ESSAYS ON THE MEASUREMENT OF WASTE AND PROJECT EVALUATION By ATSUSHI TSUNEKI B.A., The U n i v e r s i t y o f Tokyo, Tokyo, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES THE DEPARTMENT OF ECONOMICS We a c c e p t t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1987 CcT) ^Atsushi Tsuneki, 1987 In presenting requirements of British it freely agree for this for an available that I understood that financial by his or June reference and study. I extensive her or shall f l , copying granted by the of publication not be allowed Columbia /9£? of make further this head representatives. of 30 University shall The U n i v e r s i t y o f B r i t i s h 1956 Main M a l l V a n c o u v e r , Canada V6T 1Y3 Date the the Library permission. Department at of the may b e copying gain degree fulfilment that for purposes or partial agree for permission scholarly in advanced Columbia, department for thesis It this without thesis of my is thesis my written ABSTRACT Harberger's methodology for the measurement of deadweight loss i s reformulated in a general equilibrium context with adopting the A l l a i s Debreu-Diewert approach and i s applied to various problems with imperfect markets. class of We also develop second best project evaluation rules for the same economies. Chapter 1 i s devoted to the survey of various welfare i n d i c a t o r s . especially discuss the two welfare indicators due to A l l a i s , We Debreu, Diewert and Hicks, Boiteux i n r e l a t i o n to Bergson-Samuelsonian s o c i a l welfare function. We f i r s t show that these two measures generate a Pareto inclusive ordering across various s o c i a l states, but they are rarely w e l f a r i s t , so that both are unsatisfactory as Bergson-Samuelsonian s o c i a l welfare functions. We next show that second order approximations to the Allais-Debreu-Diewert measure of waste can be computed from l o c a l information observable at the equilibrium, whereas second order approximations to the Hicks-Boiteux measure of welfare or to the Bergson -Samuelsonian s o c i a l welfare function require information on the marginal u t i l i t i e s of income of households, unavailable with ordinal u t i l i t y theory. which i s F i n a l l y , we give a diagrammatic exposition of the two measures and t h e i r approximations to give an i n t u i t i v e insight into the economic implications of the two measures. Chapter 2 and Chapter 3 study an economy with public goods. In Chapter 2, we compute an approximate deadweight loss measure for the whole economy when the endogenous choice of public goods by the government i s nonoptimal and the government revenue i s raised by distortionary taxation by extending the Allais-Debreu-Diewert approach discussed i n Chapter 1. measure of waste i s related to i n d i r e c t tax rates, public goods, The r e s u l t i n g net marginal benefits of and the derivatives of aggregate demand and supply functions evaluated at an equilibrium. In Chapter 3, cost-benefit rules for the p r o v i - sion of a public good are derived when there exist tax d i s t o r t i o n s . derive the rules as giving s u f f i c i e n t We conditions for Pareto improvement, but we also discuss when these rules are necessary conditions for an i n t e r i o r s o c i a l optimum. When i n d i r e c t taxes are f u l l y f l e x i b l e but lump-sum trans- fers are r e s t r i c t e d , we recommend a rule which generalized the cost-benefit rule due to Atkinson and Stern (1974) to a many-consumer economy. i n d i r e c t taxes and lump-sum transfers are f l e x i b l e , i s based on Diamond and Mirrlees' When both we suggest a rule which (1971) productive efficiency p r i n c i p l e . When only lump-sum transfers are variable, we obtain a version of the Harberger (1971)-Bruce-Harris (1982) cost-benefit rules. Chapters 4 and 5 study an economy with