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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 of Tokyo, Tokyo, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES THE DEPARTMENT OF ECONOMICS We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA June 1987 CcT) ^ A t s u s h i Tsuneki, 1987 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r b y h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main M a l l V a n c o u v e r , C a n a d a V6T 1Y3 D a t e June 30 f l , / 9 £ ? ABSTRACT Harberger's methodology for the measurement of deadweight loss is reformulated in a general equilibrium context with adopting the A l l a i s -Debreu-Diewert approach and is applied to various problems with imperfect markets. We also develop second best project evaluation rules for the same class of economies. Chapter 1 is devoted to the survey of various welfare indicators. We especially discuss the two welfare indicators due to A l l a i s , Debreu, Diewert and Hicks, Boiteux in relation to Bergson-Samuelsonian social welfare func-t ion. We f i r s t show that these two measures generate a Pareto inclusive ordering across various social states, but they are rarely welfarist, so that both are unsatisfactory as Bergson-Samuelsonian social welfare functions. We next show that second order approximations to the Allais-Debreu-Diewert measure of waste can be computed from local information observable at the equilibrium, whereas second order approximations to the Hicks-Boiteux measure of welfare or to the Bergson -Samuelsonian social welfare function require information on the marginal u t i l i t i e s of income of households, which is unavailable with ordinal u t i l i t y theory. Final ly , we give a diagrammatic exposition of the two measures and their approximations to give an intui t ive 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 is nonoptimal and the government revenue is raised by distortionary taxation by extending the Allais-Debreu-Diewert approach discussed in Chapter 1. The resulting measure of waste is related to indirect tax rates, net marginal benefits of public goods, and the derivatives of aggregate demand and supply functions evaluated at an equilibrium. In Chapter 3, cost-benefit rules for the provi-sion of a public good are derived when there exist tax distortions. We derive the rules as giving sufficient conditions for Pareto improvement, but we also discuss when these rules are necessary conditions for an inter ior social optimum. When indirect taxes are ful ly f lexible but lump-sum trans-fers are restricted, we recommend a rule which generalized the cost-benefit rule due to Atkinson and Stern (1974) to a many-consumer economy. When both indirect taxes and lump-sum transfers are f lexible , we suggest a rule which is based on Diamond and Mirrlees' (1971) productive efficiency principle . 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 in 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 is 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 i v d i s c u s s various l i m i t a t i o n s of our approach and the r e l a t i o n between our work and t h a t of H o t e l l i n g (1938). In Chapter 5, we consider 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 the presence of imperfect competition. We show t h a t the index number approach due t o Negishi (1962) and H a r r i s (1978) can be extended to handle s i t u a t i o n s with imperfect competition. V TABLE OF CONTENTS Page Abstract i i L i s t of Figures v i i Acknowledgements v i i i Chapter 1 APPLIED WELFARE ECONOMICS 1 1-1. The Mesures of Deadweight Loss 2 1-2. The Approximation Approach to the Measurement of Waste 11 1- 3. A Diagrammatic Exposition 26 Footnotes 35 Appendices 38 2 THE MEASUREMENT OF WASTE IN A PUBLIC GOODS ECONOMY 50 2- 1. Introduction 50 2-2. The Model 51 2-3. An Allais-Debreu-Diewert Measure of Waste 55 2-4. Second Order Approximations 57 2- 5. Conclusion 63 Footnotes 66 3 PROJECT EVALUATION RULES FOR THE PROVISION OF PUBLIC GOODS 70 3- 1. Introduction 70 3-2. The Model 72 3-3. Pigovian Rules Reconsidered 76 3-4. Cases where Lump-Sum Transfers are Available 85 v i 3- 5. Conclusion 92 Footnotes 94 Appendices 95 4 INCREASING RETURNS, IMPERFECT COMPETITION AND THE MEASUREMENT OF WASTE 99 4- 1. Introduction 99 4-2. The Model 101 4-3. The Allais-Debreu-Diewert Measure of Waste 104 4-4. Second Order Approximations 108 4- 5. Conclusion 115 Footnotes 117 5 PROJECT EVALUATION RULES FOR IMPERFECTLY COMPETITIVE ECONOMIES 121 5- 1. Introduction 121 5-2. Compensation C r i t e r i a for Cost-Benefit Analysis Reconsidered 123 5-3. The Model 127 5-4. Project Evaluation Rules 129 5-5. Conclusion 132 Footnotes 133 REFERENCES 135 v i i LIST OF FIGURES Page Figure 1. The ADD Measure of Waste 44 Figure 2. The HB Measure of Welfare 45 Figure 3. The ADD Measure: A One-Consumer Two-Goods Economy 46 Figure 4. The ADD Measure and i t s Approximations: A One-Consumer Two-Goods Economy 47 Figure 5. The HB Measure: A One-Consumer Two-Goods Economy 48 Figure 6. The HB Measure and i t s Approximations: A One-Consumer Two-Goods Economy 49 Figure 7. The ADD Measure in a Public Goods Economy 67 Figure 8. The ADD Measure and i t s Approximations i n a Public Goods Economy 68 Figure 9. An Example where Approximations of the ADD Measure are Inaccurate 69 Figure 10. The ADD Measure with Increasing Returns to Scale 119 Figure 11. The ADD Measure and i t s Approximations with Increasing Returns to Scale 120 v i i i A C K N O W L E D G E M E N T S F i r s t o f a l l , I s h o u l d l i k e t o t h a n k E r w i n D i e w e r t , my p r i n c i p a l a d v i s e r a n d c h a i r m a n o f my t h e s i s c o m m i t t e e f o r m a n y i m p o r t a n t s u g g e s t i o n s a n d k i n d e n c o u r a g e m e n t i n c l a s s e s a n d i n p r i v a t e d i s c u s s i o n s , a n d d e t a i l e d c o m m e n t s o n s e v e r a l e a r l i e r v e r s i o n s o f my t h e s i s , w h i c h m a d e i t p o s s i b l e t o c o m p l e t e t h e t h e s i s . I s h o u l d a l s o l i k e t o t h a n k C h a r l e s B l a c k o r b y a n d J o h n W e y m a r k , t h e o t h e r t w o m e m b e r s o f my t h e s i s c o m m i t t e e , a n d D a v i d D o n a l d s o n f o r r e p e a t e d v a l u a b l e d i s c u s s i o n s a n d c o m m e n t s o n my t h e s i s w h i c h h e l p e d t o i m p r o v e t h e q u a l i t y o f my t h e s i s c o n s p i c u o u s l y . I a p p r e c i a t e J o h n W e y m a r k f o r g i v i n g e x t e n s i v e c o m m e n t s , n o t o n l y o n t h e c o n t e n t b u t a l s o o n t h e e x p r e s s i o n o f my t h e s i s a n d c o r r e c t i n g my E n g l i s h c a r e f u l l y . I am a l s o t h a n k f u l t o K e i z o N a g a t a n i f o r m a n y s t i m u l a t i n g d i s c u s s i o n s a n d f o r k i n d h e l p f o r me t o a d j u s t t o N o r t h A m e r i c a n l i f e . I m u s t b e t h a n k f u l t o t h e o t h e r f a c u l t y m e m b e r s a n d g r a d u a t e s t u d e n t s o f t h e D e p a r t m e n t o f E c o n o m i c s a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a w h o m a d e my s t u d y t h e r e d u r i n g 1 9 8 3 - 1 9 8 7 b o t h c o m f o r t a b l e a n d c h a l l e n g i n g . I s h o u l d a l s o e x p r e s s my d e e p t h a n k s t o t h e p r o f e s s o r s o u t s i d e U B C w h o g a v e me m a n y i m p o r t a n t c o m m e n t s a n d w a r m e n c o u r a g e m e n t w h i l e I w a s w r i t i n g t h e t h e s i s , w h i c h i n c l u d e : K o i c h i H a m a d a , T a t s u o H a t t a , T s u n e o I s h i k a w a , M o t o s h i g e I t o h , K e i m e i K a i z u k a , Y o s h i t s u g u K a n e m o t o , R y u t a r o K o m i y a , T a k a s h i N e g i s h i , a n d Y u k i h i d e O k a n o . I a p p r e c i a t e f i n a n c i a l s u p p o r t f r o m t h e G o v e r n m e n t o f C a n a d a A w a r d s t o F o r e i g n N a t i o n a l s a d m i n i s t e r e d b y W o r l d U n i v e r s i t y S e r v i c e o f C a n a d a o n b e h a l f o f t h e D e p a r t m e n t o f E x t e r n a l A f f a i r s , a n d J o h n W e y m a r k w h o e m p l o y e d me a s a t e a c h i n g a s s i s t a n t . I t h a n k M s . J e a n e t t e P a i s l e y f o r t y p i n g my t h e s i s e x c e l l e n t l y . F i n a l l y , b u t n o t l e a s t , I m u s t e x p r e s s my d e e p a p p r e c i a t i o n t o my p a r e n t s w h o a l w a y s h e l p e d me g e n e r o u s l y a n d p a t i e n t l y t o c o m p l e t e t h i s f i r s t w o r k i n my a c a d e m i c c a r e e r . 1. CHAPTER 1 APPLIED WELFARE ECONOMICS The purpose of t h i s chapter 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 or welfare from s e v e r a l viewpoints and choose one c r i t e r i o n which s u i t s our purpose best. In doing so, we present 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 other approaches to a p p l i e d welfare economics. In s e c t i o n 1, we introduce 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 Allais-Debreu-Diewert measure of s o c i a l waste and the H i c k s - Boiteux measure of s o c i a l welfare. 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 consider whether they can serve as P a r e t o - i n c l u s i v e and 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 ) s o c i a l welfare f u n c t i o n s . We show t h a t these measures are 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 preference r e s t r i c t i o n i s s a t i s f i e d or the production p o s s i b i l i t i e s s et i s l i n e a r . The Allais-Debreu-Diewert measure of waste i s a f f e c t e d by the choice of the reference bundle of goods i n terms of which the s c a l e of e f f i c i e n c y l o s s i s determined whereas the Hicks-Boiteux measure of welfare i s a f f e c t e d by the choice of the optimal a l l o c a t i o n of r e a l income. This means th 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 the v a l u a t i o n of each i n d i v i d u a l i n the measure of s o c i a l welfare. Thus, the Allais-Debreu-Diewert measure i s a pure e f f i c i e n c y waste measure whereas the Hicks-Boiteux measure shows a change of s o c i a l welfare i n c l u d i n g both e f f i c i e n c y and e q u i t y aspects. In the 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 elaborated by t a k i n g a second order approximation to the two measures when tax d i s t o r t i o n s p r e v a i l . We show that the Allais-Debreu-Diewert measure is computable from the second order derivatives of expenditure functions and profit functions evaluated at the observed equilibrium while the Hicks-Boiteux measure or the Bergson-Samuelsonian social 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 social importance to evaluate the equity loss. Since this information is not available with ordinal u t i l i t y theory, i t is d i f f i c u l t to use the Hicks-Boiteux measure in applied welfare economics. This provides the main reason why we use the Allais-Debreu-Diewert measure in this essay. The pros and cons of the approximation approach we adopt in this essay are next compared with an alternative inf luent ia l approach, applied (or numerical) general equilibrium models. A numerical general equilibrium model computes the exact value of social welfare indicators by restrict ing 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 social welfare indicators from more general functional forms and observable information. F inal ly , in l ight of the measurement of waste approach for welfare economics, we reconsider the theory of second best. Our conclusion here is that this theory is not a replacement for the measurement of deadweight loss, even though several positive results derived in second best theory are useful. F inal ly , in section 3, in order to give insight into the economic implications of our approach, we give diagrammatic expositions of the two measures and their 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 in this thesis) which includes Hotelling (1938), Hicks (1941-2), A l la i s (1943, 1977), Boiteux (1951), Debreu (1951, 1954), Harberger (1964, 1971), and Diamond-McFaddon (1974), two types of welfare c r i t e r i a are chiefly used: the Allais-Debreu-Diewert measure of waste (the ADD measure hereafter) and the Hicks-Boiteux measure of surplus (the HB measure hereafter).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.*1 where x*1 = (x^1, . . . ,x^) T i s a consumption vector of goods 1 , . . . , N by the hth consumer. The i n i t i a l endowment vector of the hth consumer is given by x*1, h = 1 , . . . , H . There are k k K firms and firm k produces y using the production poss ib i l i t i es set S , k = 1 , . . . , K . We can define the ADD measure in terms of a primal programming problem?: ( 1 ) ADD s r ~= m a X r , x h r y k { r = Eh=1X + P ' r 1 E k = l y + Eh=1X ; f h (x h ) > U j J , h = 1, . . . , H ; y K e S k , k = 1 , . . . ,K} , where p = (p^,. . . ,p^) > 0^ is an arb i t rar i ly chosen reference bundle of commodities. To interpret this problem we rewrite (1) in an alternative N . manner. The following notation is used. R + i s the N-dimensional nonnegative K k k orthant. E K = 1 S i s t n e direct sum of the production poss ib i l i t i es sets S . S(u 1) = {x : E h f . ,x h 1 x; f h (x h ) > UjJ, h = 1,...,H} is the Scitovsky set 1 1 1 corresponding to a u t i l i t y allocation u = ( u . , . . . , u „ ) . Now (1) can be I H rewritten as 4. (2) max r {r : 6 r e Q = Z ^ x h + E k ^ S k - S ( u 1 ) } . (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 : maximize the s c a l e of the reference set of goods i n Q where Q i s the set of goods producible from the aggregate production p o s s i b i l i t i e s plus endowments which give consumers at l e a s t the 1 u t i l i t y v e c t o r 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 . In F i g . 1 we d e p i c t the ADD measure, i n a two goods economy where p i s a support p r i c e of the programming problem (1). p^ i s determined up to 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 that we can choose p^ = 1; i . e . , the optimal p r i c e of the f i r s t good i s u n i t y without l o s s of g e n e r a l i t y . Furthermore, we OT OT can choose the s c a l e of 8 so that p 8 = 1 . Then, r = (p 8)r equals AB OT s i n c e p pr i s the d i f f e r e n c e between the value of production minus consump-t i o n evaluated at p^. Note t h a t the choice of the reference 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 measure (see Diewert ( 1985a;50)) . 3 Let us now t u r n to the HB measure. We begin from an a t t a i n a b l e and 0 0 0 T 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 (u.,...,u ) . We a l s o assume t h a t t n 0 0 0 T there e x i s t s a p r i c e vector p = (p^,...,P N) which supports the s o c i a l l y optimal a l l o c a t i o n of resources. Then we can define the HB measure L..,. as H D f o l l o w s : (3) = E, - E, where we d e f i n e the expenditure f u n c t i o n : ^ (4) rah(p,uh) = min h { p T x h : f h ( x h ) > u,}, where p > 0 N and e Range f The measure L H B defined by (3) can be interpreted as the sum of the negative of the equivalent variations obtained in moving from a social ly 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 in units of the f i r s t good in a two good economy by choosing = 1 is i l lus trated in F ig . 2. Generally, the desirable properties of the ordering of social states are summarized in the Bergson-Samuelsonian social welfare function (BSSWF here-after) . (See Samuelson (1956) for a discussion of the BSSWF and i t s proper-ties l i s ted below.) We f i r s t assume that the underlying social ordering is compatible with the Pareto part ia l ordering ( i . e . , i f a l l individual u t i l i -ties increase, then so does social welfare) so that the resulting BSSWF becomes Pareto-inclusive. Suppose also that the evaluation of social states is ind iv idual i s t i c (or welfarist); i . e . , the u t i l i t y vector u prevailing at the state is the only information used in 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 is applied to get the BSSWF, W(u). Pareto-inclusiveness implies that W is monotone increasing in 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 It may, therefore, be interesting to ask whether these measures are Pareto-inclusive^ and i n d i -v idual i s t i c ; i . e . , whether they work as a kind of BSSWF. The f i r s t question 6 . may be answered easi ly. F i r s t , notice the definit ion (2) of the ADD measure. Suppose that u a is preferred to u b in terms of the Paretian part ia l ordering, then S(u a) is a subset of S(u b ) . Noting that production poss ib i l i t i e s are fixed, Q(u a) i s a subset of Q(ub) and hence r (u a ) < r ( u b ) . In the case of the HB measure, Pareto inclusiveness direct ly follows from the nondecreasingness of the expenditure function with respect to u (see Diewert (1982;541)) and i t s def init ion (3). The other question is 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 ind iv idual i s t i c ; i . e . , i t is always affected by the choice of B, which is not related to i n d i -viduals' welfare. We extend the concept of an ' indiv idual i s t ic ' evaluation by saying that r is ordinally indiv idual 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 def init ion is formalized as follows: (5) r ( u a ,p a ) > r ( u b ,p a ) i f f r ( u a ,p b ) > r ( u b ,p b ) for a l l p a > 0 N and p b > 0 N . k The profit function ir , k = 1 , . . . , K is defined as (6) i r k (p) = maxx {pTy : y e S k}, k = 1 , . . . , K , where p > 0 „ . 7 . The regularity properties of the profit function are summarized in Diewert (1982;580-1). We assume below that the production poss ib i l i t i es sets are convex and preferences are quasiconcave. Then, (1) is equivalent to the following dual max min problem: (7) L A D D ( u ,p ) = r(u,p) = maxr m i n p > Q {r(1-p Tp)+E h^ 1p Tx h+E k^ 1ir k(p) - E h f 1 m h (p ,u h )} , The proof is an application of the Uzawa ( 1958;34)-Karlin (1959;201) Saddle Point Theorem7 (see Appendix II) . If we further assume that r i s twice continuously differentiable at the relevant values of u and p, then (5) is equivalent to requiring u to be separable^ in r(u,p); that i s , r(u,p) satisf ies (8) 8(| J L7! E-)/ap = 0 for a l l i , j = 1 , . . . , H and a l l n = 1 , . . . , N . ou. ou . n r D 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^ = - dm1(p^,u^)/9u^, i = 1 , . . . , H . Sub-st i tut ing i t into (8) and using the relation: 92m1(p,u^)/9u^3p f f l = [ d x ^ ( p , y ) / 3 y ^ ] [ d m 1 ( p ° , u ^ ) / 3 u ^ ] for i = 1 , . . . , H and m = 1 r . . . , N , we have (9) E.I1[axi(p0fyJ)/ay.-8xj(p0,y5)/ay.]Op2/8Pll) = 0 for a l l i , j = 1 , . . . , H and a l l n = 1 , . . . , N , 8. where x^"(p,y.), i = 1 , . . . , H is the ordinary demand function for the nth good m i for the i th consumer and y? = m 1 ( p ° , u ^ ) . Conditions (9) are satisf ied either i f Gorman's (1953;73) res tr ic t ion on preferences is satisf ied; i . e . , preferences are quasi-homothetic and their Engel curves are paral le l to each other, (since the f i r s t term in the left-hand side of (9) i s 0 for a l l i , j ,m) or i f the production poss ib i l i t i es sets are linear (since the second vector is 0 for a l l m and n). (9) has the following meaning: when we increase any one reference good P n , then the scarcity of the nth good increases so that the system of shadow prices associated with (7) p^ changes, and this 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 satisf ied globally except for the two cases above l i s ted . We now turn to the HB measure L . By the same token as the ADD H D measure, L „ n ( u \ u ^ , p ^ ) is ordinally indiv idual i s t ic i f and only i f u^ is separable in L U D . Remember that u^ is one Pareto optimal u t i l i t y al location H D and p^ is i ts supporting price vector. Therefore p^ is a function of u^ (and 1 other parameters of the general equilibrium) so that separability of u is equivalent to the condition. (10) (—^f/—*Hf) /duf = 0 for a l l i , j = 1 , . . . , H and a l l h = 1 , . . . , H . u. 9u. r 3 ^HB i 0 1 1 Using definit ion (3), „ 1 = - 3m (p ,ui)/du{. Substitute this into ou. l (10) and we find the following equivalent conditions: 9. f o r a l l i , j = 1,...,H and a l l h = 1,...,H. Conditions (11) seem analogous to (9), except f o r the d i f f e r e n c e between dp°/dp i n (9) and d^P/bvP i n (11). The former i s the change of the support m n m n p r i c e s of the Allais-Debreu-Diewert optimum w i t h respect t o an i n c r e a s e of the nth good i n the reference bundle, while the l a t t e r i s the change of the support p r i c e s of the reference Pareto optimal a l l o c a t i o n w i t h respect t o an increase of the u t i l i t y of the hth household. Therefore, as i n the ADD measure, there does not 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 guarantee the HB measure to 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 the two c o n d i t i o n s c i t e d above; i . e . , Gorman's preference r e s t r i c t i o n or l i n e a r production p o s s i b i l i t i e s . Up t o now we have learned t h a t both measures are Pareto 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 i n general. The c o n d i t i o n s necessary to make welfare p r e s c r i p t i o n s by the 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 are as s t r i n g e n t as those needed by the HB measure. However, the economic i m p l i c a t i o n s of the two measures are completely d i f f e r e n t . The ADD index measures pure t e c h n i c a l e f f i c i e n c y i n terms of the reference bundle of goods, and the HB index measures the l o s s of both e f f i c i e n c y and equity by i n d i c a t i n g the monetary value of the d i f f e r e n c e between the s o c i a l optimum and the observed e q u i l i b r i u m . 