ^ ir (p) T  p > Q  T  h  h  k  1  k  1  - E f m (p,u )}, h  h  1  h  The proof i s an a p p l i c a t i o n of the Uzawa ( 1958;34)-Karlin (1959;201) Saddle Point Theorem (see Appendix I I ) .  If we further assume that r i s twice  7  continuously d i f f e r e n t i a b l e at the relevant values of u and p, then (5)  is  equivalent to requiring u to be separable^ i n r ( u , p ) ; that i s , r(u,p) satisfies  (8)  8(| 7! -)/ap = 0 for a l l i , j = 1 , . . . , H and a l l n = 1 , . . . , N . ou. ou . n r D JL  E  We assume that the f i r s t order necessary conditions for the max min problem (7) are equalities and define the solution as (r^,p^).  Then the well-known  envelope theorem implies that or/3u^ = - dm (p^,u^)/9u^, i = 1 , . . . , H . 1  s t i t u t i n g i t into (8) and using the r e l a t i o n :  9 m (p,u^)/9u^3p 2  1  ffl  =  [ d x ^ ( p , y ) / 3 y ^ ] [ d m ( p ° , u ^ ) / 3 u ^ ] for i = 1 , . . . , H and m = 1 . . . , N , we have 1  r  (9)  E.I [axi(p yJ)/ay.-8xj(p ,y5)/ay.]Op2/8P ) = 0 0  1  0  f  ll  for a l l i , j = 1 , . . . , H and a l l n = 1 , . . . , N ,  Sub-  8.  where x^"(p,y.), m i for  i = 1 , . . . , H i s the ordinary demand function for the nth good  the i t h consumer and y? = m ( p ° , u ^ ) . 1  Conditions (9) are s a t i s f i e d  i f Gorman's (1953;73) r e s t r i c t i o n on preferences i s s a t i s f i e d ;  either  i.e.,  preferences are quasi-homothetic and t h e i r Engel curves are p a r a l l e l to each other,  (since the f i r s t term i n the left-hand side of (9) i s 0 for a l l i , j , m )  or i f the production p o s s i b i l i t i e s i s 0 for a l l m and n).  sets are linear (since the second vector  (9) has the following meaning:  when we increase any  one reference good P , then the s c a r c i t y of the nth good increases so that n  the system of shadow prices associated with (7) p^ changes,  and t h i s change  must be orthogonal to the difference of the gradients of the Engel curves for any two consumers at the optimum.  This condition does not seem to me to be  s a t i s f i e d globally except for the two cases above l i s t e d . We now turn to the HB measure L  .  By the same token as the ADD  HD  measure, L „ ( u \ u ^ , p ^ ) n  separable i n L  U D  HD  .  i s o r d i n a l l y i n d i v i d u a l i s t i c i f and only i f u^ i s  Remember that u^ i s one Pareto optimal u t i l i t y a l l o c a t i o n  and p^ i s i t s supporting price vector.  Therefore p^ is a function of u^ (and 1  other parameters of the general equilibrium) so that s e p a r a b i l i t y of u equivalent to the condition.  (10)  ( — ^ f / — * H f ) /duf = 0 f o r a l l i , j = 1 , . . . , H and a l l h = 1 , . . . , H . u. 9u. r 3  ^HB i 0 1 1 „ 1 = - 3m (p ,ui)/du{. Substitute t h i s ou. l (10) and we find the following equivalent conditions:  Using d e f i n i t i o n (3),  into  is  9.  f o r a l l i , j = 1,...,H and a l l h = 1,...,H.  C o n d i t i o n s (11) seem analogous t o (9), except f o r t h e d i f f e r e n c e between dp°/dp i n (9) and d^P/bvP i n (11). m n m n  The former i s t h e change o f t h e s u p p o r t  p r i c e s o f t h e A l l a i s - D e b r e u - D i e w e r t optimum w i t h r e s p e c t t o an i n c r e a s e o f the n t h good i n t h e r e f e r e n c e bundle,  w h i l e t h e l a t t e r i s t h e change o f t h e  support p r i c e s o f t h e r e f e r e n c e P a r e t o o p t i m a l a l l o c a t i o n w i t h r e s p e c t t o an i n c r e a s e o f t h e u t i l i t y o f t h e h t h household.  T h e r e f o r e , as i n t h e ADD  measure, t h e r e does n o t seem t o e x i s t p l a u s i b l e c o n d i t i o n s t o g u a r a n t e e t h e HB measure t o be o r d i n a l l y i n d i v i d u a l i s t i c except  f o r t h e two c o n d i t i o n s  c i t e d above; i . e . , Gorman's p r e f e r e n c e r e s t r i c t i o n o r l i n e a r  production  possibilities. Up t o now we have l e a r n e d t h a t b o t h measures a r e P a r e t o i n c l u s i v e b u t not i n d i v i d u a l i s t i c i n g e n e r a l .  The c o n d i t i o n s n e c e s s a r y  t o make w e l f a r e  p r e s c r i p t i o n s by t h e ADD measure o r d i n a l l y i n d i v i d u a l i s t i c a r e as s t r i n g e n t as those needed by t h e HB measure.  However, t h e economic i m p l i c a t i o n s o f t h e  two measures a r e c o m p l e t e l y d i f f e r e n t .  The ADD i n d e x measures pure t e c h n i c a l  e f f i c i e n c y i n terms o f t h e r e f e r e n c e bundle o f goods, and t h e HB  index  measures t h e l o s s o f b o t h e f f i c i e n c y and e q u i t y by i n d i c a t i n g t h e monetary v a l u e of t h e d i f f e r e n c e between t h e s o c i a l optimum and t h e observed equilibrium.  Although  based on pure e f f i c i e n c y c o n s i d e r a t i o n s , u s i n g t h e ADD  measure t o rank s o c i a l s t a t e s means t h a t i m p l i c i t l y i t i s b e i n g used as a measure o f s o c i a l w e l f a r e ( i n s t e a d o f as j u s t an e s t i m a t e o f t h e r e s o u r c e a l l o c a t i o n waste o f one observed  e q u i l i b r i u m ) , and as I have shown, t h i s  10. method of valuing s o c i a l states i s affected by the choice of reference bundle of goods.  Therefore, to add equity aspects to the ADD measure, we have to  choose a reference bundle so that goods which are s o c i a l l y valuable are weighted more heavily.  However, i t i s d i f f i c u l t to determine what these  goods are, and what weights s h a l l be attached to them.  In contrast, the HB  measure i s a sum of money-metric scaling u t i l i t y functions and i t has a natural interpretation as a BSSWF, provided a reference price vector fixed.  is  Another drawback of the ADD measure i s that i t cannot be an  appropriate welfare indicator i f there i s technological change ( i . e . ,  it  is  not w e l f a r i s t i n the sense that i t depends on technological parameters). HB measure i s free of t h i s defect, (see Section  i f the reference price vector i s  The  fixed  5.2)  Let us compare these measures from another viewpoint.  Are these  measures useful when the shadow price vector does not e x i s t because of nonconvexities or externalities?  We w i l l show i n the l a t e r chapters of t h i s  essay that the ADD measure i s a very powerful tool to analyze deadweight loss under such market imperfections.  It seems that we can also use the HB  measure equally well to study deadweight loss i n such circumstances.  When we choose a reference Pareto optimal a l l o c a t i o n u^,  we find both the optimal shadow prices p^ for priced goods and the optimal demands q^ of external goods or nonpriced goods.  A l l we need i s to compare  the sum of the negative of the equivalent variations m^(p^,q^,u^) - m (p ,q , u ) , where m (p,q,u ) i s a r e s t r i c t e d expenditure function h  (see  Diewert (1986;170-6)). Note that the c a l c u l a t i o n of the two measures necessitates global computation of the optimal equilibrium which is very d i f f i c u l t to implement empir-  11.  ically.  Therefore i n t h i s essay, we concentrate on the study of approximate  measures of welfare.  In the following section, we compare the approximate  ADD measure and HB measure and discuss which one i s more implementable i n empirical research.  1-2.  The A p p r o x i m a t i o n Approach t o the Measurement o f Waste  This section i s devoted to an introduction to our approximation approach to the measurement of waste.  We f i r s t derive a second order approximation to  the ADD measure of waste (1).  This approximate measure depends on the eco-  nomic environment and types of d i s t o r t i o n s . are complete,  We assume i n i t i a l l y that markets  technologies are convex and that the only source of  i s i n d i r e c t taxes levied on consumers.  distortions  Extensions of these assumptions are a  main theme of the l a t e r chapters, so that we only work with the prototype model in t h i s chapter.  Given these assumptions,  (1) i s equivalent to  (6).  At t h i s point, we use the concept of the overspending function B which w i l l be f u l l y u t i l i z e d i n t h i s essay which is defined as  _ „  .  p  B(q,P,u) = E  H h = 1  h .  »  r « H T " - h  m (q u ) - C (  h  h = ( )  q x  n  K  k ,  »  - E _ » (P) • k  0  In Appendix I, B i s restated with i t s economic interpretation and i t s properties are summarized. concisely as  (12)  Using the d e f i n i t i o n (A.1), (7) may be rewritten  follows:  r ° = max  min  {r(1-p p) N T  n  useful  B(p,p,u )}. 1  12. Using the Uzawa (1958) - K a r l i n (1959) Theorem i n reverse,  (12)  i s also  equivalent to:  (13)  - max  {B(p,p,u ) 1  : p p > 1}. T  N  If  (p°,r°)  solves (12),  p ° solves (13) with r ° being i t s  associated  Lagrangean m u l t i p l i e r . In order to obtain a second order approximation to r^, we assume: (p^,r^) solves (12);  (ii)  hold with equalities  so that p ° » 0 „ ; ( i i i )  d i f f e r e n t i a b l e at ( p ° , p ° ) ; sufficient  the f i r s t order necessary conditions for  (iv) Samuelson's  (i)  (12)  B ( q , p , u ) i s twice continuously 1  (1947;361) strong second order  conditions hold for (13) when the inequality constraints are  replaced by e q u a l i t i e s . Let us consider the following system of equations i n N+1 unknowns p and r which are functions of a scalar variable z, for 0 < z < 1:  (14)  - V B(p(z) + t z , p(z), u )  (15)  1 - p(z) B = 0.  1  q  p  + t z , p(z), u ) 1  - r(z)p = 0,  T  When z = 0, (12)  - 7 B(p(z)  i f p(0)  (14) and (15) coincide with the f i r s t order conditions for = p ° and r(0) = r ° .  Suppose p(1)  producer prices normalized by (15) i n d i r e c t tax rates t .  Setting r(1)  = p  1  i s the set of observed  i n a tax-distorted equilibrium with = 0, when z = 1 (14) i s then the set of  equations characterizing the equality of demand and supply i n the taxdistorted equilibrium.  If we assume that appropriate lump-sum transfers  13. from the government to consumers are chosen, then there exist budget constraints for the H consumers compatible with (14) and (15).  From these  equations, i t i s also the case that s a t i s f a c t i o n of the government budget constraint i s implied. Let us d i f f e r e n t i a t e  B (16)  2  qq  (14) - (15) t o t a l l y with respect to z.  + B ,  P'U)  Z  PP  B  We have  t qq z  r' (z)  z 2 where q = p + tz i s the f i r s t set of arguments for B and B ^ = B(p(z) 1 . . z + t z , p(z), u ) for I,} = q , p , u . Note that B = 0„ „ . Also note that the qp N «N left-hand side matrix of (16) and Woodland (1977)). (iii),  is non-singular by assumption (iv)  (see Diewert  Therefore, using the d i f f e r e n t i a b i l i t y assumptions  by the I m p l i c i t Function Theorem there exist once continuously  d i f f e r e n t i a b l e functions p(z) and r(z) at z close to 0 that s a t i s f y (15).  (17)  We show i n Appendix III that the following equation i s  - r'(z) = - z t  T  (14) and  satisfied.  BJi (p'(z) + t ) . qq  We readily have  (18)  from (17).  r'(0)  = 0  Using (17),  equation follows.  i t i s shown i n Appendix IV that the following  14. (19)  - r"(0) = - (p'(0) + t )  T  B J (p'(0) + t) - p ' ( 0 )  T  B  0  p'(0) > 0  where the inequality comes from the concavity of B with respect to q and p (see Appendix I ) .  A second order Taylor approximation to the ADD measure i s  given by (noting that r(1) = 0),  (20)  L  A D D  = r(0) - r(1) - MP'(0)  T  where we use (18) and (19).  =  B ° p'(0) + [p'(0) + t ]  T  B ° [p'(0) + t]}  > 0,  Equation (16) i s used to compute p ' ( 0 ) .  Information we need to evaluate (ii)  r(0) - (r(0) + r ' ( 0 ) + J,r'"(0)) =  (20)  is:  (i)  the set of i n d i r e c t taxes t,  the second order derivatives of the overspending function with respect  to prices which equals the producers' aggregate substitution matrix and the consumers' aggregate compensated substitution matrix respectively,  evaluated  at the optimum equilibrium. The remarkable advantage of t h i s approximation approach i s that i t can be implemented from the derivatives of the overspending function evaluated at the optimum equilibrium, so that we need not know global functional forms for u t i l i t y and production functions.  However, as long as we must know the  derivatives at the optimum as i n (20),  we must actually know the optimal  prices so that we must compute the optimum or we must depend on some 'guessing'  process about the values at the optimum.  Harberger (1964) suggested  replacing these (unobservable) derivatives by those which are evaluated at the observed distorted equilibrium, since they can be calculated using data p r e v a i l i n g at the observed equilibrium.  This method can be j u s t i f i e d more  15. rigorously by Diewert's ( 1976; 118) Quadratic Approximation Lemma which showed that the approximation  (21)  L  = r(0) - r(1) = r(0) - {r(0)  A D D  + Jjr'tO) + J j r ' d ) }  i s also exact as the approximation (20) when the functional form i s quadratic (see also Diewert (1985(b);238)).  Evaluating (16) at z = 1 and using  (17),  we can show that - r ' ( 1 ) i s i d e n t i c a l with -r"(0) i n (19) except that a l l the relevant functions are evaluated at z = 1; i . e . , -r'(1) and  at the observed equilibrium;  i s nonnegative due to the semidefiniteness  consumer substitution matrices.  (22)  L  -Jftip (1) 1  A D D  £  T  p'(1)  properties of the producer  Using also (18), we find  + [p'(1) + t ]  T  B g [p'(1) + t]} 2 0. g  This approximation uses only information observable at the prevailing equilibrium as Harberger o r i g i n a l l y required, so that i t i s highly valuable in  empirical a n a l y s i s . The  next task i s to compute an approximation of the HB measure for the  same economy and compare i t with the approximation of the ADD measure.  To  begin with, we must c l a r i f y which reference optimal equilibrium to pick from a set of Pareto optimal allocations to calculate the HB measure or i t s approximations.  According to Negishi's (1960) theorem, every competitive  equilibrium i s a solution of the maximum of a linear s o c i a l welfare function H h h T T.^-^ f for some set of weights a = (a^,...,a^) given resource constraints a  and  production p o s s i b i l i t i e s of the economy, where i t i s assumed that f* ,  1 , . . . , H are concave functions.  1  h=  In our model, this means that for some vector  16. a, a perfectly competitive equilibrium i s a solution of the following programming problem:  (23)  Max {E ? a f (x )s.t. x ,y h  h  k  h  h  h  1  E ^ x N  E j ^ y * + E^x* ; 1  y  k  e S , k  k = 1,...,K}.  Using the Uzawa-Karlin Saddle-point Theorem using the d e f i n i t i o n (4), (A.1), we can rewrite (23) as follows  (6) and  (the c a l c u l a t i o n is analogous to the  derivation of (7) i n Appendix II):  (24)  Max Min  {a u T  B( p,u)}. P(  N  We assume that (i)  (u^,p^) solves (24),  (24) hold with equality so that p ° » 0 , N  (ii)  the f i r s t order conditions for  (iii)  B i s twice continuously  d i f f e r e n t i a b l e at the optimum, and (iv) B ^ + B ^ i s negative d e f i n i t e . qq PP  assumptions  (i) and ( i i ) ,  we find the f i r s t order conditions for (24)  (25)  a = 7 B(p,p,u),  (26)  - V B(p,p,u) - V B(p,p,u) = 0.  From  are:  Condition (26) i s the equality of demand and supply at the optimum while  (25)  i s the rule to equate the marginal s o c i a l importance of each person to the inverse of his marginal u t i l i t y of income (see Negishi (1960)).9  Note that  the solution depends on a which i s equivalent to picking a reference  17.  equilibrium.  We have to pick one reference equilibrium from various  competitive e q u i l i b r i a corresponding to various a.  Varian (1974,  1976)  persuasively discussed the welfare significance of the equal d i v i s i o n equilibrium, which i s a perfectly competitive equilibrium obtained from the equal d i v i s i o n of i n i t i a l endowments across i n d i v i d u a l s .  Varian (1976),  following the approach of Negishi (1960), also examined the r e l a t i o n s h i p between his theory of fairness and more t r a d i t i o n a l welfare economics based on the concept of a s o c i a l welfare function, which we followed i n t h i s section.  By Negishi s theorem, the equal d i v i s i o n equilibrium i s 1  also  characterized as a solution to a nonlinear programming (23) for some choice of a.  By finding this a and associated reference price vector p^, we can  find the HB measure. We now compute the second order approximation to the HB measure around the optimal equilibrium i n an analogous way as we computed the approximation to the ADD measure.  F i r s t we construct a z-equilibrium:  (27)  V B(p(z) + t z , p(z), u(z))  (28)  - 7 B(p(z) + t z , p(z), u(z))  u  g  When z = 0,  = a + Xz ;  - V B(p(z) + t z , p(z),u(z)) = 0. p  (27) and (28) coincide with the f i r s t order conditions for  the maximum of s o c i a l welfare (25) and (26), s p°.  When z = 1,  i f we define u(0)  = u ° and p(0)  (28) i s a set of equations to show the market clearing  conditions at the tax-distorted equilibrium, i f u(1) = u^ and p(1) the values p r e v a i l i n g at the observed distorted equilibrium.  = p^ are  If we assume  that the l e v e l of lump-sum transfers from the government to consumers are  18.  appropriately chosen, there e x i s t budget constraints for consumers compatible with (27) and (28).  (28) and these budget constraints imply the budget  balance of the government.  When z = 1,  (27) quantifies the  d i s t o r t i o n s at the observed equilibrium; i . e . ,  'equity'  -X^ shows the difference  between the marginal s o c i a l importance of the hth person and the inverse of his marginal u t i l i t y of income.  It must be noted that both a and the  marginal u t i l i t y of income are not invariant to a monotone transformation of f^(x^).  However, they are adjusted proportionally so that (25) i s v a l i d .  We  must also adjust X^ proportionally to h's marginal u t i l i t y of income and a so that (27)  is  valid.  Now d i f f e r e n t i a t e  3  ,  uq  uu  (29)  (27) and (28) with respect to z;  B , qu'  B  Z  Z  qq  + B  Z  pp  u' (z)  -X + B t uq  P' (z)  B t qq  z 2 where B ^ = V^B(p(z) + t z , p(z), u(z)) 0  H)<N  .  Z  for i , j = q , p , u .  Note that B  z u p  =  Assumptions ( i i i ) and (iv) guarantee, v i a the I m p l i c i t Function  Theorem, that once continuously d i f f e r e n t i a b l e functions u(z) and p(z) satisfying  (29) e x i s t at z close to 0.  sides of (29) and using property ( i i i )  T T Premultiplying [ 0 „ , p(z) ] to both H  of the overspending function i n  Appendix I, we can derive  (30)  V m (p(z) + t z , u ( z ) ) u£ (z) h  u  h  = z [ t B ( p ' ( z ) + t) + t T  z  VJVJ  T  B  z  VJ u  u'(z)],  1  19. analogously to the derivation of (17) z =0,  i n Appendix III.  Evaluating (30)  at  we get  (31)  h=lV  [  h ( p  °' h h U  ) u  ( 0 )  =  °"  Analogously to the derivation of (19) i n Appendix IV, we next  differentiate  (30) with respect to z, and evaluate at z = 0 to obtain  (32)  U'(0)  T  + E ! V  B ° u'(0)  h  u  - P'(0)  Premultiplying (29)  h { P  1  T  °' h U  ) U  h  (  0  )  =  t T B  qq  ( p  '  ( 0 )  +  fc)  B Ju'(O).  toLp'(0) ] and adding the T  evaluated at z = 0 by  n  resulting i d e n t i t y to (32),  we have  u'(0) B V(0) + E " V m (p°,u°)u^(0)  (33)  T  = - p'(0) B °p'(0)  h  h  u  1  T  u  - [p'(0) + t ]  T  p  B ° [p (0) + t] 1  g  > 0.  A second order Taylor approximation to the HB measure (3) at z = 0 i s as follows:  ( 3 4 )  L  HB  5  -  E  h=lV °' 2 h h(p  E  Substituting (31) and (33)  u  )u  (0)  " H[u-(0)V\r(0)  h-lV °' S> h(p  u  U  h  into (34) we have  ( 0 )  ^  J  +  20.  (35)  L £ - J i { p ' ( 0 ) V p ' ( 0 ) + [p'(0) + t ] B ° [ p ' ( 0 ) + t]} HB pp qq T  > 0.  To compute (35), we could again replace B?^ by B ^ i n (35) and (29) since the B^j are observable,  again following H a r b e r g e r .  10  I t i s interesting to com-  pare (35) with the second order approximation to the gain i n s o c i a l  welfare  using the linear welfare function i n moving to the optimum from the distorted equilibrium, E " a [ f ( x ) - f ( x h  h  ( 3 6 )  l  L = HB £  h  h 0  h  1  +  J  h 1  ) ] = L ^ . We find that  4»'(0) B Su'(0) > L T  u  H B  where the t i l d e shows i t i s an approximation of the o r i g i n a l measure and the inequality comes from the positive semidefiniteness of B ^ , which i s implied by the concavity of the u t i l i t y functions.  According to Varian (1976;257),  the l i n e a r u t i l i t y function does not count the problem of equity.  Therefore,  when moving from the equitable equilibrium to market distorted equilibrium, only measures efficiency  loss and does not evaluate i t s equity l o s s .  t h i s sense, L may be taken as a lower bound of the welfare r  change.  In  11  Li  However,  (36) shows that L  H R  i s even smaller than L ^ .  This i s because, with  diminishing marginal u t i l i t y of income, increasing the inequality i n terms of u t i l i t y (or r e a l income) holding the (weighted) sum of u t i l i t y constant tends to increase the aggregate expenditure necessary to a t t a i n the reference u t i l i t y allocation.  This problem of inequity i n the HB measure may not a r i s e  i f we adopt money metric u t i l i t y scaling so that u 1,...,H. 2 1  H, = L  H B  = m (p^,u ), n  h  h  h=  i f t h i s i s the case, v B ( p ° , u ) = 1„ and B = 0 „ „ so that u H uu H«H and = L ^ . With t h i s assumption we can regard the HB measure as 0  B  21 summing the change of u t i l i t i e s of i n d i v i d u a l s ; i . e . ,  i t is a u t i l i t a r i a n  measure of welfare. We now compare the empirical implementability of L and L  H B  i n (35).  