increasing returns to scale i n production and imperfect competition. In Chapter 4, we discuss a methodology for computing an approximate deadweight loss due to imperfect regulation of monopolistic industries by extending the Allais-Debreu-Diewert approach to incorporate the nonconvex technology. With the assumption of the quasi-con- cavity of production functions and fixed number of firms, we can derive an approximate deadweight loss formula which i s related to markup rates of firms, and the derivatives of aggregate demand functions, factor supply and demand functions and the derivatives of marginal cost functions. We also iv d i s c u s s v a r i o u s l i m i t a t i o n s o f our approach and t h e r e l a t i o n between our work and t h a t o f H o t e l l i n g (1938). I n Chapter 5, we c o n s i d e r c o s t - b e n e f i t r u l e s of a l a r g e p r o j e c t a p p l i c a b l e i n t h e presence of i m p e r f e c t c o m p e t i t i o n . We show t h a t t h e i n d e x number approach due t o N e g i s h i (1962) and H a r r i s (1978) can be extended t o h a n d l e s i t u a t i o n s w i t h i m p e r f e c t competition. V TABLE OF CONTENTS Page Abstract i i L i s t of Figures v i i Acknowledgements viii Chapter 1 2 3 APPLIED WELFARE ECONOMICS 1 1-1. The Mesures of Deadweight Loss 2 1-2. The Approximation Approach 1- 3. A Diagrammatic t o the Measurement of Waste Exposition 11 26 Footnotes 35 Appendices 38 THE MEASUREMENT OF WASTE IN A PUBLIC GOODS ECONOMY 50 2- 1. Introduction 50 2-2. The Model 51 2-3. An A l l a i s - D e b r e u - D i e w e r t Measure o f Waste 55 2-4. Second 57 2- 5. Conclusion Order Approximations 63 Footnotes 66 PROJECT EVALUATION RULES FOR THE PROVISION OF PUBLIC GOODS 70 3- 1. Introduction 70 3-2. The Model 72 3-3. P i g o v i a n Rules R e c o n s i d e r e d 76 3-4. Cases where Lump-Sum T r a n s f e r s a r e A v a i l a b l e 85 vi 3- 5. 4 5 Conclusion 92 Footnotes 94 Appendices 95 INCREASING RETURNS, IMPERFECT COMPETITION AND THE MEASUREMENT OF WASTE 99 4- 1. Introduction 99 4-2. The Model 4-3. The A l l a i s - D e b r e u - D i e w e r t 4-4. Second Order Approximations 108 4- 5. Conclusion 115 Footnotes 117 101 Measure o f Waste 104 PROJECT EVALUATION RULES FOR IMPERFECTLY COMPETITIVE ECONOMIES 121 5- 1. Introduction 121 5-2. Compensation C r i t e r i a f o r Cost-Benefit A n a l y s i s Reconsidered 123 5-3. The Model 127 5-4. P r o j e c t E v a l u a t i o n Rules 129 5-5. Conclusion 132 Footnotes 133 REFERENCES 135 vii LIST OF FIGURES Page Figure 1. The ADD Measure of Waste 44 F i g u r e 2. The HB Measure of Welfare 45 F i g u r e 3. The ADD Measure: 46 F i g u r e 4. The ADD Measure and i t s Approximations: A One-Consumer Two-Goods Economy A One-Consumer Two-Goods Economy 47 F i g u r e 5. The HB Measure: 48 F i g u r e 6. The HB Measure and i t s Approximations: A One-Consumer Two-Goods Economy A One-Consumer Two-Goods Economy 49 F i g u r e 7. The ADD Measure i n a P u b l i c Goods Economy 67 F i g u r e 8. The ADD Measure and i t s Approximations i n a Public Goods Economy F i g u r e 9. 68 An Example where Approximations of the ADD Measure are I n a c c u r a t e 69 Figure 10. The ADD Measure w i t h I n c r e a s i n g Returns t o S c a l e Figure 11. The ADD Measure and i t s Approximations with Returns t o S c a l e 119 Increasing 120 v i i i ACKNOWLEDGEMENTS F i r s t and of chairman a l l , of encouragement several other I two valuable q u a l i t y i n should t h e s i s c l a s s e s should a l s o of of my t h e s i s comments, and i n my thesis, to thank t h e s i s not comments only my on E n g l i s h l i f e . I graduate students of the Department Columbia who my study gave w r i t i n g the Ishikawa, Komiya, from I a l s o Itoh, Keimei and John of must and express Kaizuka, f o r my p a t i e n t l y of Canada who on deep my t h