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 , using the ADD measure to 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 being used as a measure of s o c i a l w e l f a r e ( i n s t e a d of as j u s t an estimate of the resource a l l o c a t i o n waste of 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 social states is 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 social ly 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 shall 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 is fixed. Another drawback of the ADD measure is that i t cannot be an appropriate welfare indicator i f there is technological change ( i . e . , i t is not welfarist in the sense that i t depends on technological parameters). The HB measure is free of this defect, i f the reference price vector is fixed (see Section 5.2) Let us compare these measures from another viewpoint. Are these measures useful when the shadow price vector does not exist because of nonconvexities or externalities? We wi l l show in the later chapters of this essay that the ADD measure is 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 in such circumstances. When we choose a reference Pareto optimal al location 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 is to compare the sum of the negative of the equivalent variations m^(p^,q^,u^) - m (p ,q , u h ) , where m (p,q,u ) i s a restricted expenditure function (see Diewert (1986;170-6)). Note that the calculation of the two measures necessitates global compu-tation of the optimal equilibrium which is very d i f f i c u l t to implement empir-11. i c a l l y . Therefore in this 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 is more implementable in empirical research. 1-2. The Approximation Approach to the Measurement of 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 distortions. We assume i n i t i a l l y that markets are complete, technologies are convex and that the only source of distortions is indirect taxes levied on consumers. Extensions of these assumptions are a main theme of the later chapters, so that we only work with the prototype model in this chapter. Given these assumptions, (1) is equivalent to (6). At this point, we use the concept of the overspending function B which w i l l be ful ly u t i l i zed in this essay which is defined as _ „ . p H h . » r « H T " - h n K k , » B(q,P,u) = E h = 1 m (q ( u h ) - C h = ( ) q x - E k _ 0 » (P) • In Appendix I, B is restated with i ts economic interpretation and i t s useful properties are summarized. Using the definition (A.1), (7) may be rewritten concisely as follows: (12) r ° = max min n {r(1-pTp) - B(p,p,u 1 )}. N 12. Using the Uzawa (1958) - Karl in (1959) Theorem in reverse, (12) i s also equivalent to: (13) - max {B(p,p,u 1) : pTp > 1}. N If ( p ° , r ° ) solves (12), p° solves (13) with r ° being i t s associated Lagrangean mult ipl ier . In order to obtain a second order approximation to r^, we assume: (i) (p^,r^) solves (12); ( i i ) the f i r s t order necessary conditions for (12) hold with equalities so that p° » 0 „ ; ( i i i ) B(q,p,u 1 ) i s twice continuously differentiable at ( p ° , p ° ) ; (iv) Samuelson's (1947;361) strong second order sufficient conditions hold for (13) when the inequality constraints are replaced by equalities. Let us consider the following system of equations in N+1 unknowns p and r which are functions of a scalar variable z, for 0 < z < 1: (14) - V qB(p(z) + tz, p(z), u 1 ) - 7pB(p(z) + tz, p(z), u 1) - r(z)p = 0, (15) 1 - p(z) TB = 0. When z = 0, (14) and (15) coincide with the f i r s t order conditions for (12) i f p(0) = p° and r(0) = r ° . Suppose p(1) = p 1 i s the set of observed producer prices normalized by (15) in a tax-distorted equilibrium with indirect tax rates t . Setting r(1) = 0, when z = 1 (14) i s then the set of equations characterizing the equality of demand and supply in the tax-distorted 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 is also the case that satisfaction of the government budget constraint i s implied. Let us differentiate (14) - (15) tota l ly with respect to z. We have (16) B 2 + B Z , qq P P P ' U ) r' (z) B z t qq z 2 where q = p + tz is the f i r s t set of arguments for B and B ^ = B(p(z) 1 . . z + tz , 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) is non-singular by assumption (iv) (see Diewert and Woodland (1977)). Therefore, using the d i f ferent iab i l i ty assumptions ( i i i ) , by the Implicit Function Theorem there exist once continuously differentiable functions p(z) and r(z) at z close to 0 that satisfy (14) and (15). We show in Appendix III that the following equation is sat is f ied. (17) - r'(z) = - z t T BJi (p'(z) + t ) . qq We readily have (18) r'(0) = 0 from (17). Using (17), i t i s shown in Appendix IV that the following equation follows. 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 is given by (noting that r(1) = 0), (20) L A D D = r(0) - r(1) = r(0) - (r(0) + r ' (0) + J,r'"(0)) = - M P ' ( 0 ) T B ° p '(0) + [p '(0) + t ] T B ° [p '(0) + t]} > 0, where we use (18) and (19). Equation (16) i s used to compute p ' (0) . Information we need to evaluate (20) i s : (i) the set of indirect taxes t, ( i i ) 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 this approximation approach is 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 in (20), we must actually know the optimal prices so that we must compute the optimum or we must depend on some 'guess-ing' 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 prevailing at the observed equilibrium. This method can be just i f ied more 15. rigorously by Diewert's ( 1976; 118) Quadratic Approximation Lemma which showed that the approximation (21) L A D D = r(0) - r(1) = r(0) - {r(0) + Jjr'tO) + Jjr'd)} i s also exact as the approximation (20) when the functional form is quadratic (see also Diewert (1985(b);238)). Evaluating (16) at z = 1 and using (17), we can show that -r'(1) is identical with -r"(0) in (19) except that a l l the relevant functions are evaluated at z = 1; i . e . , at the observed equilibrium; -r'(1) is nonnegative due to the semidefiniteness properties of the producer and consumer substitution matrices. Using also (18), we find (22) L A D D £ -Jftip1 (1) T p'(1) + [p'(1) + t ] T B g g [p'(1) + t]} 2 0. This approximation uses only information observable at the prevailing equilibrium as Harberger or ig inal ly required, so that i t is highly valuable in empirical analysis. 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 lar i fy 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 is a solution of the maximum of a linear social welfare function H h h T T.^-^ a f for some set of weights a = (a^,... ,a^) given resource constraints and production poss ib i l i t i es of the economy, where i t is assumed that f*1, h = 1 , . . . , H are concave functions. In our model, this means that for some vector 16. a, a perfectly competitive equilibrium is a solution of the following programming problem: (23) Max h k { E h ? 1 a h f h ( x h ) s . t . E ^ x N E j ^ y * + E^x* 1 ; y k e S k , x , y k = 1 , . . . ,K} . Using the Uzawa-Karlin Saddle-point Theorem using the definit ion (4), (6) and (A.1), we can rewrite (23) as follows (the calculation is analogous to the derivation of (7) in Appendix II): (24) Max Min {aTu - B( P ( p,u)} . N We assume that (i) (u^,p^) solves (24), ( i i ) the f i r s t order conditions for (24) hold with equality so that p° » 0 N , ( i i i ) B i s twice continuously differentiable at the optimum, and (iv) B ^ + B ^ is negative def inite . From qq PP assumptions (i) and ( i i ) , we find the f i r s t order conditions for (24) are: (25) a = 7 B(p,p,u), (26) - V B(p,p,u) - V B(p,p,u) = 0. Condition (26) i s the equality of demand and supply at the optimum while (25) is the rule to equate the marginal social 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 equi l ibr ia corresponding to various a. Varian (1974, 1976) persuasively discussed the welfare significance of the equal divis ion equilibrium, which is a perfectly competitive equilibrium obtained from the equal divis ion of i n i t i a l endowments across individuals. Varian (1976), following the approach of Negishi (1960), also examined the relationship between his theory of fairness and more tradit ional welfare economics based on the concept of a social welfare function, which we followed in this section. By Negishi 1s theorem, the equal divis ion equilibrium is 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 in an analogous way as we computed the approximation to the ADD measure. F i r s t we construct a z-equilibrium: (27) V uB(p(z) + tz , p(z), u(z)) = a + Xz ; (28) - 7 gB(p(z) + tz , p(z), u(z)) - V pB(p(z) + tz, p(z),u(z)) = 0. When z = 0, (27) and (28) coincide with the f i r s t order conditions for the maximum of social welfare (25) and (26), i f we define u(0) = u° and p(0) s p ° . When z = 1, (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) = p^  are the values prevailing at the observed distorted equilibrium. If we assume that the level of lump-sum transfers from the government to consumers are 18. appropriately chosen, there exist 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 'equity' distortions at the observed equilibrium; i . e . , -X^ shows the difference between the marginal social 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) is va l id . We must also adjust X^ proportionally to h's marginal u t i l i t y of income and a1 so that (27) i s va l id . Now differentiate (27) and (28) with respect to z; (29) 3 , uu uq B Z , B Z + B Z qu' qq pp u' (z) P' (z) -X + B t uq B Z t qq z 2 z where B ^ = V^B(p(z) + tz , p(z), u(z)) for i , j = q,p,u. Note that B u p = 0 H ) < N . Assumptions ( i i i ) and (iv) guarantee, via the Implicit Function Theorem, that once continuously differentiable functions u(z) and p(z) T T satisfying (29) exist at z close to 0. Premultiplying [ 0 „ , p(z) ] to both H sides of (29) and using property ( i i i ) of the overspending function in Appendix I, we can derive (30) V um h(p(z) + tz , u h (z)) u£ (z) = z[t T B z (p' (z) + t) + t T B z u'(z)] , VJVJ VJ u 19. analogously to the derivation of (17) in Appendix III. Evaluating (30) at z =0, we get (31) [ h = l V h ( p ° ' U h ) u h ( 0 ) = °" Analogously to the derivation of (19) in Appendix IV, we next differentiate (30) with respect to z, and evaluate at z = 0 to obtain (32) U ' ( 0 ) T B u ° u'(0) + E h ! 1 V h { P ° ' U h ) U h ( 0 ) = t T B q q ( p ' ( 0 ) + fc) - P'(0) T B Ju ' ( O ) . Premultiplying (29) evaluated at z = 0 by toLp'(0)T] and adding the n resulting identity to (32), we have (33) u ' ( 0 ) T B uV ( 0 ) + E h " 1 V u m h ( p ° , u ° ) u ^ ( 0 ) = - p ' ( 0 ) T B p ° p ' ( 0 ) - [p'(0) + t ] T B g ° [p 1(0) + t] > 0. A second order Taylor approximation to the HB measure (3) at z = 0 i s as follows: ( 3 4 ) LHB 5 - E h = l V h ( p°' u2 ) u h ( 0 ) " H [ u - ( 0 ) V \ r ( 0 ) + E h-lV h ( p°' uS> U h ( 0 ) ^ J Substituting (31) and (33) into (34) we have 20. (35) L £ - Ji{p'(0)Vp'(0) + [p'(0) + t ] T B ° [ p ' ( 0 ) + t]} > 0. HB pp qq To compute (35), we could again replace B?^ by B ^ in (35) and (29) since the B^j are observable, again following Harberger. 1 0 It i s interesting to com-pare (35) with the second order approximation to the gain in social welfare using the linear welfare function in moving to the optimum from the distorted equilibrium, E h " 1 a h [ f h ( x h 0 ) - f h ( x h 1 ) ] = L^. We find that ( 3 6 ) l L = £HB + J 4»'(0) T B u Su'(0) > L H B where the t i lde shows i t is an approximation of the original measure and the inequality comes from the positive semidefiniteness of B ^, which is implied by the concavity of the u t i l i t y functions. According to Varian (1976;257), the linear 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 ts equity loss. In this sense, L r may be taken as a lower bound of the welfare change. 1 1 Li However, (36) shows that L H R is even smaller than L^. This is because, with diminishing marginal u t i l i t y of income, increasing the inequality in terms of u t i l i t y (or real income) holding the (weighted) sum of u t i l i t y constant tends to increase the aggregate expenditure necessary to attain the reference u t i l i t y al locat ion. This problem of inequity in the HB measure may not arise i f we adopt money metric u t i l i t y scaling so that u h = m n (p^,u h ) , h = 1 , . . . , H . 1 2 i f this is the case, v B ( p ° , u ) = 1„ and B 0 = 0„ „ so that u H uu H«H H, = L H B and = L ^ B . With this assumption we can regard the HB measure as 21 summing the change of u t i l i t i e s of individuals; 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 A Q D in (20) and (22) and L H B in (35). Though (20) and (35) look ident ical , their meanings are completely different. F i r s t , the substitution matrices are evaluated at distorted level of u t i l i t i e s in (20) while they are evaluated at optimal level of u t i l i t i e s in (35). Second, p'(0) is calculated from different sets of equations, (16) and (29). The f i r s t difference is inessential, since, as was already stressed, we replace these matrices with matrices evaluated at an observed distorted equilibrium. However, the second difference matters 1 1 cruc ia l ly . In (22), 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 (22). In (35), we need both the substitution matrices B \ B ^ and income PP qq effect matrices B ^, tax rates t and the distributional distortion parameters X so that the informational requirements are much higher. Though i t is 1 1 possible to calculate B and B from local 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 prices, this does not give information on the marginal u t i l i t y of income at the observed equilibrium, and this is what we real ly require. If we adopt money metric scaling at the observed distorted prices 1 1 . 1 1 1 p + t, then B is easy to calculate since v B(p + t ,p ,u ) = 1„. In this qu u H case we also have B = 0„ „ . However, we s t i l l cannot compute X from (27) UU n * H since now we do not have 1„ = v B ( p ° , p ° , u 0 ) ; i . e . , a is not a vector of ones M U anymore. McKenzie (1983, chapter 3) studied the methodology for calculating 22. the money metrics, and he correctly 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 is based on normalizing the marginal u t i l i t y of income at one price vector, but in 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 is 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 social importance and the inverse of the marginal u t i l i t y of income), instead of the measurement of hypothetical income. In l ight of these observations, we must conclude that the approximate HB measure lacks empirical operat iona l ly without a knowledge of the original u t i l i t y functions whereas the ADD measure is free from this problem. Note that this cr i t ic i sm w i l l also apply even i f we compute the waste using the Bergson-Samuelsonian social welfare function. It is chiefly for this reason that we adopt the ADD measure as our welfare cr i ter ion . Needless to say, however, the informational advantage of using the ADD measure does not mean that i t is 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 in 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 in this con-text we have found a remarkable property of the ADD measure: i t is comput-able from local 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 equi l ibr ia i s large. The numerical general equilibrium approach by Shoven and Whalley (1972, 1973, 1977) chooses an alternative way to compute equi l ibr ia direct ly corresponding to various tax and expenditure policies so that a more exact welfare evaluation seems available. 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 corollary of this fact, our approximate measure can be derived from any set of f lexible functional forms using information based on the observed equilibrium. On the contrary, in the numerical general e q u i l i -brium approach, very restr ict ive functional forms are adopted to make global computation possible, and these restrict ions are easily rejected in econo-metric tests using more general functional forms (see Jorgenson (1984;140)). Moreover, the approximation approach does not involve any numerical computa-tions that are more complicated than a single matrix inversion, whereas there are often substantial numerical d i f f i cu l t i e s involved in computing general equi l ibr ia . 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 is universally superior to the o t h e r . 1 3 The measurement of waste is prominently a practical subject. As i s pointed out by Harberger (1964;58), the comparison of welfare measures is the only constructive way to give a policy prescription under the 'nth best' 24. situation,1* i . e . , by comparing the amount of waste corresponding to various feasible policies we can give a ranking among them even i f there are various other distortions. 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 triangle approximation may change considerably by changing the choice of approximation point. In this context, they cr i t i c i zed Feldstein (1978) who measured the net benefit of capital income tax reform by comparing Harberger's (1964) measure at two taxed equi l ibr ia . Green and Sheshinski noted that there exist differences 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 similar cr i t ic i sm 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 ike 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 equ i l ibr ium. 1 5 However, due to the complexity of the resulting formula, we have omitted this derivation. Therefore, this approximation error may lead to reversals in the true ranking of policies 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 policies either by approximation or by equilibrium compu-tation while we have to reach some decision on the choice or reform of eco-nomic pol ic ies . In the second best theory approach originated by Lipsey and Lancaster (1956), recommendations for policies or their part ia 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 attain the f i r s t best is restr icted. This approach has successfully derived many inter-esting results in optimum taxation theory, piecemeal policy recommendations and cost-benefit analysis .16 However, we have to note at least two basic drawbacks of this approach. F i r s t , in contrast to the f i r s t best solution, general second best solutions cannot be decentralized in a simple principle (see Guesnerie (1979)) so that the poss ibi l i ty of meaningful policy recom-mendations i s quite restricted except under rather simplified second best situations as in an optimal taxation economy . Second, since most of the second best results depend on local necessary conditions for optimality, they suffer from theoretical crit icisms from the viewpoint of general equilibrium theory. As i s shown by Foster and Sonnenschein (1970) and Hatta (1977), multiple equi l ibr ia and ins tab i l i ty can easily occur in a well-behaved economy with tax-distortions. Harris (1977) pointed out that the sufficiency of the necessary conditions for second best optimality depends on the third order derivatives so that the interpretation of these sufficiency conditions is not easy. In contrast, tax reform approach due orig inal ly to Meade (1955) avoids the problem by restr ic t ing i t s attention to the local area around the observed distorted equilibrium. Various authors, represented by Dixit (1975) and Hatta (1977), derived sufficiency conditions for welfare improvement by some policy changes. Unfortunately, these conditions depend on many restr ic t ive assumptions. Especially, the assumption that the policy maker can change the set of taxes incrementally is often irrelevant, since i t s reform alternatives are discrete changes of taxation. By the same token, i t is often the case that the reform alternatives are inst i tut ional ly restricted to the ones which are short of fu l ly satisfying the sufficient conditions. In these cases, this approach cannot t e l l anything about the ranking of 26. pol ic ies , but our approach can. Furthermore, the sequence of local improvements may not converge to global optimum, but may stay on a local optimum or some stationary point. These problems seem to give limitations on the use of local optimality or improvement conditions for policy recommendations. Considering these defects, we seem to be obliged to conform to a conven-tional view on second best; i . e . , i f conditions on propositions are met, implement the prescribed policy. 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. Part icularly , even i f conditions are not met for positive second best propositions, this does not justify the status quo in any way, since even in 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 in any given nonoptimal position is of high pract ical importance when we cannot know how to make the best of a bad situation. 