A Q D  i n (20) and (22)  Though (20) and (35) look i d e n t i c a l , t h e i r meanings are  completely d i f f e r e n t .  First,  the substitution matrices are evaluated at  distorted l e v e l of u t i l i t i e s i n (20) while they are evaluated at optimal l e v e l of u t i l i t i e s i n (35). of equations,  (16) and (29).  was already stressed,  Second, p'(0)  i s calculated from d i f f e r e n t sets  The f i r s t difference i s i n e s s e n t i a l ,  since,  as  we replace these matrices with matrices evaluated at an  observed distorted equilibrium.  However, the second difference matters  1 1 the substitution matrices, B , B , tax rates t and pp qq reference bundles 8 are a l l information required to compute p'(1) and hence  crucially.  (22).  In (22),  In (35),  we need both the substitution matrices B \ B ^ and income PP qq effect matrices B ^, tax rates t and the d i s t r i b u t i o n a l d i s t o r t i o n parameters X so that the informational requirements are much higher.  Though i t  is  1 1 possible to calculate B and B from l o c a l information on ordinary demand pp qq curves and supply curves at the distorted equilibrium, we have to know the 1 1 1 1 marginal u t i l i t y of income V B(p +t,p ,u ) to compute X from (27) or B from ordinary demand curves.  Even i f we adopt the money metric scaling convention  using the optimal p r i c e s , this does not give information on the marginal u t i l i t y of income at the observed equilibrium, and t h i s i s what we r e a l l y require. p  1  + t,  If we adopt money metric scaling at the observed distorted prices 1 . then B i s easy to calculate since v B(p qu u  case we also have B  UU  = 0„ „ . n*H  0  anymore.  In t h i s  However, we s t i l l cannot compute X from (27)  since now we do not have 1„ = v B ( p ° , p ° , u ) ; M  1 1 1 + t , p ,u ) = 1 „ . H  i.e.,  a i s not a vector of ones  U  McKenzie (1983, chapter 3) studied the methodology for c a l c u l a t i n g  22.  the money metrics, and he c o r r e c t l y pointed out that the marginal u t i l i t y of income i s not an operational concept without knowing the u t i l i t y function. His approach i s based on normalizing the marginal u t i l i t y of income at one price vector, but i n our case, we have to know i t at two sets of prices p and p + t,  and we cannot normalize twice.  Diewert (1984;36) already pointed out  that his approximate HB measure depends on the hypothetical income vector at the optimum which i s d i f f i c u l t to obtain.  Though we adopted a different  method of approximation, the same problem seems to occur by the measurement of marginal u t i l i t y of income (more rigorously, the difference between the marginal s o c i a l importance and the inverse of the marginal u t i l i t y of income), instead of the measurement of hypothetical income. these observations,  In l i g h t of  we must conclude that the approximate HB measure lacks  empirical o p e r a t i o n a l l y without a knowledge of the o r i g i n a l u t i l i t y functions  whereas the ADD measure i s free from this problem.  Note that  this  c r i t i c i s m w i l l also apply even i f we compute the waste using the Bergson-Samuelsonian s o c i a l welfare function.  It i s c h i e f l y for t h i s reason  that we adopt the ADD measure as our welfare c r i t e r i o n .  Needless to say,  however, the informational advantage of using the ADD measure does not mean that i t i s a superior measure to either the HB measure or the BSSWF.  As long  as we can measure the difference between the weight of a linear BSSWF and the inverse of the marginal u t i l i t y of income, the same type of analysis as  is  presented i n Chapters 2 and 4 for the ADD measure can be carried out using the HB measure or a BSSWF. We have compared the informational requirements for the approximations of the ADD and the HB measures to be empirically computable, and i n this context we have found a remarkable property of the ADD measure:  i t i s comput-  able from l o c a l information on supply curves and ordinary demand curves at the observed equilibrium.  A natural defect of our approximation approach i s  that the approximation might deviate from i t s true value considerably when the 'gap' between two e q u i l i b r i a i s large.  The numerical general equilibrium  approach by Shoven and Whalley (1972, 1973,  1977)  chooses an alternative way  to compute e q u i l i b r i a d i r e c t l y corresponding to various tax and expenditure policies  so that a more exact welfare evaluation seems a v a i l a b l e .  However,  an obvious drawback of the numerical general equilibrium approach i s that we must have information on global functional forms of u t i l i t y and production functions.  In contrast,  our approximation approach requires only second  order derivatives of these functions evaluated at the observed equilibrium. As an important c o r o l l a r y of t h i s fact, derived from any set of f l e x i b l e the observed equilibrium.  our approximate measure can be  functional forms using information based on  On the contrary, i n the numerical general e q u i l i -  brium approach, very r e s t r i c t i v e functional forms are adopted to make global computation possible,  and these r e s t r i c t i o n s are easily rejected i n econo-  metric tests using more general functional forms (see Jorgenson (1984;140)). Moreover, the approximation approach does not involve any numerical computations that are more complicated than a single matrix inversion, whereas there are often substantial numerical d i f f i c u l t i e s equilibria.  involved i n computing general  Therefore, these two competing programs have their own pros and  cons so that i t would be d i f f i c u l t to judge which one i s u n i v e r s a l l y superior to the  other.  1 3  The measurement of waste i s prominently a p r a c t i c a l subject.  As i s  pointed out by Harberger (1964;58), the comparison of welfare measures i s only constructive way to give a policy prescription under the 'nth best'  the  24.  situation,1* i . e . , feasible  by comparing the amount of waste corresponding to various  p o l i c i e s we can give a ranking among them even i f there are various  other d i s t o r t i o n s .  However, as long as we use approximations, we cannot  avoid approximation errors which might cause erroneous policy assessment. For example, Green and Sheshinski (1979) pointed out that Harberger's t r i a n g l e approximation may change considerably by changing the choice of approximation point.  In t h i s context, they c r i t i c i z e d Feldstein (1978) who  measured the net benefit of c a p i t a l income tax reform by comparing Harberger's (1964) measure at two taxed e q u i l i b r i a . noted that there e x i s t differences  Green and Sheshinski  between Feldstein's Harberger measure and  a second order approximation of income gain evaluated at the i n i t i a l tax equilibrium.  A s i m i l a r c r i t i c i s m also applies to Turunen (1986) who applied  the approximate ADD measure for the numerical assessment of gains from t a r i f f reform.  It would be possible to derive Green-Sheshinski l i k e approximate  gains formula for tax reform which i s a second order approximation to the change of the ADD measure evaluated at an i n i t i a l tax e q u i l i b r i u m .  1 5  However, due to the complexity of the r e s u l t i n g formula, we have omitted derivation.  this  Therefore, t h i s approximation error may lead to reversals i n the  true ranking of p o l i c i e s based on the exact amount of waste. We have to admit a dilemma that we cannot get an exact welfare measure for various sets of p o l i c i e s either by approximation or by equilibrium computation while we have to reach some decision on the choice or reform of economic p o l i c i e s .  In the second best theory approach originated by Lipsey and  Lancaster (1956), recommendations for p o l i c i e s or t h e i r p a r t i a l reforms are given using the programming method under the constraint that some of the optimality conditions are not met, or some of the instruments to a t t a i n the  f i r s t best i s r e s t r i c t e d .  This approach has successfully derived many i n t e r -  esting r e s u l t s i n optimum taxation theory, piecemeal p o l i c y recommendations and cost-benefit  analysis.16  drawbacks of this approach.  However, we have to note at least two basic First,  i n contrast to the f i r s t best s o l u t i o n ,  general second best solutions cannot be decentralized i n a simple p r i n c i p l e (see Guesnerie (1979)) so that the p o s s i b i l i t y of meaningful policy recommendations i s quite r e s t r i c t e d except under rather s i m p l i f i e d second best situations as i n an optimal taxation economy .  Second, since most of the  second best r e s u l t s depend on l o c a l necessary conditions for optimality, they suffer from t h e o r e t i c a l c r i t i c i s m s from the viewpoint of general equilibrium theory.  As i s shown by Foster and Sonnenschein (1970) and Hatta (1977),  multiple e q u i l i b r i a and i n s t a b i l i t y can e a s i l y occur i n a well-behaved economy with t a x - d i s t o r t i o n s .  Harris (1977) pointed out that the  sufficiency  of the necessary conditions for second best optimality depends on the t h i r d order derivatives so that the i n t e r p r e t a t i o n of these sufficiency is not easy.  In contrast,  conditions  tax reform approach due o r i g i n a l l y to Meade (1955)  avoids the problem by r e s t r i c t i n g i t s attention to the l o c a l area around the observed d i s t o r t e d equilibrium.  Various authors, represented by D i x i t (1975)  and Hatta (1977), derived sufficiency conditions for welfare improvement by some p o l i c y changes.  Unfortunately, these conditions depend on many  r e s t r i c t i v e assumptions.  E s p e c i a l l y , the assumption that the policy maker  can change the set of taxes incrementally i s often i r r e l e v a n t , since reform a l t e r n a t i v e s are discrete changes of taxation.  its  By the same token,  it  i s often the case that the reform alternatives are i n s t i t u t i o n a l l y r e s t r i c t e d to the ones which are short of f u l l y s a t i s f y i n g the s u f f i c i e n t  conditions.  In these cases, t h i s approach cannot t e l l anything about the ranking of  26. policies,  but our approach can.  Furthermore, the sequence of l o c a l  improvements may not converge to global optimum, but may stay on a l o c a l optimum or some stationary point.  These problems seem to give l i m i t a t i o n s on  the use of l o c a l optimality or improvement conditions for policy recommendations. Considering these defects, we seem to be obliged to conform to a convent i o n a l view on second best; i . e . ,  i f conditions on propositions are met,  implement the prescribed p o l i c y .  If the actual economic situations do not  coincide with the conditions, or we do not have enough information to judge whether i t i s actually the case, we cannot t e l l anything from the second best theory.  P a r t i c u l a r l y , even i f conditions are not met for positive  best propositions,  second  this does not j u s t i f y the status quo i n any way,  since  even i n this case, the deadweight loss of the economy could be too large to neglect.  Following Harberger (1964), "The Economics of nth Best," to measure  the deadweight loss associated with the economy's being i n any given nonoptimal position i s of high p r a c t i c a l importance when we cannot know how to make the best of a bad s i t u a t i o n .  1-3.  A Diagraamatic Exposition In t h i s section we i l l u s t r a t e diagrammatically the ADD measure and the  HB measure and their approximations using a simple model i n order to c l a r i f y the i n t u i t i v e content of the discussions  i n the previous  section.  We assume that there i s one good and one production factor  (labour).  One aggregate firm produces the good y using labour v according to the production function y <. g(v).  We also assume that there i s a single consumer  who enjoys u t i l i t y u from the consumption of the good x and l e i s u r e L by  27.  means of the u t i l i t y function f ( x , L ) .  The i n i t i a l endowment of labour i s v  and there i s a zero endowment of the good. We f i r s t specify the tax-distorted observed equilibrium. labour as numeraire so that i t s p r i c e , w = 1 .  We choose  We assume that there i s a  s p e c i f i c tax t on the good levied for consumption so that i t s producer price i s p whereas i t s consumer price i s p + t .  It i s also assumed that the  s p e c i f i c tax revenue i s transferred to the consumer as a lump-sum transfer. Then, using the p r o f i t function w(1,p) dual to y < g(v) and the expenditure function dual to f ( x , L ) ,  the observed equilibrium i s characterized by the  market c l e a r i n g conditions for the good and labour;  (37)  V TT(1,P) - v m ( 1 , p+t,  (38)  V TT(1,P) - v m(1,p+t,u) + v = 0. W  u) = 0  w  1  1  We assume that (p ,u ) solves (37) and (38) uniquely.  From the homogeneity  properties of TT and m, we can deduce (39)  i.e.,  v + i r ( 1 , p ) + t V m , ( 1 , p + t , u ) = m( 1 , p + t , u ) ; 1  1  1  1  1  p  the budget constraint of the representative i n d i v i d u a l i s  satisfied.  Now we define the ADD measure of waste i n t h i s simple model.  We assume  that the surplus of the economy i s measured by the numeraire good, labour. Therefore,  the general primal programming problem ( 1 ) and i t s dual (7) are  s i m p l i f i e d respectively i n t h i s model as  follows:  28  ADD  ( 4 0 )  L  (41)  5  M a X  = Min  y,v,L ^ " {  ~  L  V  :  y  1  {TT(1,P) - m(1,p,u ) 1  p > Q  '  x  y  1  g  (  v  )  '  f  (  x  '  L  )  1  "  }  + v}.  A We assume that p = p > 0 i s a unique solution of the f i r s t order condition:  (42)  V TT(1, ) - 7 m ( 1 , p \ u ) A  1  P  tr  Therefore,  = 0.  tr  (40)  and (41)  can be rewritten as  (43)  v + 7 w(1, ) - V m ( 1 , p \ u ) = L  (44)  v + ii(1,p )  A  w  A  Note that (43)  1  P  w  = L  and (44)  properties of TT and m.  + m(1,p ,u ). A  & n n  A D D  1  are equivalent by using (42)  and the  homogeneity  We can i l l u s t r a t e the ADD measure of waste  diagrammatically i n F i g . 3.  The program (40)  b o i l s down to searching for a  point where the horizontal length of the lens-shaped area formed by the 1 . production p o s s i b i l i t i e s maximal.  set and the indifference curve with u = u  This maximum i s characterized by an equal slope 1/p  is  of the two  curves. In t h i s simple example, we can also express the ADD measure of waste as a more familiar Hotelling-Harberger-like c u r v i l i n e a r t r i a n g l e ABC i n F i g . 4. This can be proven as follows.  The area ABC i s defined from F i g . 4 as  (45)  ABC = ;  From t h i s we have  ABC = m(1,p +t,u )-mn,p ,u )+Tr(1,p )-ir(1,p )-tV m( 1,p +t,u ), P 1  1  A  = v - m( 1, p , u ) A  = L  1  1  A  1  + ir(1,p )  1  (from  A  1  (39))  (from (44)).  ADD  In F i g . 4, we have also drawn two triangles ABC and ABC".  ABC i s a  linear approximation to ABC using the slopes of the Hicksian demand curve and the supply curve at the optimum point whereas ABC" i s a l i n e a r approximation to ABC using the slopes of the two curves at the distorted equilibrium. These two triangles of waste (20)  correspond to the two approximations of the ADD measure  and (22)  i n t h i s simple example.  To show t h i s ,  l e t us  first  construct a z-equilibrium as i n the previous section for t h i s simple model as follows:  (46)  V i(1,p(z)) - v m(1,p(z)+tz,u ) = 0,  (47)  V ir(1,p(z)) - V m(1,p(z)+tz,u ) + v = r ( z ) , w w  1  p  1  where 0 < z < 1 and p(0) z = 0,  (46)  and (47)  = p , p(1) A  = p , r(0)  correspond to (42)  1  = L  and (43),  A  m  )  and r(1)  = 0.  When  and when z = 1, they  30. correspond to (37) and (38).  T o t a l l y d i f f e r e n t i a t i n g (46) and (47) with  respect to z we can compute r ' ( z ) .  From t h i s r ' ( 0 ) ,  r"(0) and r'(1)  are also  computable so that we can calculate the two approximations to the ADD measure of waste (20) and (21) as follows (see Appendix V ) :  th P °P  (48) 2  ^ oPoP "  t E 2  (49) 2(E  1  pp 1  PP  s  S  0  PP  O PP  1  PP  - s  1  PP  )  where E = V m( 1 ,p(z)+tz,u ) and S = V ir(1,p(z)) for z = 0,1 pp pp pp pp irv Z  2  1  z  2  As AB = t and the height of the t r i a n g l e ABC i s  0  (tE  PP  (48) equals the area ABC' while the height of ABC" i s  S  (tE  1  PP  so that (49) equals the area ABC".  (See Appendix V.)  - S ° ),  ° ) / ( E °  PP  PP  S  1  ) / ( E  PP  - S  1  PP  PP  1  )  PP  Note that the slope of  A C equals the slope of the demand curve at C while the slope of B C equals the slope of the supply curve at the point C.  Therefore, i n t h i s simple  model, our triangular expression of the deadweight loss corresponds to that by Harberger (1964) except that we allowed for nonlinear production possibilities  set.  We next turn to a diagrammatic interpretation of the HB measure of welfare and i t s approximations.  For t h i s purpose, we f i r s t have to find a  price vector which supports the s o c i a l optimum.  In a single consumer  economy, i t may be defined as the price vector which corresponds to the  31 .  u t i l i t y maximum given resource and technology constraints.  Therefore, i t  is  B B the p r i c e solution (p ,w ) to the concave programming problem below:  (50)  Max  {f (x,L)  (51)  = Max M i n u  : y ) x, v ) v H ,  p > 0 w > ( )  y < g(v)}  { u - m(w,p,u) + n(w,p) + wv}.  B  B B  We assume that an i n t e r i o r optimum point (u ,p ,w ) solves (51) uniquely with B p  B > 0 and w  > 0.  conditions for  It i s a solution to the following f i r s t order necessary  (51):  (52)  v m(w ,p ,u ) = 1,  (53)  V i(w ,p )  (54)  v + V ir(w ,p )  u  B  - v m ( w , p u ) = 0,  B  B  p 1  B  p  B  B  w  B  (  - V m ( w , p , u ) = 0. B  B  B  w  As  (53) and (54) are unchanged by a proportional change of w and p, we set  w  =1.  For t h i s normalization, we can assume that (52)  i s always met by  choosing a money-metric normalization of the u t i l i t y function at the reference p r i c e (1,p ).  Therefore we can delete (52) from the system and B  assume that (53) and (54) determine u normalization,  B and p  B from w = 1 .  (53) and (54) imply the following budget constraint of the  representative consumer for the optimum price vector (1,p (55)  Using t h i s  v + ir(1,p ) D  = m(1,p ,u ). D  D  ):  32.  Now the HB measure of welfare (3) i n t h i s simple model may be defined as follows:  (56)  L  Fig.  = m(1,p ,u ) B  H B  - m(1,p ,u )  B  B  1  5. i l l u s t r a t e s the HB measure for t h i s simple model.  This i s nothing  but a Hicksian compensating v a r i a t i o n when moving from a tax-distorted equilibrium to a s o c i a l optimum. We also i l l u s t r a t e L _ using a Hotelling-Harberger-like expression i n u  rib Fig.  6.  This figure i s the same as F i g .  4 for the Hicksian compensated  demand curve for the good V m(1,p,u ) using the tax-distorted u t i l i t y l e v e l P u^ and the supply curve v i ( 1 , p ) . We also include the compensated demand P 1  curve for the good for the s o c i a l l y optimum u t i l i t y l e v e l , Fig.  Vpm(1,p,u ).  6 corresponds to the case where the good i s normal so that Vpin(1,p,u )  i s above V m(1,p.u**).  If the good i s i n f e r i o r ,  the former curve i s below the  l a t t e r curve and i f the good changes from a normal to an i n f e r i o r good then the two curves i n t e r s e c t . above.  Using F i g .  Our results below apply to a l l cases l i s t e d  6, the HB measure can be shown to be equal to the sum of  two c u r v i l i n e a r t r i a n g l e s AFE and FBD.  To show t h i s ,  f i r s t note that L „ can D  Ho  be decomposed as follows: (57) L = {m(1,p ,u ) - m ( 1 , + t , u ) } + {m(1,p +t,u ) - m ( 1 , p , u ) } . B  H B  B  1  1  1  1  B  1  P  Substituting (39) and (55) into the f i r s t term on the right-hand side of (57),  L  H  B  may be further rewritten as  33.  (58)  However, the sum of the areas AFE and FBD, denoted as AFBCDE,  is  (59) P  1  p  1  Performing the integration i n (59) yields the expression i n  (58).  For this simple model, the triangles ABG and ABC" drawn i n F i g . 6 correspond to the approximation to the HB measure where ABG corresponds to (35) and ABC" i s i t s variant where observed information is used. shown as follows.  (60)  First,  v T(1,P(Z))  (61)  construct a z-equilibrium:  - V m(1,p(z) + t z ,  tr  v + V »(1,p(z)) w  u(z))  = 0,  tr  - v m(1,p(z) + t z , u(z)) w  where 0 <. z < 1 and p(0) = p , p(1) = p , u(0) B  z = 0,  This may be  1  = 0,  = u  B  and u(1) = u . g 1  When  (60) and (61) correspond to (53) and (54) with w = 1 and when z = 1  they correspond to (37) and (38). respect to z, we can derive u ' ( z ) .  T o t a l l y d i f f e r e n t i a t i n g (60) and (61) with From this u'(0)  and u"(0)  can also be  computed so that we can calculate the second order approximation to the HB measure (34) as (see Appendix V I ) .  (62)  34.  where E * = V J m ( 1 , p ( z ) PP PP  + t z , u ( z ) ) and S  = 7 ir(1,p(z)) for z = PP  0,1.  2  z  PP  Note t h a t E ^ and S ^ a r e d i f f e r e n t from t h e analogous  e x p r e s s i o n i n (48)  s i n c e , i n ( 6 2 ) , t h e d e r i v a t i v e s a r e e v a l u a t e d a t the s o c i a l l y optimum p o i n t B  B  (1,p ,u ).  As AB = t and t h e h e i g h t of t h e t r i a n g l e ABG i s  (tE J S ?)/([  J - S J),  (62) e q u a l s t h e a r e a ABG.  