e s i s me the t h i s other at the both to I as the teaching e x c e l l e n t l y . f i r s t parents work of i n and outside while I Ryutaro of support F i n a l l y , by E x t e r n a l a s s i s t a n t . my was Tsuneo administered who and B r i t i s h f i n a n c i a l Department my adjust members Hatta, Nationals my to professors Kanemoto, of Keizo comfortable Tatsuo the g i v i n g U n i v e r s i t y appreciate of a f a c u l t y encouragement Yoshitsugu a p p r e c i a t i o n to complete help f o r the the expression kind Foreign me f o r f o r Hamada, behalf employed t y p i n g to to improve to warm Okano. the the repeated t h a n k f u l thanks and Weymark on Weymark, a l s o 1983-1987 K o i c h i Awards John on complete f o r to kind comments am to deep John helped a l s o Economics my Canada Weymark P a i s l e y of Yukihide Service and and to a d v i s e r and d e t a i l e d Donaldson which I t h a n k f u l and David but p r i n c i p a l i t p o s s i b l e appreciate comments Motoshige Negishi, and content express important my suggestions Blackorby t h e s i s during i n c l u d e : Government generously there which Jeanette least, be t h e s i s , U n i v e r s i t y A f f a i r s , Ms. many Takashi the World me must made c a r e f u l l y . American who my I the North UBC which s t i m u l a t i n g d i s c u s s i o n s should Diewert, important Charles on to I many committee, f o r challenging. f o r Erwin p r i v a t e d i s c u s s i o n s , and Nagatani made thank conspicuously. c o r r e c t i n g many of and to committee l i k e my d i s c u s s i o n s l i k e and v e r s i o n s members extensive thesis my e a r l i e r thesis. I but always academic I thank not helped career. me 1. CHAPTER 1 APPLIED WELFARE ECONOMICS The purpose of t h i s c h a p t e r i s t o compare a l t e r n a t i v e c r i t e r i a f o r s o c i a l waste o r w e l f a r e from s e v e r a l v i e w p o i n t s and choose one which s u i t s our purpose b e s t . criterion I n d o i n g so, we p r e s e n t our b a s i c s t r a t e g i e s f o r the measurement of deadweight l o s s and d i s c u s s t h e i r pros and cons compared w i t h o t h e r approaches t o a p p l i e d w e l f a r e economics. In s e c t i o n 1, we i n t r o d u c e two c r i t e r i a f o r measuring deadweight l o s s ; t h a t i s , the A l l a i s - D e b r e u - D i e w e r t measure of s o c i a l waste and the H i c k s B o i t e u x measure o f s o c i a l w e l f a r e . A f t e r e x p l a i n i n g t h e i r i n t u i t i v e meanings by i l l u s t r a t i o n s we c o n s i d e r whether t h e y can s e r v e as P a r e t o - i n c l u s i v e and individualistic (or w e l f a r i s t ) s o c i a l w e l f a r e f u n c t i o n s . We show t h a t t h e s e measures a r e P a r e t o - i n c l u s i v e , but not i n d i v i d u a l i s t i c except when e i t h e r Gorman's p r e f e r e n c e r e s t r i c t i o n i s s a t i s f i e d o r the p r o d u c t i o n set i s l i n e a r . possibilities The A l l a i s - D e b r e u - D i e w e r t measure of waste i s a f f e c t e d by the c h o i c e of t h e r e f e r e n c e bundle of goods i n terms o f which t h e s c a l e of e f f i c i e n c y l o s s i s d e t e r m i n e d whereas the H i c k s - B o i t e u x measure of w e l f a r e i s a f f e c t e d by t h e c h o i c e o f t h e o p t i m a l a l l o c a t i o n o f r e a l income. T h i s means t h a t the former measure i s a f f e c t e d by the v a l u a t i o n of each good f o r s o c i a l e f f i c i e n c y w h i l e the l a t t e r i s a f f e c t e d by t h e v a l u a t i o n o f each i n d i v i d u a l i n the measure o f s o c i a l w e l f a r e . Thus, the A l l a i s - D e b r e u - D i e w e r t measure i s a pure e f f i c i e n c y waste measure whereas the H i c k s - B o i