1-3. A Diagraamatic Exposition In this section we i l lus tra te diagrammatically the ADD measure and the HB measure and their approximations using a simple model in order to c lar i fy the intui t ive content of the discussions in the previous section. We assume that there is 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 leisure 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 is v and there i s a zero endowment of the good. We f i r s t specify the tax-distorted observed equilibrium. We choose labour as numeraire so that i t s price, w = 1 . We assume that there i s a specif ic tax t on the good levied for consumption so that i t s producer price is p whereas i t s consumer price is p + t. It i s also assumed that the specific tax revenue i s transferred to the consumer as a lump-sum transfer. Then, using the prof i t function w(1,p) dual to y < g(v) and the expenditure function dual to f (x ,L) , the observed equilibrium is characterized by the market clearing conditions for the good and labour; (37) V TT(1,P) - v m(1, p+t, u) = 0 (38) VWTT(1,P) - vwm(1,p+t,u) + v = 0. 1 1 We assume that (p ,u ) solves (37) and (38) uniquely. From the homogeneity properties of TT and m, we can deduce (39) v + i r ( 1 , p 1 ) + t V p m , ( 1 , p 1 + t , u 1 ) = m( 1 ,p 1 +t ,u 1 ) ; i . e . , the budget constraint of the representative individual i s sat isf ied. Now we define the ADD measure of waste in this simple model. We assume that the surplus of the economy is measured by the numeraire good, labour. Therefore, the general primal programming problem ( 1 ) and i t s dual (7) are simplified respectively in this model as follows: 28 ( 4 0 ) LADD 5 M a X y , v , L { ^ " L ~ V : y 1 x ' y 1 g ( v ) ' f ( x ' L ) 1 " } (41) = M i n p > Q {TT(1,P) - m(1,p,u 1) + v}. A We assume that p = p > 0 is a unique solution of the f i r s t order condition: (42) V TT(1,P A) - 7 m(1 ,p \u 1 ) = 0. tr tr Therefore, (40) and (41) can be rewritten as (43) v + 7 ww(1, P A) - V w m(1,p \u 1 ) = L A D D (44) v + ii(1,pA) = L & n n + m(1,p A ,u 1 ) . Note that (43) and (44) are equivalent by using (42) and the homogeneity properties of TT and m. We can i l lus tra te the ADD measure of waste diagrammatically in F ig . 3. The program (40) boils down to searching for a point where the horizontal length of the lens-shaped area formed by the 1 . production poss ib i l i t i e s set and the indifference curve with u = u is maximal. This maximum is characterized by an equal slope 1/p of the two curves. In this simple example, we can also express the ADD measure of waste as a more familiar Hotelling-Harberger-like curvilinear triangle ABC in F ig . 4. This can be proven as follows. The area ABC is defined from Fig . 4 as (45) ABC = ; From this we have ABC = m(1,p 1+t,u 1)-mn,p A,u 1)+Tr(1,p A)-ir(1,p 1)-tV m( 1,p1+t,u1), P = v - m( 1 , p A , u 1 ) + ir(1,p A) (from (39)) = L ADD (from (44)). In F i g . 4, we have also drawn two triangles ABC and ABC". ABC is a linear approximation to ABC using the slopes of the Hicksian demand curve and the supply curve at the optimum point whereas ABC" is a l inear approximation to ABC using the slopes of the two curves at the distorted equilibrium. These two triangles correspond to the two approximations of the ADD measure of waste (20) and (22) in this simple example. To show this , let us f i r s t construct a z-equilibrium as in the previous section for this simple model as follows: (46) V i(1,p(z)) - v pm(1,p(z)+tz,u 1) = 0, (47) V ir(1,p(z)) - V m(1,p(z)+tz,u 1) + v = r (z) , w w where 0 < z < 1 and p(0) = p A , p(1) = p 1 , r(0) = L A m ) and r(1) = 0. When z = 0, (46) and (47) correspond to (42) and (43), and when z = 1, they 30. correspond to (37) and (38). Totally differentiating (46) and (47) with respect to z we can compute r ' ( z ) . From this 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): (48) (49) th ° S 0 PP P P 2 ^ o o " O P P P P t 2 E 1 s 1  pp P P 1 1 2(E - s ) P P PP where E Z = V 2 m( 1 ,p(z)+tz,u 1) and S z = V 2ir(1,p(z)) for z = 0,1 pp pp pp pp i r v As AB = t and the height of the triangle ABC is ( t E 0 S ° ) / ( E ° - S° ), PP P P P P P P (48) equals the area ABC' while the height of ABC" is ( t E 1 S 1 ) / ( E 1 - S 1 ) PP P P PP P P so that (49) equals the area ABC". (See Appendix V.) Note that the slope of AC equals the slope of the demand curve at C while the slope of BC equals the slope of the supply curve at the point C. Therefore, in this simple model, our triangular expression of the deadweight loss corresponds to that by Harberger (1964) except that we allowed for nonlinear production poss ib i l i t i es set. We next turn to a diagrammatic interpretation of the HB measure of welfare and i t s approximations. For this purpose, we f i r s t have to find a price vector which supports the social 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 i s B B the price solution (p ,w ) to the concave programming problem below: (50) Max {f (x,L) : y ) x, v ) v H , y < g(v)} (51) = Maxu M i n p > 0 w > ( ) { u - m(w,p,u) + n(w,p) + wv}. B B B We assume that an interior optimum point (u ,p ,w ) solves (51) uniquely with B B p > 0 and w > 0. It i s a solution to the following f i r s t order necessary conditions for (51): (52) vum(w ,p ,u ) = 1, (53) V p 1i(w B , p B ) - v pm(w B , p B (u B) = 0, (54) v + V ir(w B , p B ) - V m(w B ,p B,u B) = 0. w w As (53) and (54) are unchanged by a proportional change of w and p, we set w = 1 . For this normalization, we can assume that (52) is always met by choosing a money-metric normalization of the u t i l i t y function at the reference price (1,p ). Therefore we can delete (52) from the system and B B B assume that (53) and (54) determine u and p from w = 1 . Using this normalization, (53) and (54) imply the following budget constraint of the representative consumer for the optimum price vector (1,p ): (55) v + ir(1,p D ) = m(1,p D ,u D ) . 32. Now the HB measure of welfare (3) in this simple model may be defined as follows: (56) L H B = m(1,p B ,u B) - m(1,p B ,u 1 ) F ig . 5. i l lus trates the HB measure for this simple model. This is nothing but a Hicksian compensating variation when moving from a tax-distorted equilibrium to a social optimum. We also i l lu s t ra te L u _ using a Hotelling-Harberger-like expression in 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 1) using the tax-distorted u t i l i t y level P u^  and the supply curve v i ( 1 , p ) . We also include the compensated demand P curve for the good for the social ly optimum u t i l i t y level , Vpm(1,p,u ). F ig . 6 corresponds to the case where the good is normal so that Vpin(1,p,u ) is above V m(1,p.u**). If the good i s infer ior , the former curve is below the latter curve and i f the good changes from a normal to an infer ior good then the two curves intersect. Our results below apply to a l l cases l i s ted above. Using Fig . 6, the HB measure can be shown to be equal to the sum of two curvil inear triangles AFE and FBD. To show this , f i r s t note that L„ D can H o be decomposed as follows: (57) L H B = {m(1,pB,uB) - m(1, P 1+t,u 1)} + {m(1,p1+t,u1) - m(1,p B ,u 1)}. 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) 1 p P 1 Performing the integration in (59) yields the expression in (58). For this simple model, the triangles ABG and ABC" drawn in Fig . 6 correspond to the approximation to the HB measure where ABG corresponds to (35) and ABC" is i t s variant where observed information is used. This may be shown as follows. F i r s t , construct a z-equilibrium: (60) v T ( 1 , P ( Z ) ) - V m(1,p(z) + tz , u(z)) = 0, tr tr (61) v + V » ( 1 , p ( z ) ) - v m(1,p(z) + tz , u(z)) = 0, w w where 0 <. z < 1 and p(0) = p B , p(1) = p 1 , u(0) = u B and u(1) = u 1 . When g z = 0, (60) and (61) correspond to (53) and (54) with w = 1 and when z = 1 they correspond to (37) and (38). Totally differentiating (60) and (61) with respect to z, we can derive u'(z). 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 VI). (62) 34. where E * = V J m(1,p(z) + t z , u(z)) and S z = 7 2 i r ( 1 , p ( z ) ) f o r z = 0,1. PP PP PP PP Note t h a t E ^ and S ^ are d i f f e r e n t from the analogous expression i n (48) s i n c e , i n (62), the d e r i v a t i v e s are evaluated at the s o c i a l l y optimum p o i n t B B (1,p ,u ). As AB = t and the height of the t r i a n g l e ABG i s ( t E J S ? ) / ( [ J - S J ) , (62) equals the area 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 the d e r i v a t i v e s E ° and S 0 by those observable d e r i v a -PP PP 1 •) t i v e s E and S , then t h i s approximation i s i d e n t i c a l to (49) which i s a trkr trtr suggested approximation of the ADD measure. (49) i s i l l u s t r a t e d i n F i g . 6 as ABC", which i s a l s o shown i n F i g . 4. In t h i s simple model, the ADD measure and the HB measure c o i n c i d e i f the two points C and D c o i n c i d e i n F i g . 6; i . e . , the ADD optimum and the s o c i a l optimum c o i n c i d e . (48) and (62) (or ABC i n F i g . 4 and ABG i n F i g . 6), which are second order approximations of the ADD measure, and the HB measure c o i n c i d e i f the curvatures of the compensated demand f u n c t i o n s and the supply f u n c t i o n at points C and D are the same. However, the f u r t h e r approximations to these f u n c t i o n s depending on the d e r i v a t i v e s of the supply and compensated demand fu n c t i o n s a t the 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 simple model as the t r i a n g l e ABC". I t i s , however, c l e a r from the d i s c u s s i o n of the previous 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 general many-consumer model. F i n a l l y , a l l of these approximations 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. This i s the case where there i s no P P income e f f e c t f o r the good and, i n t h i s case, the M a r s h a l l i a n consumer's surplus c o i n c i d e s w i t h the ADD and the HB measures (see Hicks (1946;38-41)). 35. F O O T N O T E S F O R C H A P T E R 1 1 These two measures were examined comparatively by Diewert (1981, 1984, 1985a). 2 x » 0 N means that each element of the vector x is s t r i c t l y positive, x > 0„ means that each element of x i s nonnegative, and x > 0„ means x > 0., — N N — N but x T* 0 N > A superscript T means transpose. 3 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 is discussed in this section. * See Diewert (1982;554) for the regularity properties that must be satisf ied by the functions m h. 5 Most welfare evaluation methods cannot even generate orderings. For example, the Kaldor (1939)-Hicks (1939)-Scitovsky (1941-2(a)) test is neither complete nor transit ive (see Gorman (1955)). Aggregate Hicksian (1941-2) compensating and equivalent variations cannot be transit ive i f the base price is 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 variation 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 across various tax schemes even in a single-consumer economy, i . e . , not Pareto inclusive. See Kay (1980) and Pazner and Sadka (1980). 7 We also assume that the Slater constraint qual i f icat ion condition applies in this economy; i . e . , we require that a feasible solution for (1) exists that sat isf ies the f i r s t N inequality constraints s t r i c t l y . 8 The definitions of various concepts of separability and their economic applications are surveyed in Geary and Morishima (1973) and Blackorby, Primont and Russell (1978). Pages 52-61 of the latter book are important for our analysis. 9 With appropriate lump-sum transfers across households, the budget constraints of individuals are satisf ied and the government budget constraint is 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 or ig inal ly to Arrow (1951). 1 0 It i s d i f f i c u l t in this case to interpret this further approximation 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 later, we can show that -^{p 1 (1) T B p p p'(1) + [p'(1)+t]TB q q[p'(1)+t]} + Jau'(1)TBu^ u'(1) i s also accurate for quadratic functions as (35) by this Lemma. 1 1 If we assume that there exists a concave Bergson-Samuelsonian social welfare function which is maximized at the equal divis ion equilibrium, we can show that the second order approximation to the difference of the BSSWF, evaluated at an optimum or distorted equilibrium L R S , satisf ies the inequality L R S 2 L L -1 2 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). We assumed that mh(p^,u^) is s t r i c t l y increasing in u h . Its sufficient condition was given by Weymark (1985). We also assume that m h (p^,f h (x h )) i s concave in x1 for the reference price p^, but this i s not guaranteed in general. See Blackorby and Donaldson (1986). Applications of money metrics to applied welfare economics are given by King (1983) and McKenzie (1983). 1 3 Most computable general equilibrium models adopt neo-classical perfect market assumptions. However, Pigott and Whalley (1982) incorporated public goods and Harris (1984) introduced increasing returns to scale by allowing fixed costs in numerical general equilibrium models. 1 * In contributions collected in Harberger (1974), he applied his methodology in various policy assessments. Many studies use the HB measure or Marshallian consumer surpluses for the same purpose (see Currie-Murphy-Schmitz (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 in 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 profits of firms facing prices p. It inherits many useful properties of expenditure functions and profit functions which are exhibited in Diewert (1982). We col lect several important properties for later use. An overspending function i s defined by (A.1) B(q,p,u) = E h " 1 m h (q,u h ) - E h " 0 q T x h - E ^ 0 i T k ( p ) . It has the following properties. (i) B is concave with respect to p and q. ( i i ) If B(q,p,u) is once continuously differentiable with respect to q and p at (q,p,u), then V qB(q,p,u) is the aggregate net consumption vector and -V p B(q,p,u) i s the aggregate net production vector, ( i i i ) The following identit ies are val id for any (q,p,u) i f B i s twice continuously differentiable at (q,p,u): (A.2) (A.3) T T q \ u ° ( , u B | T * <3»l(<!.u,>/3u1 8nH(q,uH)/3uH), 3 9 . (A.4) PP where . = V,j.B(q,p,u) for i , j = q,p,u. Property (i) follows from the fact that an expenditure function is concave with respect to prices and a profit function is convex with respect to prices. Property ( i i ) i s a straightforward consequence of Hotell ing's (1932;594) lemma and the Hicks (1946;331)-Shephard (1953;11) lemma. Property ( i i i ) i s a consequence of the linear homogeneity of an expenditure function and a profit function with respect to prices. Appendix I I In this 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 qualif ication holds. In (1), the set {x : f (x ) 2 u n ) i s convex from the quasi-concavity of f h ( x h ) , S k i s also assumed to be convex and the inequalities are l inear. Therefore, (1) is a concave programming problem and the Uzawa-Karlin Saddle Point Theorem is applicable. Rewrite (1) as: (A.5) 0 {r + P T [ E H y k + E H i h - E H h x - pr] r = max mm P>0, N k=1 h=1 h=1 f h (x h ) 2 u j f h = 1 , . . . , H ; y k e S k , k = 1 , . . . ,K} , 40. (A.6) = max rmin p > 0 {r(1-pTB) + E ^ p V + £ , , ! ! , [ m a x - p T x h : f h ( x h ) > UjJ] ~ N + E^^max p T y k : y k e Sk]} = max rmin p 2 0^{r (1 - pT8) + E ^ P ^ 1 1 + E k ^ i r k ( p ) - E h" 1m h(P,uJ)} using the de f i n i t i o n s (4) and (6). Appendix III In this Appendix, we derive (17). Premultiply both sides of (16) by T [p(z) ,0]. Using (15), we have: (A.7) p(z) TB qq(p"(z) + t) + p(z ) T B p p p' (z ) = - r ' ( z ) . 1 From (A.2) and (A.4) evaluated at (q,p,u) = (p(z) + tz, 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 this Appendix, we derive (19). Differentiate (17) with respect to z and evaluate at z = 0, and we have (A.8) - r"(0) = -t T B q g(P'(0) + t ) . Next premultiply both sides of (16) evaluated at z = 0, by [p*(0) T ,r ' (0)] . We obtain 41 . (A.9) - p ' l O l V t p M O ) + t) - p'(0) TB V (O) qq pp = 0. Adding (A.8) and (A.9), we have (19). Appendix V Total differentiat ion of (46), (47) gives the following: (A.10) s 2 - E 2 PP LPP S 2 - E 2 wp wp , o P'(z) PP , -1 r'(z) E 2 t wp z 2 1 z 2 where E - = V i.m(1,p(z)+tz,u ) and = .ir( 1 ,p(z)) for i , j , = p,w. Premultiplying both sides by [p(z),1], using the identit ies S z + p(z )S z = 0, E 2 + ( p ( z ) + t z ) E 2 = 0, wp PP wp ' pp ' 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 p 2 t ) / ( S p z - E p z ) Substituting i t into (A.11), we have 42. (A.12) r'(z) = (zt 2S Z E Z ) / (S Z - E Z ) . PP PP PP PP Using (A.12), we can compute the two approximations (20) and (21) which correspond to (48) and (49) respectively. Now draw a perpendicular l ine from point C to AB and define the cross point with AB, H. Then the height of the triangle is C H . We have C H = - E ^-AH = S ^-(t-AH). From these two equations, we can solve C H = PP PP ( t E °S ° ) / ( E ° - S ° ) . The proof of ABC" is perfectly analogous. PP PP PP PP Appendix VI Total ly differentiating (60) and (61) with respect to z, we have (A.13) -T z S Z - E z PU' pp pp u' (z) E Z t PP - E Z , S Z - E 2 wu wp wp P'(z) E z t wp z 2 z 2 where E— = V i .m( 1, p(z)+tz, u(z)) , = V^irt 1 ,p(z)) for i , j = w,p,u. We compute u'(z) by inverting the left-hand side matrix of (A.13). F i r s t , the determinant of the matrix D is (A. 14) D = E z (S Z - E z ) - E Z(S Z - E * ) . wu pp pp pu wp wp Using the l inear homogeneity properties of m and TT, 43. E w u + ( P ( z ) + t z ) E P u = V { 1 , p ( z ) + t z , u ( z ) ) ' E Z +(p (z)+tz)E Z = 0, S +p(z)S Z = 0, wp pp wp ^  pp we can rewrite (A.14) as (A.15) D = - 7 m ( 1 , p ( z ) + t z , u ( z ) ) ( E Z -S Z ) - z t E ZS Z . u v PP PP pu pp The numerator of u'(z) , defined as N, is given by (A.16) N = {S Z - E z> E z t - (S I - E z ) E z t wp wp pp pp pp wp = {-p(z)S z + ( p ( z ) + t z ) E Z } E Z t + (S Z - E z ) ( p ( z ) + t z ) E Z t PP pp pp pp pp pp = z t 2 s Z E 2 pp PP From (A.15) and (A.16) we have (A.17) u ' ( z ) = - { z t 2 S p 2 E p Z } / { V u m ( 1 , p ( z ) + t z , u ( z ) ) ( E p 2 - S p z ) + z t E p Z S p 2 } . From (A.17) we have, u'(0> = 0 and u-'(0) = - { t 2 S p ° E p ° > / { V u m ( 1 , P ° , u ° ) ( E p ° - S p ° ) > Substituting them into (34), we have (62). We can show analogously as Appendix V that (62) coincides with the area ABG. 44. pr i s the reference bundle AB is the ADD measure Fig . 1 The ADD Measure of Waste 45 . X 2 Scitovsky set Scitovsky set 1 0 S(u ) S(u ) AB is the HB measure F ig . 2 The HB Measure of Welfare F i g . 3 The ADD Measure: A One-Consumer Two-Goods Economy 47 . V nm(1,p A lu) = 7 n(1,pA) x ' y Xr P Fig . 4 The ADD Measure and i t s Approximations: A One-Consumer Two-Goods Economy F i g . 5 The HB Measure: 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 indirect taxation has been the main concern of this l i terature . This section proposes a methodology for measuring the waste due to an externality, which seems to be an alternative and equally important situation involving a market fa i lure . Though our methodology is applicable to other externalit ies, here we focus on the problem of public goods. Consider a government which collects revenue from both lump-sum and i n -direct taxation and provides public goods. This economy exhibits the waste due to a price distort ion 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 social benefit and social cost of public goods in addition to the set of indirect 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 this public good economy, and we show that the approximate deadweight loss can be expressed in terms of the derivatives of restricted expenditure functions and restricted profit functions evaluated at the observed equilibrium as long as we know the marginal benefits of pub-l i c goods for consumers. In deriving the approximate waste, we need not assume local l inear i ty of the production poss ib i l i t i es set as in Harberger (1964) and we need not assume restr ict ive functional forms for u t i l i t y and production function as in the numerical or applied general equilibrium l i t e r -ature. The waste to be studied is 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 in this section. The next section is devoted to the description of our model of a public goods economy, while section 3 defines the Allais-Debreu-Diewert measure of waste in this 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 interpret 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. 2-2. The Model Our model i s similar to the one used in 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 poss ib i l i t ies sets of T firms. A quantity vector of public goods is denoted as G = ( G 1 , . . . , G I ) > Oj. There are K profit 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 poss ib i l i t i es set S for k = 1 , . . . , K , i . e . , i f (y , - G) e S , then the k k k T vector of net outputs y = (y^ , . . . , y N ) is producible by sector k using the k vector of public goods G. The sector k restricted profit function ir , which k is dual to the production poss ib i l i t ies set S , i s : (1) ir K (p,G) = maxy {pTy : (y, - G) e S k }, k = 1 K, 52. where p > 0^. We assume that either G or an entrepreneurial factor i s a l imiting factor of production, so that S exhibits decreasing returns to scale when G is fixed. (See Meade (1952) for the def init ion of an 'unpaid factor" public input.) The vector of public goods i s produced by the government, k = 0, which has the production poss ib i l i t i es set S^. If (y,G) e S ,^ the government produces G using the input vector y. If some component of y is positive, the government i s jo int ly producing the corresponding private good with G. The government restricted profit function TT^ , which i s dual to S ,^ i s : where p > 0 M . 