I f (62) i s f u r t h e r  approx-  imated by r e p l a c i n g t h e d e r i v a t i v e s E ° and S by t h o s e o b s e r v a b l e d e r i v a PP PP 1 •) tives E and S , t h e n t h i s a p p r o x i m a t i o n i s i d e n t i c a l t o (49) which i s a 0  trkr  trtr  suggested a p p r o x i m a t i o n of t h e ADD 6 as ABC",  measure.  (49) i s i l l u s t r a t e d i n F i g .  which i s a l s o shown i n F i g . 4.  I n t h i s s i m p l e model, t h e ADD  measure and t h e HB measure c o i n c i d e i f t h e  two p o i n t s C and D c o i n c i d e i n F i g . 6; i . e . , the ADD optimum c o i n c i d e . a r e second  (48) and  (62) (or A B C  optimum and t h e  i n F i g . 4 and ABG  o r d e r a p p r o x i m a t i o n s of the ADD  social  i n F i g . 6), which  measure, and the HB measure  c o i n c i d e i f t h e c u r v a t u r e s of the compensated demand f u n c t i o n s and the s u p p l y f u n c t i o n a t p o i n t s C and D a r e the same. t o t h e s e f u n c t i o n s depending  However, the f u r t h e r  approximations  on the d e r i v a t i v e s of t h e s u p p l y and  compensated  demand f u n c t i o n s a t t h e observed e q u i l i b r i u m c o i n c i d e f o r t h i s s i m p l e model as the t r i a n g l e ABC".  I t i s , however, c l e a r from t h e d i s c u s s i o n of t h e  p r e v i o u s s e c t i o n t h a t t h i s i d e n t i t y cannot go through f o r a g e n e r a l many-consumer model.  F i n a l l y , a l l of t h e s e a p p r o x i m a t i o n s c o i n c i d e  1 B i f V m(1,p,u ) = 7 m(1,p,u ) f o r a l l p. P  T h i s i s t h e case where t h e r e i s no  P  income e f f e c t f o r t h e good and, i n t h i s case, t h e M a r s h a l l i a n consumer's s u r p l u s c o i n c i d e s w i t h t h e ADD  and the HB measures (see H i c k s  (1946;38-41)).  35.  FOOTNOTES  1  FOR CHAPTER  1  These two measures were examined comparatively by Diewert (1981,  1984,  1985a). 2 x » 0  means that each element of the vector x i s s t r i c t l y p o s i t i v e ,  N  x > 0„ means that each element of x i s nonnegative, and x > 0„ means x > 0., —  N  N  but x T* 0 3  N>  N  —  A superscript T means transpose.  Note also that the solution to (1) may not correspond to a Pareto  optimal point.  This does not, however, contradict the Pareto inclusiveness  of the ADD measure which i s discussed i n this  section.  * See Diewert (1982;554) for the r e g u l a r i t y properties that must be s a t i s f i e d by the functions m . h  5  Most welfare evaluation methods cannot even generate orderings.  example, the Kaldor (1939)-Hicks  (1939)-Scitovsky  complete nor t r a n s i t i v e (see Gorman (1955)).  For  (1941-2(a)) test i s neither  Aggregate Hicksian (1941-2)  compensating and equivalent variations cannot be t r a n s i t i v e i f the base price i s not fixed (see,  for example, Mohring (1971;365-7) or Blackorby and  Donaldson (1985;256-7)) . 6  A widely adopted welfare measure by Diamond and McFadden (1974) can be  shown to be an equivalent v a r i a t i o n where tax-distorted prices are base prices.  Therefore, i t cannot give a consistent ranking of u t i l i t i e s  various tax schemes even i n a single-consumer economy, i . e . , inclusive. 7  across  not Pareto  See Kay (1980) and Pazner and Sadka (1980).  We also assume that the Slater constraint q u a l i f i c a t i o n condition  applies i n t h i s economy; i . e . ,  we require that a feasible solution for (1)  exists that s a t i s f i e s the f i r s t N inequality constraints s t r i c t l y .  The d e f i n i t i o n s of various concepts of separability and t h e i r economic  8  applications are surveyed i n Geary and Morishima (1973) and Blackorby, Primont and Russell (1978).  Pages 52-61 of the l a t t e r book are important for  our a n a l y s i s . With appropriate lump-sum transfers across households,  9  the budget  constraints of individuals are s a t i s f i e d and the government budget constraint i s implied by them and (26).  Combined with Negishi's theorem,  the  program (23) and i t s interpretation may be regarded as a restatement of the second fundamental theorem of welfare economics, due o r i g i n a l l y to Arrow (1951). It i s d i f f i c u l t i n t h i s case to interpret t h i s further approximation  1 0  by adopting the Quadratic Approximation Lemma in the same manner as with the ADD measure.  However, using the money-metric u t i l i t y scaling adopted l a t e r ,  we can show that - ^ { p ( 1 ) B p ' ( 1 ) + [p'(1)+t] B [p'(1)+t]} + Jau'(1) B ^ 1  T  T  pp  u'(1)  T  qq  u  i s also accurate for quadratic functions as (35) by this Lemma. If we assume that there exists a concave Bergson-Samuelsonian s o c i a l  1 1  welfare function which i s maximized at the equal d i v i s i o n equilibrium, we can show that the second order approximation to the difference of the BSSWF, evaluated at an optimum or distorted equilibrium L inequality L 1 2  R S  R S  , satisfies  2L L  The term money metric u t i l i t y was introduced into economics by  Samuelson (1974), but the concept dates back to McKenzie (1957). that m (p^,u^) h  i s s t r i c t l y increasing i n u .  given by Weymark (1985). x  1  the  h  We assumed  Its s u f f i c i e n t condition was  We also assume that m ( p ^ , f ( x ) ) i s concave i n h  h  h  for the reference price p^, but t h i s i s not guaranteed i n general.  Blackorby and Donaldson (1986).  See  Applications of money metrics to applied  welfare economics are given by King (1983) and McKenzie (1983).  Most computable general equilibrium models adopt n e o - c l a s s i c a l  1 3  market assumptions.  perfect  However, Pigott and Whalley (1982) incorporated public  goods and Harris (1984) introduced increasing returns to scale by allowing fixed costs i n numerical general equilibrium models. 1  * In contributions collected i n Harberger (1974), he applied his  methodology i n various policy assessments.  Many studies use the HB measure  or Marshallian consumer surpluses for the same purpose (see Currie-MurphySchmitz (1971)). 1 5 Needless to say, both second order approximations as well as a mean value of the two f i r s t order derivatives are exact approximations for quadratic functions. 1 6  We do not survey these studies i n this paper.  Excellent surveys were  provided by Auerbach (1985;86-118), Mirrlees (1986) and Dreze and Stern ( 1986).  38. Appendices for Chapter 1  Appendix I :  The Properties of the Overspending Function.  An overspending function, introduced into economics by Bhagwati, Brecher and Hatta (1983;608) summarizes the general equilibrium relations of an economy within one equation.  It may be interpreted as the aggregate net  expenditure of consumers facing prices q minus the aggregate p r o f i t s of firms facing prices p.  It i n h e r i t s many useful properties of expenditure functions  and p r o f i t functions which are exhibited i n Diewert (1982).  We c o l l e c t  several important properties for l a t e r use. An overspending function i s defined by  (A.1)  B(q,p,u) = E " m ( q , u )  - E " q x  h  h  1  T  h  h  h  0  - E^ iT (p) . k  0  I t has the following properties. (i) (ii)  B i s concave with respect to p and q. If B(q,p,u) i s once continuously d i f f e r e n t i a b l e with respect to q and p at ( q , p , u ) , then V B(q,p,u) i s the aggregate net consumption q  vector and -V B(q,p,u) i s the aggregate net production vector, p  (iii)  The following i d e n t i t i e s are v a l i d for any (q,p,u) i f B i s twice continuously d i f f e r e n t i a b l e at  T  (A.2)  (A.3)  q  \ u  (q,p,u):  T  °  ( ,  u  B | T  * <3» (<!.u,>/3u l  8n (q,u )/3u ), H  1  H  H  3 9 .  (A.4)  where  PP  . = V,j.B(q,p,u) for i , j = q , p , u .  Property (i)  follows from the fact that an expenditure function i s  concave with respect to prices and a p r o f i t function i s convex with respect to p r i c e s .  Property ( i i )  i s a straightforward consequence of H o t e l l i n g ' s  (1932;594) lemma and the Hicks (1946;331)-Shephard (iii)  (1953;11) lemma.  Property  i s a consequence of the l i n e a r homogeneity of an expenditure function  and a p r o f i t function with respect to p r i c e s .  Appendix I I In t h i s Appendix, we show that (1) and (7) are equivalent given quasiconcave u t i l i t y functions and convex production sets, provided the Slater constraint q u a l i f i c a t i o n holds.  In (1),  convex from the quasi-concavity of f ( x ) , h  and the i n e q u a l i t i e s are l i n e a r .  S  h  k  the set {x  : f (x ) 2 u ) n  i s also assumed to be convex  Therefore, (1) i s a concave programming  problem and the Uzawa-Karlin Saddle Point Theorem i s applicable.  Rewrite (1)  as:  (A.5)  r  0  = max  {r + mm P>0, N  T P  [E  f (x ) 2 uj h  is  h  f  y k=1 H  k  + E  H  ih - E  h=1  h = 1,...,H; y  k  H  h x - pr]  h=1  e S , k = 1,...,K}, k  40.  (A.6)  = max min {r(1-p B) + ~ N T  r  p>0  E ^ p V +£,,!!,[max  + E^^max p y T  = max min r  ^ { r ( 1 - p 8) + E ^ P ^ T  p20  1 1  :y  k  k  - p x T  h  : f (x ) > h  h  e S ]}  + E ^ir (p) k  k  k  - E " m (P,uJ)} h  h  1  using the d e f i n i t i o n s (4) and (6).  Appendix III In t h i s Appendix, we derive (17).  Premultiply both sides of (16) by  T [p(z)  ,0].  (A.7)  Using (15),  we have:  p(z) B q(p"(z) + t) + p ( z ) B p ' ( z ) = - r ' ( z ) . T  T  q  pp  1  From (A.2) and (A.4) evaluated at (q,p,u) = (p(z) + t z , p ( z ) , u ) , we have T z T z T z T p(z) B = - zt B and p(z) B = 0 „ . Substituting these equations into qq qq pp N (A.7), we have  (17).  Appendix IV In t h i s Appendix, we derive (19).  Differentiate (17) with respect to z  and evaluate at z = 0, and we have  (A.8)  - r"(0) = - t B g ( P ' ( 0 ) T  q  + t).  Next premultiply both sides of (16) evaluated at z = 0, by We obtain  [p*(0) ,r'(0)]. T  UjJ  41 .  (A.9)  - p'lOlVtpMO) qq  + t) - p'(0) B T  Adding (A.8) and (A.9), we have  V ( O ) = 0. pp  (19).  Appendix V Total d i f f e r e n t i a t i o n of (46),  o  P'(z)  S - E wp wp , -1  r'(z)  s (A.10)  2  , - E PP 2  PP 2  L  2  z 2 1 where E - = V .m(1,p(z)+tz,u ) and  (47) gives the following:  PP E  z  i  Premultiplying both sides by [ p ( z ) , 1 ] ,  S  z  wp  + p(z)S  z  PP  E  = 0,  2  2 .ir( 1 ,p(z)) for i , j , = p,w.  =  using the  + (p(z)+tz)E  2  wp  t wp  2  = 0,  ' pp  identities  '  we have  (A.11)  r ' ( z ) = ztE *(p'(z)+t). srtr  Inverting the right-hand side matrix of (A.10), we have  P'(z) = ( E  2 p  t)/(S  z p  -E  z p  )  Substituting i t into (A.11), we have  42.  (A.12)  r'(z)  = (zt S 2  E )/(S - E ) . PP PP PP PP Z  Z  Z  Z  Using (A.12), we can compute the two approximations (20) and (21) which correspond to (48) and (49)  respectively.  Now draw a perpendicular l i n e from point C to AB and define the cross point with AB, H.  Then the height of the t r i a n g l e i s C H .  - E ^-AH = S ^-(t-AH). PP PP  From these two equations,  ( t E °S ° ) / ( E ° - S ° ) . PP PP PP PP  We have C H =  we can solve C H =  The proof of ABC" i s perfectly  analogous.  Appendix VI Totally differentiating  -T  S PU' pp  -E  ,S - E wu wp wp  z  Z  E  z  pp  (60) and (61) with respect to z, we have  u' (z)  E  t PP  P'(z)  E  t wp  Z  (A.13) Z  Z  2  z 2 where E— = V .m( 1, p(z)+tz, u(z)) ,  z  i  We compute u'(z)  z  2 = V^irt 1 ,p(z)) for i , j = w,p,u.  by inverting the left-hand side matrix of (A.13).  the determinant of the matrix D i s  (A. 14)  D =  E  z  wu  (S  Z  - E  pp  z  )  pp  -  E  (S - E * ) . pu wp wp  Z  Z  Using the l i n e a r homogeneity properties of m and TT,  First,  43.  E  w u  (P  +  ( z ) + t z  )  E P  u  V  =  {  1  ,  p  (  z  )  +  t  z  ,  u  (  z  )  '  )  E +(p(z)+tz)E = 0, S +p(z)S = 0, wp pp wp ^ pp Z  Z  Z  we can rewrite (A.14) as  (A.15)  D = - 7 m(1,p(z)+tz,u(z))(E -S ) - z t E S . u PP PP p u pp Z  Z  Z  Z  v  The numerator of u ' ( z ) , defined as N, i s given by  (A.16)  N = {S  - E > E t - (S I - E ) E t wp pp pp pp wp z  Z  wp  z  z  z  = {-p(z)S + ( p ( z ) + t z ) E } E t + (S - E ) ( p ( z ) + t z ) E PP pp pp pp pp Z  z  = zt s 2  Z  Z  z  Z  t pp  E pp P P Z  2  From (A.15) and (A.16) we have  (A.17)  u'(z) = -{zt S  2  2  p  E  Z p  }/{V m(1,p(z) tz,u(z))(E +  u  2 p  -S  z p  + ztE  )  Z p  S  2 p  }.  From (A.17) we have,  u'(0>  = 0 and u-'(0) = - { t S ° E ° > / { V m ( 1 , ° , u ° ) ( E ° - S ° ) >  Substituting them into (34),  2  p  we have  p  u  P  p  (62).  We can show analogously as Appendix V that (62) ABG.  p  coincides with the area  44.  pr i s the reference bundle AB i s the ADD measure Fig. 1 The ADD Measure of Waste  45 .  X  2  Scitovsky set  1 S(u )  Scitovsky  set  0 S(u )  AB i s the HB measure Fig. 2 The HB Measure of Welfare  Fig. 3  The ADD Measure:  A One-Consumer Two-Goods Economy  47 .  V m(1,p u) A  n  l  = 7 n(1,p )  Xr  A  P  x  '  y  Fig. 4 The ADD Measure and i t s Approximations:  A One-Consumer Two-Goods Economy  Fig.  The HB Measure:  5  A One-Consumer Two-Goods Economy  Fig. 6 The HB Measure and Its Approximations:  A One-Consumer Two-Goods Economy  50. CHAPTER 2 THE MEASUREMENT OF WASTE IN A PUBLIC GOODS ECONOMY  2-1.  Introduction In the long history of the study on the measurement of deadweight loss  in applied welfare economics, the waste due to i n d i r e c t taxation has been the main concern of t h i s l i t e r a t u r e .  This section proposes a methodology for  measuring the waste due to an externality, which seems to be an alternative and  equally important s i t u a t i o n involving a market f a i l u r e .  methodology i s applicable to other e x t e r n a l i t i e s ,  Though our  here we focus on the  problem of public goods. Consider a government which c o l l e c t s revenue from both lump-sum and i n d i r e c t taxation and provides public goods.  This economy exhibits the waste  due to a price d i s t o r t i o n and to an incomplete market at the same time.  As  was already suggested by Harberger (1964;73), the deadweight loss of the whole economy depends on the difference between the s o c i a l benefit and s o c i a l cost of public goods i n addition to the set of i n d i r e c t taxes (or mark-up rates of noncompetitive firms).  We derive approximations to the A l l a i s -  Debreu-Diewert measure of waste of t h i s public good economy, and we show that the approximate deadweight loss can be expressed i n terms of the derivatives of  r e s t r i c t e d expenditure functions and r e s t r i c t e d p r o f i t functions evaluated  at the observed equilibrium as long as we know the marginal benefits of public  goods for consumers.  In deriving the approximate waste, we need not  assume l o c a l l i n e a r i t y of the production p o s s i b i l i t i e s set as i n Harberger (1964) and we need not assume r e s t r i c t i v e functional forms for u t i l i t y and production function as i n the numerical or applied general equilibrium l i t e r ature.  The waste to be studied i s due to the simultaneous existence of  51 . distortionary taxes and the nonoptimal provision of public goods.  Needless  to say, a simple sum of these two types of waste cannot even approximate the simultaneous loss measured i n this section. The next section i s devoted to the description of our model of a public goods economy, while section 3 defines the Allais-Debreu-Diewert measure of waste i n t h i s economy.  In section 4, we compute second order approximations  to the ADD measure to gain more insight about the nature of the waste.  We  also i n t e r p r e t the empirical significance of the approximate ADD measure.  In  section 5, some drawbacks to our approximate ADD measures are discussed and a diagrammatic exposition of our analysis i s presented.  The Model  2-2.  Our model i s similar to the one used i n Section 1-1 and 1-2 except that we now introduce public goods into the model.  There are N private goods  . . T which are traded at positive prices p = ( p ^ , . . . , p ^ ) and I public goods which affect both consumers' u t i l i t i e s and the production p o s s i b i l i t i e s sets of T firms.  A quantity vector of public goods i s denoted as G = ( G , . . . , G ) 1  I  > Oj.  There are K p r o f i t maximizing private firms which produce goods and services by u t i l i z i n g both private and public inputs using the production k k k p o s s i b i l i t i e s set S for k = 1 , . . . , K , i . e . , i f (y , - G) e S , then the k k kT vector of net outputs y = ( y ^ , . . . , y ) i s producible by sector k using the k vector of public goods G. The sector k r e s t r i c t e d p r o f i t function ir , which k i s dual to the production p o s s i b i l i t i e s set S , i s : N  (1)  ir (p,G) = max {p y : (y, - G) e S K  T  y  k  },  k = 1  K,  52. where p > 0^.  We assume that either G or an entrepreneurial factor i s a  l i m i t i n g factor of production, so that S scale when G i s fixed.  exhibits decreasing returns to  (See Meade (1952) for the d e f i n i t i o n of an 'unpaid  factor" public input.) The vector of public goods i s produced by the government, k = 0, which has the production p o s s i b i l i t i e s  set S^.  produces G using the input vector y.  If (y,G) e S^, the government  If some component of y i s p o s i t i v e ,  government i s j o i n t l y producing the corresponding private good with G. government r e s t r i c t e d p r o f i t function TT^, which i s dual to S^,  (2)  the  The  is:  0 T O TT (p,G) = max {p y : (y,G) e S }, y  where p > 0 . 1 N M  Let us now look at the consumer side of our model. are H i n d i v i d u a l s , h = 1 , . . . , H ,  i n the economy.  We assume that there  The preference of  individual h can be represented by a quasi-concave u t i l i t y function f*  1  defined over a translated orthant i n R * , N +  Q* . 1  Define the i n d i v i d u a l h  h h r e s t r i c t e d expenditure function m , which i s dual to f , for h = 1 , . . . , H , by:  (3)  where p > 0  (x,G) e Q } h  N  and U  r  e Range f .  We suppose that each i n d i v i d u a l h possesses  nonnegative endowment vector of private goods, x  h  > 0 „ , for h = 1 , . . . , H . N  also allow the government, which i s h = 0, to have an i n i t i a l endowment vector x^ -> 0N „.  We  As i n section 1-2,  the government raises revenue by the set of i n d i r e c t  T taxes t = ( t . | , . . . , t )  to provide the public goods.  N  make a net transfer g  to i n d i v i d u a l h.  h  lump-sum tax c o l l e c t e d from person h. consumers face p + t > 0  The government can also  If g^ < 0, -g^ i s the amount of Producers face prices p > 0  N  whereas  at the observed distorted equilibrium.  N  We use the overspending function defined by (4)  B(q,p,G,u) = E " m ( q , G , u ) h  h  1  h  h=o  q  x  to characterize our general equilibrium system.  " k=0 E  f  (  P  ,  G  )  Diewert ( 1986;131-155,  170-176) showed that the properties of a p r o f i t function and an expenditure function are v a l i d i n t h e i r r e s t r i c t e d functional form. properties of an overspending function (i) Chapter 1 are v a l i d for (4).  - (iii)  This means that the  l i s t e d i n Appendix I to  Diewert (1986) also showed that:  (iv) a  r e s t r i c t e d p r o f i t function i s concave with respect to G i f the production possibilities  set i s convex and a r e s t r i c t e d expenditure function i s  convex  with respect to G, so that B i s convex with respect to G, (v) -V_m^(q,G,u, ), o n for  h = 1,...,H,  i s the marginal benefit vector of consumer h for the  v public goods; V v (p,G) for k = 1 , . . . , K i s the marginal benefit vector of firm k for the public goods, and -V_ir^(p,G) i s a marginal cost vector for the o n  public goods, public goods.  so that -V^B(q,p,G,u) shows the aggregate net benefit vector of From the l i n e a r homogeneity of B with respect to p r i c e s ,  identity  (5)  qT  V  T +  P  T  B P  G  = (  V  G  B  )  holds i n addition to (1.A.2) -  (1.A.4).  the  The system of equations characterizing the observed equilibrium i s now hk stated i n a f a i r l y simple manner where a i s defined as the f r a c t i o n of a hk firm k held by i n d i v i d u a l h, with 0 < a  < 1 for h = 1 , . . . , H and k = 1 , . . . , K  and EjJ^a* * = 1 for k = 1 , . . . , K . 1  m (p+t G,u )  (7)  V B(p+t,p,G,u) + V B(p+t,p,G,u)  (8)  -V B(p+t,p,G,u)  Here (6)  shows the budget constraints  h  (  h  = g  q  + (p+t) x  + E ^a Tr (p,G), h = 1,...,H,  (6)  T  h  h  h K  p  V = d.  G  for the H individuals and (7)  equality of demand and supply for goods 1 , . . . , N . s t r a i n t i s implied by (6) and (7).  k  k  The government budget con-  From the property (v),  the net marginal benefit vector of the public goods. s i s t e n t with the well-known Samuelson (1954)-Kaizuka  d i n (8)  If d = 0^,  (8)  public goods are not supplied optimally.  i s con-  (1965) conditions  We assume that the  for  distortions  parameter d arises because of the limited a b i l i t y of the government  We regard (6)  defines  Therefore, d f 0j means that the  the optimal provision of public goods.  provide public goods  shows the  to  efficiently.  - (8) as a general equilibrium system which determines  P n « - - - P « r d , u and one component of t and g given the remaining components of t and g, with p^ = 1 as numeraire and G fixed. 1 1 1 distorted equilibrium (u ,p ,G , t , d , g )  exists.  We assume that an observed  55.  2-3.  An Allais-Debreu-Diewert Measure of Waste An Allais-Debreu-Diewert measure of waste that was defined and discussed  i n Chapter 1 i s now u t i l i z e d to measure the waste due to the public good externalities. T Pick a nonnegative reference vector of private goods p = (P^,...,P^) T . > 0 and consider the following primal programming problem: N  (9)  r ° = max r,x  {r :  (i) E x h=1 H  ,y , G .  (ii) (iii)  where u  = (u.,...,u„)  + pr < E  v k=0 K  k  + E  i h=0  H  f ( x , G ) > u^ ; (x ,G) e Q , h = 1 , . . . , H ; h  h  h  (y ,-G) e S , k = 1,...K; k  k  h  (y°,G)  observed d i s t o r t e d equilibrium defined i n the previous section. interpretation of L ^ repeat i t here.  ;  h  e S° }  i s the u t i l i t y vector which corresponds to the  n  1  h  The  = r^ i s discussed i n Chapter 1 so that we w i l l not  D D  For s i m p l i c i t y of computation, our reference bundle does not  include public goods. numeraire good, i . e . ,  A l l a i s proposed to measure the waste i n terms of a T . • i n our context p = ( 1 , 0 , . . . , 0 ) . Debreu's c o e f f i c i e n t  of resource u t i l i z a t i o n model (which assumed that p was proportional to the economy's t o t a l endowment vector) is also consistent with our present model since we assumed that there were no endowments of public goods. Given the l e v e l of G, (9) i s a concave programming problem so that we can derive i t s dual equivalent problem-.  (10)  r ° = max [max m i n G  r  {r(1-p p) T  p > 0  - B(p,p,G,u)}].2  (The process of derivation i s analogous to that i n Appendix 1-II.)  56.  If G°, r ° and p ° solve (10),  then p r ° i s a measure of the resources that can  be extracted from the economy while maintaining households at t h e i r d i s t o r t e d equilibrium u t i l i t y levels and  i s a corresponding "optimal" level of  public goods and p ° i s a vector of private goods prices which supports the efficient  equilibrium.  Note that i n t h i s "optimal" equilibrium, not only are  public goods being provided e f f i c i e n t l y ,  but also a l l commodity tax  d i s t o r t i o n s have been removed. Given the l e v e l of G, (10) may be rewritten by using the Uzawa-Karlin Saddle Point Theorem i n reverse as  (11)  r ° = - max  {B(p,p,G,u ) : p p > 1} N 1  n  T  where B i s the overspending function defined by (4). solves (10),  If G°, r ° and p °  then p^ solves (11) and r^ i s the associated Lagrangean  m u l t i p l i e r for the constraint i n (11). and p^ solve (10),  It i s also the case that i f G°, r °  then G^ i s the solution to the following unconstrained  maximization problem:  (12)  max  G  {r°(1-p p) 0 T  -  B(p°,p ,G,u )} 0  Our expressions for the ADD measure,  1  (10) and (11),  approach to the measurement of deadweight loss.  present our basic  However, these abstract  expressions do not indicate how the magnitude of the loss depends on the size of d i s t o r t i o n parameters t and d.  Furthermore, the global computation of  57.  (10) i s very d i f f i c u l t as was discussed i n Chapter 1.  Therefore, we turn to  the computation of second order approximations to the ADD measure.  2-4.  Second Order Approximations To obtain a second order approximation to the loss measure, we require  some stronger assumptions.  Suppose that:  (i) B i s twice continuously  d i f f e r e n t i a b l e with respect to q, p and G at the optimum of (10); Oj, p^ » 0  ( i i ) G° »  so that the f i r s t order necessary conditions for the max min  N  problem (10) hold with equality; ( i i i )  Samuelson's (1947;361) strong second  order s u f f i c i e n t conditions hold for (11) when the inequality constraints are replaced by e q u a l i t i e s ,  and these conditions also hold for  (12).  Consider the following system of equations i n the N + I + 1 unknowns, p, G and r , regarded as functions of a scalar parameter z defined for 0 <. z <. 1:  (13)  V B(p(z)+tz,p(z),G(z),u )+V B(p(z)+tz,p(z),G(z),u )+pr(z)  (14)  V B ( p ( z ) + t z , p ( z ) , G ( z ) , u ) = - zd,  (15)  1 - p(z) fi = 0.  1  1  q  p  = 0 , N  1  G  T  When z = 0, define p(0) = p ° , G(0) = G° and r(0) = r ° .  Then (13) -  become the f i r s t order conditions for the max min problem (10). t i v e l y , when z = 1, define p(1) = p \  G(1) = G and r(1) = 0. 1  the reference waste bundle 8 s a t i s f i e s the normalization  (15)  AlternaSuppose that  58 .1T  (16)  p = 1  by choosing the scale of p appropriately, which seems quite innocuous. (13)  - (15) coincide with (7),  (8) and (16).  Therefore, (13)  t e r i z e s the observed distorted equilibrium when z = 1. and (8) are s a t i s f i e d ,  Then,  - (15) charac-  Note that when (7)  (6) i s also s a t i s f i e d for the observed choice of g, . n  Therefore, we can safely conclude that (13) - (15) maps the Allais-DebreuDiewert reference equilibrium into the observed distorted equilibrium as z i s adjusted from 0 to 1. D i f f e r e n t i a t i n g the system (13) - (15) with respect to z and evaluating at z = 0, we obtain  °  B + B , B ° + B qq pp qG pG 0  (17)  B  Gq  0  +  B  r  , p  Gp ' GG B  P'(0)  B ° t qq  G' (0)  B ° t + d  Gq r' (0)  , o  where the second order derivatives of the overspending function B^?, i , j = q, p, G are evaluated at the optimum z = 0. follows:  The meaning of the B^9 are as  B  q q  i s an aggregate consumers' compensated substitution matrix  whereas - B  p p  i s an aggregate producers' substitution matrix evaluated at the  optimum; B ^ shows the change of aggregate compensated demands with respect qo to an increase of public goods and B -  p G  shows the change of aggregate net  supply of goods for firms with respect to an increase i n the public good supply.  59. Now regard (17) assumptions  (iii)  as an i d e n t i t y i n z, v a l i d for z close to 0.  Our  introduced at the outset of t h i s section imply that an  inverse exists for the matrix on the left-hand side of Diewert-Woodland (1977, Appendix I ) ) .  (13)  - (15)  Premultiply both sides of  (See  Hence, by the i m p l i c i t Function  Theorem, there e x i s t once continuously d i f f e r e n t i a b l e and r(z) which s a t i s f y  (17).  functions p(z),  G(z)  i n a neighbourhood of z = 0.  (17)  evaluated at z close to 0 by [p(z)  (1.A.2),  (1,A.4), and (5) evaluated at the  T ,  T 0 , 0].  Using i d e n t i t i e s ,  z-equilibrium, and then using (14) and (15) we get (18)  r'(z).= z[t B T  ( p ' ( z ) + t) + t B ! G ' ( Z ) + d G ' ( z ) ] . qq qG Z  T  (The process for deriving (18) From (18)  (19)  T  i s s i m i l a r to the one i n Appendix 1 - I I I . ) .  we readily have  r'(0)  = 0.  Now d i f f e r e n t i a t e  (18)  with respect to z, evaluate at z = 0, and adding the T  i d e n t i t y derived by premultiplying [p'(0) (17),  (20)  T , - G'(0)  ,0] to both sides of  we find  - r"(0)  = G'(0) B ° G ' ( 0 ) uu T  - p'(0)\V(0) -  [p (0)+t] B 1  pp  (The d e r i v a t i o n i s analogous to the one i n Appendix 1-IV.)  T  °[p'(0)+t] qq  60. Note that the l a s t two terms i n the right-hand side of (20) are nonnegative because of the concavity of B with respect to p r i c e s .  We also  assume that B ^ i s p o s i t i v e semidefinite; t h i s assumption i s s a t i s f i e d i f the production p o s s i b i l i t i e s sets are a l l convex, but i t i s much milder than assuming global convexity i n production.  I n t u i t i v e l y , i t means that the  concavity of the u t i l i t y functions outweighs any nonconvexity i n aggregate production with respect to public goods i n the neighbourhood of the optimum. Given t h i s assumption, -r"(0)  > 0 i s implied.  The Allais-Debreu-Diewert measure of waste r(0) may be written as  (  2  1  )  ADD  L  =  I  since r(1) = 0 .  (  0  "  )  r  (  1  )  A second order approximation to L^p i s obtained by using a D  Taylor series expansion evaluated at z = 0,  (22)  L  A [ ) D  * - [r'(0) + Jjr"(0)] = - Jir"(0)  Using (19) we therefore have the following theorem.  Theorem 1  ADD * " ^ "  L  r  ( 0 )  1  0  where the inequality i s v a l i d from (20) and i t s following discussion. i s quadratic,  (22) provides an exact expression for I «  use the expression for r"(0) given i n (20).  A n n  -  If r  To compute (22),  The vectors of derivatives  61 . p'(0)  and G'(0) i n (20) can be calculated by inverting the matrix on the  left-hand side of (17).  Therefore, the information required to calculate the  approximate ADD measure i s the reference bundle P, the d i s t o r t i o n parameters (t,  d) and the second order derivatives of the overspending function evalu-  ated at the optimum. Let us s c r u t i n i z e the informational requirements for computing (22) more carefully.  The vectors t and p are d i r e c t l y observable.  d, we must know the consumers' marginal benefits at the observed consumer p r i c e s .  To know the vector  from public goods evaluated  This means that we must overcome the w e l l -  known preference revelation problem for public goods.  Furthermore, to  estimate the matrix B„^, we need to know the derivatives of the net marginal oo benefits  for public goods for both consumers and producers.  To calculate the  other second order derivatives of the overspending function, we need to know the f i r s t order derivatives of the net supply functions of firms for private goods and the compensated demand functions of consumers, which depend on both prices and public goods. the second set i s not. elasticities  Though the f i r s t set of functions i s observable, It i s well-known, however, that the compensated price  can be computed from data on the ordinary demand functions using  the derivatives with respect to both prices and income i n the Slutsky equation.  (See,  for example, Diewert (1982;572).)  S i m i l a r l y , the derivatives of  the compensated demand functions with respect to public goods can also be computed from market demand functions using ' S l u t s k y - l i k e ' equations Wildasin (1984;230)).  (see  The fact that we need information on the second order  derivatives of the overspending function evaluated at the optimum considerably decreases the usefulness of (22),  since these values are not observable  62. (at the market d i s t o r t e d equilibrium) and, i n general, are d i f f e r e n t from the values observed i n the distorted equilibrium. An a l t e r n a t i v e approach to approximating L  f t D D  Diewert's (1976;118) Quadratic Approximation Lemma.  can be developed using This Lemma demonstrates  that r(0) - r(1) can be appproximated by - ( 1 / 2 ) ( r ' ( 0 ) + r ' ( 1 ) ) , with the approximation being exact i f r i s quadratic.  Note that t h i s approximation  formula does not employ second-order derivatives of r . Suppose that (17)  i s v a l i d for z close to 1 (instead of our previous  assumption that i t i s v a l i d for z close to 0). obtain that r'(1)  Setting z = 1 i n (18), we  i s equal to the right-hand side of (20) evaluated at z = 1  instead of at z = 0.  Using (19) and Diewert's Quadratic Approximation Lemma,  we have the following c o r o l l a r y :  Corollary  (23)  1.1  L  A D D  * -(1/2) r'(1)  > 0.  A desirable a t t r i b u t e of t h i s approximation i s that i t only u t i l i z e s l o c a l information at the observed equilibrium. We thus see that both of our approximations to the deadweight loss measure r^ can be calculated from the derivatives up to second order of the overspending function evaluated at the reference equilibrium i n the case of (22) and evaluated at the observed equilibrium i n the case of (23).  In  p a r t i c u l a r , i t i s not necessary to make any assumptions concerning the functional form of B or place any r e s t r i c t i o n s on the values of observed economic v a r i a b l e s , other than the general r e s t r i c t i o n s used i n describing  our model.  On the contrary, to calculate r , as opposed to an approximation  to r ° , i t would be necessary to adopt s p e c i f i c  (and possibly  restrictive)  functional forms i n order to solve the max-min problem (10) g l o b a l l y .  2-5. Conclusion This chapter has discussed the measurement of waste and i t s  local  approximations for an economy facing distortions due to i n d i r e c t taxation and nonoptimal levels of public good production.  Use has been made of the ADD  measure defined i n Chapter 1 and two l o c a l approximations to the exact measure were calculated. tion on an overspending  These approximations only required l o c a l informafunction.  Figs. 7 - 9 i l l u s t r a t e the diagrammatic interpretation of the ADD measure of waste i n a public goods economy and i t s approximations. that there i s one private good and one public good. an aggregate production p o s s i b i l i t i e s  Suppose  In F i g . 7, we have drawn  set that transforms the private good  into the public good and the indifference  curve of the  representative  consumer corresponding to the u t i l i t y l e v e l received at the observed distorted equilibrium.  Though we cannot introduce a distortionary  taxation  in this one private good economy, the observed equilibrium i s not optimal because of the distortionary provision of the public good, and i t  is  expressed by the discrepancy of the marginal rate of substitution and the marginal rate of transformation at the equilibrium.  By choosing the r e f e r -  ence bundle to consist only of the private good, the ADD measure, as shown i n F i g . 7, i s a maximum surplus of the private good with holding the u t i l i t y level of the consumer and satisfying  the production p o s s i b i l i t i e s  set.  The  point where the surplus good i s maximized i s characterized by the equality of  the m a r g i n a l r a t e o f s u b s t i t u t i o n and the m a r g i n a l r a t e of t r a n s f o r m a t i o n . The ADD  measure can be r e i n t e r p r e t e d i n a H o t e l l i n g - H a r b e r g e r way  s i m p l e model as i n F i g . 8.  in this  The m a r g i n a l b e n e f i t o f t h e p u b l i c good i s t h e  m a r g i n a l r a t e of s u b s t i t u t i o n a t u = u  1  as a f u n c t i o n of the amount of the  p u b l i c good, and t h e m a r g i n a l c o s t of t h e p u b l i c good i s t h e m a r g i n a l r a t e of t r a n s f o r m a t i o n as a f u n c t i o n of the amount of the p u b l i c good. mum  At the o p t i -  t h e y c o i n c i d e , but t h e former i s h i g h e r than t h e l a t t e r a t t h e d i s t o r t e d  e q u i l i b r i u m , and t h e i r d i s c r e p a n c y i s denoted as d.  We  measure of waste i s shown as a c u r v i l i n e a r t r i a n g l e ABC approximations  (22) and  can show t h a t the  ADD  and t h a t i t s two  (23) c o i n c i d e w i t h the t r i a n g l e s ABC  and A B C ' .  The  d e r i v a t i o n i s analogous t o Appendix V of c h a p t e r 1 f o r the i n t e r p r e t a t i o n of the t a x l o s s as shown i n F i g . 4.  In F i g . 8, the a p p r o x i m a t i o n s  are rather  a c c u r a t e i n comparison t o t h e t r u e amount of waste, but i t i s d i f f i c u l t t o t e l l how general.  w e l l the a p p r o x i m a t i o n s  can approximate the t r u e amount of waste i n  I n F i g . 9, we show an example of one consumer economy w i t h l i n e a r  p r o d u c t i o n p o s s i b i l i t i e s s e t where two a p p r o x i m a t i o n s even i n t h i s s i m p l e model.  Tsuneki  can be q u i t e i n a c c u r a t e  (1987a) g i v e s a more e x t e n s i v e  d i s c u s s i o n on t h i s n u m e r i c a l example and concludes  t h a t the  approximations  can g i v e a t l e a s t an o r d e r of magnitude e s t i m a t e of t h e t r u e amount of waste and the a p p r o x i m a t i o n s  can work q u i t e w e l l as l o n g as the optimum and  the  d i s t o r t e d e q u i l i b r i u m a r e not f a r a p a r t . To c o n c l u d e t h i s c h a p t e r :  we can i n c o r p o r a t e t h e c h o i c e of p u b l i c goods  by governments (which a r e used both by consumers and p r o d u c e r s ) t i o n a l general e q u i l i b r i u m Harberger-type  measurement of deadweight l o s s  framework by a d o p t i n g t h e A l l a i s - D e b r e u - D i e w e r t approach. more g e n e r a l than Harberger's  in a tradi-  Our approach i s  a n a l y s i s i n t h e sense t h a t i t a l l o w s f o r ( i )  65.  the choice of f l e x i b l e functional forms (instead of linear ones as i n Harberger or CES-type ones i n the numerical general equilibrium l i t e r a t u r e ) for the production sectors and ( i i )  the loss due to i n d i r e c t taxation and the  nonoptimal provision of public goods i s evaluated simultaneously.  FOOTNOTES FOR CHAPTER 2  1 Assuming that there i s a single government production sector involves no loss of generality. 2 Formula (10)  See Tsuneki (1987a) for more d e t a i l s .  follows using d e f i n i t i o n s  (1958)-Karlin (1959) Saddle Point Theorem. constraint q u a l i f i c a t i o n condition applies.  (1) - (3) and the Uzawa  We assume that S l a t e r ' s  67 .  Fig. 7 The ADD Measure i n a Public Goods Economy  68.  Fig. 8 The ADD Measure and i t s Approximations i n a Public Goods Economy  69 .  Marginal Benefit of P u b l i c Good  Marginal Cost o f P u b l i c Good  0  1—f  Optimum  Distorted Equilibrium  Public Good  Fig. 9 An Example where Approximations  o f the ADD Measure a r e I n a c c u r a t e  70.  CHAPTER 3 PROJECT EVALUATION RULES FOR THE PROVISION OF PUBLIC GOODS  3-1.  Introduction The  tion,  t h e o r y of the p r o v i s i o n of a p u b l i c good w i t h d i s t o r t i o n a r y  first  s e t f o r t h by Pigou  r u l e t o equate the sum  (1947), m a i n t a i n s  taxa-  t h a t the Samuelsonian  (1954)  of m a r g i n a l b e n e f i t s t o i t s marginal c o s t cannot  be  an  a p p r o p r i a t e r u l e f o r the maximum of s o c i a l w e l f a r e . A main o b j e c t i v e of the p r e s e n t c h a p t e r i s t o f o r m u l a t e some c o s t - b e n e f i t r u l e s f o r the p r o v i s i o n of a p u b l i c good which d e f i n i t e l y improves the w e l f a r e of a l l the i n d i v i d u a l s w i t h i n the economy.  T h i s means  t h a t our approach c o n s i d e r s s u f f i c i e n t c o n d i t i o n s f o r the e x i s t e n c e of a Pareto improvement when the p u b l i c good i s p r o v i d e d i n a d i s t o r t i o n a r y f a s h i o n , and transfers,  ( i ) i n d i r e c t tax r a t e s ,  (iii)  ( i i ) i n d i r e c t taxes r a t e s and  lump-sum t r a n s f e r s , are a l l o w e d t o vary w i t h the  lump-sum provision  of p u b l i c good. In case  (i),  we  suggest a G e n e r a l i z e d P i g o v i a n Rule which i s a many-  person g e n e r a l i z a t i o n of the second-best good p r o v i s i o n which i s due and  (iii)  we  t o A t k i n s o n and  public  S t e r n (1974), w h i l e i n cases ( i i )  suggest a G e n e r a l i z e d Samuelsonian Rule and a M o d i f i e d  Harberger-Bruce-Harris Samuelsonian  o p t i m a l i t y c o n d i t i o n f o r the  Rule, where a l l of them more or l e s s d i f f e r  from  the  rule.  S i n c e our r u l e s are v a l i d when the e q u i l i b r i u m i s away from the  second-  b e s t optimum, our approach c o n t r a s t s w i t h the p r e v i o u s l i t e r a t u r e on  project  e v a l u a t i o n r u l e s f o r p u b l i c goods by S t i g l i t z and Dasgupta (1971),  Atkinson  and  King  S t e r n (1974),  Diamond (1975),  A t k i n s o n and  Stiglitz  (1980) and  71  (1986).  They analyze the f i r s t order necessary conditions for ( i n t e r i o r )  second best s o c i a l welfare optima.  Another objective of t h i s chapter i s ,  however, to reconcile these two apparently different approaches and to together strands of previous discussions within our framework. the cost-benefit  1  tie  We show that  rules i n this chapter are v a l i d both as necessary and  s u f f i c i e n t conditions i f the manipulable taxation scheme i s optimized. After describing our model i n the next section,  3-3 studies an economy  where distortionary commodity taxes are used to finance the provision of public goods; lump-sum transfers are not a v a i l a b l e . (1974) cost-benefit  Atkinson and Stern's  rule for public goods provision, which generalized the  r e s u l t in Pigou's (1947) pioneering study, i s extended to a heterogeneousconsumers' economy i n t h i s section.  In Atkinson and Stern's (1974) model,  the marginal u t i l i t y of income does not equal the marginal s o c i a l cost of r a i s i n g one d o l l a r by i n d i r e c t taxation; t h i s difference arises because there i s a welfare cost due to i n d i r e c t taxation and there i s an income effect due to taxation on tax revenue.  The f i r s t d i s t o r t i o n i s emphasized by Pigou, but  the second one i s neglected by him.  When we extend the Atkinson and Stern  r e s u l t to a heterogeneous-consumers'  economy, two differences a r i s e .  First,  the income effect of taxation is the sum of individual income effects with the hth weight being the share of tax revenue paid by the hth i n d i v i d u a l . Second, the change i n the income d i s t r i b u t i o n that results from increased taxation affects the s o c i a l cost of taxation; e . g . ,  i f the tax i s levied on  people with high s o c i a l importance, the s o c i a l cost of taxation w i l l be higher. Section 3-4 discusses the cases where lump-sum transfers are available to finance an increased supply of public goods.  If we can perturb both  72. i n d i r e c t tax rates and lump-sum transfers at the same time, a generalization of a t r a d i t i o n a l Samuelsonian r u l e , generalized Samuelsonian rule applies for the project evaluation.  However, when there exists unchangeable i n d i r e c t tax  d i s t o r t i o n s , we derive a Modified Harberger-Bruce-Harris rule for evaluating the public good.  This approach proceeds by using the lump-sum tranfers to  keep everyone on t h e i r i n i t i a l indifference curves when the supply of a public good i s increased.  