t e u x measure shows a change o f s o c i a l w e l f a r e i n c l u d i n g b o t h e f f i c i e n c y and e q u i t y a s p e c t s . I n t h e second s e c t i o n , t h i s p o i n t i s f u r t h e r e l a b o r a t e d by t a k i n g a second o r d e r a p p r o x i m a t i o n t o the two measures when t a x d i s t o r t i o n s prevail. We show that the Allais-Debreu-Diewert measure i s computable from the second order derivatives of expenditure functions and p r o f i t functions evaluated at the observed equilibrium while the Hicks-Boiteux measure or the BergsonSamuelsonian s o c i a l welfare function requires information on the difference between the inverse of the marginal u t i l i t y of income and the marginal s o c i a l importance to evaluate the equity l o s s . Since t h i s information i s not available with ordinal u t i l i t y theory, i t i s d i f f i c u l t to use the Hicks-Boiteux measure i n applied welfare economics. This provides the main reason why we use the Allais-Debreu-Diewert measure in this essay. and The pros cons of the approximation approach we adopt i n this essay are next compared with an a l t e r n a t i v e i n f l u e n t i a l approach, applied (or numerical) general equilibrium models. A numerical general equilibrium model computes the exact value of s o c i a l welfare indicators by r e s t r i c t i n g the functional forms of production and u t i l i t y to overly simple forms. Our approach, on the other hand, computes approximate values of s o c i a l welfare indicators from more general functional forms and observable information. of Finally, in light the measurement of waste approach for welfare economics, we reconsider the theory of second best. Our conclusion here i s that this theory i s not a replacement for the measurement of deadweight loss, even though several positive r e s u l t s derived i n second best theory are useful. Finally, i n section 3, in order to give insight into the economic implications of our approach, we give diagrammatic expositions of the two measures and t h e i r approximations for a one-consumer two-goods economy. 1-1. The Measure of Deadweight Loss In a long series of papers on the measurement of deadweight loss (or 'welfare cost' or 'waste,' terms which are used interchangeably i n this thesis) which includes H o t e l l i n g (1938), Hicks (1941-2), A l l a i s (1943, 1977), Boiteux (1951), Debreu (1951, 1954), Harberger (1964, 1971), and DiamondMcFaddon (1974), two types of welfare c r i t e r i a are c h i e f l y used: the Allais-Debreu-Diewert measure of waste (the ADD measure hereafter) and the Hicks-Boiteux measure of surplus (the HB measure h e r e a f t e r ) . 1 Let us set up the model of our economy to discuss these two measures. There are H consumers having quasi-concave u t i l i t y functions f (x ), h = 1, . . . , H defined over a translated orthant P.* where x* = (x^ , . . . ,x^) 1 1 1 consumption vector of goods 1 , . . . , N by the hth consumer. T is a The i n i t i a l endowment vector of the hth consumer i s given by x* , h = 1 , . . . , H . There are 1 k k K firms and firm k produces y using the production p o s s i b i l i t i e s set S , k = 1,...,K. problem? We can define the ADD measure i n terms of a primal programming : ADD ( 1 ) s ~= r m a X r , h x r y k = h=1 { r E f (x ) h + P ' r 1 E k=l y > U j J , h = 1, . . . , H ; y + K E h=1 X ; e S , k = 1,...,K}, k > 0^ i s an a r b i t r a r i l y chosen reference bundle of where p = ( p ^ , . . . , p ^ ) commodities. h X To i n t e r p r e t t h i s problem we rewrite (1) i n an a l t e r n a t i v e N . The following notation i s used. R i s the N-dimensional nonnegative K k k orthant. E i d i r e c t sum of the production p o s s i b i l i t i e s sets S . manner. + S s t n e K = 1 S(u ) 1 = {x : E f . , x 1 x; f ( x ) h h h h > UjJ, h = 1,...,H} i s the Scitovsky set corresponding to a u t i l i t y a l l o c a t i o n u rewritten as 1 1 1 = (u.,...,u„). I H Now (1) can be 4. (2) max {r r : 6 r e Q = Z^x + E ^S h k (2) has a s t r a i g h t f o r w a r d i t e r p r e t a t i o n : set k - S(u )}. 