1 N Let us now look at the consumer side of our model. We assume that there are H individuals, h = 1 , . . . , H , in the economy. 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 in R N + * , Q*1. Define the individual h h h restricted expenditure function m , which is dual to f , for h = 1 , . . . , H , by: where p > 0 N and U r e Range f . We suppose that each individual h possesses nonnegative endowment vector of private goods, x h > 0 „ , for h = 1 , . . . , H . We N also allow the government, which is h = 0, to have an i n i t i a l endowment vector x^ > 0 „ . (2) 0 T O TT (p,G) = maxy {p y : (y,G) e S }, (3) (x,G) e Qh} - N As in section 1-2, the government raises revenue by the set of indirect T taxes t = ( t . | , . . . , t N ) to provide the public goods. The government can also make a net transfer g h to individual h. If g^ < 0, -g^ i s the amount of lump-sum tax collected from person h. Producers face prices p > 0 N whereas consumers face p + t > 0 N at the observed distorted equilibrium. We use the overspending function defined by to characterize our general equilibrium system. Diewert ( 1986;131-155, 170-176) showed that the properties of a profit function and an expenditure function are va l id in their restricted functional form. This means that the properties of an overspending function (i) - ( i i i ) l i s ted in Appendix I to Chapter 1 are val id for (4). Diewert (1986) also showed that: (iv) a restricted profit function is concave with respect to G i f the production poss ib i l i t i es set i s convex and a restricted expenditure function i s convex with respect to G, so that B is convex with respect to G, (v) -V_m^(q,G,u, ), o n for h = 1 , . . . , H , is the marginal benefit vector of consumer h for the v public goods; Vnv (p,G) for k = 1 , . . . , K is 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 public goods, so that -V^B(q,p,G,u) shows the aggregate net benefit vector of public goods. From the l inear homogeneity of B with respect to prices, the identity (4) B(q,p,G,u) = E h " 1 m h (q ,G,u h ) h = o q x " Ek=0 f ( P , G ) (5) T T T q V + P B P G = ( V G B ) holds in addition to (1.A.2) - (1.A.4). The system of equations characterizing the observed equilibrium is now hk stated in a f a i r l y simple manner where a is defined as the fraction of a hk firm k held by individual h, with 0 < a < 1 for h = 1 , . . . , H and k = 1 , . . . , K and EjJ^a*1* = 1 for k = 1 , . . . , K . (6) m h(p+t (G,u h) = g h + (p+t)Tx h + E k ^ a h K T r k ( p , G ) , h = 1 , . . . , H , (7) V qB(p+t,p,G,u) + V pB(p+t,p,G,u) V (8) -VGB(p+t,p,G,u) = d. Here (6) shows the budget constraints for the H individuals and (7) shows the equality of demand and supply for goods 1 , . . . , N . The government budget con-straint i s implied by (6) and (7). From the property (v), d in (8) defines the net marginal benefit vector of the public goods. If d = 0^, (8) is con-sistent with the well-known Samuelson (1954)-Kaizuka (1965) conditions for the optimal provision of public goods. Therefore, d f 0j means that the public goods are not supplied optimally. We assume that the distortions parameter d arises because of the limited ab i l i t y of the government to provide public goods e f f ic ient ly . We regard (6) - (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. We assume that an observed 1 1 1 distorted equilibrium (u ,p ,G ,t ,d,g) exists. 55. 2-3. An Allais-Debreu-Diewert Measure of Waste An Allais-Debreu-Diewert measure of waste that was defined and discussed in Chapter 1 i s now ut i l i zed to measure the waste due to the public good externalit ies . T Pick a nonnegative reference vector of private goods p = (P^,...,P^) T . > 0N and consider the following primal programming problem: (9) r ° = max {r : (i) E H x h + pr < E K v k + E H i h ; r ,x ,y ,G. h=1 k=0 h=0 ( i i ) f h (x h ,G) > u^ ; (x h,G) e Q h , h = 1 , . . . , H ; ( i i i ) (y k , -G) e S k , k = 1 , . . . K ; (y° ,G) e S° } where u = ( u . , . . . , u „ ) is the u t i l i t y vector which corresponds to the 1 n observed distorted equilibrium defined in the previous section. The interpretation of L ^ D D = r^ i s discussed in Chapter 1 so that we wi l l not repeat i t here. For simplicity of computation, our reference bundle does not include public goods. A l la i s proposed to measure the waste in terms of a T . • numeraire good, i . e . , in our context p = (1 ,0 , . . . ,0) . Debreu's coefficient of resource u t i l i za t ion model (which assumed that p was proportional to the economy's total endowment vector) is also consistent with our present model since we assumed that there were no endowments of public goods. Given the level of G, (9) is a concave programming problem so that we can derive i t s dual equivalent problem-. (10) r ° = maxG[maxr m i n p > 0 {r(1-pTp) - B(p,p,G,u)}].2 (The process of derivation i s analogous to that in Appendix 1-II.) 56. If G°, r ° and p° solve (10), then pr° i s a measure of the resources that can be extracted from the economy while maintaining households at their distorted equilibrium u t i l i t y levels and is a corresponding "optimal" level of public goods and p° i s a vector of private goods prices which supports the eff ic ient equilibrium. Note that in this "optimal" equilibrium, not only are public goods being provided ef f ic ient ly , but also a l l commodity tax distortions have been removed. Given the level of G, (10) may be rewritten by using the Uzawa-Karlin Saddle Point Theorem in reverse as (11) r ° = - max n {B(p,p,G,u 1) : pTp > 1} N where B is the overspending function defined by (4). If G°, r ° and p° solves (10), then p^ solves (11) and r^ is the associated Lagrangean multiplier for the constraint in (11). It is also the case that i f G°, r ° and p^ solve (10), then G^ is the solution to the following unconstrained maximization problem: (12) maxG { r ° ( 1 - p 0 T p ) - B ( p ° , p 0 , G , u 1 ) } Our expressions for the ADD measure, (10) and (11), present our basic approach to the measurement of deadweight loss. However, these abstract expressions do not indicate how the magnitude of the loss depends on the size of distort ion parameters t and d. Furthermore, the global computation of 57. (10) is very d i f f i c u l t as was discussed in 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 is twice continuously differentiable with respect to q, p and G at the optimum of (10); ( i i ) G° » Oj, p^ » 0 N so that the f i r s t order necessary conditions for the max min problem (10) hold with equality; ( i i i ) Samuelson's (1947;361) strong second order sufficient conditions hold for (11) when the inequality constraints are replaced by equalit ies, and these conditions also hold for (12). Consider the following system of equations in the N + I + 1 unknowns, p, G and r, regarded as functions of a scalar parameter z defined for 0 <. z <. 1: (13) V qB(p(z)+tz,p(z),G(z),u 1)+V pB(p(z)+tz,p(z),G(z),u 1)+pr(z) = 0 N , (14) V G B(p(z)+tz,p(z),G(z),u 1 ) = - zd, (15) 1 - p(z)Tfi = 0. When z = 0, define p(0) = p ° , G(0) = G° and r(0) = r ° . Then (13) - (15) become the f i r s t order conditions for the max min problem (10). Alterna-t ive ly , when z = 1, define p(1) = p \ G(1) = G 1 and r(1) = 0. Suppose that the reference waste bundle 8 satisf ies the normalization 58 (16) .1T p = 1 by choosing the scale of p appropriately, which seems quite innocuous. Then, (13) - (15) coincide with (7), (8) and (16). Therefore, (13) - (15) charac-terizes the observed distorted equilibrium when z = 1. Note that when (7) and (8) are sat is f ied, (6) is also satisf ied for the observed choice of g, . n Therefore, we can safely conclude that (13) - (15) maps the Allais-Debreu-Diewert reference equilibrium into the observed distorted equilibrium as z i s adjusted from 0 to 1. Differentiating the system (13) - (15) with respect to z and evaluating at z = 0, we obtain (17) B 0 + B 0 , B ° + B °r , p qq pp qG pG BGq + BGp ' BGG , o P'(0) G' (0) r' (0) B ° t qq B ° t + d Gq where the second order derivatives of the overspending function B^?, i , j = q, p, G are evaluated at the optimum z = 0. The meaning of the B 9^ are as follows: 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 in the public good supply. 59. Now regard (17) as an identity in z, val id for z close to 0. Our assumptions ( i i i ) introduced at the outset of this section imply that an inverse exists for the matrix on the left-hand side of (17). (See Diewert-Woodland (1977, Appendix I)) . Hence, by the implic i t Function Theorem, there exist once continuously differentiable functions p(z), G(z) and r(z) which satisfy (13) - (15) in a neighbourhood of z = 0. T Premultiply both sides of (17) evaluated at z close to 0 by [p(z) , T 0 , 0]. Using identi t ies , (1.A.2), (1,A.4), and (5) evaluated at the z-equilibrium, and then using (14) and (15) we get (18) r'(z) .= z[t T B Z (p'(z) + t) + t T B !G ' (Z ) + d T G'(z) ] . qq qG (The process for deriving (18) i s similar to the one in Appendix 1 - I I I . ) . From (18) we readily have (19) r'(0) = 0. Now differentiate (18) with respect to z, evaluate at z = 0, and adding the T T identity derived by premultiplying [p'(0) , - G'(0) ,0] to both sides of (17), we find (20) - r"(0) = G'(0)TB ° G ' ( 0 ) - p '(0)\V(0) - [p 1(0)+t]TB ° [ p ' ( 0 ) + t ] uu pp qq (The derivation is analogous to the one in Appendix 1-IV.) 60. Note that the last two terms in the right-hand side of (20) are nonnegative because of the concavity of B with respect to prices. We also assume that B ^ i s positive semidefinite; this assumption is satisf ied i f the production poss ib i l i t i e s sets are a l l convex, but i t is much milder than assuming global convexity in production. Intuit ively, i t means that the concavity of the u t i l i t y functions outweighs any nonconvexity in aggregate production with respect to public goods in the neighbourhood of the optimum. Given this assumption, -r"(0) > 0 i s implied. The Allais-Debreu-Diewert measure of waste r(0) may be written as ( 2 1 ) LADD = I ( 0 ) " r ( 1 ) since r(1) =0 . A second order approximation to L^p D i s obtained by using a 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 LADD * " ^ r " ( 0 ) 1 0 where the inequality i s val id from (20) and i t s following discussion. If r is quadratic, (22) provides an exact expression for I « A n n - To compute (22), use the expression for r"(0) given in (20). The vectors of derivatives 61 . p'(0) and G'(0) in (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 is the reference bundle P, the distortion parameters (t, d) and the second order derivatives of the overspending function evalu-ated at the optimum. Let us scrutinize the informational requirements for computing (22) more carefully. The vectors t and p are direct ly observable. To know the vector d, we must know the consumers' marginal benefits from public goods evaluated at the observed consumer prices. This means that we must overcome the well-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. Though the f i r s t set of functions is observable, the second set is not. It is well-known, however, that the compensated price e las t i c i t i e s can be computed from data on the ordinary demand functions using the derivatives with respect to both prices and income in the Slutsky equa-t ion. (See, for example, Diewert (1982;572).) Similarly, the derivatives of the compensated demand functions with respect to public goods can also be computed from market demand functions using 'Slutsky-l ike' equations (see Wildasin (1984;230)). The fact that we need information on the second order derivatives of the overspending function evaluated at the optimum consider-ably decreases the usefulness of (22), since these values are not observable 62. (at the market distorted equilibrium) and, in general, are different from the values observed in the distorted equilibrium. An alternative approach to approximating L f t D D can be developed using Diewert's (1976;118) Quadratic Approximation Lemma. 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 is quadratic. Note that this approximation formula does not employ second-order derivatives of r . Suppose that (17) is val id for z close to 1 (instead of our previous assumption that i t i s va l id for z close to 0). Setting z = 1 in (18), we obtain that r'(1) 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 corollary: Corollary 1.1 (23) L A D D * -(1/2) r'(1) > 0. A desirable attribute of this approximation is that i t only u t i l i ze s local 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 in the case of (22) and evaluated at the observed equilibrium in the case of (23). In part icular, i t i s not necessary to make any assumptions concerning the functional form of B or place any restrict ions on the values of observed economic variables, other than the general restrictions used in describing our model. On the contrary, to calculate r , as opposed to an approximation to r ° , i t would be necessary to adopt specific (and possibly restr ict ive) functional forms in order to solve the max-min problem (10) globally. 2-5. Conclusion This chapter has discussed the measurement of waste and i t s local approximations for an economy facing distortions due to indirect taxation and nonoptimal levels of public good production. Use has been made of the ADD measure defined in Chapter 1 and two local approximations to the exact measure were calculated. These approximations only required local informa-tion on an overspending function. Figs. 7 - 9 i l lus tra te the diagrammatic interpretation of the ADD measure of waste in a public goods economy and i ts approximations. Suppose that there is one private good and one public good. In Fig . 7, we have drawn an aggregate production poss ib i l i t ies 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 level received at the observed distorted equilibrium. Though we cannot introduce a distortionary taxation in this one private good economy, the observed equilibrium is 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 refer-ence bundle to consist only of the private good, the ADD measure, as shown in Fig . 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 poss ib i l i t ies set. The point where the surplus good i s maximized is characterized by the equality of the marginal r a t e of s u b s t i t u t i o n and the marginal 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 Hotelling-Harberger way i n t h i s simple model as i n F i g . 8. The marginal b e n e f i t of the p u b l i c good i s the marginal 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 the marginal cost of the p u b l i c good i s the marginal r a t e of tr 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. At the o p t i -mum they c o i n c i d e , but the former i s higher than the l a t t e r a t the 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 discrepancy i s denoted as d. We can show th a t the ADD 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 and t h a t i t s two approximations (22) and (23) c o i n c i d e with the t r i a n g l e s ABC and ABC'. The d e r i v a t i o n i s analogous to Appendix V of chapter 1 f o r the i n t e r p r e t a t i o n of the tax l o s s as shown i n F i g . 4. In F i g . 8, the approximations are rather accurate i n comparison to the t r u e amount of waste, but i t i s d i f f i c u l t to t e l l how w e l l the approximations can approximate the true amount of waste i n general. In F i g . 9, we show an example of one consumer economy with l i n e a r production p o s s i b i l i t i e s set where two approximations can be q u i t e i n a c c u r a t e even i n t h i s simple model. Tsuneki (1987a) gives a more extensive d i s c u s s i o n on t h i s numerical example and concludes t h a t the approximations can give a t l e a s t an order of magnitude estimate of the t r u e amount of waste and the approximations can work q u i t e w e l l as long 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 are not f a r apart. To conclude t h i s chapter: we can i n c o r p o r a t e the choice of p u b l i c goods by governments (which are used both by consumers and producers) i n a t r a d i -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 adopting the Allais-Debreu-Diewert approach. Our approach i s more general than Harberger's a n a l y s i s i n the sense t h a t i t allows f o r ( i ) 65. the choice of f lexible functional forms (instead of linear ones as in Harberger or CES-type ones in the numerical general equilibrium l iterature) for the production sectors and ( i i ) the loss due to indirect 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. See Tsuneki (1987a) for more detai ls . 2 Formula (10) follows using definitions (1) - (3) and the Uzawa (1958)-Karlin (1959) Saddle Point Theorem. We assume that Slater's constraint qual i f icat ion condition applies. 67 . F ig . 7 The ADD Measure in a Public Goods Economy 68. Fig. 8 The ADD Measure and i t s Approximations in a Public Goods Economy 69 . 0 Optimum 1—f Distorted Equilibrium Marginal Cost of Public Good Marginal Benefit of Public Good Public Good F i g . 9 An Example where Approximations of the ADD Measure are Inaccurate 70. CHAPTER 3 PROJECT EVALUATION RULES FOR THE PROVISION OF PUBLIC GOODS 3-1. Introduction The theory of the provision of a public good with d i s t o r t i o n a r y taxa-t i o n , f i r s t set f o r t h by Pigou (1947), maintains that the Samuelsonian (1954) rul e to equate the sum of marginal benefits to i t s marginal cost cannot be an appropriate rule for the maximum of s o c i a l welfare. A main objective of the present chapter i s to formulate some cost-benefit rules for the provision of a public good which d e f i n i t e l y improves the welfare of a l l the i n d i v i d u a l s within the economy. This means that our approach considers s u f f i c i e n t conditions for the existence of a Pareto improvement when the public good i s provided i n a d i s t o r t i o n a r y fashion, and (i) i n d i r e c t tax rates, ( i i ) i n d i r e c t taxes rates and lump-sum trans f e r s , ( i i i ) lump-sum transfers, are allowed to vary with the provision of public good. In case ( i ) , we suggest a Generalized Pigovian 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 optimality condition for the public good pr o v i s i o n which i s due to Atkinson and Stern (1974), while i n cases ( i i ) and ( i i i ) we suggest a Generalized Samuelsonian Rule and a Modified Harberger-Bruce-Harris Rule, where a l l of them more or less d i f f e r from the Samuelsonian r u l e . Since our rules are v a l i d when the equilibrium i s away from the second-best optimum, our approach contrasts with the previous l i t e r a t u r e on project evaluation rules f o r public goods by S t i g l i t z and Dasgupta (1971), Atkinson and Stern (1974), Diamond (1975), Atkinson and S t i g l i t z (1980) and King 71 (1986). They analyze the f i r s t order necessary conditions for ( interior) second best social welfare optima. Another objective of this chapter i s , however, to reconcile these two apparently different approaches and to t ie together strands of previous discussions within our framework.1 We show that the cost-benefit rules in this chapter are val id both as necessary and sufficient conditions i f the manipulable taxation scheme is optimized. After describing our model in 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 available. Atkinson and Stern's (1974) cost-benefit rule for public goods provision, which generalized the result in Pigou's (1947) pioneering study, is extended to a heterogeneous-consumers' economy in this section. In Atkinson and Stern's (1974) model, the marginal u t i l i t y of income does not equal the marginal social cost of raising one dollar by indirect taxation; this difference arises because there is a welfare cost due to indirect taxation and there is an income effect due to taxation on tax revenue. The f i r s t distortion is emphasized by Pigou, but the second one is neglected by him. When we extend the Atkinson and Stern result to a heterogeneous-consumers' economy, two differences arise . F i r s t , 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 individual . Second, the change in the income distribution that results from increased taxation affects the social cost of taxation; e.g. , i f the tax is levied on people with high social importance, the social 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. indirect tax rates and lump-sum transfers at the same time, a generalization of a tradit ional Samuelsonian rule, generalized Samuelsonian rule applies for the project evaluation. However, when there exists unchangeable indirect tax distortions, 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 their i