The induced change i n the net supply of private  goods i s then evaluated using Harberger's generalized weighted-average  shadow  prices for fixed i n d i r e c t tax d i s t o r t i o n s . Since we adopted the approach of searching for s u f f i c i e n t conditions for a Pareto improvement, our cost-benefit  rules can be implemented with know-  ledge of the i n i t i a l demand and supply vectors and of the derivatives of the aggregate demand and supply functions evaluated at the observed equilibrium value, as long as preferences for public goods can be determined.  Our  approach may be contrasted with an alternative approach which searches for necessary conditions for an i n t e r i o r welfare optimum. cost-benefit  In this approach, the  rules depend on the derivatives of the aggregate demand and  supply functions evaluated at the optimum point.  3-2.  The Model The model we u t i l i z e i n t h i s chapter i s i d e n t i c a l with the one we used  in the previous chapter to characterize the observed distorted equilibrium, (2.6)  - (2.8).  We assume for s i m p l i c i t y that p r o f i t income i s  taxed away, following Diamond and Mirrlees (1971).  completely  This assumption can be  relaxed by assuming that the entrepreneurial factors are additional commodit i e s (see Diewert (1978) and D i x i t (1979)). restate (2.6)  and (2.7)  as  follows:  With these assumptions, we can  73.  h = 1, . . . , H  (1)  (2)  0,  t  1  endogenously given p  -  1  = 1, t  T  T = (u-|,...,u ) and T T = ( t , . . . , t ) , g = (g.,,...,g ) and G =  We assume that (1) and (2) determine p  =  2  (P2F--- PN) (  ».  u  N  H  H  T (G^,...,G ) N  .  Note again that by Walras' law (1) and (2) imply the budget  constraint of the government i s s a t i s f i e d .  When the equality i n (2)  replaced by the inequality (<.) by assuming free disposal, we c a l l i t inequality version of  is "the  (2)."  The Pigovian cost-benefit comparative s t a t i c s exercise  problem we study i n t h i s chapter i s simply a  in which at the i n i t i a l observed equilibrium we  perturb G and some of the available tax variables.  We assume that G i s a  scalar (or a l t e r n a t i v e l y , we assume that only the production of the f i r s t public good i s varied while the other public goods are held f i x e d ) .  Three  a l t e r n a t i v e rules are derived depending on which taxation instruments we can change. (3)  D i f f e r e n t i a t i n g (1) and (2) t o t a l l y ,  assuming that  h =  which i s implied by money metric u t i l i t y scaling (see Samuelson (1974)), we obtain:  74.  —  —  -X  -X  du =  dp +  B qu  -B qq  h  -B ~  -B  PP  qq  1  + -B qq  1  w dg +  o  dG " qG ' B  _ N«H_  B P  G  where the net demand matrix of consumers  X =  dt  is:  ] (H«N matrix, with X H « 1 and X ,H«(N-1)) V  where the hth row shows the net demand vector of the hth consumer and  W = (W ...,W ) 1 f  T  H  = (-V m (p+t,G,u ),..., 1  G  1  -V m (p+t,G,u )) H  G  T  H  i s a vector of the marginal benefits of the public good for the consumers. The  scalar  M C=  " k=oV ' E  k ( p  c )  i s the net aggregate marginal cost of the public good, L  i s an H»H unit  n  matrix and 0 „ „ i s an N«H matrix consisting of zeros.  A l l the derivatives of  the overspending function B are evaluated at the observed equilibrium point (p+t,p,G,u).  75. Throughout the chapter, we assume that  2 (5)  PP  = V-.. B(p+t,p,G,u) i s negative PP  definite.  We express (3) in a d i f f e r e n t way for later use:  (6)  Adu = B.jdp  + B dt 2  + B dt  1  + B dg + B^dG.  3  4  When we refer to "the inequality version of (6)" we mean that the H + 1 , . . . , H+Nth equalities i n (6) are replaced by i n e q u a l i t i e s  (<.).  This case  u t i l i z e s the assumption that an excess supply of goods can be freely disposed.  We assume that [ A , - B ^ , - B ] 2  1  exists, so that we can l o c a l l y  solve for u, p, and t^ as functions of the exogenous variables, using the I m p l i c i t Function Theorem.  This a n a l y t i c a l technique closely follows Diewert  (1983b). F i n a l l y , we have to define our welfare c r i t e r i a .  In a many-consumer  economy, we have to d i s t i n g u i s h between two c r i t e r i a for a welfare improvement.  The f i r s t c r i t e r i o n i s the s t r i c t Pareto c r i t e r i o n .  improvement occurs i f each person's u t i l i t y i s increased.  A s t r i c t Pareto The second  T H T c r i t e r i o n makes e x p l i c i t use of the s o c i a l welfare function 8 u = E. „B,u. h=1 h h T where 8 > 0 „ . The l i n e a r function 8 u can be thought of as a l o c a l l i n e a r n r  p  approximation to a general quasiconcave s o c i a l welfare function evaluated at the i n i t i a l u t i l i t y vector u.  In this chapter, we consider a d i f f e r e n t i a l  effect of the various sets of tax-expenditure instruments with respect to s o c i a l welfare. satisfying  If a set of available instruments i s f u l l y perturbed with  (4) and du » 0  occurred, then we define i t as a d i f f e r e n t i a l l y  s t r i c t Pareto improvement.  If available tools are f u l l y perturbed with  T satisfying  (4) and p  du > 0 occurred, then we define i t as a d i f f e r e n t i a l l y  s t r i c t welfare improvement.  (These d e f i n i t i o n s  follow Diewert (1983b).)  Obviously, a d i f f e r e n t i a l l y s t r i c t Pareto improvement ( i . e . , du 2> 0 „ ) i s a H d i f f e r e n t i a l l y s t r i c t welfare improvement for any nonnegative, u t i l i t y weight vector p. for  but nonzero,  Therefore, i f we can find a s u f f i c i e n t  condition  the existence of a d i f f e r e n t i a l l y s t r i c t Pareto improvement, then there  exists a d i f f e r e n t i a l l y s t r i c t welfare improvement as well.  Note also that  T T P du > 0 implies the improvement of s o c i a l welfare p u i n a l o c a l sense but the opposite i s not true i n general, since i t i s possible that there exists T an i n f l e x i o n point of p u with respect to the set of instruments so that the T improvement of s o c i a l welfare occurs even i f p du = 0. The same argument applies for the change of individual u t i l i t y . We also define p-optimality . . . . T with respect to some set of instruments as an equilibrium i n which p u i s maximized with respect to the instruments.  3-3.  Pigovian Rules Reconsidered Most papers on cost-benefit  rules for public goods provision follow the  Pigovian t r a d i t i o n and suppose that the government can vary i n d i r e c t tax rates t simultaneously with changes i n the production of the public good dG 2 0; however,  lump-sum transfers g are fixed.  Atkinson and Stern (1974) gave  the most elegant formula for such a cost-benefit consumers have i d e n t i c a l preferences and wealth.  rule by assuming that a l l The purpose of t h i s  section  i s to extend t h e i r formula, which we c a l l a Generalized Pigovian Rule to a heterogeneous-consumers'  economy and to compare the economic implications of  t h i s new rule with that of Atkinson and Stern. this section as  follows:  We state our main theorem i n  Theorem 3.1  Suppose that public good production i s i r r e v e r s i b l e so that dG > 0 and  2  the government can perturb t a r b i t r a r i l y .  Suppose also that  (7)  and  there i s no solution a  to a  u  T T > 0 „ and a X = 0 „ H u N  u  T that the i n d i r e c t tax revenue R = t v B i s nonzero. q  T Then i f a l l for -y > 0„ for which no d i f f e r e n t i a l l y s t r i c t improvement of ^ u H with respect to i n d i r e c t tax rates e x i s t s ,  (8) <rW(1 + T  "j^B  h  ^f -  ~C l ^ q u F ^ l ^ i r  3  i s s a t i s f i e d where R* = t x 1  hth  O)  T  b  h 1 >  M  C  "  E ! t (ax (p+t G I )/aG) T  f t  h  1  l  f  i s the amount of i n d i r e c t tax revenue paid by  person and  s || = a x ( p + t , G , u ) / a u h  h  = ax (p+t,G,i )/ai , h  h  h  h  h = I,...,H  i s a vector of income effects for the hth i n d i v i d u a l , then there exists a d i f f e r e n t i a l l y s t r i c t Pareto improvement du » 0 „ .  h  78.  (ii)  If the pre-project equilibrium i s 8-optimal with respect to the  of t,  and i f  no) fro *^-  hls\»h<Ah > MC -  is satisfied,  choice  E ^ t  1  (9x (p+t,G,I )/r3G) h  h  then (10) i s a necessary condition for a d i f f e r e n t i a l l y  strict  T  increase i n s o c i a l welfare B du > 0.  PROOF:  (i) (11)  A Pareto improvement with dG 2 0 exists i f and only i f there e x i s t dG > 0, du, dt such that du » 0 version of  (6) i s s a t i s f i e d  T there does not exist an a [a a ] T  r  and the inequality  with dg = 0^.  Applying Motzkin's Theorem (see Appendix I ) ,  (12)  H  this i s equivalent  T -T = [ a , a ^ , a ] such that u  > o j , a [ B , B , B ] = 0 _*, a A > o j , T  T  1  2  3  to  2N  and a B T  If a Pareto improving i n d i r e c t tax perturbatin i s possible, always s a t i s f i e d with dG = 0 and the problem i s vacuous. assume that such an improvement does not e x i s t .  5  < 0.  then (11) i s  Therefore, we  Then there exists an a  T  =  79.  [ a J , a a ] such that [ a a ] > 0^, a [ B B , B ] = 0 T  T  1 (  T  v  i ;  2  3  _ , a A > 0^. T  2 N  T  For any a  • • T T that s a t i s f i e s t h i s condition, we define t = a A. We would l i k e to show that for such a, (12) i s s a t i s f i e d .  Suppose (8) holds, and also suppose, T  contrary to the theorem, a solution to (12) e x i s t s . T  T  Subtracting a B = 0 ^ 3  T  from a B =  using the i d e n t i t y (1.A.4) and using the supposition (5),  1  we have (see Appendix II) (13)  a = a^.  Using (13), the i d e n t i t y (2.5), and the definitions of W and MC, we get  (14)  a B  = (aj + a  T  5  1  f*)W - a ^ C + a ^ B .  T Suppose that a^ = 0. Then, a = 0 from (13). Therefore, a [ B B ] T T T T = 0 „ implies a X = 0 „ . Furthermore, since a A > 0., , we have a > 0 „ . This N u N ' H u H N  1  1 (  contradicts the supposition (7), so that (12) i s s a t i s f i e d . > 0.  2  Suppose that a^  T T T Now postmultiply t^ and t to a B = 0 and a B = 0 _^ respectively, add 2  3  N  them together and using (13) and (1.A.2) we have (see Appendix III)  H  (15)  a  n  L / \  h=1 a  1  T  K  7 R  = 1 1 t/R. qq  T T We also have a A = t = (  ,  1  H  y,...,'r)so  show that (see Appendix III)  using (13) and (1.A.3) and (3), we can  80. h h • u T h — = — - 1 + tS , a a qu 3  (16)  h=1,...,H.  n  x  1  1  Substituting  (16)  "1  into (15),  we find  t B t  Now substituting  (16)  U  and (17)  T  U V . R  W^  into (14)  and using the S l u t s k y - l i k e  equation  by Wildasin (1984;230),3 we get  T (18)  t  \  . a ^ W . U  1  - E J A u  ^  ?"C/«R  )  - MC + E " t ( 9 x ( p + t , G , I ) / 3 G ) } . T  h  Therefore,  (ii)  (8)  T implies a B  1  c  h  h  > 0 and we have a contradiction.  If t i s chosen optimally at the pre-project equilibrium then i t i s a  solution to the problem:  (19)  T max ~ {B u : (1) and the inequality version of u,p,t satisfied}.  The f i r s t order Kuhn-Tucker conditions for (19)  are:  (2)  are  81  (20)  there e x i s t [a.,,a ] > OjJ, such that a A = 8 , a [ B B , B ] = 0 _^ T  T  T  T  r  2  2N  3  since the Mangasarian-Fromovitz constraint q u a l i f i c a t i o n conditions are implied by the existence of [ A , - B - B ] 1 f  -1  2  (see Mangasarian (1969; 172-3)).  A d i f f e r e n t i a l l y s t r i c t improvement i n s o c i a l welfare exists i f and only if  (11)  is satisfied  with du » 0  H  T replaced by p du > 0.  i s given by replacing a A > 0^ i n (12) T  by a A = p . T  T  Its dual condition  This dual condition, and  T (20) imply a B establish  5  > 0, which i s equivalent to (10) using the argument to  (8) from (12).  Q. E. D.  We now have to consider the economic implications of Theorem 3.1.  The  assumption that i n d i r e c t tax revenue i s nonzero i s standard i n the optimal tax l i t e r a t u r e .  Assumption (7)  i s more subtle,  but i t may well be j u s t i f i e d ,  since i t i s implied by the existence of a Diamond and Mirrlees' good (1971 ;23).  More generally,  (7) i s the condition for the existence of Pareto  improving price changes ignoring production constraints,  and equivalently  there exists a Hicksian composite good i n net demand (or net supply) by a l l consumers.  Then, lowering (raising)  consumers better off call  (8) and (10)  the price of the Hicksian good makes a l l  (see Weymark (1979)).  With these assumptions,  the Generalized Pigovian Rules (GPR hereafter)  many-person Pigovian rules for the provision of public goods.  we may  or the  There are  several i n t e r e s t i n g interpretations of these two formulae. Let  us f i r s t consider the r e l a t i o n between (8) and (10).  Obviously, the  only difference between the two formulae i s that we must consider any semipositive  u t i l i t y weight vector for which a s o c i a l welfare improving tax  perturbation does not exist i n the former, while we specify the weight p i n the l a t t e r .  This may be explained as follows.  We f i r s t assume that i n d i r e c t  82. taxes are set so that we cannot make a d i f f e r e n t i a l l y s t r i c t Pareto improvement with dG = 0.  Otherwise, the problem i s t r i v i a l .  However, once  the i n d i r e c t taxes are set i n this manner, there exists at least one weight vector f (and probably many) so that the i n d i r e c t taxes are set such that T increasing i u i s impossible  (see D i x i t (1979, 152)).  Therefore we can use T  the f i r s t order necessary conditions of the maximal s o c i a l welfare f u with respect to i n d i r e c t taxes, and hence the rest of the problem i s an extension of Atkinson and Stern's  (1974) r e s u l t on s o c i a l l y optimal provision of public  good with optimal taxes to a many-consumer economy.  Furthermore, i f we  specify t = 6 assuming that the economy i s at the 8-optimum, then we can get (10). We now discuss how to extend the Atkinson and Stern's cost-benefit rules to a many-consumer economy. i.e.,  at a 8 optimum, _  (1974,  122)  With taxes set optimally,  (14) has the following interpretation.  At the p-optimum, a|j, h = 1 , . . . , H , from the programming (19).  and a^ are the Lagrange m u l t i p l i e r s  As d ( 8 u ) / d g T  = a , ajj i s a net benefit of b  h  giving hth person one unit of numeraire good by r a i s i n g the i n d i r e c t taxes. T -1 It i s also the case that d(8 u)/dx^ = a^, a.^ i s the s o c i a l gam of the society to have one more unit of the numeraire good (so that i n d i r e c t taxes are reduced).  Therefore,  (a  y  + a^)/a^ i s a gross benefit i n terms of s o c i a l  value of the numeraire of giving hth person one unit of numeraire good, and hence i t i s Diamond's (1975;341) s o c i a l marginal u t i l i t y of income a^, h = 1, T . . . . ...,H. Therefore, we can rewrite a B > 0 using the d e f i n i t i o n of a, , h = 1, c  . . . , H i t i s equivalent  to  D  h  83 .  (  2  1  )  E  h=1 h h > a  W  M  - ^ q C  C  where the left-hand side i s the s o c i a l value of the public good while the right-hand side i s the net s o c i a l cost of the public good both measured i n terms of the s o c i a l value of numeraire.  a  (22)  h  T T can also be rewritten from a A = B using (13)  a  =  h  + t S  , qu'  T  a  h = 1, . . . ,H  h  1  as  which coincides with Diamond's (1975;341) o r i g i n a l formula. (22)  (23)  into (21),  By substituting  we have  I l A ^+  t S T  h=1 a^ v  )W qu' h  h  > MC - t B qG T  Using Wildasin's (1984;231) S l u t s k y - l i k e equation for public goods footnote 3),  (24)  (23)  may be further rewritten as  (E " B W )/a h  h  1  (see  h  > MC - t ( 9 E " x ( p + t , G , I ) / a G ) . T  1  The left-hand side of (24)  h  h  1  h  i s the weighted sum of the marginal willingness  to  pay for public goods discounted by the shadow cost of r a i s i n g one d o l l a r by i n d i r e c t taxation.  The right-hand side i s the marginal cost of the public  good minus the complementarity effect of public goods provision which means the effect of public good provision on tax revenue due to the complementarity  84.  between public and private goods.  Therefore,  (24)  extends the formula (3) of  Atkinson and Stern (1974; 122) to a many consumer context.  What was  emphasized by Atkinson and Stern was that 1/a^ may not necessarily smaller than unity, i n spite of Pigou's be seen from our formula (17)  for 1/a.j.  between our formula and t h e i r s . L  (1947;34) conjecture.  be  This may also  There are two main differences  First,  the revenue effect of  taxation  H T h h T h .t S R /R i s a weighted sum of the i n d i v i d u a l revenue effect t S h=1 qu ' * . qu  where the hth weight i s the share of t o t a l taxes paid by the hth i n d i v i d u a l . T When there i s only one person this expression i s simply t S and  Stern (1974;123)).  q u  (see Atkinson  Second, i n a many consumer context one also has  d i s t r i b u t i o n a l effects to consider.  Raising one d o l l a r by taxation  involves  changing the d i s t r i b u t i o n of income proportionately to the tax shares of individuals. This d i s t r i b u t i o n a l effect i s reflected in the term H h h h E _ - | 8 (R /R) • If the tax i s levied on people with high s o c i a l importance 8 , n  then t h i s expression increases as does a^; i . e . ,  the s o c i a l cost of r a i s i n g  one d o l l a r i s higher because of the increase of s o c i a l inequity. concerns are summarized i n the GPR (10).  These  To see the d i s t r i b u t i v e concern i n  (10) more f u l l y , we define the covariance term following Feldstein (1972);  (25)  H <p_ = E * G  (26)  H _ 1  <p = E A R n  _  S W ^ * - / H, 8W n  ^  1  h  ?^ / H.  8 R  where p, R and W are defined as p = E f P / H , h  h  1  R = R/H and W = E f W / H . h  h  1  the c o r r e l a t i o n between the s o c i a l importance and the d i s t r i b u t i o n of  As  85 . marginal willingnesses to pay or of tax burdens increases, increase.  Substituting (25) and (26)  (27)  into (10)  - Eh^Sgu  n>_ and tp  yields:  > MC -  t Ox (p+t,G,I )/3G). T  h  h  This formula e x p l i c i t l y shows the importance of d i s t r i b u t i o n a l concern i n a <P many-person GPR by the term — . If the d i s t r i b u t i o n of the public goods R G  benefits are regressive or the d i s t r i b u t i o n of the tax burden i s  progressive,  the s o c i a l welfare of the public good must be valued higher than the simple sum of the marginal willingnesses to pay.  Before closing t h i s section,  we should mention the r e l a t i o n between our  model and the recent work by King (1986).  Our formula (21) with the  h interpretation of a  by (22)  i s obviously indentical with his formula (31)  King (1986;281) so that i t i s possible to interpret (21)  i n his way.  in  His  result i s more general than ours i n the sense that he i s not assuming the Pareto e f f i c i e n t  i n d i r e c t taxation, but our approach i s more complete than  his i n the sense that he i s not deriving the e x p l i c i t formula and i n t e r p r e t a tion of the shadow price of government revenue l i k e (17) of ours, for i t u t t e r l y depends on the a r b i t r a r y structure of i n d i r e c t taxation i n his model. 3-4.  Cases Where Lump-sum Transfers Are Available In contrast to the previous section where lump-sum transfers cannot be  changed, the conventional Samuelsonian project evaluation rule which equates the sum of the marginal willingnesses to pay with the marginal cost of the  86 . public good has a strong i n t u i t i v e appeal when lump-sum taxes are available for financing the public good.  We show i n this section that the Samuelson  rule i s appropriate with some generalizations i f both i n d i r e c t taxes and lump-sum transfers are v a r i a b l e , whereas i t i s not appropriate i f there exits unchangeable d i s t o r t i o n s due to i n d i r e c t taxation. Let us f i r s t consider the case where we can change i n d i r e c t taxes and lump-sum tranfers at the same time.  Theorem 3.2  (i) i.e.,  Suppose that the government can change t dG > 0.  If,  E!W + t  (28)  h  and g when G i s increased,  1  h  T  B  q G  > MC,  then there exists a s t r i c t Pareto improvement du » 0 „ . n  (ii)  Suppose that t  a B-optimum.  and g are chosen so that the pre-project equilibrium i s  Then (28) i s also necessary for the existence of a d i f f e r e n T  t i a l l y s t r i c t increase of s o c i a l welfare 8  du > 0.  PROOF: (i)  A s u f f i c i e n t condition for the existence of a Pareto improvement with  dG > 0 i s :  87 .  (29)  there exists dG > 0, dt, dg, such that the inequality version of (6) i s s a t i s f i e d with du > 0 .  H  By Motzkin's Theorem, this i s equivalent  T (30)  to:  T  ~T  there does not exist an a = [ a , a^, a ] such that: [a a ] 2 oJ,a [B B ,B ,B ] = 0 _ * , a A > ojj, and a B u  T  T  v  T  1 (  2  3  4  T  2 N + H  The argument used to show the equivalence of (29) and (30) that used to show the equivalence of (11) and (12)  5  < 0.  i s s i m i l a r to  in Appendix I, so i s  omitted.  Suppose (28) holds, but also suppose, contrary to the theorem, a . . T T . solution to (30) e x i s t s . The conditions a B, = 0 implies a =0... In the 4 H u H T T TJ  proof of Theorem  3.1,  i t i s shown that a [B^B^] = 0 ^_ 2  T can rewrite a B by using a = 0 and (13) 5 u H c  (31)  If a  a B T  1  5  TI  = a/(-B  T > 0, then a B  5  p G  -B  q(J  h +  N  We  t B T  q G  - MC) .  . . a contradiction. T  If a^ = 0, a = 0 _^ from (13).  implies (13).  as  ) = a, ( E ^ W  > 0 by (28),  2  T  Therefore, a A = 0^ and again we have a  contradiction. (ii)  If t  and g are optimally chosen at the pre-project equilibrium, then  they are a solution to the problem:  (32)  T max ~ . {8 u : (1) and the inequality version of (2) are u,p,t,g satisfied}.  88.  The f i r s t order Kuhn-Tucker conditions for (32)  (33)  ~T, there exists [ a . , a ]  a  Suppose (33)  , -T , ,. ' T, „T > Q , such that a A = 6 ,  [B B ,B B 3  T  r  is satisfied  are:  2  3 (  but (28)  4  = 0  i s not.  T 2N+H-1"  The argument following  (30)  then  T establishes that a Bj < 0,  so (30)  i s not s a t i s f i e d .  Consequently,  (28)  i s also necessary for the existence of a d i f f e r e n t i a l l y s t r i c t increase of s o c i a l welfare 8 du > 0 at a 8-optimum for t and g. Q. E. D.  To understand the implications of the Generalized Samuelsonian Rule (GSR hereafter)  (28),  l e t us assume that i n d i r e c t taxes and transfers are  Pareto e f f i c i e n t l y ,  set  so that we cannot make a d i f f e r e n t i a l l y s t r i c t Pareto  improvement with dG = 0.  Pareto e f f i c i e n t  that the economy i s i n f i r s t best.  i n d i r e c t taxes and transfers imply  It i s well-known that the proportional  commodity tax rates t = 8(p+t) for some r e a l number 8 i s f i r s t best with some appropriate lump-sum transfers.  Substituting this r e l a t i o n into (28) and  using (p+t) B  equals  T  (34)  (1-8)  = - E f » , h  E ° W h  1  h  1  h  (28)  > MC.  This means that the sum of marginal willingnesses to pay for the public good deflated by 8 (which i s a r a t i o between producer and consumer prices) must be compared with the marginal cost.  Needless to say, i f  no i n d i r e c t taxes, then the Samuelsonian rule applies.  8 = 0 so that there are  89 . Though proportional i n d i r e c t taxes are always f i r s t best, there may e x i s t some other f i r s t best taxes depending on the structure of the economy. For example, i f there i s no room for technological s u b s t i t u t a b i l i t y among private goods so that B = 0, , then any i n d i r e c t taxes can be p-optimal N«N PP I with apropriate lump-sum transfers (see Diewert (1978)). It i s obvious i n t h i s case that the use of the simple Samuelsonian rule is erroneous and we have to use the GSR (28). We now move to an alternative case where we can perturb g and G while holding the commodity tax d i s t o r t i o n s t  fixed.  We c a l l the r e s u l t i n g rule  within the following proposition, a Modified Harberger-Bruce-Harris Rule (MHBHR hereafter),  since i t i s an a p p l i c a t i o n of Harberger (1971) and  Bruce-Harris (1982) to a project evaluation approach to the production of a public good (see also Diewert (1983b)).  Theorem 3.3  (i)  Suppose that the government can change only the transfer vector g  when G i s increased; i . e . , Rule^ i s  (35)  where  (36)  T  e  ~T  = [CM ] = 1  dG > 0.  Then, the Modified Harberger-Bruce-Harris  90. Condition (35) du » 0  (ii)  fl  is sufficient  for the existence of a Pareto improvement  .  If i n the tax-distorted pre-project economy, g was chosen optimally,  then (35)  is also a necessary condition for a small increase i n public good  production to lead to a d i f f e r e n t i a l l y s t r i c t welfare improvement.  PROOF:  (i)  (37)  A sufficient  condition for a Pareto improvement i s :  there exists dg and dG _> 0 such that the inequality version of  (6)  i s s a t i s f i e d and du > 0 „ .  n  Condition (37)  i s equivalent to the following Motzkin dual condition:  T (38)  ~T  T  there does not exist [a , a a ] = a  oj,  1 f  such that  0 J,  oj.  aA > a B < 0, a [ B B , B ] = N+ [a aj] > Suppose (35) holds, but also suppose that, contrary to the theorem, a T T solution to (38) e x i s t s . The conditions a B^ = 0^ imply a = 0^. Hence we T T can rewrite a B. = 0„ . as: 1 N-1 T  T  T  5  V  2  4  v  y  (39)  a = a ^ p + e)  using (1.A.2) and (1.A.4)  (see Appendix IV).  Using a  = 0 , (36) and (39),  91 .  a B T  = a, (P + ) [ - B  - B  T  5  E  p G  q G  ].  T If a  1  (39) .  (ii)  > 0,  (35)  > 0, a contradiction. If a = 0, a = 0 from T T With a„ = 0„ we have a A = 0 „ , and again we have a contradiction. U  implies a B  5  1  n  N 1  H  If g i s optimally chosen at the pre-project equilibrium, g i s a  solution to the problem.  (40)  max  T - . {B u ; (1) and the inequality version of (2) are u,p,t ,g 1  satisfied}.  The f i r s t order Kuhn-Tucker conditions for (40)  (41)  there exists [a^,a ] _> 0 , N  Suppose (41)  is satisfied  T establishes that a B  c  are:  such that a A = 8 ,a [B^,B2,B ] = 0  but (35) i s not.  4  N + H >  The argument following (38)  . . . < 0, so (38) i s not s a t i s f i e d .  Consequently,  (35)  then is  also necessary for a d i f f e r e n t i a l l y s t r i c t increase of s o c i a l welfare at a B-optimum for g.  Q. E . D.  The economic i n t u i t i o n behind the two Propositions i n this section i s as follows.  Given the pre-project l e v e l s of u t i l i t y , increasing the provision  of the public good permits a reduction i n the consumption of private goods but requires additional inputs for the increased public good production. appropriately offsetting  the marginal benefits of the public good ( i . e . ,  By the  92 .  externality)  by changing lump-sum transfers to keep consumers at t h e i r  o r i g i n a l u t i l i t y l e v e l s , i t i s only necessary to evaluate the r e s u l t i n g change i n the quantities of the private goods by appropriate shadow p r i c e s . If  the vector of tax rates t i s v a r i a b l e , the production price vector i s  appropriate shadow price vector.  See (31)  i s a version of the production efficiency (1971).  behind a GSR (28)  .  the  This r e s u l t  theorem i n Diamond and Mirrlees  If t i s fixed, a MHBHR (35) must be adopted which uses a Harberger-  Bruce-Harris shadow price vector.  3-5.  Conclusion Our present chapter has derived project evaluation formulae for the  provision of public goods i n various second-best situations. three cases.  (1) the case where i n d i r e c t tax rates can be varied; (2)  case where both lump-sum transfers and i n d i r e c t tax-rates the case where lump-sum transfers are varied.  the  can be varied  (3)  We showed that the the use of  a GPR, a GSR and a MHBHR are suggested for cases (1), tively.  We considered  (2) and (3)  respec-  Our basic point i s that project evaluation rules must vary depending  on what instruments we can change when we a l t e r the supply of public goods. We have to note that there are severe limitations i n u t i l i z i n g our costbenefit  rules; i . e . ,  we have ignored the preference revelation problem for  public goods i n measuring the marginal willingnesses to pay W for consumers. Once t h i s d i f f i c u l t y i s overcome, our rules can be implemented by using only l o c a l information observable at the pre-project equilibrium, that i s , l e v e l of taxes, public goods,  the  prices, incomes, and the f i r s t order d e r i v a -  t i v e s of the ordinary demand functions and the net supply functions  for  93. private goods (which depend on both prices and public goods). and B 2.4.  Note that B  can be computed from ordinary demand functions as we pointed out i n Note further that information on W i s necessary to use the MHBHR (35)  as we need to compute B - from data on the ordinary demand functions. go  There-  fore, this rule i s also vulnerable to the f r e e - r i d e r problem. We have shown that i t i s f a i r l y easy to obtain s u f f i c i e n t  conditions for  the existence of a small Pareto improvement corresponding to an increase i n public goods production, given that various taxation instruments are a v a i l able.  It seems that t h i s approach i s more useful compared to the t r a d i t i o n a l  approach which derives the f i r s t order conditions for an i n t e r i o r second-best welfare optimum.  Our results also show that conventional cost-benefit  for the provision of public goods are not always correct.  rules  94. FOOTNOTES FOR CHAPTER 3  1  Our approach draws on the methodology found i n the tax reform  literature, e.g.,  Guesnerie (1977), Diewert (1978), D i x i t (1979) and Weymark  (1979), and the project evaluation study by Diewert (1983b).  Wildasin (1984)  also worked with a framework similar to ours, but his paper has various r e s t r i c t i v e assumptions; e . g . , all  only one commodity tax rate i s variable and  other goods are untaxed. 2 If we evaluate a possible reduction i n the production of the public  good dG <. 0, a l l we need i s to reverse the d i r e c t i o n of the i n e q u a l i t i e s the cost-benefit omitted.  rule.  in  The proof is straightforward and hence may be  The same comment applies to a l l cost-benefit  formulae i n t h i s  chapter. 3  It i s given by  \S^.  9 x ( p + t , G , I ) / 3 G = {3x (p+t,G,u )/9G} + n  h  h  h  T Premultiplymg  by t  E ^t (9x ( T  and i n summation over h, we have  h  h  P +  t,G,I )/9G)  which i s used to derive * If t = 0  N  h  = t B T  E ^t W S T  q G  +  h  h  b q  ,  (18).  so that there are no pre-existing tax d i s t o r t i o n s ,  then the  Modified Harberger-Bruce-Harris Rule i s i d e n t i c a l with the t r a d i t i o n a l Samuelsonian r u l e .  The proof i s straightforward.  APPENDICES FOR CHAPTER 3  Appendix I :  The derivation of (12).  Motzkin's Theorem i s as follows:  Either Ex » 0, Fx > 0, Gx = 0 has a solution x where E i s a nonvacu1T  2T  3T  ous matrix, F and G are matrices and x i s a vector or v E + v F + v G = T 1 2 1 2 3 0 , v > 0, v 2 0 has a solution where v , v and v are vectors, but not both.  See Mangasarian (1969).  We now apply i t to rewrite (11). = 0 . U  Decompose A, B. (i=1,2,3,5)  H  B^dg can be dropped from (6), for dg  between A*, B?, which are the top H rows  I  and A**, B?*, which are the bottom N rows. ' l  x = [du, dp , d t  E  F =  [ I  1  Define  dt , dG] ,  r  H ' °H«(2N+H) ' ]  -A** A  o**  n**  n**  1 ' 1 ' °2 ' 3 ' B  B  n** ti  5  '2N+H  where e  2 N + H  i s a unit vector with unity i n 2N+Hth row, and  G = [-A*,B*,B*,B*,B*].  96.  Then, the primal condition of the Motzkin's Theorem i s i d e n t i c a l with  (11).  ^ "j* "* T T T Defining v . j d - H row vector), v = [ a ^ a ,v] where v i s a scalar, v = a , the 2  3  dual condition i s :  there i s no solution v^, a^, a, v, a v  v  > 0, a  1  such that  2 0, a 2 0, v 2 0,  1  - a A = 0, a [ B B , B ] = 0  T  y  T  T  1 (  2  which i s i n turn i d e n t i c a l with  3  2 N  _  T  , a B T  5  + v = 0,  (12).  Appendix I I :  The derivation of (13). T T T Subtracting a B = 0 _^ from a B^ = 3  (A.1)  - a.B 1  N  T  "  e  have  - - a B~~ = 0„ f. PP N-1 T  From (1.A.4),  (A. 2)  where p  (A.3)  B  - + p B-~ = 0„ « pp N-1 T  p^p  1  = 1.  ( a /  Substituting (A.2) into (A.1), we have  - a )  By assumption (5),  T  B  p p  =  0^.  B«-~ i s nonsingular, which implies  (13).  u  97 Appendix III:  The derivation of (15) and (16)  T T Postmultiply t^ and t to a Bj = 0 and a B  3  T = 0^_y and adding them  together, we have  (A.4)  a*Xt + [ a a ] B t T  1 f  Substituting  (A.5)  q q  = 0.  (13) and rewriting the f i r s t term of  E A a V h=i u  + a , p B t = 0. 1 qq T  1  Substituting the i d e n t i t y  (1.A.2)  rearranging terms, we get  (15).  (A.6)  =  aj + [ a a ] B T  1 (  Substituting  (A.7)  a  m  (13)  q u  into the second term of (A.5), and T T We can rewrite a A = f as  .  into (A.6), we have  m  m  + a„p B = -y • u r qu 1  However, from (1.A.3)  and (3) we get  (A.S)  =  (p + t ) B T  Substituting  <»•»  q u  t  J•  (A.8) into (A.7) we have  «J - 7  T  (A.4), we get  - a, l „ • a , t \ . u  98  From the d e f i n i t i o n of B and S i n (9), qu qu  Appendix IV: T a B  (A.10)  1  The derivation of N  -a.[B  U  (A.11)  (39).  T = 0 _ can be rewritten as  1  using a  (A.9) i s i d e n t i c a l with  = 0„. n  B  1  - + B -] - a [B-~ + B~~] = 0 „ « q«q PiP qq PP N-1 T  From (1.A.4) and (1.A.2), we get  - + p B~~ = 0 „ 1 T  P-,P  N-1  PP  and  (A. 12)  t B - + B -+ p B~~ = 0 qq q^ qq N-1' T  T  M  respectively.  (A. 13) ^  Therefore, adding up (A.11) and (A.12), we have  B - + B q-iq  ~ = - t B - - p B~~ - p B ~ qq qq PP T  Substituting (A.13) into  (A. 14)  T  T  (A.10),  a . [ t B - + p B ~ + p B-~] = a [ B ~ + B~~] 1 qq qq pp qq PP T  T  T  T  1  J  Inverting the matrix [B~~ + B - ~ ] , and using d e f i n i t i o n (36), qq pp follows.  (39)  (16)  99 . CHAPTER 4 INCREASING RETURNS, IMPERFECT COMPETITION AND THE MEASUREMENT OF WASTE  4-1  Introduction In the presence of increasing returns to scale i n production, i t  is  well-known that Pareto optimal e q u i l i b r i a may not be decentralized through perfect competition and moreover, imperfect competition prevails frequently. Therefore, both positive and normative analysis of resource a l l o c a t i o n with increasing returns to scale becomes an important topic i n applied welfare economics.  The normative problem of developing mechanisms to support Pareto  optima i n the presence of increasing returns to scale has been discussed by many authors, including Arrow and Hurwicz (1960), Guesnerie (1975) and Brown and Heal (1980).  The second best p r i c i n g problem of public u t i l i t i e s  facing  a revenue constraint i s discussed by the optimal p r i c i n g and taxation l i t e r a ture beginning with Boiteux (1956).  There have been numerous positive analy-  ses of o l i g o p o l i s t i c markets in the vast l i t e r a t u r e on strategic  interactions  among incumbent firms or among incumbent firms and potential entrants.  More-  over, there i s a large l i t e r a t u r e on Chamberlinian (1962) monopolistic competition.  In contrast,  the measurement of waste due to imperfect competition  with increasing returns to scale i s a r e l a t i v e l y less developed area, although the important seminal paper by Hotelling (1938) dealt with this topic.  The aim of this chapter i s to consider this measurement of waste  problem. Let us f i r s t review the problem discussed by Hotelling (1938) and l i s t the points which seem to c a l l for extensions.  First,  Hotelling claimed that  f i r s t best optimality i s characterized by the marginal cost p r i n c i p l e ,  i.e.,  100. the price of the product should equal i t s marginal cost, for increasing returns to scale firms.  However, i t was pointed out by Arrow and Hurwicz  (1960) that this solution is not necessarily optimal with a general nonconvex technology,  and Silberberg (1980) pointed out that Hotelling (1938) i s  actually not proving the optimality of marginal cost p r i c i n g .  Therefore, in  the l i t e r a t u r e on the measuremnt of deadweight l o s s , which includes Debreu (1954), Harberger (1964) and Diewert (1981, 1983(a), 1985(a)) i n order to avoid t h i s d i f f i c u l t y i t i s assumed that a l l firms have a convex  technology.  Therefore, i n order to compute the deadweight loss, we f i r s t characterize the optimality i n nonconvex economy rigorously.  Second, H o t e l l i n g ' s (1938)  measure of waste does not seem to be correct i n a general equilibrium sense, and furthermore, requires the computation of an optimum equilibrium which necessitates global information on consumer preferences and technology,  so  that we would l i k e to derive a measure of waste which can be evaluated using only l o c a l information on preferences and technology,  so that the measure is  more useful i n empirical research on the measurement of waste. In this chapter, we show that these problems can be solved i n a s a t i s factory way, at least i n our s i m p l i f i e d model. Our findings i n this chapter may be summarized as follows.  We can  derive a Hotelling-Harberger type general equilibrium approximate deadweight loss measure due to imperfect competition allowing for quite general d i f f e r entiable functional forms for production and u t i l i t y functions,  including  production functions that exhibit increasing returns to scale.  This approxi-  mate measure can be implemented from l o c a l information up to the second order obtained at an observed distorted equilibrium.  There are different waste  measures depending on the types of increasing returns to scale,  since the  101 characterization of the optimum depends on these types of increasing returns to scale. In the next section, we construct a model employing the assumptions that production functions are quasiconcave, factor markets are competitive, and the number of firms i n one production sector i s fixed.  We characterize the  imperfectly competitive general equilibrium by a system of equations.  In  4-3, we derive an Allais-Debreu-Diewert measure of waste with increasing returns to scale and show that the corresponding optimum equilibrium is characterized by the marginal cost p r i n c i p l e .  In 4-4, we compute a second order  approximation to the ADD loss measure, discuss i t s informational requirements, and show how our measure generalizes H o t e l l i n g ' s o r i g i n a l approach and other works on deadweight loss which assume technologies are convex.  We also  discuss various relaxations of our assumptions, and l i m i t a t i o n s on applying our approach to empirical studies of various market imperfections.  Section  4-5 concludes with a diagrammatic interpretation of our approximate measures.  4-2.  The Model We assume that there are N goods i n the economy, where the corresponding T  price vector i s p E ( p ^ , - . . , P ^ )  » 0 , N  and that only sector n produces the  nth good for n = 1 , . . . , N by combining the other goods and M nonproducible factors.  This vector of primary factors has the vector of factor prices  w = (w ...,w ) r  M  T  » 0 . M  Each production unit i s assumed to have a quasi-concave production function f ( x , . . . , x , n  1  N  v  1 (  . . . , v ) ; that i s , M  for a given l e v e l of output y ,  marginal rates of technical substitution between inputs are d i m i n i s h i n g . This assumption i s weaker than global convexity i n production; the  1  102 .  p o s s i b i l i t y of increasing returns to scale i s allowed for when we change the l e v e l of output i n t h i s c h a r a c t e r i z a t i o n . 2 We define the sector n cost function C ( p , w , y ) as n  n  (1)  C (p,w,y ) = rain {p x + w v : f ( x , v ) > y }, n = 1 , . . . , N . n x>0. ,v>0„ ' - n' ' ' ' —N M n  T  v A  C l  i  T  n  r  2  T  C  n  is i d e n t i c a l to the expenditure function m* defined by (1.4), except that 1  the u t i l i t y l e v e l i s replaced by the production l e v e l .  We assume that the  regularity conditions l i s t e d i n Diewert (1982;554) are s a t i s f i e d .  There are  H households i n t h i s economy and t h e i r demands are characterized i n terms of the expenditure  functions  h T T h m (p,w,u. ) = min ,{p a+w b : f (a,b) n a, D  (2)  h where P. i s a (translated)  h > u, ,(a,b)eQ }, n  h = 1,...,H,  N+M orthant of R defined as i n (1.4).  We assume  _  that the hth household holds the vector of i n i t i a l endowments Y = ,-hT  (a  rhT.T  ,b  ) .  To characterize the general equilibrium, we u t i l i z e the overspending function B defined by:  (3)  B(y,p,w,u) = E ^ { m ( p , w , u ) - p a h  h  T  1  h  h  - wb} T  h  - [ ,{p y - C (p,w,y )}, n=1 n n n N  where y = ( y , . . . , y ) 1  N  T  and u =  n  (u ,...,u ) . T  1  H  Compared with the overspending functions i n previous chapters,  consumers  and producers are facing the same prices i n (3) so that we no longer have two  103 .  D e f i n i t i o n ( 3 ) may be s i m p l i f i e d by defining  sets of prices as arguments. T TT Q = (p ,w ) as  follows:  B(y Q,u)  (4)  (  = E " {m (Q,u ) h  h  1  h  - Q Y} - ^ { p ^ T  h  - C (Q,y )}. n  n  In the same manner as we derived the properties of an overspending function in Appendix I of chapter 1, we can e a s i l y derive the following properties for the new overspending function:  (i) B i s concave with respect to Q ; ( i i )  i s once continuously d i f f e r e n t i a b l e with respect to prices, the vector of excess demands; ( i i i ) prices.  From t h i s ,  ( 5 )  Q T B  V^Bty.QjU)  if B  equals  B i s l i n e a r l y homogeneous with respect  the r e s u l t i n g i d e n t i t i e s  are  satisfied:  Q Q = °N M' +  and  (6)  Q B T  where B  Q y  = V B(y,Q,u)  = V B(y,Q,u) 2  Q Q  T  y  Q  and B  = V B(y,Q,u). 2  Q y  Q  Note that - V B ( y , Q , u ) y  is a  vector whose i t h component i s the difference between the price and marginal cost of the i t h good. Now using the above r e l a t i o n s , we characterize the general equilibrium 1 1 1 1 1T 1T T (y , Q ,u ) where Q = (p ,w ) as  (7)  follows:  h, 1 „ 1 . ^IT^h , N h n 1 1 „ n , - 1 1., m (Q , u ) = Q Y + E a { P y - C (Q ,y )} ft  r  h  f  n = 1  n  n  n  , + g , h  , . h = 1,...,H,  to  104 .  (8)  (9)  N+M  where a  hn  i s the share of the nth firm held by the hth i n d i v i d u a l and  H hn Ej^a = 1, for n = 1 , . . . , N .  The number g^, h = 1 , . . . , H ,  lump-sum transfer given to the hth i n d i v i d u a l and t = (t^  shows the net T ,...,t^)  where  t  R  i s the monopolistic mark-up imposed by firm n on his sales. We can show that (7) and (9) imply that the sum of the transfers g^, h = 1,...,H,  equals zero.  the H i n d i v i d u a l s .  The equations i n (7) are the budget constraints of  The equations i n (8) state that the difference between  the price of the i t h good and i t s marginal cost i s equal to the mark-up t^.  For perfectly competitive firms t = 0^, but with imperfect competition  we expect t » 0^.  With increasing returns to scale,  firms must charge prices  larger than t h e i r marginal costs i n order to a t t a i n nonnegative  profits.  This does not necessarily mean that the monopolistic markup i s fixed for monopolists.  We just define t ex-post at the equilibrium as the  between consumer prices and marginal costs.  difference  Noting that v^B equals the  vector of excess demands, the equations i n (9) are the market clearing conditions for the equilibrium.  