n i t i a l indifference curves when the supply of a public good is increased. The induced change in the net supply of private goods is then evaluated using Harberger's generalized weighted-average shadow prices for fixed indirect tax distortions. Since we adopted the approach of searching for sufficient 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 interior welfare optimum. In this approach, the cost-benefit 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 in this chapter is identical with the one we used in the previous chapter to characterize the observed distorted equilibrium, (2.6) - (2.8). We assume for simplicity that profit income is completely taxed away, following Diamond and Mirrlees (1971). This assumption can be relaxed by assuming that the entrepreneurial factors are additional commodi-ties (see Diewert (1978) and Dixit (1979)). With these assumptions, we can restate (2.6) and (2.7) as follows: 73. (1) h = 1, . . . , H (2) 0, T T We assume that (1) and (2) determine p = ( P 2 F - - - ( PN ) ». u = (u- | , . . . ,u H ) and - T T t 1 endogenously given p 1 = 1, t = ( t 2 , . . . , t N ) , g = (g. , , . . . ,g H ) and G = T ( G ^ , . . . , G N ) . Note again that by Walras' law (1) and (2) imply the budget constraint of the government is sat isf ied. When the equality in (2) is replaced by the inequality (<.) by assuming free disposal, we c a l l i t "the inequality version of (2)." The Pigovian cost-benefit problem we study in this chapter is simply a comparative statics exercise 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 is a scalar (or alternatively, we assume that only the production of the f i r s t public good is varied while the other public goods are held fixed). Three alternative rules are derived depending on which taxation instruments we can change. Differentiating (1) and (2) tota l ly , assuming that (3) h = which is implied by money metric u t i l i t y scaling (see Samuelson (1974)), we obtain: 74. — — -X -X du = dp + dt 1 + B -B - -B ~ -B -B -qu qq PP qq 1 qq h dg + o _ N«H_ w "BqG ' B P G dG where the net demand matrix of consumers i s : X = ] (H«N matrix, with X V H«1 and X ,H«(N-1)) where the hth row shows the net demand vector of the hth consumer and W = ( W 1 f . . . , W H ) T = ( -V G m 1 (p+t ,G,u 1 ) , . . . , -V G m H (p+t,G,u H )) T i s a vector of the marginal benefits of the public good for the consumers. The scalar MC = " E k = o V 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„ „ is 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) = V-.. B(p+t,p,G,u) i s negative definite. PP PP We express (3) in a different way for later use: (6) Adu = B.jdp + B 2 d t 1 + B 3 dt + B4dg + B^dG. When we refer to "the inequality version of (6)" we mean that the H+1, . . . , H+Nth equalities in (6) are replaced by inequalities (<.). This case ut i l i zes 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 local ly solve for u, p, and t^ as functions of the exogenous variables, using the Implicit Function Theorem. This analytical technique closely follows Diewert (1983b). F ina l ly , we have to define our welfare c r i t e r i a . In a many-consumer economy, we have to distinguish between two c r i t e r i a for a welfare improve-ment. The f i r s t cr i ter ion is the s t r i c t Pareto cr i ter ion. A s t r i c t Pareto improvement occurs i f each person's u t i l i t y is increased. The second T H T cr i ter ion makes expl ic i t use of the social welfare function 8 u = E. „B,u. r h=1ph h T where 8 > 0 „ . The l inear function 8 u can be thought of as a local l inear n approximation to a general quasiconcave social 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 ferent ia l effect of the various sets of tax-expenditure instruments with respect to social welfare. If a set of available instruments is ful ly perturbed with satisfying (4) and du » 0 occurred, then we define i t as a d i f ferent ia l ly s t r i c t Pareto improvement. If available tools are fu l ly perturbed with T satisfying (4) and p du > 0 occurred, then we define i t as a d i f ferent ia l ly s t r i c t welfare improvement. (These definitions follow Diewert (1983b).) Obviously, a d i f ferent ia l ly s t r i c t Pareto improvement ( i . e . , du 2> 0„) is a H di f ferent ia l ly s t r i c t welfare improvement for any nonnegative, but nonzero, u t i l i t y weight vector p. Therefore, i f we can find a sufficient condition for the existence of a d i f ferent ia l ly s t r i c t Pareto improvement, then there exists a d i f ferent ia l ly s t r i c t welfare improvement as well. Note also that T T P du > 0 implies the improvement of social welfare p u in a local sense but the opposite is not true in general, since i t is possible that there exists T an inflexion point of p u with respect to the set of instruments so that the T improvement of social 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 in which p u is maximized with respect to the instruments. 3-3. Pigovian Rules Reconsidered Most papers on cost-benefit rules for public goods provision follow the Pigovian tradit ion and suppose that the government can vary indirect tax rates t simultaneously with changes in 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 rule by assuming that a l l consumers have identical preferences and wealth. The purpose of this section is to extend their formula, which we c a l l a Generalized Pigovian Rule to a heterogeneous-consumers' economy and to compare the economic implications of this new rule with that of Atkinson and Stern. We state our main theorem in this section as follows: Theorem 3.1 Suppose that public good production is irreversible so that dG > 0 2 and the government can perturb t a r b i t r a r i l y . Suppose also that T T (7) there i s no solution a to a > 0„ and a X = 0„ u u H u N T and that the indirect tax revenue R = t v B is nonzero. q T Then i f a l l for -y > 0„ for which no d i f ferent ia l ly s t r i c t improvement of ^ u H with respect to indirect tax rates exists, "j^B h h (8) <rTW(1 + ^ f3- ~ C l ^ q u F ^ l ^ i r 1 > M C " E f t ! 1 t T ( a x h ( p + t l G f I h ) / a G ) i s sat isf ied where R*1 = t T x b i s the amount of indirect tax revenue paid by hth person and O ) s || = a x h (p+t ,G ,u h ) / a u h = a x h ( p + t , G , i h ) / a i h , h = I,...,H i s a vector of income effects for the hth individual , then there exists a d i f ferent ia l ly s t r i c t Pareto improvement du » 0 „ . 78. ( i i ) If the pre-project equilibrium is 8-optimal with respect to the choice of t, and i f no) fro *^- hls\»h<Ah > MC - E ^ t 1 (9xh(p+t,G,Ih)/r3G) is sat isf ied, then (10) i s a necessary condition for a d i f ferent ia l ly s t r i c t T increase in social welfare B du > 0. PROOF: (i) A Pareto improvement with dG 2 0 exists i f and only i f (11) there exist dG > 0, du, dt such that du » 0 H and the inequality version of (6) is satisfied with dg = 0^. Applying Motzkin's Theorem (see Appendix I), this is equivalent to T T - T (12) there does not exist an a = [a u ,a^,a ] such that [ a r a T ] > o j , a T [ B 1 , B 2 , B 3 ] = 02N_*, aTA > o j , and a T B 5 < 0. If a Pareto improving indirect tax perturbatin i s possible, then (11) i s always satisf ied with dG = 0 and the problem is vacuous. Therefore, we T assume that such an improvement does not exist. Then there exists an a = 79. [ a J , a 1 ( a T ] such that [ a v a T ] > 0^, a T [ B i ; B 2 , B 3 ] = 0 2 N _ T , aTA > 0^. For any a • • T T that sat isf ies this condition, we define t = a A. We would l ike to show that for such a, (12) i s sat isf ied. Suppose (8) holds, and also suppose, T T contrary to the theorem, a solution to (12) exists. Subtracting a B 3 = 0 ^ T T from a B 1 = using the identity (1.A.4) and using the supposition (5), we have (see Appendix II) (13) a = a ^ . Using (13), the identity (2.5), and the definitions of W and MC, we get (14) a T B 5 = (aj + a 1 f*)W - a^C + a ^ B . T Suppose that a^  = 0. Then, a = 0 N 1 from (13). Therefore, a [ B 1 ( B 2 ] 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 contradicts the supposition (7), so that (12) is sat isf ied. Suppose that a^ T T T > 0. Now postmultiply t^  and t to a B 2 = 0 and a B 3 = 0N_^ respectively, add them together and using (13) and (1.A.2) we have (see Appendix III) H a n K T (15) L / \ 7 = 1 1 t/R. h=1 a 1 R qq T T 1 H We also have a A = t =( , y,...,'r)so using (13) and (1.A.3) and (3), we can show that (see Appendix III) 80. 3 h h • u T h (16) — = — - 1 + txS n , h = 1 , . . . , H . a 1 a 1 qu Substituting (16) into (15), we find "1 t B t U T W ^ U V . R Now substituting (16) and (17) into (14) and using the Slutsky-like equation by Wildasin (1984;230),3 we get T (18) t \ . a ^ W . U 1 ^ - E J A u ? " C / « R ) - MC + E h " 1 t T (9x h (p+t,G,I h ) /3G)}. T Therefore, (8) implies a B c > 0 and we have a contradiction. ( i i ) If t i s chosen optimally at the pre-project equilibrium then i t is a solution to the problem: T (19) max ~ {B u : (1) and the inequality version of (2) are u ,p , t satisfied}. The f i r s t order Kuhn-Tucker conditions for (19) are: 81 (20) there ex i s t [a.,,aT] > OjJ, such that a TA = 8 T , a T [ B r B 2 , B 3 ] = 0 2 N_^ since the Mangasarian-Fromovitz constraint qualif ication conditions are -1 implied by the existence of [ A , - B 1 f - B 2 ] (see Mangasarian (1969; 172-3)). A d i f ferent ia l ly s t r i c t improvement in social welfare exists i f and only T i f (11) is satisf ied with du » 0H replaced by p du > 0. Its dual condition is given by replacing aTA > 0^ in (12) by aTA = p T . This dual condition, and T (20) imply a B 5 > 0, which i s equivalent to (10) using the argument to establish (8) from (12). Q. E. D. We now have to consider the economic implications of Theorem 3.1. The assumption that indirect tax revenue is nonzero is standard in the optimal tax l i terature . Assumption (7) is more subtle, but i t may well be just i f ied , since i t i s implied by the existence of a Diamond and Mirrlees' good (1971 ;23). More generally, (7) is the condition for the existence of Pareto improving price changes ignoring production constraints, and equivalently there exists a Hicksian composite good in net demand (or net supply) by a l l consumers. Then, lowering (raising) the price of the Hicksian good makes a l l consumers better off (see Weymark (1979)). With these assumptions, we may c a l l (8) and (10) the Generalized Pigovian Rules (GPR hereafter) or the many-person Pigovian rules for the provision of public goods. There are several interesting interpretations of these two formulae. Let us f i r s t consider the relation between (8) and (10). Obviously, the only difference between the two formulae is that we must consider any semipositive u t i l i t y weight vector for which a social welfare improving tax perturbation does not exist in the former, while we specify the weight p in the lat ter . This may be explained as follows. We f i r s t assume that indirect 82. taxes are set so that we cannot make a d i f ferent ia l ly s t r i c t Pareto improvement with dG = 0. Otherwise, the problem is t r i v i a l . However, once the indirect taxes are set in this manner, there exists at least one weight vector f (and probably many) so that the indirect taxes are set such that T increasing i u i s impossible (see Dixit (1979, 152)). Therefore we can use T the f i r s t order necessary conditions of the maximal social welfare f u with respect to indirect taxes, and hence the rest of the problem is an extension of Atkinson and Stern's (1974) result on social ly optimal provision of public good with optimal taxes to a many-consumer economy. Furthermore, i f we specify t = 6 assuming that the economy is at the 8-optimum, then we can get (10). We now discuss how to extend the Atkinson and Stern's (1974, 122) cost-benefit rules to a many-consumer economy. With taxes set optimally, i . e . , at a 8_optimum, (14) has the following interpretation. At the p-optimum, a|j, h = 1 , . . . , H , and a^  are the Lagrange multipliers from the programming (19). As d(8 T u)/dg h = a b , ajj is a net benefit of giving hth person one unit of numeraire good by raising the indirect taxes. T -1 It is also the case that d(8 u)/dx^ = a^, a.^ i s the social gam of the society to have one more unit of the numeraire good (so that indirect taxes are reduced). Therefore, (a y + a^)/a^ is a gross benefit in terms of social value of the numeraire of giving hth person one unit of numeraire good, and hence i t is Diamond's (1975;341) social marginal u t i l i t y of income a^, h = 1, T . . . . . . . , H . Therefore, we can rewrite a B c > 0 using the definit ion of a, , h = 1, D h . . . , H i t is equivalent to 83 . ( 2 1 ) Eh=1 ahWh > M C - ^ q C where the left-hand side is the social value of the public good while the right-hand side is the net social cost of the public good both measured in terms of the social value of numeraire. h T T a can also be rewritten from a A = B using (13) as (22) a h = + t T S h , h = 1, . . . ,H a 1 qu' which coincides with Diamond's (1975;341) original formula. By substituting (22) into (21), we have (23) I l A ^ + t T S h )W h > MC - t T B h=1va^ qu' qG Using Wildasin's (1984;231) Slutsky-like equation for public goods (see footnote 3), (23) may be further rewritten as (24) ( E h " 1 B h W h ) / a 1 > MC - t T ( 9 E h " 1 x h ( p + t , G , I h ) / a G ) . The left-hand side of (24) is the weighted sum of the marginal willingness to pay for public goods discounted by the shadow cost of raising one dollar by indirect taxation. The right-hand side is 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 be smaller than unity, in spite of Pigou's (1947;34) conjecture. This may also be seen from our formula (17) for 1/a.j. There are two main differences between our formula and theirs . F i r s t , the revenue effect of taxation H T h h T h L . t S R /R is a weighted sum of the individual revenue effect t S h=1 qu ' * . qu where the hth weight is the share of total taxes paid by the hth individual . T When there is only one person this expression is simply t S q u (see Atkinson and Stern (1974;123)). Second, in a many consumer context one also has distr ibutional effects to consider. Raising one dollar by taxation involves changing the distr ibution of income proportionately to the tax shares of individuals. This distributional effect is reflected in the term H h h h E n _- |8 (R /R) • If the tax i s levied on people with high social importance 8 , then this expression increases as does a^; i . e . , the social cost of raising one dollar is higher because of the increase of social inequity. These concerns are summarized in the GPR (10). To see the distributive concern in (10) more fu l ly , we define the covariance term following Feldstein (1972); H S n Wh (25) <p_ = E * ^ * - / H, G H _ 1 8 W (26) <p = E A ^ ?^ / H. R n _ 1 8 R where p, R and W are defined as p = E h f 1 P h / H , R = R/H and W = E h f 1 W h / H . As the correlation between the social importance and the distribution of 85 . marginal willingnesses to pay or of tax burdens increases, n>_ and tp increase. Substituting (25) and (26) into (10) yields: (27) - E h ^ S g u > MC - t T Ox h (p+t ,G,I h ) /3G) . This formula expl i c i t ly shows the importance of distributional concern in a <PG many-person GPR by the term — . If the distribution of the public goods R benefits are regressive or the distribution of the tax burden is progressive, the social welfare of the public good must be valued higher than the simple sum of the marginal willingnesses to pay. Before closing this section, we should mention the relation between our model and the recent work by King (1986). Our formula (21) with the h interpretation of a by (22) is obviously indentical with his formula (31) in King (1986;281) so that i t is possible to interpret (21) in his way. His result is more general than ours in the sense that he is not assuming the Pareto eff ic ient indirect taxation, but our approach is more complete than his in the sense that he i s not deriving the expl ic i t formula and interpreta-tion of the shadow price of government revenue l ike (17) of ours, for i t utterly depends on the arbitrary structure of indirect taxation in 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 intui t ive appeal when lump-sum taxes are available for financing the public good. We show in this section that the Samuelson rule is appropriate with some generalizations i f both indirect taxes and lump-sum transfers are variable, whereas i t is not appropriate i f there exits unchangeable distortions due to indirect taxation. Let us f i r s t consider the case where we can change indirect taxes and lump-sum tranfers at the same time. Theorem 3.2 (i) Suppose that the government can change t and g when G is increased, i . e . , dG > 0. If, (28) Eh!1Wh + t T B q G > MC, then there exists a s t r i c t Pareto improvement du » 0„ . n ( i i ) Suppose that t and g are chosen so that the pre-project equilibrium is a B-optimum. Then (28) is also necessary for the existence of a differen-T t i a l l y s t r i c t increase of social welfare 8 du > 0. PROOF: (i) A sufficient 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) is satisf ied with du > 0 . H By Motzkin's Theorem, this is equivalent to: T T ~T (30) there does not exist an a = [a u , a^, a ] such that: [ a v a T ] 2 oJ , a T [ B 1 ( B 2 , B 3 , B 4 ] = 0 2 N + H _*, aTA > ojj, and a T B 5 < 0. The argument used to show the equivalence of (29) and (30) is similar to that used to show the equivalence of (11) and (12) in Appendix I, so is omitted. Suppose (28) holds, but also suppose, contrary to the theorem, a . . T T . solution to (30) exists. The conditions a B, = 0TJ implies a =0... In the 4 H u H T T proof of Theorem 3.1, i t is shown that a [B^B^] = 0 2^_ 2 implies (13). We T can rewrite a B c by using a = 0TI and (13) as 5 u H (31) a T B 5 = a / ( - B p G -B q ( J) = a, ( E ^ Wh + t T B q G - MC) . T . . If a 1 > 0, then a B 5 > 0 by (28), a contradiction. T T If a^  = 0, a = 0N_^ from (13). Therefore, a A = 0^  and again we have a contradiction. ( i i ) If t and g are optimally chosen at the pre-project equilibrium, then they are a solution to the problem: T (32) 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) are: (33) there exists [a. ,a ] > Q , such that a A = 6 , ~T, , -T , , . ' T, „T a T [ B r B 2 , B 3 ( B 4 3 = 0 T 2N+H-1" Suppose (33) is sat isf ied but (28) is not. The argument following (30) then T establishes that a Bj < 0, so (30) is not satisf ied. Consequently, (28) i s also necessary for the existence of a d i f ferent ia l ly s t r i c t increase of To understand the implications of the Generalized Samuelsonian Rule (GSR hereafter) (28), le t us assume that indirect taxes and transfers are set Pareto ef f ic ient ly , so that we cannot make a d i f ferent ia l ly s t r i c t Pareto improvement with dG = 0. Pareto eff ic ient indirect taxes and transfers imply that the economy is in f i r s t best. It is well-known that the proportional commodity tax rates t = 8(p+t) for some real number 8 is f i r s t best with some appropriate lump-sum transfers. Substituting this relation into (28) and using (p+t)TB = - E h f 1 » h , (28) equals This means that the sum of marginal willingnesses to pay for the public good deflated by 8 (which i s a ratio between producer and consumer prices) must be compared with the marginal cost. Needless to say, i f 8 = 0 so that there are no indirect taxes, then the Samuelsonian rule applies. social welfare 8 du > 0 at a 8-optimum for t and g. Q. E. D. (34) (1-8) E h ° 1 W h > MC. 89 . Though proportional indirect taxes are always f i r s t best, there may exist some other f i r s t best taxes depending on the structure of the economy. For example, i f there is no room for technological subst i tutabi l i ty among with apropriate lump-sum transfers (see Diewert (1978)). It is obvious in this 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 distortions t fixed. We c a l l the resulting rule within the following proposition, a Modified Harberger-Bruce-Harris Rule (MHBHR hereafter), since i t is an application 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 is increased; i . e . , dG > 0. Then, the Modified Harberger-Bruce-Harris Rule^ is private goods so that B = 0, PP I N«N , then any indirect taxes can be p-optimal (35) where (36) T ~T e = [CM 1] = 90. Condition (35) is sufficient for the existence of a Pareto improvement du » 0 f l . ( i i ) If in the tax-distorted pre-project economy, g was chosen optimally, then (35) is also a necessary condition for a small increase in public good production to lead to a d i f ferent ia l ly s t r i c t welfare improvement. PROOF: (i) A sufficient condition for a Pareto improvement i s : (37) there exists dg and dG _> 0 such that the inequality version of (6) is satisf ied and du > 0 „ . n Condition (37) is equivalent to the following Motzkin dual condition: T ~T T (38) there does not exist [a , a 1 f a ] = a such that aTA > oj, a T B 5 < 0, a T [B VB 2,B 4] = 0N+J, [ a v a j ] > oj. Suppose (35) holds, but also suppose that, contrary to the theorem, a T T solution to (38) exists. The conditions a B^ = 0^  imply a y = 0^. Hence we T T can rewrite a B. = 0„ . as: 1 N-1 (39) a = a^p + e) using (1.A.2) and (1.A.4) (see Appendix IV). Using a = 0 , (36) and (39), 91 . a T B 5 = a, (P + E ) T [ - B p G - B q G ] . T If a 1 > 0, (35) implies a B 5 > 0, a contradiction. If a 1 = 0, a = 0 N 1 from T T (39) . With a„ = 0„ we have a A = 0 „ , and again we have a contradiction. U n H ( i i ) If g is optimally chosen at the pre-project equilibrium, g is a solution to the problem. T (40) max - . {B u ; (1) and the inequality version of (2) are u , p , t 1 , g satisfied}. The f i r s t order Kuhn-Tucker conditions for (40) are: (41) there exists [a^,a ] _> 0 N , such that a A = 8 ,a [B^,B2,B 4] = 0 N + H > Suppose (41) i s satisf ied but (35) i s not. The argument following (38) then T . . . establishes that a B c < 0, so (38) i s not sat isf ied. Consequently, (35) i s also necessary for a d i f ferent ia l ly s t r i c t increase of social welfare at a B-optimum for g. Q. E. D. The economic intui t ion behind the two Propositions in this section is as follows. Given the pre-project levels of u t i l i t y , increasing the provision of the public good permits a reduction in the consumption of private goods but requires additional inputs for the increased public good production. By appropriately offsetting the marginal benefits of the public good ( i . e . , the 92 . externality) by changing lump-sum transfers to keep consumers at their original u t i l i t y levels, i t i s only necessary to evaluate the resulting change in the quantities of the private goods by appropriate shadow prices. If the vector of tax rates t i s variable, the production price vector is the appropriate shadow price vector. See (31) behind a GSR (28) . This result is a version of the production efficiency theorem in Diamond and Mirrlees (1971). 