Therefore, (7) to (9) characterize an  imperfectly competitive general equilibrium, as elaborated by Negishi (1960-1), Arrow and Hahn (1971, Ch. 6) and Roberts and Sonnenschein (1977). 4-3.  The Allais-Debreu-Diewert Measure of Waste Let us f i r s t take an N + M dimensional nonnegative reference bundle of  goods and factors A = ( a , 8 ) T  T  T  >0  and each consumer's u t i l i t y l e v e l u},  105.  h = 1,...,H,  i n the imperfectly competitive equilibrium, and consider the  following primal planning problem:  0  nn\ (  1  0  )  r  , H  r  a  a  x  h , h  n  n  ...  { r  :  (  1  a ,b ,y .x ,v  )  E  - H  h=1  h , - N  a  +  E  n=1  X  n , +  a  r  ,r,H-h  .  *  y  r  - h=1  +  E  h=1 ' a  n  h=1  E  E  n=1  V  P  '  E  ( i i i ) f ( a , b ) 2 UjJ, (a ,b )eQ h = 1 , . . . , H , h  h  h  h  h  h  (iv) f ( x , v ) 2 y , n = 1,...,N}. n  n  n  n  The solution to (10) defines  the ADD measure of waste L , ^ ^ = ADD  .  Problem  (10) may be interpreted as maximizing the number of multiples r of the given reference bundle X that can be obtained while maintaining consumers' u t i l i t i e s at u^, h = 1 , . . . , H , technology constraints.  and satisfying the materials balance and  We assume that a f i n i t e maximum exists for  We can also derive a dual expression to (10) as follows.  (10).  First,  let us  T f i x y = (y.|, . . . ,Yjj) .  From the d e f i n i t i o n of quasi-concavity, the sets  f ( x , v ) 2 Y ( n = 1 , . . . , N ) are convex sets belonging to R ^ n  n  n  n  +M  .  Then, the  remaining programming problem becomes a concave programming so that we can rewrite (10) using the Uzawa (1958)-Karlin (1959) Saddle Point Theorem as 3  (11)  r  0  = max  [max min N  using d e f i n i t i o n s  (1)  N+M  (2) and (4),  T 1 (r(1-Q X ) - B(y,Q,u )}]  where Q i s the vector of Lagrangean  m u l t i p l i e r s associated with the resource constraints,  (i) and ( i i ) .  The  max-min problem within the squared bracket of (11) can be rewritten using the Uzawa-Karlin Theorem i n reverse as  106 .  (12)  -  m a x n  >n  W Y I Q ' " ) **.t. Q X > 1} 1  T  N+M  For the given l e v e l of y, the solution of the max-min problem within becomes r(y) and Q(y) which are functions of y.  Then (11)  (11)  can also be  written as  13)  r ° = max  {r(y)(1-Q(y) X) T  -  B(y,Q(y),u )} 1  N  The global programming problem (10) and (11) define the ADD measure of 1 0 waste when the observed u t i l i t i e s are u , but i t i s d i f f i c u l t to compute r using this approach since we need global information on preferences and technologies.  To get more insight about the amount of waste i n r e l a t i o n to  the degree of monopoly, and bridge the gap between conventional  deadweight  loss measures and our ADD measure, we derive a second order approximation to the ADD measure of waste. assumptions as (i)  For this purpose, we have to strengthen our  follows:  (y°,r ,Q )  solves (11) with y ° » 0 , N conditions for (11) hold with e q u a l i t y ; 0  0  M  4  h = 1,...,H, n  M  M  are twice continuously d i f f e r e n t i a b l e with respect to Q at  0 1 (Q ' ^ ; ( i i i ) u  Q° » 0 ^ so that the f i r s t order N+M ( i i ) the expenditure functions m*\  n the cost functions C , n = 1 , . . . , N , are twice continuously  d i f f e r e n t i a b l e at (Q^,y^); (iv) Samuelson's  (1947) strong second order  conditions hold for the two problems (12) and (13) when the inequality constraint i n (12)  i s replaced by an equality.  The regularity condition (i) implies that there are no free goods and a l l firms are useful.  Conditions ( i i )  and ( i i i )  are d i f f e r e n t i a b i l i t y  107 . assumptions, (iv)  which are natural for a l o c a l analysis such as ours.  Condition  i s an assumption which guarantees that the maximum of the planning  problem (10)  is l o c a l l y unique.  Our regularity conditions on (12)  imply the  bordered Hessian  -B° -x QQ' *  (14)  T -A  where  =  V^B(y®  A  0  definite  0  ,  ,Q®  are evaluated at z = 0.  (15)  i s positive  ) and the superscript 0 means that the  derivatives  By defining  , -B yy 0  and  (16)  B°=[-B °,0 ] y  N  where the superscript 0 means that B ^ and B ^ are evaluated at the optimum, yy yQ our condition i n (iv) i s equivalent to the following condition:  (17)  0 0 0 -1 OT A - B (C ) B is negative  The condition (17)  definite.  i s much weaker than assuming marginal costs are  increasing, which requires A to be negative d e f i n i t e ,  for C° i s  definite  we are admitting the  0  by (14).  By merely requiring condition ( i v ) ,  positive  108 . p o s s i b i l i t y of a downward sloping marginal cost curve, which follows  the  s p i r i t of Hotelling (1938;255-6) .5 It follows from assumption (i) that an i n t e r i o r solution exists to (11).  The f i r s t order conditions are given by:  18)  -v B(y°,Q°,u ) = 0 , 1  y  N  (19)  -Ar° - V B ( y ° , 0 ° , u ) = 0  (20)  1 - Q A = 0,  1  Q  N + M  ,  0T  where (18)  i s a marginal cost p r i c i n g p r i n c i p l e for monopolistic firms, (19)  are resource balance equations for goods and factors with Ar"* > 0^ being the vector of surplus goods and factors,  and (20)  i s a normalization rule for the  optimal p r i c e s .  4-4.  Second Order Approximations Now comparing the market equilibrium conditions and the f i r s t order  conditions for the optimum, we construct a z-equilibrium which depends on a scalar parameter z (0 <. z <. 1);  (21)  -v B(y(z),Q(z),u )  (22)  - V B ( y ( z ) , Q ( z ) , u ) - Ar(z) = 0  (23)  1 - Q(z) A = 0.  1  y  = tz,  1  Q  T  N + M  ,  109 .  If we define (y(0), Q(0), r(0))  = (y ,Q ,i ), then (21) - (23) coincide with  the optimality conditions (18) - (20) when z = 0. (yd),  Q ( D , r(1))  In contrast, i f we define  = ( y , Q , 0 ) , then (21) and (22) coincide with (8) and (9) 1  respectively when z = 1.  1  In this case,  (7) i s also s a t i s f i e d  appropriate choice of transfers g^, h = 1 , . . . , H .  for an  From condition (23)  at  z = 1, we also assume that the market prices satisfy the normalization,  (24)  1 = Q X, 1 T  by choosing the scale of X appropriately. z-equilibrium (21)  Thus we can conclude that the  - (23) maps the optimal equilibrium into the imperfectly  competitive equilibrium as z i s adjusted from zero to one. Equation (21)  that maps the marginal cost p r i c i n g condition (18)  into  the monopolistic markup equilibrium condition (8) may seem unnatural because markups are decreasing l i n e a r l y , but the change i n t may be nonlinear depending on the behaviour of monopolists. tions,  Even i n the case of t a x - d i s t o r -  however, i t is possible to choose some nonlinear path of the change of  tax rates as the equilibrium i s adjusted and the r e s u l t i n g magnitude of waste depends on this choice of path.  (This problem i s also related to the  t r a d i t i o n a l problem of path independence i n consumers' surplus analysis.)  As  i t i s d i f f i c u l t to overcome this a r b i t r a r i n e s s within our framework, we have just assumed that there i s a uniform reduction of monopoly d i s t o r t i o n s . The main theorem i n t h i s section i s as follows:  1 10. Theorem 1:  A second order approximation to the ADD measure of waste (10)  is  given by  V(A  (25)  PROOF:  0  - B°(C°)-  B°V t 1  1  > 0.  Differentiate (21) - (23) with respect to z and we have  A ,  B  Z  t  y' (z)  Z  (26)  =  Q'iz) B  Z T  , C  °N+M  r' (z)  Z  0  where A , B , C are the matrices A , B z  z  z  and C defined by (15),  (16) and  (14) evaluated at z, rather than 0. T  T  Premultiplying (26) by [0 ,Q(z)  ,0], we have  N  (27)  - Q ( z ) B y ' ( z ) - Q ( z ) B Q ' ( z ) - Q ( z ) A r ' ( z ) = 0. T  Z  T  Q  Substituting (5), (28)  Z  T  Q  (6),  and (23),  and then (21) into (27),  we obtain  r'(z) = z t y ' ( z ) . T  Noting that r(1) = 0, by using a Taylor series expansion the ADD measure of waste L n A n  (29)  =  r<  ^  =  r° - r  r  1  (0)  ~ d) r  can be approximated by  = r ° - { r ° + r'(0)  However, from (28),  r'(0)  + hr"(0)} = - r'(0)  = 0 and r"(0) = t y ' ( 0 ) . T  - Jjr"(0)  Therefore, we have  111.  (30)  r° - r  1  .=  - J^tV (0).  Evaluating (26) at z = 0 and inverting the left-hand side matrix yields y'(0)  = (A^ - B^(C^) ^B^ ) ^t. T  result  (25)  follows.  Substituting t h i s expression into (30),  The inequality i n (25)  follows  from (17).  the  Q. E . D.  The formula (25) gives a general formula of deadweight loss applicable to either a convex or nonconvex economy.  This formula i s i d e n t i c a l to the  Debreu (1954)-Diewert (1985a) approximate deadweight loss formula when the technologies are convex.  However, the converse i s not true, since  the  optimal shadow price (or i n t r i n s i c price to use  Debreu's (1951, 1954)  may not exist with increasing returns to scale.  This problem i s overcome by  our two-stage optimization procedure (11)  term)  for the characterization of the  optimum, an approach which was suggested by Arrow and Hurwicz (1960) and Guesnerie (1975).  Our r e s u l t i n g approximate loss formula (25)  not from the derivatives of supply functions,  is  calculated  but from the derivatives of  r e s t r i c t e d factor demand functions and marginal cost functions evaluated at the optimal l e v e l of output. As our work i s preceded by Hotelling (1938), i t i s important to his work in r e l a t i o n to ours.  discuss  Hotelling's contributions i n t h i s paper are  known to be that (i) he showed the optimality of the marginal cost p r i c i n g p r i n c i p l e of the regulated firms, and that ( i i )  he derived the approximate  deadweight loss formula deviating from the optimality above. For the f i r s t point, Silberberg (1980) pointed out that H o t e l l i n g ' s proof i s not a v a l i d one.  In section 3 we gave a rigorous proof based on  programming that the marginal cost p r i c i n g p r i n c i p l e i s necessary  for o p t i -  112.  mality i f technologies  are quasi-concave.  As Arrow and Hurwicz (1960)  showed, t h i s condition i s not necessary for general nonconvex technology. Suppose that we a l t e r n a t i v e l y consider Guesnerie's (1975) type 3 firm; i s a firm's technology i s convex i f some input i s given.  that  By applying results  i n Guesnerie ( 1975;12-13), i t i s straightforward to show that optimality i s characterized by the competitive maximization of ' r e s t r i c t e d ' p r o f i t given the level of the input which causes the increasing returns to scale, the equality of the marginal value product with the factor p r i c e . Aoki (1971).)6  and by (See also  For the second point, H o t e l l i n g (1938;254) derived a similar  deadweight loss measure to our formula (30).  Similar to his f i r s t point,  however, the derivation of his loss measure lacks true general equilibrium considerations and cannot be v a l i d despite his own conjecture. (1987b)).  (See Tsuneki  Furthermore, even i f we interpret his measure as i n (30),  not useful without knowing how to compute y'(0) Function Theorem.  it  is  using (26) v i a the Implicit  We must also note that our approach for the measurement of  waste can be applied to an economy including type-3 firms which was not considered by Hotelling (1938).  As we have seen, we can derive f i r s t order  necessary conditions for the optimality.  Then, comparing the optimum with a  market equilibrium which includes mark-up rates i n either product or factor markets, we can derive a deadweight loss measure using the methodology employed above. (25).  However, the r e s u l t i n g approximate measure i s different from  What matters now are derivatives of r e s t r i c t e d p r o f i t functions,  given  the input that causes the nonconvexity, instead of the derivatives of cost functions. The drawback of our approach i s that these derivatives are not observable at the distorted observed equilibrium. following c o r o l l a r y :  It i s somewhat overcome by the  113. Corollary 1.1:  T  (31)  The approximate ADD measure  1  1  -1  1  -Jjt ( A - B (C ) 1  1T  B  -1  )  t  i s also accurate for quadratic functions as PROOF:  According to Diewert's  (25).  (1976;118) Quadratic Approximation Lemma, both  - ( r ' ( 0 ) + Jjr'tO)) and -^(r'(0) + r'(1)) give the exact value of r(0) - r(1) if  r i s quadratic.  The former was adopted to derive (25).  Now using the  l a t t e r approximation and using (28) we have  0 (32)  r  1  - r  T  = -^ty'(1).  Evaluating (26) at z = 1, computing y'(1) matrix and substituting i t into (32),  The remarkable property of (31)  by inverting the left-hand side  we get  (31).  Q. E. D.  i s that we can compute the  deadweight  loss of the economy from the l o c a l derivatives of demand and supply (cost) functions evaluated at the observed equilibrium. of this observation, i s that (31)  One important consequence  can be computed using f l e x i b l e  forms for u t i l i t y and production functions,  functional  so that we need not assume  r e s t r i c t i v e functional forms to calculate the global optimum point, as  is  usual i n the numerical general equilibrium l i t e r a t u r e . To derive our approximate loss formulae (25) and (31), several r e s t r i c t i v e assumptions.  we maintained  The assumption of competitive  factor  markets can be dropped by introducing mark-up rates on factor prices, even though the resulting formulae become more complicated.  The assumption that  114.  the production functions  must be quasi-concave  was required to guarantee the  optimality of the marginal cost p r i n c i p l e , and we already discussed how to extend our approach when we dropped the  assumption.  We have assumed that each industry i s monopolized. the result to the case of an o l i g o p o l i s t i c  It i s easy to extend  industry i f we know the mark-up  rates of firms and that the number of firms within one industry i s fixed for a l l industries.  However, i t i s d i f f i c u l t to introduce entry-exit  behaviour,  since the f i r s t order s o c i a l optimality and market equilibrium conditions incumbents and entrants are characterized by inequalities ities,  for the number of firms changes discontinuously  for  rather than equal-  as e q u i l i b r i a are  adjusted from the observed equilibrium to the optimum as is shown i n the l i m i t pricing literature.  Therefore, i t i s d i f f i c u l t to apply our approach  based on the Implicit Function Theorem.1  The only case with entry that we  can deal with within our framework i s a Chamberlinian (1962)  monopolistic  competition with each product produced by homogeneous producers with respect to market shares, product q u a l i l t y and technology. number of firms i s continuous.  Suppose also that the  Then, the long-run equilibrium i s  characterized by the z e r o - p r o f i t conditions of firms,  i.e.  e q u a l i t i e s where  the number of firms i s also endogenous, and Chamberlinian excess capacities cause deadweight l o s s .  The optimality conditions are characterized by the  marginal cost p r i c i n g p r i n c i p l e and the optimum number of firms i s at the point where the marginal cost equals average cost.  However,  determined this  model may be incomplete as a monopolistic competition model, since product d i v e r s i t y i s exogenous i n our model. with much more s i m p l i f i e d models, Stiglitz  (1977).  To make i t endogenous, we must work  as adopted i n Spence (1976), and D i x i t and  115. 4-5.  Conclusion This chapter has reconsidered the methodology for the measurement of  waste due to imperfect competition i n the presence of increasing returns to scale.  We noted H o t e l l i n g ' s (1938) confusion about the optimality of the  marginal cost p r i n c i p l e and the derivation of his deadweight loss formula and rederived his formula as (30).  The drawback to H o t e l l i n g ' s measure (30)  that i t cannot be computed without finding the optimum beforehand.  is  This  drawback was corrected by our measure (25) and (31) where we required only l o c a l information i n order to measure the deadweight loss. for the loss measure defined by (31),  In p a r t i c u l a r ,  only information observable at the  distorted equilibrium i s required to measure the dead loss. Fig.  10 shows a single-consumer economy with one good y and one non-  producible production factor v, labour for example. i t y set OA exhibits increasing returns to scale, equilibrium cannot e x i s t . M =  (y^' [Y|) v  c  a  n  The production p o s s i b i l -  so that a competitive  However, an imperfectly competitive equilibrium  exist where the marginal rate of substitution between the  good and labour i n consumption i s different from the rate of substitution i n production.  The ADD optimum point D = ( Y Q I Q ) i s V  a  point where surplus  labour is maximized given the u t i l i t y level at the observed distorted equilibrium where the reference bundle 6 consists only of labour.  The point  D i s characterized by the equality of the marginal rates of substitution i n consumption (or marginal benefit of the good) and the marginal rates of substitution i n production (or marginal cost of the good). these two curves as MB and MC.  F i g . 11 shows  The true amount of deadweight loss i s shown  by the c u r v i l i n e a r t r i a n g l e ABC while the approximate measure (25) by the t r i a n g l e ABC and (31) i s shown by A B C ' .  i s shown  The proof that ABC, A B C ,  116. ABC" r e a l l y correspond to (11),  (25),  (31)  for t h i s simple economy i s  analogous to the derivation and construction of (1.45),  (1.48),  (1.49) i n  Chapter 1. Given the l i m i t a t i o n s and assumptions l i s t e d within the chapter, we can apply our generalized H o t e l l i n g ' s measure to various models of imperfect competition and to p u b l i c l y regulated markets when increasing returns to scale are present.  We hope that the theoretical foundation provided here for  Hotelling's measure w i l l stimulate future empirical research and policy evaluation using i t .  117. FOOTNOTES FOR CHAPTER 4  1  For the production function f ( x ^ , . . . , x , v , . . . , v ) , n  N  3f /3x = 0. ' n n  1  M  Therefore, the cost function C dual to f ' n  n  we assume that  has the derivative  3C /9p = 0. ' n n  r  2  For example, the increasing returns to scale technology obtained by  combining a convex production p o s s i b i l i t i e s be dealt with within our framework.  set with a large fixed cost can  (See Negishi (1962).)  Aoki (1971) also  used a similar technological assumption to the one adopted here. 3  To apply the theorem, we need to assume that the Slater constraint  q u a l i f i c a t i o n condition holds; that i s , for  (10)  we assume that a feasible  exists that s t r i c t l y s a t i s f i e s the f i r s t N+M inequality  * With increasing returns to scale,  0  0  0> .  constraints.  a l o c a l optimum that s a t i s f i e s  f i r s t order conditions may not be globally optimal. , 0  solution  the  We assume that  . . .  (r ,y ,p ,w ) i s a global optimum. 5  Increasing returns to scale i s usually defined as a more than  proportionate increase of output when a l l the inputs are proportionately increased.  Baumol, Panzar and W i l l i g (1982;18-21) propose a weaker notion of  increasing returns to scale,  i.e.,  decreasing average cost, and showed that  i t i s implied by decreasing marginal cost. 6 Arrow and Hurwicz (1960) and Arnott and Harris (1976) gave examples where cost minimization and the marginal cost p r i n c i p l e r e s u l t i n productive i n e f f i c i e n c y i n a type-3 economy. 7  According to the recent study of contestable markets by Baumol, Panzar  and W i l l i g (1982), these strategic aspects are immaterial when the fixed cost i s not sunk.  Since a natural monopoly must set the price equal to  its  118. average cost for a sustainable equilibrium, the mark-up rates t the difference between the average cost and marginal cost, approach i s applicable.  equal  so that our  119.  F i g . 10 The ADD Measure with Increasing Returns to Scale  120.  Marginal Benefit of y  U  V  M  yo  Fig.  y  11  The ADD Measure and i t s Approximations with Increasing Returns to Scale  121 CHAPTER 5 PROJECT EVALUATION RULES FOR IMPERFECTLY COMPETITIVE ECONOMIES  5-1  Introduction In t h i s chapter, we are interested i n evaluating the net benefit of  introducing a new technology i n the presence of pre-existing d i s t o r t i o n s . This problem, we c a l l project evaluation, may be defined as follows.  Given a  pre-project general equilibrium where consumers and firms follow some behavioural rules and demand and supply are equal, consider introducing a net output vector, c a l l e d a project.  Both consumers and firms adjust to t h i s  change and the economy moves to a post-project equilibrium.  Project evalua-  tion means to determine whether the project increased or decreased s o c i a l welfare.  The evaluation of a small project when there i s perfect competition  with tax d i s t o r t i o n s was surveyed by Diewert (1983b) and we applied his approach to evaluate the benefit of public goods when there are tax d i s t o r tions i n Chapter 3.  Therefore, a natural way to proceed seems to be to  extend t h i s approach to the evaluation of a small project i n an imperfectly competitive economy. looks at f i r s t .  However, this approach may not be as promising as  it  Commenting on Davis and Whinston's (1965) use of a perceived  demand curve i n the second best theory of imperfect competition, Negishi (1967) pointed out that the second best policy of a public firm i s indeterminate unless the perceived demand curves of the imperfect competitors are known.  Therefore, we have to follow a different avenue.  