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 in various second-best situations. We considered three cases. (1) the case where indirect tax rates can be varied; (2) the case where both lump-sum transfers and indirect tax-rates can be varied (3) the case where lump-sum transfers are varied. We showed that the the use of a GPR, a GSR and a MHBHR are suggested for cases (1), (2) and (3) respec-t ive ly . Our basic point i s that project evaluation rules must vary depending on what instruments we can change when we alter the supply of public goods. We have to note that there are severe limitations in u t i l i z i n g our cost-benefit rules; i . e . , we have ignored the preference revelation problem for public goods in measuring the marginal willingnesses to pay W for consumers. Once this d i f f i cu l ty is overcome, our rules can be implemented by using only local information observable at the pre-project equilibrium, that i s , the level of taxes, public goods, prices, incomes, and the f i r s t order deriva-tives of the ordinary demand functions and the net supply functions for 93. private goods (which depend on both prices and public goods). Note that B and B can be computed from ordinary demand functions as we pointed out in 2.4. Note further that information on W is necessary to use the MHBHR (35) as we need to compute B - from data on the ordinary demand functions. There-go fore, this rule is also vulnerable to the free-rider problem. We have shown that i t is f a i r l y easy to obtain sufficient conditions for the existence of a small Pareto improvement corresponding to an increase in public goods production, given that various taxation instruments are ava i l -able. It seems that this approach is more useful compared to the tradit ional approach which derives the f i r s t order conditions for an interior second-best welfare optimum. Our results also show that conventional cost-benefit rules for the provision of public goods are not always correct. 94. FOOTNOTES FOR CHAPTER 3 1 Our approach draws on the methodology found in the tax reform l i terature , e.g. , Guesnerie (1977), Diewert (1978), Dixit (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 restr ic t ive assumptions; e.g. , only one commodity tax rate is variable and a l l other goods are untaxed. 2 If we evaluate a possible reduction in the production of the public good dG <. 0, a l l we need is to reverse the direction of the inequalities in the cost-benefit rule. The proof is straightforward and hence may be omitted. The same comment applies to a l l cost-benefit formulae in this chapter. 3 It i s given by 9x n (p+t,G,I h )/3G = {3xh(p+t,G,uh)/9G} + \S^. T Premultiplymg by t and in summation over h, we have E h ^ t T ( 9 x h ( P + t , G , I h ) / 9 G ) = t T B q G + E h ^ t T W h S q b , which is used to derive (18). * If t = 0N so that there are no pre-existing tax distortions, then the Modified Harberger-Bruce-Harris Rule is identical with the tradi t ional Samuelsonian rule. The proof is straightforward. APPENDICES FOR CHAPTER 3 Appendix I: The derivation of (12). Motzkin's Theorem is as follows: Either Ex » 0, Fx > 0, Gx = 0 has a solution x where E is a nonvacu-1 T 2 T 3 T ous matrix, F and G are matrices and x is 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). B^dg can be dropped from (6), for dg = 0 U . Decompose A, B. (i=1,2,3,5) between A*, B?, which are the top H rows H I 1 and A**, B?*, which are the bottom N rows. Define ' l x = [du, dp , d t r dt , dG] , E [ I H ' °H«(2N+H) ] ' F = -A** o** n * * n * * n * * A1 ' B1 ' °2 ' B3 ' ti5 '2N+H where e 2 N + H i s a unit vector with unity in 2N+Hth row, and G = [-A*,B*,B*,B*,B*]. 96. Then, the primal condition of the Motzkin's Theorem is identical with (11). ^ "j* "* T T T Defining v . jd-H row vector), v 2 = [a^a ,v] where v is a scalar, v 3 = a u , the dual condition i s : there is no solution v^, a^, a, v, a y such that v 1 > 0, a 1 2 0, a 2 0, v 2 0, v T - aTA = 0, a T [ B 1 ( B 2 , B 3 ] = 0 2 N _ T , a T B 5 + v = 0, which is in turn identical with (12). Appendix I I : The derivation of (13). T T T T Subtracting a B 3 = 0N_^ from a B^  = " e have (A.1) - a.B - - aTB~~ = 0„ f. 1 PP N-1 From (1.A.4), (A. 2) B - + pTB-~ = 0„ « p^p pp N-1 where p 1 = 1. Substituting (A.2) into (A.1), we have (A.3) ( a / - aT) B p p = 0 ^ . By assumption (5), B«-~ i s nonsingular, which implies (13). 97 Appendix III: The derivation of (15) and (16) T T T Postmultiply t^  and t to a Bj = 0 and a B 3 = 0^_y and adding them together, we have (A.4) a*Xt + [ a 1 f a T ] B q q t = 0. Substituting (13) and rewriting the f i r s t term of (A.4), we get (A.5) E A a V 1 + a ,p T B t = 0. h=i u 1 qq Substituting the identity (1.A.2) into the second term of (A.5), and T T rearranging terms, we get (15). We can rewrite a A = f as (A.6) aj + [ a 1 ( a T ] B q u = . Substituting (13) into (A.6), we have m m m (A.7) a + a„p B = -y • u r qu 1 However, from (1.A.3) and (3) we get (A.S) (p + t ) T B q u = t J • Substituting (A.8) into (A.7) we have <»•» «J - 7 T - a, l „ • a , t \ u . 98 From the definit ion of B and S in (9), (A.9) is identical with (16) qu qu Appendix IV: The derivation of (39). T T a B 1 = 0 N_ 1 can be rewritten as (A.10) -a.[B - + B -] - aT[B-~ + B~~] = 0„ « 1 q«q P i P qq PP N-1 using a = 0 „ . From (1.A.4) and (1.A.2), we get U n (A.11) B - + pTB~~ = 0„ 1 P-,P PP N-1 and (A. 12) t T B - + B -+ pTB~~ = 0M qq q ^ qq N-1' respectively. Therefore, adding up (A.11) and (A.12), we have (A. 13) ^ B - + B ~ = - t T B - - pTB~~ - p T B ~ q-iq qq qq PP Substituting (A.13) into (A.10), (A. 14) a . [ t T B - + p T B ~ + pTB-~] = a T [ B ~ + B~~] 1 qq qq pp 1 qq PPJ Inverting the matrix [B~~ + B-~], and using definit ion (36), (39) qq pp follows. 99 . CHAPTER 4 INCREASING RETURNS, IMPERFECT COMPETITION AND THE MEASUREMENT OF WASTE 4-1 Introduction In the presence of increasing returns to scale in production, i t i s well-known that Pareto optimal equi l ibr ia may not be decentralized through perfect competition and moreover, imperfect competition prevails frequently. Therefore, both positive and normative analysis of resource al location with increasing returns to scale becomes an important topic in applied welfare economics. The normative problem of developing mechanisms to support Pareto optima in 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 pricing problem of public u t i l i t i e s facing a revenue constraint is discussed by the optimal pricing and taxation l i t e r a -ture beginning with Boiteux (1956). There have been numerous positive analy-ses of o l igopol is t ic markets in the vast l i terature on strategic interactions among incumbent firms or among incumbent firms and potential entrants. More-over, there is a large l i terature on Chamberlinian (1962) monopolistic compe-t i t i o n . In contrast, the measurement of waste due to imperfect competition with increasing returns to scale is a relatively less developed area, although the important seminal paper by Hotelling (1938) dealt with this topic. The aim of this chapter is 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 ca l l for extensions. F i r s t , Hotelling claimed that f i r s t best optimality is characterized by the marginal cost principle , 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) is actually not proving the optimality of marginal cost pricing. Therefore, in the l i terature on the measuremnt of deadweight loss, which includes Debreu (1954), Harberger (1964) and Diewert (1981, 1983(a), 1985(a)) in order to avoid this d i f f i cu l ty i t is assumed that a l l firms have a convex technology. Therefore, in order to compute the deadweight loss, we f i r s t characterize the optimality in nonconvex economy rigorously. Second, Hotelling's (1938) measure of waste does not seem to be correct in 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 ike to derive a measure of waste which can be evaluated using only local information on preferences and technology, so that the measure is more useful in empirical research on the measurement of waste. In this chapter, we show that these problems can be solved in a satis-factory way, at least in our simplified model. Our findings in 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 fer-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 local 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 in one production sector is 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 char-acterized by the marginal cost principle . In 4-4, we compute a second order approximation to the ADD loss measure, discuss i ts informational require-ments, and show how our measure generalizes Hotelling's original approach and other works on deadweight loss which assume technologies are convex. We also discuss various relaxations of our assumptions, and limitations 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 in the economy, where the corresponding T price vector is 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 r . . . , w M ) T » 0M. Each production unit is assumed to have a quasi-concave production function f n ( x 1 , . . . , x N , v 1 ( . . . , v M ) ; that i s , for a given level of output y , marginal rates of technical substitution between inputs are diminishing. 1 This assumption is weaker than global convexity in production; the 1 0 2 . poss ib i l i ty of increasing returns to scale is allowed for when we change the level of output in this characterization.2 We define the sector n cost function C n (p,w,y n ) as (1) C n(p,w,y ) = rain v A {pTx + wTv : f n (x ,v) > y }, n = 1 , . . . , N . C l n x>0. T ,v>0„ i r ' - 2n' ' ' ' — N M C n is identical to the expenditure function m*1 defined by (1.4), except that the u t i l i t y level is replaced by the production leve l . We assume that the regularity conditions l i s ted in Diewert (1982;554) are sat isf ied. There are H households in this economy and their demands are characterized in terms of the expenditure functions h T T h h (2) m (p,w,u. ) = min ,{p a+w b : f (a,b) > u, ,(a,b)eQ }, h = 1 , . . . , H , n a, D n h N+M where P. is a (translated) orthant of R defined as in (1.4). We assume _ that the hth household holds the vector of i n i t i a l endowments Y = ,-hT r h T . T (a ,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 h ^ 1 {m h (p,w,u h ) - p T a h - wTbh} - [ N ,{p y - C n(p,w,y )}, n=1 n n n where y = ( y 1 , . . . , y N ) T and u = ( u 1 , . . . , u H ) T . Compared with the overspending functions in previous chapters, consumers and producers are facing the same prices in (3) so that we no longer have two 103 . sets of prices as arguments. Definition ( 3 ) may be simplified by defining T T T Q = (p ,w ) as follows: (4) B(y ( Q ,u ) = E h " 1 { m h ( Q , u h ) - QTYh} - ^ { p ^ - C n ( Q , y n ) } . In the same manner as we derived the properties of an overspending function in Appendix I of chapter 1, we can easily derive the following properties for the new overspending function: (i) B is concave with respect to Q ; ( i i ) i f B is once continuously differentiable with respect to prices, V ^ B t y . Q j U ) equals the vector of excess demands; ( i i i ) B is l inearly homogeneous with respect to prices. From this , the resulting identit ies are satisf ied: ( 5 ) Q T BQQ = °N+M' and (6) Q T B Q y = V y B ( y , Q , u ) T where B Q Q = V Q 2 B ( y , Q , u ) and B Q y = V Q 2 B(y ,Q,u) . Note that - V y B(y ,Q , u ) is a vector whose i th component is the difference between the price and marginal cost of the i th good. Now using the above relations, we characterize the general equilibrium 1 1 1 1 1T 1T T (y , Q ,u ) where Q = (p ,w ) as follows: (7) h, f t1 „ 1 . ^IT^h , r N hn f 1 1 „ n , - 1 1., , , . m (Q ,u h) = Q Y + E n = 1 a {P n y n -C (Q ,y n)} + g h , h = 1 , . . . , H , 104 . (8) (9) N+M hn where a is the share of the nth firm held by the hth individual and H hn E j ^ a = 1, for n = 1 , . . . , N . The number g^, h = 1 , . . . , H , shows the net T lump-sum transfer given to the hth individual and t = (t^ , . . . , t^ ) where t R is 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 equations in (7) are the budget constraints of the H individuals. The equations in (8) state that the difference between the price of the i th good and i t s marginal cost is 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 their marginal costs in order to attain nonnegative prof i ts . This does not necessarily mean that the monopolistic markup is fixed for monopolists. We just define t ex-post at the equilibrium as the difference between consumer prices and marginal costs. Noting that v^B equals the vector of excess demands, the equations in (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 T , 8 T ) T > 0 and each consumer's u t i l i t y level u}, 105. h = 1 , . . . , H , in the imperfectly competitive equilibrium, and consider the following primal planning problem: nn\ 0 , . . . - H h , - N n , . , r , H - h ( 1 0 ) r H r a a x h , h n n { r : ( 1 ) Eh=1 a + En=1X + a r * y + E h = 1 a ' a ,b ,y n.x ,v Eh=1 En=1V P r - Eh=1 ' ( i i i ) f h ( a h,b h) 2 UjJ, (a h,b h)eQ h h=1, . . . ,H, (iv) f n ( x n , v n ) 2 y n , n = 1, . . . ,N}. The solution to (10) defines the ADD measure of waste L , ^ ^ = . Problem ADD (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 , and satisfying the materials balance and technology constraints. We assume that a f in i te maximum exists for (10). We can also derive a dual expression to (10) as follows. F i r s t , let us T fix y = (y.|, . . . ,Yjj) . From the definit ion of quasi-concavity, the sets f n ( x n , v n ) 2 Y n (n = 1, . . . ,N) are convex sets belonging to R^ + M . Then, the remaining programming problem becomes a concave programming so that we can rewrite (10) using the Uzawa (1958)-Karlin (1959) Saddle Point Theorem3 as 0 T 1 (11) r = max [max min (r(1-Q X ) - B(y,Q,u )}] N N+M using definitions (1) (2) and (4), where Q is the vector of Lagrangean multipliers 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 in reverse as 106 . (12) - m a x n > n W Y I Q ' " 1 ) **.t. Q TX > 1} N+M For the given level of y, the solution of the max-min problem within (11) becomes r(y) and Q(y) which are functions of y. Then (11) can also be written as 13) r ° = max {r(y)(1-Q(y)TX) - 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 is 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 in relation 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. For this purpose, we have to strengthen our assumptions as follows: (i) ( y ° , r 0 , Q 0 ) solves (11) with y° » 0M, Q° » 0M^M so that the f i r s t order N N+M conditions for (11) hold with equal i ty; 4 ( i i ) the expenditure functions m*\ h = 1 , . . . , H , are twice continuously differentiable with respect to Q at 0 1 n (Q ' u n ^ ; ( i i i ) the cost functions C , n = 1 , . . . , N , are twice continuously differentiable at (Q^,y^); (iv) Samuelson's (1947) strong second order conditions hold for the two problems (12) and (13) when the inequality constraint in (12) is 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 erent iab i l i ty 107 . assumptions, which are natural for a local analysis such as ours. Condition (iv) is an assumption which guarantees that the maximum of the planning problem (10) is local ly unique. Our regularity conditions on (12) imply the bordered Hessian (14) -B° -x QQ' * T - A , 0 is positive definite where = V^B(y® ,Q® ) and the superscript 0 means that the derivatives are evaluated at z = 0. By defining (15) A 0 , -B 0 yy and (16) B ° = [ - B y ° , 0 N ] where the superscript 0 means that B ^ and B ^ are evaluated at the optimum, yy yQ our condition in (iv) is equivalent to the following condition: 0 0 0 -1 OT (17) A - B (C ) B is negative definite. The condition (17) is much weaker than assuming marginal costs are increasing, which requires A 0 to be negative definite, for C° i s positive definite by (14). By merely requiring condition ( iv) , we are admitting the 108 . poss ib i l i ty 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 interior solution exists to (11). The f i r s t order conditions are given by: 18) - v y B ( y ° , Q ° , u 1 ) = 0 N , (19) -Ar° - V Q B ( y ° , 0 ° , u 1 ) = 0 N + M , (20) 1 - Q 0 T A = 0, where (18) is a marginal cost pricing principle 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) is a normalization rule for the optimal prices. 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 y B(y(z) ,Q(z) ,u 1 ) = tz , (22) -V Q B(y(z) ,Q(z),u 1 ) - Ar(z) = 0 N + M , (23) 1 - Q(z)TA = 0. 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. In contrast, i f we define ( y d ) , Q ( D , r(1)) = ( y 1 , Q 1 , 0 ) , then (21) and (22) coincide with (8) and (9) respectively when z = 1. In this case, (7) is also satisf ied for an appropriate choice of transfers g^, h = 1 , . . . , H . From condition (23) at z = 1, we also assume that the market prices satisfy the normalization, (24) 1 = Q 1 T X , by choosing the scale of X appropriately. Thus we can conclude that the z-equilibrium (21) - (23) maps the optimal equilibrium into the imperfectly competitive equilibrium as z is adjusted from zero to one. Equation (21) that maps the marginal cost pricing condition (18) into the monopolistic markup equilibrium condition (8) may seem unnatural because markups are decreasing l inearly , but the change in t may be nonlinear depending on the behaviour of monopolists. Even in the case of tax-distor-tions, however, i t is possible to choose some nonlinear path of the change of tax rates as the equilibrium is adjusted and the resulting magnitude of waste depends on this choice of path. (This problem is also related to the tradit ional problem of path independence in consumers' surplus analysis.) As i t is d i f f i c u l t to overcome this arbitrariness within our framework, we have just assumed that there is a uniform reduction of monopoly distortions. The main theorem in this section is as follows: 1 10. Theorem 1: A second order approximation to the ADD measure of waste (10) is given by (25) V ( A 0 - B ° ( C ° ) - 1 B°V1t > 0. PROOF: Differentiate (21) - (23) with respect to z and we have (26) A Z , B Z y' (z) t Q'iz) = °N+M B Z T , C Z r' (z) 0 where A z , B z , C z are the matrices A , B and C defined by (15), (16) and (14) evaluated at z, rather than 0. T T Premultiplying (26) by [0N,Q(z) ,0], we have (27) -Q(z) T B Q Z y'(z) - Q(z) T B Q Z Q'(z) - Q(z) T Ar'(z) = 0. Substituting (5), (6), and (23), and then (21) into (27), we obtain (28) r'(z) = z t T y ' ( z ) . Noting that r(1) = 0, by using a Taylor series expansion the ADD measure of waste L A n n = r <^ = r (0) ~ r d ) can be approximated by (29) r ° - r 1 = r ° - { r ° + r'(0) + hr"(0)} = - r'(0) - Jjr"(0) However, from (28), r'(0) = 0 and r"(0) = t T y' (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. Substituting this expression into (30), the result (25) follows. The inequality in (25) follows from (17). Q. E. D. The formula (25) gives a general formula of deadweight loss applicable to either a convex or nonconvex economy. This formula is identical to the Debreu (1954)-Diewert (1985a) approximate deadweight loss formula when the technologies are convex. However, the converse is not true, since the optimal shadow price (or in tr ins i c price to use Debreu's (1951, 1954) term) may not exist with increasing returns to scale. This problem is overcome by our two-stage optimization procedure (11) for the characterization of the optimum, an approach which was suggested by Arrow and Hurwicz (1960) and Guesnerie (1975). Our resulting approximate loss formula (25) is calculated not from the derivatives of supply functions, but from the derivatives of restricted factor demand functions and marginal cost functions evaluated at the optimal level of output. As our work is preceded by Hotelling (1938), i t i s important to discuss his work in relation to ours. Hotelling's contributions in this paper are known to be that (i) he showed the optimality of the marginal cost pricing principle 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 Hotelling's proof i s not a val id one. In section 3 we gave a rigorous proof based on programming that the marginal cost pricing principle is necessary for opt i -112. mality i f technologies are quasi-concave. As Arrow and Hurwicz (1960) showed, this condition i s not necessary for general nonconvex technology. Suppose that we alternatively consider Guesnerie's (1975) type 3 firm; that is a firm's technology i s convex i f some input is given. By applying results in Guesnerie ( 1975;12-13), i t is straightforward to show that optimality is characterized by the competitive maximization of 'restricted' profi t given the level of the input which causes the increasing returns to scale, and by the equality of the marginal value product with the factor price. (See also Aoki (1971).)6 For the second point, Hotelling (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 valid despite his own conjecture. (See Tsuneki (1987b)). Furthermore, even i f we interpret his measure as in (30), i t i s not useful without knowing how to compute y'(0) using (26) via the Implicit Function Theorem. 