In project evaluation, i t i s often the case that a new project has effects which are too large to be approximated by d i f f e r e n t i a l changes so that a shadow-pricing approach must be given up.  Project evaluation rules  122 . for large projects have been studied by Negishi (1962) and Harris (1978) for the case of perfect competition with an increasing returns to scale technology due to a large fixed cost.  Negishi (1962) studied the welfare i m p l i -  cations of the entry of a new firm which i s either a perfect competitor but has a large fixed cost technology or i s the only firm which deviates perfect competition.  from  Some of Negishi's results were extended and some new  rules were developed by Harris (1978), who also considered economies with distortionary taxation and public goods. Negishi's  However, Harris (1978) kept  (1962) assumptions about perfect competition and a convex technol-  ogy with a fixed cost. The purpose of this chapter i s to extend the Negishi and Harris results to an imperfect market economy.  This extension to an imperfectly competitive  economy may be important considering the above mentioned indeterminacy of the optimum policy when there i s imperfect competition. Our results in this chapter may be summarized as follows.  First,  the  Harris and Negishi results hold even i f the assumption of a convex technology with a large fixed cost i s replaced by general nonconvex technology, provided i t i s assumed that pre and post-project e q u i l i b r i a e x i s t . the extensions of Negishi's  Second, some of  (1962) results by Harris (1978) depend on an  i m p l i c i t weakening of the c r i t e r i o n for welfare improvement made by Harris compared with Negishi's o r i g i n a l welfare c r i t e r i o n .  T h i r d l y , but most  importantly, most of their rules can be applied i n imperfectly competitive economies generally, again as long as pre and post-project e q u i l i b r i a are assumed to e x i s t . In the next section, we discuss welfare c r i t e r i a for cost-benefit analysis.  We discuss some confusion which exists concerning the use of the  123 . compensation p r i n c i p l e and show that the c r i t e r i o n adopted by Negishi (1962) for the acceptance of a project i s more s t r i c t than that by Harris (1978). We cannot judge which c r i t e r i o n i s superior to the other.  However, when we  develop project evaluation rules, we simply have to be e x p l i c i t on which c r i t e r i o n each rule i s based.  After presenting the model i n section 3, i n  section 4 we reconsider the rules l i s t e d by Harris (1978), which include Negishi's  (1962) o r i g i n a l rules, and show that most of them are applicable in  an imperfectly competitive economy.  Economic implications and informational  requirements for extending project evaluation rules to imperfectly competitive environments are also discussed.  5-2.  Section 5 concludes.  Compensation C r i t e r i a for Cost-Benefit Analysis Reconsidered In chapter 1, we analyzed the properties of the ADD measure and the HB  measure as a s o c i a l welfare function.  It was shown that they are consistent  with the Pareto quasiordering, but they are not t y p i c a l l y w e l f a r i s t .  We also  suggested the use of deadweight loss measures for policy evaluation; a l t e r n a t i v e p o l i c i e s are ranked by the associated  level of deadweight  loss.  The c r u c i a l assumption for using t h i s procedure was that production possibilities  sets remain unchanged by these p o l i c i e s .  Therefore,  this  approach works for changes i n tax or regulation p o l i c i e s with a fixed technology.  We should note,  however, that i t i s impossible to compare the  values of these measures when production p o s s i b i l i t i e s the introduction of new projects, w e l f a r i s t assumptions.  sets are changed by  since these measures do not  satisfy  For example, the d e f i n i t i o n of Pareto optimal  allocations takes as given the production p o s s i b i l i t i e s Pareto optimal a l l o c a t i o n a.  sets.  Consider a  Suppose there i s a change i n technology which  124.  permits the attainment of a new a l l o c a t i o n a which i s preferred by a l l consumers to a.  Using the ADD measure, a i s always measured to exhibit at  least as much deadweight loss as a, as a l l Pareto optimal a l l o c a t i o n s have ADD measures of zero.  As a consequence,  i t i s inappropriate to use this  deadweight loss measure when the technology i s not fixed.  The optimal  reference equilibrium on which the HB measure i s based depends on technology so i f the introduction of a new project changes the technology,  then a unique  reference price cannot be determined to calculate the HB measure.  Therefore,  the topic of this chapter, the evaluation of a new project, necessitates an a l t e r n a t i v e c r i t e r i o n for s o c i a l welfare and especially the problem of u t i l i t y comparison.  We recommend two c r i t e r i a ; the f i r s t i s Bergson  Samuelson s o c i a l welfare function and the other i s Hicks-Kaldor compensation principle.1  In either c r i t e r i o n , we show i n this chapter that our project  evaluation c r i t e r i a can be related to aggregate quantities of i n d i v i d u a l consumption bundles. One natural way to proceed i s to suppose that there exists a Paretoi n c l u s i v e social welfare function by assuming that either there exists an omniscient planner who d i s t r i b u t e s income optimally at any point  (see  Samuelson (1956)) or consumers' preferences s a t i s f y Gorman's (1953) r e s t r i c t i o n of quasi-homotheticity.  It i s obvious that a Bergson-Samuelsonian  s o c i a l welfare function can serve as a welfare indicator to evaluate states corresponding to d i f f e r e n t technologies Unfortunately,  the  i n a consistent manner.  i t i s d i f f i c u l t to come to a consensus as to what an appro-  priate functional form for the s o c i a l welfare function i s .  Also the assump-  tion of quasihomothetic preferences i s empirically r e s t r i c t i v e . An alternative method t r a d i t i o n a l l y adopted for the evaluation of projects i s the Kaldor (1939)-Hicks  (1939, 1940) compensation p r i n c i p l e , which  125 .  states that a move from one state to another should be made i f a potential Pareto improvement can be made.2  However, there are several versions of the  compensation p r i n c i p l e and we have to be careful i n d i s t i n g u i s h i n g t h e i r d i f ferent meanings. 1  We suppose that two states of the economy (z,Y^,x^) and  1  (z,Y ,x ) are compared where z i s a vector of i n i t i a l resources which i s fixed, x  1  Y i s an aggregate production p o s s i b i l i t i e s set i n state i = 0,1, and 1  i s an aggregate consumption bundle i n state i = 0,1. T  u t i l i t y levels for the H households u = ( u ^ , . . . , u )  We also define the  and the Scitovsky sets  H  SCu ) for i = 0,1 corresponding to the u t i l i t y functions f ; 1  h  the u t i l i t y  functions are assumed to be continuous from above, quasiconcave, and nonsatiated. {x : E f n  1  x  The Scitovsky set for period i i s defined as S(u ) = x  1 x, f ( x )  h  h  h  ) uj, h = 1,...,H},  where u  1  = (u*,...,u^) .  assumptions ensure that S ( u ) i s convex (see Scitovsky (1941-2(b)). 1  T  Our  Now we  can define the four types of compensation t e s t . 1 £ p 0 (read state 1 i s preferred to state 0 by the Kaldor strong R  S  (V  principle) i f f x^ e S(u^). 1 R  (2)  0 (read state 1 i s preferred to state 0 by the Kaldor weak  1 . 0 p r i n c i p l e ) i f f z + Y intersects with S(u ).  1 R (3)  K w p  UCT  , 0 (read state 1 i s preferred to state 0 by the Hicks strong  nor  p r i n c i p l e ) i f f x ° f. S ( u ) . 1  1 R p 0 (read state 1 i s preferred to state 0 by the Hicks weak HW  (4) p r i n c i p l e ) i f f z + Y° and S(u ) are d i s j o i n t . 1  126 .  What i s called Scitovsky's  (1941-2(a)) double c r i t e r i o n i s that 0 i s prefer-  red to 1 i f f both the Hicks and Kaldor c r i t e r i a are met i n either weak or strong form.  The following two propositions are obvious.  PROPOSITION 1:  If 1 R  K g p  0, then 1 R ^ 0, but not vice versa.  PROPOSITION 2:  If 1 R  R W p  0, then 1 R  Negishi (1962;88) wrote that i f 0 Rjj^p 1  (  H S p  (z+Y ) and S ( u ° ) are d i s j o i n t , 1  then state 1 i s not recommended.  and S(u ) are d i s j o i n t , i . e . , 1  1 R  H w p  0, but not vice versa.  i.e.,  In page 89, he wrote i f z + Y^  0 then state 1 i s preferred to 0.  Therefore, we may conclude that Negishi adopted the Hicks weak compensation c r i t e r i o n (4) as his project acceptance c r i t e r i o n . (1978;412) suggested that i f x p. 414,  1  In contrast,  Harris  £ S ( u ° ) then 0 i s preferred to 1 and that on  i f x^ i S(u^) then 1 i s preferred to 0.  strong compensation c r i t e r i o n (3).  Therefore, he u t i l i z e d Hicks  By Proposition 2, we deduce that i f  project 1 i s accepted by Negishi's c r i t e r i o n , 1 i s also accepted by Harris' criterion,  but not vice versa.  In economic terms, state 1 meets Harris'  acceptance c r i t e r i o n i f the pre-project aggregate consumption bundle x^ cannot be redistributed so as to make everyone as well off as u \  whereas  state 1 meets Negishi's acceptance c r i t e r i o n i f everyone cannot be made as well off as u^ even when the best production plans and income d i s t r i b u t i o n policy are executed using the i n i t i a l endowment z and technology Y^. These two project rules are equivalent under the following condition.  127 . PROPOSITION 3:  We define perfect competition as an equilibrium where  there exits a price vector that equilibrates the markets and consumers are maximizing u t i l i t i e s and producers are maximizing p r o f i t s given the prices. If  consumers' preferences are quasiconcave and quasihomothetic  i s perfectly competitive, then 1 R„ , cr  are  0 implies 1 R  and state 0  3  0, i . e . ,  m 7 n  HSP and HWP  equivalent.  PROOF:  From d e f i n i t i o n (3),  competitive, there exists p and p  0 T  x° 2 P°  T  x ° £ S(u ).  If state 0 i s  1  2 0^ such that p  y for a l l y e Y° + z.  x 2 P  x  perfectly  for a l l x E S(u )  (See Debreu (1951, 1959)).  . . 0 1 . 1 Since the Gorman aggregation conditions are met, x f. S(u ) implies x e S ( u ° ) and S(u ) i s a subset of S ( u ° ) , 1  Therefore,p x > p 0T  0 T  x°  but x  1  i s not on the boundary of S ( u ° ) .  > p y for a l l y e Y° + z and for a l l x e S ( u ) . 0 T  1  Therefore, S(u ) and Y° + z are d i s j o i n t and from d e f i n i t i o n (4) 1  follows.  Therefore,  1 R  H W p  0  Q. E . D.  for perfectly competitive economies i n which the Gorman  aggregation conditions hold, the two Hicksian c r i t e r i a are equivalent.  If  these two conditions are not met, Negishi's c r i t e r i o n i s stronger than that of Harris, so that we have to c l a r i f y whether a project acceptance rule i s based on the Negishi or Harris c r i t e r i o n .  5-3.  The Model We now sketch the model of Harris (1978).  and H consumers, h = 1 , . . . , H . u t i l i t y functions f (x ) h  There are N goods n = 1 , . . . N ,  Consumers' preferences are represented by  where x e Q , a transformed orthant R . h  +  As for  128 . production, an aggregate closed production set and a net output vector belonging to i t for the private firm sector are denoted by Y and y.  A new  firm introduced by the government has an operating technology and a net output vector G and g, respectively. properties of the technologies  We do not assume anything about the  Y and G except closedness (which i s harmless  from an empirical point of view).  We assume the existence of e q u i l i b r i a as  follows. N A before-project equilibrium i s defined as an H + 2 tuple i n R , r o 0 , 0 0,, . . (P ,Y , ( x , . . . , x ) } such that 1  (5)  H  f (x£) 2 h  f h  (  x n  )  f  o  r  a  1  1  x  e n  (budget constraint for h under p }, 0  for h = 1 , . . . , H ,  (6)  y° e Y  (7)  0 H 0 0^ x =L x = y + z. r  h = 1  h  In the same way, we define an a f t e r - p r o j e c t equilibrium to be an H + 3 tuple i n R , {p , g , y , (x!j, . . . ,x^)} such that N  (8)  f h  1  1  1  ^ h^ - ^( ) x  f  x  f  n  for h = 1 , . . . , H ,  (9)  y  1  eY  (10)  g  1  e G  1  1  o  r  a  1  1  X  h  e  {k  u d  9  e t  constraint for h under p } 1  129  These two definitions  seem to incorporate the minimum requirements for  an imperfectly competitive e q u i l i b r i a studied by Negishi (1961-2), Arrow and Hahn (1971, Ch. 6), and Roberts and Sonnenschein (1977); i . e . , taking behaviour of consumers, and ( i i i )  (ii)  (i)  price  f e a s i b i l i t y of equilibrium production,  equality of demand and supply.  The main r e s u l t i n t h i s chapter i s  that the existence of before and after-project e q u i l i b r i a i s s u f f i c i e n t  for  the v a l i d i t y of most of the project evaluation rules developed by Negishi and Harris.  5-4.  Project Evaluation Rules Using the Hicksian strong compensation c r i t e r i o n , Harris' two main  project evaluation rules (using his numbering of the rules) can be restated within our framework as  Rule 2:  A sufficient  follows.  condition to reject a proposed project i s that the  project have a net value at before-project prices which i s less than the change i n the p r o f i t s on a l l other production a c t i v i t i e s before-project prices; i . e . ,  Rule 4: profits  A sufficient  evaluated at  OT 1 OT 0 1 the rejection c r i t e r i o n i s p g < p (y - y ).  condition to accept a project i s that minus the  (or minus the net value) of the project at post-project prices be  less than the change i n p r o f i t s i n the rest of the economy at a f t e r - p r o j e c t prices; i . e . ,  the acceptance c r i t e r i o n i s -p  IT  g  1  < p  1T  (y  1  0 - y ).  130. PROOF:  Substituting the resources constraints (7) and (11) into Rule 2  OT 0 OT 1 and Rule 4, we find that Rule 2 i s equivalent to p x > p x , and Rule 4 i s 1T 1 1T 0 0 1 equivalent to p x > p x . As the Scitovsky sets S(u ) and S(u ) are convex sets, and x^ and x^ belong to the boundary of S(u^) and S(u^), respectively from (5) and (8), Rule 2 implies x 0 1 x i S(u ).  1  £ S ( u ° ) and Rule 4 implies  . . . . . Now applying the Hicks strong compensation p r i n c i p l e (3), Rule 2  gives a s u f f i c i e n t condition for state 0 to be preferred to 1, and Rule 4 gives a s u f f i c i e n t condition for state 1 to be preferred to 0.  Q. E . D.  We proved H a r r i s ' s two main rules without making any assumptions concerning market structure. consumers are price takers.  We assumed only that markets clear and In p a r t i c u l a r , we did not assume that either the  private production sector or the government optimizes.  Therefore, Harris'  rules have a very broad a p p l i c a b i l i t y . From Proposition 3, Rule 2 i s v a l i d for Hicks' weak compensation p r i n c i p l e i f the Gorman aggregation conditions for consumers' preferences are met and the a f t e r - p r o j e c t equilibrium i s perfectly competitive.  Similarly,  Rule 4 i s v a l i d for Hicks' weak compensation p r i n c i p l e i f the aggregation of consumers' preferences conditions are met and the before-project equilibrium i s perfectly competitive.''  This i s the reason why Negishi (1962; 91) assumed  that Gorman's preference r e s t r i c t i o n s were met i n his demonstration of the v a l i d i t y of Rule 4. Harris (1978) restated Negishi's 3 i n Harris' numbering, as follows: principle  (1962) main two r u l e s , Rule 1 and Rule Referring to Hicks weak compensating  131 .  Rule 1:  A s u f f i c i e n t condition to r e j e c t a proposed project i s that i t  is  impossible for the project to show a nonnegative net value at before-project . . equilibrium prices; i . e . ,  Rule 3.  . . OT 1 the r e j e c t i o n c r i t e r i o n i s p g < 0.  A s u f f i c i e n t condition to accept a project i s that the project  show positive p r o f i t s at post-project prices; i . e . , is p  g  the acceptance c r i t e r i o n  > 0.  In general, unless the economy i s competitive,  except for the new firm  introduced by the government, Rule 1 and Rule 3 are i n v a l i d (see Negishi (1962)).5  However, i f the competition assumption i s met, then Rule 1 implies  Rule 2 and Rule 3 implies Rule 4 as the price-taking assumptions for firms OT 0 OT 1 1T 1 1T 0 imply p y 2 p y and p y 2 p y . be v a l i d i n general.  Obviously, other implications cannot  This means that the Harris rules are more complete than  Negishi's for competitive economies  (Harris (1978;413)).  Stated another way,  some project accepted by Harris' Rule 4 may not be accepted by Negishi's Rule 3 and some project rejected by Harris' Rule 2 may not be rejected by Negishi's Rule 1.  This indeterminacy of Rules 1 and 3 comes partly from the  fact that Negishi adopted Hicks' weak p r i n c i p l e as his welfare c r i t e r i o n , which i s more indeterminate than the Hicks strong p r i n c i p l e adopted by H a r r i s , but c h i e f l y i t i s because the p r o f i t a b i l i t y c r i t e r i o n i s a less exact estimate of the s o c i a l welfare change than the index number approach used i n Rule 2 and Rule 4. The drawback of Rule 2 and Rule 4 seems to be t h e i r more demanding informational requirements, i . e . ,  as long as the economy is competitive,  the  informational requirements for implementing Rules 1 and 3 seem less onerous  132. than for Rules 2 and 4.  In Rule 1, we only need to know the production  p o s s i b i l i t i e s set of the public agency. be somehow predicted.  In Rule 3, a f t e r - p r o j e c t prices must  However, i n Rules 2 and 4, a f t e r - p r o j e c t output l e v e l s  of the rest of the economy are also required, and t h i s i s d i f f i c u l t to obtain ex ante.  In summary, Rules 1 and 3 show the f i r s t - b e s t significance of a new  technology i n terms of i t s p r o f i t a b i l i t y .  They are almost always less exact  than Rule 2 and Rule 4, and cannot t e l l us anything i n second best conditions when we do not have perfect competition.  Rule 2 and Rule 4 are v a l i d i n both  f i r s t best and second best conditions, and i n a second best, i t evaluates  the  improvement of technical efficiency and market e f f i c i e n c y at the same time.  5-5.  Conclusion Harris discussed thirteen project evaluation rules i n his paper.  Excluding Rule 1 and Rule 3, which we have discussed, and Rule 8, which i s analogous to Rule 1 i n the tax-distorted economy, a l l of his rules are  effec-  t i v e for non-competitive market structures, because the proofs of a l l of them are s i m i l a r to the proofs of Rule 2 and Rule 4, or they are contrapositives of other r u l e s .  In p a r t i c u l a r , the s a t i s f a c t i o n of Samuelsonian conditions  i s not necessary to prove Harris' Rules 11, 12 and 13, which give f i t rules for supplying a public input.  cost-bene-  Although p r o f i t a b i l i t y of a new  project has a normative meaning only i n f i r s t - b e s t situations where the usual marginal conditions hold, the application of index number theorems due to Hicks (1940, 1941-2) and Samuelson (1950) are f r u i t f u l even i n a second best economy.  133 . FOOTNOTES FOR CHAPTER 5  1 Note also that, i n general, the relationship between the compensation p r i n c i p l e and the sum of equivalent or compensating variations i s ambiguous. See Boadway (1974), Smith and Stevens (1975), Foster (1976) and Boadway (1976). 2  In chapter 3, we considered a s u f f i c i e n t condition for the existence  of a Pareto improvement. principle. 3  This discussion may be related to the compensation  See Bruce and Harris (1982).  Quasi-homotheticity i s s a t i s f i e d i f Engel curves are straight  and they are p a r a l l e l for a l l consumers.  See Gorman (1953).  lines  Alternatively,  we can think of the case where income d i s t r i b u t i o n i s always optimized with respect to a Bergson-Samuelsonian s o c i a l welfare function.  In this case,  Bergson's s o c i a l indifference surfaces do not intersect and convex to the o r i g i n i f u t i l i t y functions are concave and a s o c i a l welfare function i s quasi-concave (see Gorman (1959) and Negishi (1963)).  Replacing the  Scitovsky set with the better set of Bergson's indifference surface, the of  the discussion goes through.  preferences,  rest  When we mention Gorman's r e s t r i c t i o n s on  we can also allow for this alternative case.  * Harris (1978;410) pointed out that his welfare c r i t e r i o n i s  consistent  with an ordering based on the Bergson-Samuelsonian s o c i a l welfare function, if  i t exists.  Referring to Proposition 3, Negishi's welfare c r i t e r i o n may  not be consistent with such a s o c i a l welfare function, i f the assumption of perfect competition i s dropped. 5  More exactly,  Rule 1 applies even i f the a f t e r - p r o j e c t equilibrium i s  imperfectly competitive and Rule 3 applies even i f the before-project  134. equilibrium i s imperfectly competitive, provided that the e q u i l i b r i a e x i s t . This i s obvious from the proofs of these rules by Negishi (1962).  135 . REFERENCES A l l a i s , M. Paris:  (1943). A l a recherche Imprimerie Nationale.  d'une d i s c i p l i n e  economique,  Tom I,  A l l a i s , M. (1977). "Theories of General Economic Equilibrium and Maximum E f f i c i e n c y , " pp. 129-201 i n Equilibrium and Disequilibrium i n Economic Theory, E . Schwodiauer e d . , Dordrecht: D. Reidel. Aoki, M. (1971). ing Returns."  "Investment Planning Process for an Economy with IncreasReview of Economic Studies 38, 273-80.  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