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 in either product or factor markets, we can derive a deadweight loss measure using the methodology employed above. However, the resulting approximate measure is different from (25). What matters now are derivatives of restricted profit functions, given the input that causes the nonconvexity, instead of the derivatives of cost functions. The drawback of our approach is that these derivatives are not observ-able at the distorted observed equilibrium. It is somewhat overcome by the following corollary: 113. Corollary 1.1: The approximate ADD measure T 1 1 1 - 1 1 T - 1 (31) -Jjt ( A - B (C 1) B ) t is also accurate for quadratic functions as (25). PROOF: According to Diewert's (1976;118) Quadratic Approximation Lemma, both -(r'(0) + Jjr'tO)) and -^(r'(0) + r'(1)) give the exact value of r(0) - r(1) i f r is quadratic. The former was adopted to derive (25). Now using the latter approximation and using (28) we have 0 1 T (32) r - r = - ^ t y ' ( 1 ) . Evaluating (26) at z = 1, computing y'(1) by inverting the left-hand side matrix and substituting i t into (32), we get (31). Q. E. D. The remarkable property of (31) is that we can compute the deadweight loss of the economy from the local derivatives of demand and supply (cost) functions evaluated at the observed equilibrium. One important consequence of this observation, is that (31) can be computed using f lexible functional forms for u t i l i t y and production functions, so that we need not assume restr ict ive functional forms to calculate the global optimum point, as is usual in the numerical general equilibrium l i terature . To derive our approximate loss formulae (25) and (31), we maintained several restr ict ive assumptions. 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 principle , and we already discussed how to extend our approach when we dropped the assumption. We have assumed that each industry is monopolized. It is easy to extend the result to the case of an ol igopol is t ic industry i f we know the mark-up rates of firms and that the number of firms within one industry is fixed for a l l industries. However, i t is d i f f i c u l t to introduce entry-exit behaviour, since the f i r s t order social optimality and market equilibrium conditions for incumbents and entrants are characterized by inequalities rather than equal-i t i e s , for the number of firms changes discontinuously as equi l ibr ia are adjusted from the observed equilibrium to the optimum as is shown in the l imi t pricing l i terature . Therefore, i t is 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 is a Chamberlinian (1962) monopolistic competition with each product produced by homogeneous producers with respect to market shares, product qual i l ty and technology. Suppose also that the number of firms is continuous. Then, the long-run equilibrium is characterized by the zero-profit conditions of firms, i . e . equalities where the number of firms is also endogenous, and Chamberlinian excess capacities cause deadweight loss. The optimality conditions are characterized by the marginal cost pricing principle and the optimum number of firms is determined at the point where the marginal cost equals average cost. However, this model may be incomplete as a monopolistic competition model, since product diversity is exogenous in our model. To make i t endogenous, we must work with much more simplified models, as adopted in Spence (1976), and Dixit and S t i g l i t z (1977). 115. 4-5. Conclusion This chapter has reconsidered the methodology for the measurement of waste due to imperfect competition in the presence of increasing returns to scale. We noted Hotelling's (1938) confusion about the optimality of the marginal cost principle and the derivation of his deadweight loss formula and rederived his formula as (30). The drawback to Hotelling's measure (30) is that i t cannot be computed without finding the optimum beforehand. This drawback was corrected by our measure (25) and (31) where we required only local information in order to measure the deadweight loss. In particular, for the loss measure defined by (31), only information observable at the distorted equilibrium is 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. The production poss ib i l -i ty set OA exhibits increasing returns to scale, so that a competitive equilibrium cannot exist. However, an imperfectly competitive equilibrium M = (y^' v[Y|) c a n exist where the marginal rate of substitution between the good and labour in consumption is different from the rate of substitution in production. The ADD optimum point D = (YQI VQ) is 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 is characterized by the equality of the marginal rates of substitution in consumption (or marginal benefit of the good) and the marginal rates of substitution in production (or marginal cost of the good). Fig . 11 shows these two curves as MB and MC. The true amount of deadweight loss is shown by the curvilinear triangle ABC while the approximate measure (25) is shown by the triangle ABC and (31) is shown by A B C ' . The proof that ABC, A B C , 116. ABC" really correspond to (11), (25), (31) for this simple economy is analogous to the derivation and construction of (1.45), (1.48), (1.49) in Chapter 1. Given the limitations and assumptions l i s ted within the chapter, we can apply our generalized Hotelling's measure to various models of imperfect competition and to publicly 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 n ( x ^ , . . . , x N , v 1 , . . . , v M ) , we assume that 3f n/3x = 0. Therefore, the cost function C n dual to f n has the derivative ' n ' 3C n/9p = 0. ' r n 2 For example, the increasing returns to scale technology obtained by combining a convex production poss ib i l i t i es set with a large fixed cost can be dealt with within our framework. (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 qualif ication condition holds; that i s , we assume that a feasible solution for (10) exists that s t r i c t l y satisf ies the f i r s t N+M inequality constraints. * With increasing returns to scale, a local optimum that satisf ies the f i r s t order conditions may not be globally optimal. We assume that , 0 0 0 0> . . . . (r ,y ,p ,w ) is a global optimum. 5 Increasing returns to scale is usually defined as a more than proportionate increase of output when a l l the inputs are proportionately increased. Baumol, Panzar and Wi l l ig (1982;18-21) propose a weaker notion of increasing returns to scale, i . e . , decreasing average cost, and showed that i t is implied by decreasing marginal cost. 6 Arrow and Hurwicz (1960) and Arnott and Harris (1976) gave examples where cost minimization and the marginal cost principle result in productive inefficiency in a type-3 economy. 7 According to the recent study of contestable markets by Baumol, Panzar and Wi l l ig (1982), these strategic aspects are immaterial when the fixed cost is not sunk. Since a natural monopoly must set the price equal to i t s 118. average cost for a sustainable equilibrium, the mark-up rates t equal the difference between the average cost and marginal cost, so that our approach is applicable. 1 1 9 . Fig . 10 The ADD Measure with Increasing Returns to Scale 120. Marginal Benefit of y U VM yo y Fig . 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 this chapter, we are interested in evaluating the net benefit of introducing a new technology in the presence of pre-existing distortions. 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, called a project. Both consumers and firms adjust to this change and the economy moves to a post-project equilibrium. Project evalua-tion means to determine whether the project increased or decreased social welfare. The evaluation of a small project when there i s perfect competition with tax distortions was surveyed by Diewert (1983b) and we applied his approach to evaluate the benefit of public goods when there are tax distor-tions in Chapter 3. Therefore, a natural way to proceed seems to be to extend this approach to the evaluation of a small project in an imperfectly competitive economy. However, this approach may not be as promising as i t looks at f i r s t . Commenting on Davis and Whinston's (1965) use of a perceived demand curve in the second best theory of imperfect competition, Negishi (1967) pointed out that the second best policy of a public firm is indeter-minate unless the perceived demand curves of the imperfect competitors are known. Therefore, we have to follow a different avenue. In project evaluation, i t is often the case that a new project has effects which are too large to be approximated by di f ferent ia 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 tech-nology due to a large fixed cost. Negishi (1962) studied the welfare impl i -cations of the entry of a new firm which is either a perfect competitor but has a large fixed cost technology or is the only firm which deviates from perfect competition. 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. However, Harris (1978) kept Negishi's (1962) assumptions about perfect competition and a convex technol-ogy with a fixed cost. The purpose of this chapter is 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 is imperfect competition. Our results in this chapter may be summarized as follows. F i r s t , the Harris and Negishi results hold even i f the assumption of a convex technology with a large fixed cost is replaced by general nonconvex technology, provided i t is assumed that pre and post-project equi l ibr ia exist. Second, some of the extensions of Negishi's (1962) results by Harris (1978) depend on an impl ic i t weakening of the cr i ter ion for welfare improvement made by Harris compared with Negishi's original welfare cr i ter ion . Thirdly, but most importantly, most of their rules can be applied in imperfectly competitive economies generally, again as long as pre and post-project equi l ibr ia are assumed to exist. In the next section, we discuss welfare c r i t e r i a for cost-benefit analy-s is . We discuss some confusion which exists concerning the use of the 123 . compensation principle and show that the cr i ter ion adopted by Negishi (1962) for the acceptance of a project is more s t r i c t than that by Harris (1978). We cannot judge which cr i ter ion is superior to the other. However, when we develop project evaluation rules, we simply have to be expl ic i t on which cr i ter ion each rule is based. After presenting the model in section 3, in section 4 we reconsider the rules l i s ted by Harris (1978), which include Negishi's (1962) original 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. Section 5 concludes. 5-2. Compensation Cr i t er ia for Cost-Benefit Analysis Reconsidered In chapter 1, we analyzed the properties of the ADD measure and the HB measure as a social welfare function. It was shown that they are consistent with the Pareto quasiordering, but they are not typical ly welfarist . We also suggested the use of deadweight loss measures for policy evaluation; alternative policies are ranked by the associated level of deadweight loss. The crucial assumption for using this procedure was that production poss ib i l i t i e s sets remain unchanged by these pol ic ies . Therefore, this approach works for changes in tax or regulation policies with a fixed technology. We should note, however, that i t is impossible to compare the values of these measures when production poss ib i l i t i es sets are changed by the introduction of new projects, since these measures do not satisfy welfarist assumptions. For example, the definit ion of Pareto optimal allocations takes as given the production poss ib i l i t i es sets. Consider a Pareto optimal allocation a. Suppose there is a change in technology which 124. permits the attainment of a new allocation a which is preferred by a l l consumers to a. Using the ADD measure, a is always measured to exhibit at least as much deadweight loss as a, as a l l Pareto optimal allocations have ADD measures of zero. As a consequence, i t i s inappropriate to use this deadweight loss measure when the technology is not fixed. The optimal reference equilibrium on which the HB measure is 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 alternative cr i ter ion for social 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 is Bergson Samuelson social welfare function and the other is Hicks-Kaldor compensation principle .1 In either cr i ter ion , we show in this chapter that our project evaluation c r i t e r i a can be related to aggregate quantities of individual consumption bundles. One natural way to proceed is to suppose that there exists a Pareto-inclusive social welfare function by assuming that either there exists an omniscient planner who distributes income optimally at any point (see Samuelson (1956)) or consumers' preferences satisfy Gorman's (1953) res tr ic -tion of quasi-homotheticity. It is obvious that a Bergson-Samuelsonian social welfare function can serve as a welfare indicator to evaluate the states corresponding to different technologies in a consistent manner. Unfortunately, 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 social welfare function i s . Also the assump-tion of quasihomothetic preferences is empirically res tr ic t ive . An alternative method tradi t ional ly adopted for the evaluation of pro-jects is the Kaldor (1939)-Hicks (1939, 1940) compensation principle , 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 principle and we have to be careful in distinguishing their d i f -ferent meanings. We suppose that two states of the economy (z,Y^,x^) and 1 1 (z,Y ,x ) are compared where z is a vector of i n i t i a l resources which is fixed, Y 1 i s an aggregate production poss ib i l i t i es set in state i = 0,1, and x 1 i s an aggregate consumption bundle in state i = 0,1. We also define the T u t i l i t y levels for the H households u = (u^, . . . ,u H ) and the Scitovsky sets SCu1) for i = 0,1 corresponding to the u t i l i t y functions f h ; the u t i l i t y functions are assumed to be continuous from above, quasiconcave, and nonsatiated. The Scitovsky set for period i is defined as S(u x) = {x : E n f 1 x h 1 x, f h (x h ) ) u j , h = 1, . . . ,H}, where u 1 = (u* , . . . , u^) T . Our assumptions ensure that S(u 1 ) is convex (see Scitovsky (1941-2(b)). Now we can define the four types of compensation test. (V (2) (3) (4) 1 R £ S p 0 (read state 1 is preferred to state 0 by the Kaldor strong principle) i f f x^  e S(u^). 1 R K w p 0 (read state 1 i s preferred to state 0 by the Kaldor weak 1 . 0 principle) i f f z + Y intersects with S(u ). 1 R U C T , 0 (read state 1 is preferred to state 0 by the Hicks strong nor principle) i f f x ° f. S(u 1 ) . 1 RH Wp 0 (read state 1 is preferred to state 0 by the Hicks weak principle) i f f z + Y° and S(u 1) are d is jo int . 1 2 6 . What is called Scitovsky's (1941-2(a)) double cr i ter ion is that 0 is prefer-red to 1 i f f both the Hicks and Kaldor c r i t e r i a are met in 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 H S p 0, but not vice versa. Negishi (1962;88) wrote that i f (z+Y1) and S (u° ) are dis jo int , i . e . , 0 Rjj^ p 1( then state 1 is not recommended. In page 89, he wrote i f z + Y^ and S(u 1) are disjoint , i . e . , 1 R H w p 0 then state 1 is preferred to 0. Therefore, we may conclude that Negishi adopted the Hicks weak compensation cr i ter ion (4) as his project acceptance cr i ter ion . In contrast, Harris (1978;412) suggested that i f x 1 £ S (u° ) then 0 is preferred to 1 and that on p. 414, i f x^ i S(u^) then 1 is preferred to 0. Therefore, he ut i l i zed Hicks strong compensation cr i ter ion (3). By Proposition 2, we deduce that i f project 1 is accepted by Negishi's cr i ter ion, 1 is also accepted by Harris' cr i ter ion , but not vice versa. In economic terms, state 1 meets Harris' acceptance cr i ter ion 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 cr i ter ion i f everyone cannot be made as well off as u^  even when the best production plans and income distr ibution 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 profits given the prices. If consumers' preferences are quasiconcave and quasihomothetic3 and state 0 is perfectly competitive, then 1 R„cr, 0 implies 1 R m 7 n 0, i . e . , HSP and HWP are equivalent. PROOF: From definit ion (3), x ° £ S(u 1 ) . If state 0 i s perfectly competitive, there exists p 2 0^  such that p x 2 P x for a l l x E S(u ) and p 0 T x ° 2 P ° T y for a l l y e Y° + z. (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 1) is a subset of S ( u ° ) , but x 1 i s not on the boundary of S ( u ° ) . Therefore,p 0 T x > p 0 T x ° > p 0 T y for a l l y e Y° + z and for a l l x e S(u 1 ) . Therefore, S(u 1) and Y° + z are disjoint and from definit ion (4) 1 R H W p 0 follows. Q. E. D. Therefore, for perfectly competitive economies in 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 cr i ter ion is stronger than that of Harris, so that we have to c lar i fy whether a project acceptance rule is based on the Negishi or Harris cr i t er ion . 5-3. The Model We now sketch the model of Harris (1978). There are N goods n = 1 , . . .N , and H consumers, h = 1 , . . . , H . Consumers' preferences are represented by u t i l i t y functions f (x h) where x he Q , a transformed orthant R + . 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. We do not assume anything about the properties of the technologies Y and G except closedness (which is harmless from an empirical point of view). We assume the existence of equi l ibr ia as follows. N A before-project equilibrium is defined as an H + 2 tuple in R , r o 0 , 0 0,, . . (P ,Y , ( x 1 , . . . , x H ) } such that (5) f h ( x £ ) 2 f h ( x n ) f o r a 1 1 x n e (budget constraint for h under p 0}, for h = 1 , . . . , H , (6) y° e Y 0 r H 0 0 ^ (7) x = L h = 1 x h = y + z. In the same way, we define an after-project equilibrium to be an H + 3 tuple in R N , {p1 ,g 1 , y 1 , (x!j, . . . ,x^)} such that (8) f h ^ x h ^ - f ^( x n ) f o r a 1 1 X h e { k u d 9 e t constraint for h under p1} for h = 1 , . . . , H , (9) y 1 e Y 1 (10) g 1 e G 1 129 These two definitions seem to incorporate the minimum requirements for an imperfectly competitive equi l ibria studied by Negishi (1961-2), Arrow and Hahn (1971, Ch. 6), and Roberts and Sonnenschein (1977); i . e . , (i) price taking behaviour of consumers, ( i i ) f eas ib i l i ty of equilibrium production, and ( i i i ) equality of demand and supply. The main result in this chapter is that the existence of before and after-project equi l ibr ia is sufficient for the va l id i ty of most of the project evaluation rules developed by Negishi and Harris. 5-4. Project Evaluation Rules Using the Hicksian strong compensation cr i ter ion , Harris' two main project evaluation rules (using his numbering of the rules) can be restated within our framework as follows. Rule 2: A sufficient condition to reject a proposed project is that the project have a net value at before-project prices which is less than the change in the profits on a l l other production act iv i t i es evaluated at OT 1 OT 0 1 before-project prices; i . e . , the rejection cr i ter ion is p g < p (y - y ). Rule 4: A sufficient condition to accept a project is that minus the profits (or minus the net value) of the project at post-project prices be less than the change in profits in the rest of the economy at after-project IT 1 1T 1 0 prices; i . e . , the acceptance cr i ter ion is -p g < p (y - 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 is equivalent to p x > p x , and Rule 4 is 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 1 £ S (u° ) and Rule 4 implies 0 1 . . . . . x i S(u ). Now applying the Hicks strong compensation principle (3), Rule 2 gives a sufficient condition for state 0 to be preferred to 1, and Rule 4 gives a sufficient condition for state 1 to be preferred to 0. Q. E. D. We proved Harris's two main rules without making any assumptions concerning market structure. We assumed only that markets clear and consumers are price takers. In particular, we did not assume that either the private production sector or the government optimizes. Therefore, Harris' rules have a very broad appl icabi l i ty . From Proposition 3, Rule 2 is val id for Hicks' weak compensation principle i f the Gorman aggregation conditions for consumers' preferences are met and the after-project equilibrium is perfectly competitive. Similarly, Rule 4 is val id for Hicks' weak compensation principle i f the aggregation of consumers' preferences conditions are met and the before-project equilibrium is perfectly competitive.'' This i s the reason why Negishi (1962; 91) assumed that Gorman's preference restrictions were met in his demonstration of the va l id i ty of Rule 4. Harris (1978) restated Negishi's (1962) main two rules, Rule 1 and Rule 3 in Harris' numbering, as follows: Referring to Hicks weak compensating principle 131 . Rule 1: A sufficient condition to reject a proposed project is that i t is impossible for the project to show a nonnegative net value at before-project . . . . OT 1 equilibrium prices; i . e . , the rejection cr i ter ion is p g < 0. Rule 3. A sufficient condition to accept a project is that the project show positive profits at post-project prices; i . e . , the acceptance cri ter ion is p g > 0. In general, unless the economy is competitive, except for the new firm introduced by the government, Rule 1 and Rule 3 are inval id (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 . Obviously, other implications cannot be val id in general. 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 principle as his welfare cr i ter ion, which is more indeterminate than the Hicks strong principle adopted by Harris, but chiefly i t is because the pro f i tab i l i ty cr i ter ion is a less exact estimate of the social welfare change than the index number approach used in Rule 2 and Rule 4. The drawback of Rule 2 and Rule 4 seems to be their 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 poss ib i l i t i e s set of the public agency. In Rule 3, after-project prices must be somehow predicted. However, in Rules 2 and 4, after-project output levels of the rest of the economy are also required, and this is d i f f i c u l t to obtain ex ante. In summary, Rules 1 and 3 show the f irst-best significance of a new technology in terms of i t s pro f i tab i l i ty . They are almost always less exact than Rule 2 and Rule 4, and cannot t e l l us anything in second best conditions when we do not have perfect competition. Rule 2 and Rule 4 are val id in both f i r s t best and second best conditions, and in a second best, i t evaluates the improvement of technical efficiency and market efficiency at the same time. 5-5. Conclusion Harris discussed thirteen project evaluation rules in his paper. Excluding Rule 1 and Rule 3, which we have discussed, and Rule 8, which is analogous to Rule 1 in the tax-distorted economy, a l l of his rules are effec-tive for non-competitive market structures, because the proofs of a l l of them are similar to the proofs of Rule 2 and Rule 4, or they are contrapositives of other rules. In particular, the satisfaction of Samuelsonian conditions i s not necessary to prove Harris' Rules 11, 12 and 13, which give cost-bene-f i t rules for supplying a public input. Although prof i tab i l i ty of a new project has a normative meaning only in f irst-best 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 in a second best economy. 133 . FOOTNOTES FOR CHAPTER 5 1 Note also that, in general, the relationship between the compensation principle and the sum of equivalent or compensating variations is ambiguous. See Boadway (1974), Smith and Stevens (1975), Foster (1976) and Boadway (1976). 2 In chapter 3, we considered a sufficient condition for the existence of a Pareto improvement. This discussion may be related to the compensation principle . See Bruce and Harris (1982). 3 Quasi-homotheticity is satisf ied i f Engel curves are straight l ines and they are paral le l for a l l consumers. See Gorman (1953). Alternatively, we can think of the case where income distribution i s always optimized with respect to a Bergson-Samuelsonian social welfare function. In this case, Bergson's social indifference surfaces do not intersect and convex to the origin i f u t i l i t y functions are concave and a social welfare function is quasi-concave (see Gorman (1959) and Negishi (1963)). Replacing the Scitovsky set with the better set of Bergson's indifference surface, the rest of the discussion goes through. When we mention Gorman's restrict ions on preferences, we can also allow for this alternative case. * Harris (1978;410) pointed out that his welfare cr i ter ion is consistent with an ordering based on the Bergson-Samuelsonian social welfare function, i f i t exists. Referring to Proposition 3, Negishi's welfare cr i ter ion may not be consistent with such a social welfare function, i f the assumption of perfect competition is dropped. 5 More exactly, Rule 1 applies even i f the after-project equilibrium is imperfectly competitive and Rule 3 applies even i f the before-project 134. equilibrium is imperfectly competitive, provided that the equi l ibria exist. This is obvious from the proofs of these rules by Negishi (1962). 135 . REFERENCES A l l a i s , M. (1943). A la recherche d'une disc ipl ine economique, Tom I, Paris: Imprimerie Nationale. A l l a i s , M. (1977). "Theories of General Economic Equilibrium and Maximum Efficiency," pp. 129-201 in Equilibrium and Disequilibrium in Economic  Theory, E . Schwodiauer ed. , Dordrecht: D. Reidel. Aoki, M. (1971). "Investment Planning Process for an Economy with Increas-ing Returns." Review of Economic Studies 38, 273-80. Arnott, R., and R. G. Harris. (1976). "Increasing Returns and Decentraliza-tion Rules for the Public Sector." Queen's University Discussion Paper No. 22. Arrow, K. J . (1951). "An Extension of the Basic Theorems of Classical Wel-fare Economics." pp. 507-532 in Proceedings of the Second Berkeley Sym- posium on Mathematical Stat ist ics and Probability. J . Neyman, ed. Berkeley: University of Cal i fornia Press. Arrow, K. J . , and F. H. Hahn. (1971). General Competitive Analysis. San Francisco: Holden Day. Arrow, K. J . , and L. Hurwicz. (1960). "Decentralization and Computation in Resource Allocation." pp. 34-104 in Essays in Economics and Economet- r ics . R. Pfouts, ed. University of North Carolina Press. Atkinson, A. B . , and N. H. Stern. (1974). "Pigou, Taxation and Public Goods." Review of Economic Studies, 41, 119-128. Atkinson, A. B . , and J . E. S t i g l i t z . (1980). Lectures on Public Economics. New York: McGraw-Hill. Auerbach, A. J . (1985). "The Theory of Excess Burden and Optimal Taxation." pp. 61-127 in Handbook of Public Economics Vol . 1. A. J . Auerbach, and M. S. Feldstein, eds. Amsterdam, North-Holland. Baumol, W. J . , J . C. Panzar, and R. D. Wi l l i g . (1982). Contestable Markets  and the Theory of Industry Structure. New York: Harcourt Brace and Jovanovich. Bhagwati, J . N . , R. A. Brecher, and T. Hatta (1983). "The Generalized Theory of Transfer and Welfare: Bi la tera l Transfers in a Mult i lateral World." American Economic Review 73, 606-618. Blackorby, C , and D. Donaldson. (1985). "Consumers' Surpluses and Consist-ent Cost-Benefit Tests." Social Choice and Welfare 1, 251-262. Blackorby, C , and D. Donaldson. (1986). "Money-Metric U t i l i t y : A Harmless Normalization?" U.B.C. discussion paper, 86-9. 136 . Blackorby, C , D. Primont, and R. R. Russell . (1978). Duality, Separabil- i t y and Functional Structure: Theory and Economic Applications. New York: Elsevier North-Holland. Boadway, R. W. (1974). "The Welfare Foundations of Cost-Benefit Analysis." Economic Journal 84, 426-439. Boadway, R. W. (1976). "The Welfare Foundations of Cost-Benefit Analysis—A Reply." Economic Journal 86, 359-361. Boiteux, M. (1951). "Le 'revenue distruable' et les pertes economiques." Econometrica 19, 112-133. Boiteux, M. (1956). "Sur la gestion des monopoles publics astreints a l 'equi l ibre budgetaire." Econometrica 24, 22-40. (Translation in , Journal of Economic Theory 3, 219-240.) Brown, D. J . , and G. Heal. (1980). "Two-Part Tar i f f s , Marginal Cost Pricing and Increasing Returns in a General Equilibrium Model." Journal of  Public Economics 13, 25-49. Bruce, N . , and R. G. Harris. (1982). "Cost-Benefit Cr i ter ia and the Compen-sation Principle in Evaluating Small Projects." Journal of P o l i t i c a l  Economy 90, 755-775. Chamberlin, E. H. (1962). The Theory of Monopolistic Competition, 8th edi-t ion. Harvard University Press. Currie, J . M. , J . A. Murphy, and A. Schmitz. (1971). "The Concept of Economic Surplus and Its Use in Economic Analysis." Economic Journal 81, 741-799. Davis, 0. A . , and A. B. Whinston. (1967). "Piecemeal Policy in the Theory of Second Best." Review of Economic Studies 34, 323-331. Debreu, G. (1951). "The Coefficient of Resource Ut i l i za t ion ." Economet- r i ca 19, 273-291. Debreu, G. (1954). "A Classical Tax-Subsidy Problem." Econometrica 22, 14-22. Debreu, G. (1959). Theory of Value. New York: John Wiley. Diamond, P. A. (1975). "A Many-Person Ramsey Tax Rule." Journal of Public  Economics 4, 227-244. Diamond, P. A . , and D. L. McFadden. (1974). "Some Uses of the Expenditure Function in Public Finance." Journal of Public Economics 3, 3-21. Diamond, P. A . , and J . A. Mirrlees. (1971). "Optimal Taxation and Public Production I: Production Efficiency and II: Tax Rules." American Eco- nomic Review 61, 8-27 and 261-278. 137 Diewert, W. E. (1976). "Exact and Superlative Index Numbers." Journal of  Econometrics 4, 115-145. Diewert, W. E. (1978). "Optimal Tax Perturbations." Journal of Public Eco- nomics 10, 139-177. Diewert, W. E. (1981). "The Measurement of Deadweight Loss Revisited." Econometrica 49, 1225-1244. Diewert, W. E. (1982). "Duality Approaches to Microeconomic Theory." pp. 535-599 in Handbook of Mathematical Economics Vol . 2, K. J . Arrow and M. D. Intr i l igator , eds. Amsterdam: North-Holland. Diewert, W. E. (1983a). "The Measurement of Waste Within the Production Sector of an Open Economy." Scandinavian Journal of Economics 85, 159-179. Diewert, W. E. (1983b). "Cost-Benefit Analysis and Project Evaluation: A Comparison of Alternative Approaches." Journal of Public Economics 22, 265-302. Diewert, W. E. (1984a). "The Measurement of Deadweight Loss in an Open Eco-nomy." Economica 51, 23-42. Diewert, W. E. (1984b). "On the Relationship Between the Measurement of Deadweight Loss and the Measurement of Project Benefits." U.B.C. Discussion Paper 84-32. Diewert, W. E. (1985a). "The Measurement of Waste and Welfare in Applied General Equilibrium Models." pp. 42-103 in New Developments in Applied  General Equilibrium Analysis. J . Pigott and J . Whalley eds. Cambridge: Cambridge University Press. Diewert, W. E. (1985b). "A Dynamic Approach to the Measurement of Waste in an Open Economy." Journal of International Economics 19, 213-240. Diewert, W. E. (1986). The Measurement of Economic Benefits of Infrastruc- ture Services. Lecture Notes in Economics and Mathematical Systems 278, Berl in: Springer Verlag. Diewert, W. E . , and A. D. Woodland. (1977). "Frank Knight's Theorem in Linear Programming Revisited." Econometrica 45, 375-398. Dix i t , A. K. (1975). "Welfare Effects of Tax and Price Changes." Journal  of Public Economics 4, 103-123. Dix i t , A. K. (1979). "Price Changes and Optimum Taxation in a Many-Consumer Economy." Journal of Public Economics 11, 143-157. Dixi t , A. K . , and J . E. S t i g l i t z . (1977). "Monopolistic Competition and Optimum Product Diversity." American Economic Review 67, 297-308. 138 . Dreze, J . P . , and N. H. Stern. (1986). "The Theory of Cost-Benefit Analysis," forthcoming in Handbook of Public Economics. A. J . Auerbach and M. S. Feldstein, eds. Amsterdam, North-Holland. Feldstein, M. S. (1972). "Distributional Equity and the Optimal Structure of Public Prices." American Economic Review 62, 32-36. Feldstein, M. S. (1978). "The Welfare Cost of Capital Income Taxation." Journal of P o l i t i c a l Economy 86, S29-51. Foster, E. (1976). "The Welfare Foundations of Cost-Benefit Analysis—A Comment." Economic Journal 86, 353-358. Foster, E . , and H. Sonnenschein. (1970). "Price Distortions and Economic Welfare." Econometrica 38. 281-297. Geary, P. T . , and M. Morishima. (1973). "Demand and Supply under Separabil-i ty ." pp. 87-147 in Theory of Demand: Real and Monetary. M. Morishima and others. Oxford: Clarendon. Gorman, W. M. (1953). "Community Preference Fields." Econometrica 21, 63-80. Gorman, W. M. (1955). "The Intransit ivity of Certain Cr i t er ia Used in Wel-fare Economics." Oxford Economic Papers NS7, 25-35. Gorman, W. M. (1959). "Are Social Indifference Curves Convex?" Quarterly  Journal of Economics 73, 485-496. Green, J . , and E. Sheshinski. (1979). "Approximating the Efficiency Gain of Tax Reforms." Journal of Public Economics 11, 179-195. Guesnerie, R. (1975). "Pareto Optimality in Non-Convex Economies." Econo- metrica 43, 1-30. Guesnerie, R. (1977). "On the Direction of Tax Reform." Journal of Public  Economics 7, 179-202. Guesnerie, R. (1979). "General Statements on Second-Best Pareto Optimal-i ty ." Journal of Mathematical Economics 6, 169-194. Hammond, P. (1986). "Project Evaluation by Potential Tax Reform." Journal  of Public Economics 30, 1-36. Harberger, A. C. (1964). "The Measurement of Waste." American Economic  Review 54, 58-76. Harberger, A. C. (1971). "Three Basic Postulates for Applied Welfare Economics: An Interpretive Essay." Journal of Economic Literature 9, 139 . Harberger, A. C. (1974). Taxation and Welfare. Boston: L i t t l e , Brown and Company. Harris , R. G. (1976). "A Note on Convex-Concave Demand Systems with an Application to the Theory of Optimal Taxation." Queen's University d is -cussion paper No. 197. Harris , R. G. (1978). "On the Choice of Large Projects." Canadian Journal  of Economics 11, 404-423. Harris , R. G. (1984). "Applied General Equilibrium Analysis of Small Open Economies and Imperfect Competition." American Economic Review 74, 1016-1032. Hatta, T. (1977). "A Theory of Piecemeal Policy Recommendations." Review  of Economic Studies 44, 1-21. Hicks, J . R. (1939). "The Foundation of Welfare Economics." Economic Jour- nal 49, 696-711. Hicks, J . R. (1940). "The Valuation of Social Income." Economica NS7, 105-124. Hicks, J . R. (1941-2). "Consumers' Surplus and Index Numbers." The Review  of Economic Studies 9, 126-137. Hicks, J . R. (1946). Value and Capital , second edition, Oxford: Clarendon Press. Hotell ing, H. (1932). "Edgeworth's Taxation Paradox and the Nature of Demand and Supply Functions." Journal of P o l i t i c a l Economy 40, 577-616. Hotell ing, H. (1938). "The General Welfare in Relation to Problems of Taxa-tion and of Railway and U t i l i t y Rates." Econometrica 6, 242-269. Jorgenson, D. W. (1984). "Econometric Methods for Applied General Equi l ib -rium Analysis." pp. 139-203 in Applied General Equilibrium Analysis. H. E. Scarf and J . B. Shoven eds. Cambridge: Cambridge University Press. Kaizuka, K. (1965). "Public Goods and Decentralization of Production." Review of Economics and Stat is t ics 47, 118-120. Kaldor, N. (1939). "Welfare Propositions and Interpersonal Comparisons of U t i l i t y . " Economic Journal 49, 549-552. Kar l in , S. (1959). Mathematical Methods and Theory in Games, Programming  and Economics. Reading, Mass.: Addison-Wesley. King, M. A. (1983). "Welfare Analysis of Tax Reforms Using Household Data." Journal of Public Economics 21, 183-214. 140. King, M. A. ( 1986). "A Pigovian Rule for the Optimum Provision of Public Goods." Journal of Public Economics 30, 273-291. Kay, J . A. (1980). "The Deadweight Loss from a Tax System." Journal of  Public Economics 13, 111-119. Lipsey, R. G . , and K. J . Lancaster. (1956). "The General Theory of Second Best." Review of Economic Studies 24. 11-32. Mangasarian, 0. L . (1969). Nonlinear Programming. New York: McGraw-Hill. McKenzie, G. W. (1983). Measuring Economic Welfare: New Method. London: Cambridge University Press. McKenzie, L . W. (1957). "Demand Theory Without a U t i l i t y Index." Review of  Economic Studies 24, 185-189. Meade, J . E. (1952). "External Economies and Diseconomies in a Competitive Situation." Economic Journal 62, 54-67. Meade, J . E . (1955). Trade and Welfare. Oxford: Oxford University Press. Mirrlees, J . (1986). "The Theory of Optimal Taxation." pp. 1197-1249 in Handbook of Mathematical Economics Vol . 3, K. J . Arrow and M. D. I n t r i l -igator, eds. Amsterdam: North-Holland. Mohring, H. (1971). "Alternative Welfare Gain and Loss Measures." Western  Journal of Economics 9, 349-368. Negishi, T. (1960), "Welfare Economics and Existence of an Equilibrium for a Competitive Economy." Metroeconomica 12, 92-97. Negishi, T. (1960-1). "Monopolistic Competition and General Equilibrium." Review of Economic Studies 28, 196-201. Negishi, T. (1962). "Entry and the Optimal Number of Firms." Metroeconom- ica 14, 86-96. Negishi, T. (1963). "On Social Welfare Function." Quarterly Journal of  Economics 77, 156-158. Negishi, T. (1967). "Perceived Demand Curve in the Theory of Second Best." Review of Economic Studies 34, 316-317. Negishi, T. (1978). General Equilibrium Theory and International Trade, Amsterdam: North-Holland. Pazner, E . A . , and E. Sadka. (1980). "Excess-Burden and Economic Surplus as Consistent Welfare Indicators," Public Finance 35, 439-449. 141 Pigott, J . , and J . Whalley. (1982). "General Equilibrium with Public Production and Public Provision: Computation, and Results for the U.K. Case." pp. 1-26 in Public Production. Zeitschrift fur National-okonomie, Supplementum 2, D. Bos, R. A. Musgrave and J . Wiseman, eds. Vienna: Springer Verlag. Pigou, A. C. (1947). A Study in Public Finance. Third edition. London: Macmillan. Ramsey, F. P. (1927). "A Contribution to the Theory of Taxation." Economic  Journal 37, 47-61. Roberts, D. J . , and H. Sonnenschein. (1977). "On the Foundations of the Theory of Monopolistic Competition." Econometrica 45, 101-113. Samuelson, P. A. (1947). Foundations of Economic Analysis, Cambridge, Mass.: Harvard University Press. Samuelson, P. A. (1950). "Evaluation of Real National Income." Oxford Eco- nomic Papers NS2, 1-29. Samuelson, P. A. (1954). "Pure Theory of Public Expenditure." Review of  Economics and Stat ist ics 36, 387-389. Samuelson, P. A. (1956). "Social Indifference Curves." Quarterly Journal  of Economics 70, 1-22. Samuelson, P. A. (1974). "Complementarity—An Essay on the 40th Anniversary of the Hicks-Allen Revolution in Demand Theory." Journal of Economic  Literature 4, 1255-1289. Scitovsky, T. (1941-2(a)). "A Note on Welfare Propositions in Economics." Review of Economic Studies 9, 77-88. Scitovsky, T. (1941-2(b)). "A Reconsideration of the Theory of Tar i f f s ." Review of Economic Studies 9, 89-110. Shephard, R. W. (1953). Cost and Production Functions. Princeton: Prince-ton University Press. Shoven, J . B . , and J . Whalley. (1972). "A General Equilibrium Calculation of the Effects of Differential Taxation of Income from Capital in the U.S." Journal of Public Economics 1, 281-321. Shoven, J . B . , and J . Whal-ley. (1973). "General Equilibrium with Taxes: A Computational Procedure and an Existence Proof." The Review of Economic  Studies 40, 475-489. Shoven, J . B . , and J . Whalley. (1977). "Equal Yield Tax Al t erna t ive s -General Equilibrium Computational Techniques." Journal of Public Eco- nomics 8, 211-224. 142. Silberberg, E. (1980). "Harold Hotelling and Marginal Cost Pricing." American Economic Review 70, 1054-1057. Smith, B . , and F. Stephen. (1975). "The Welfare Foundations of Cost-Benefit Analysis: Comment." Economic Journal 85, 902-905. Spence, A. M. "Product Selection, Fixed Costs and Monopolistic Competition." Review of Economic Studies 43, 217-235. S t i g l i t z , J . E . , and P. S. Dasgupta. (1971). "Differential Taxation, Public Goods and Economic Efficiency." Review of Economic Studies 38, 151-174. Tsuneki, A. (1985). "On the Choice of Large Projects: A Generalization." Canadian Journal of Economics 18, 660-664. Tsuneki, A. (1987a). "The Measurement of Waste in a Public Goods Economy." Journal of Public Economics. forthcoming. Tsuneki, A. (1987b). "Harold Hotelling's General Equilibrium Welfare Eco-nomics ." Mimeographed. Turunen, A. H. (1986). "A Note on the Measurement of Production Gain Due to a Tar i f f Reform in a Small Open Economy." Working Paper No. 86-1, Center for Economic Research, University of Texas at Austin. Uzawa, H. (1958). "The Kuhn-Tucker Theorem in Concave Programming." pp. 32-37 in Studies in Linear and Nonlinear Programming. K. J . Arrow, L. H. Hurwicz and H. Uzawa, eds. Stanford, Cal i fornia: Stanford Uni-versity Press. Varian, H. R. (1974). "Equity, Envy and Efficiency." Journal of Economic  Theory 9, 63-91. Varian, H. R. (1976). "Two Problems in the Theory of Fairness." Journal of  Public Economics 5, 249-260. Weymark, J . A. (1979). "A Reconciliation of Recent Results in Optimal Taxa-tion Theory." Journal of Public Economics 12, 171-189. Weymark, J . A. (1985). "Money-Metric U t i l i t y Functions." International  Economic Review 26, 219-232. Wildasin, D. E. (1984). "On Public Good Provision with Distortionary Taxa-t ion." Economic Inguiry 22. 227-243. 

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