1 maximize the s c a l e of the reference of goods i n Q where Q i s the s e t of goods p r o d u c i b l e from the a g g r e g a t e p r o d u c t i o n p o s s i b i l i t i e s p l u s endowments which g i v e consumers a t l e a s t the 1 u t i l i t y vector u when the goods are a p p r o p r i a t e l y d i s t r i b u t e d . 1 we d e p i c t the ADD measure, i n a two goods economy where p p r i c e of the programming problem ( 1 ) . is a In F i g . support p^ i s d e t e r m i n e d up t o a m u l t i p l i c a - t i o n by a p o s i t i v e number so t h a t we can choose p^ = 1; i . e . , the o p t i m a l p r i c e of the f i r s t good i s u n i t y w i t h o u t l o s s of g e n e r a l i t y . F u r t h e r m o r e , OT OT can choose the s c a l e of 8 so t h a t p 8 = 1 . Then, r = (p 8 ) r e q u a l s AB OT since p pr i s the d i f f e r e n c e between the v a l u e of p r o d u c t i o n minus consump- t i o n e v a l u a t e d a t p^. Note t h a t the c h o i c e of the r e f e r e n c e bundle 8 i s c r u c i a l i n the e v a l u a t i o n of the ADD L e t us now measure (see Diewert t u r n t o the HB measure. 0 ( 1985a;50)) . 3 We b e g i n from an a t t a i n a b l e and 0 0 T (u.,...,u ) . We a l s o assume t h a t t n 0 0 0 T t h e r e e x i s t s a p r i c e v e c t o r p = (p^,...,P ) which s u p p o r t s the s o c i a l l y o p t i m a l a l l o c a t i o n of r e s o u r c e s . Then we can d e f i n e the HB measure L..,. as s o c i a l l y optimal u t i l i t y a l l o c a t i o n u s N HD follows: (3) = E, - E, where we d e f i n e the e x p e n d i t u r e f u n c t i o n : ^ (4) we ra (p,u ) h h = min {p x T h h : f ( x ) > u,}, h h where p > 0 N and The measure L H B e Range f defined by (3) can be interpreted as the sum of the negative of the equivalent variations obtained i n moving from a s o c i a l l y optimal u t i l i t y vector u^ to the observed distorted u t i l i t y vector u . 1 The HB measure evaluated i n units of the f i r s t good i n a two good economy by choosing = 1 i s i l l u s t r a t e d i n F i g . 2. Generally, the desirable properties of the ordering of s o c i a l states are summarized i n the Bergson-Samuelsonian s o c i a l welfare function (BSSWF hereafter) . (See Samuelson (1956) for a discussion of the BSSWF and i t s proper- t i e s l i s t e d below.) We f i r s t assume that the underlying s o c i a l ordering i s compatible with the Pareto p a r t i a l ordering ( i . e . , t i e s increase, then so does s o c i a l welfare) becomes Pareto-inclusive. i f a l l individual utili- so that the r e s u l t i n g BSSWF Suppose also that the evaluation of s o c i a l states i s i n d i v i d u a l i s t i c (or w e l f a r i s t ) ; i.e., the u t i l i t y vector u p r e v a i l i n g at the state is the only information used i n the evaluation. Also suppose that the evaluation takes the form of a continuous ordering of u t i l i t y vectors. Then, Debreu's (1959;56) representation theorem i s applied to get the BSSWF, W(u). Pareto-inclusiveness implies that W i s monotone increasing i n u. Recalling that the ADD measure and the HB measure evaluate the states of the economy numerically, they generate orderings of the u t i l i t y vectors where the u t i l i t y vectors with smaller amounts of waste are ranked higher given the reference bundle p or the reference u t i l i t y vector u . 0 5 I t may, therefore, be i n t e r e s t i n g to ask whether these measures are Pareto-inclusive^ and i n d i vidualistic; i . e . , whether they work as a kind of BSSWF. The f i r s t question 6. may be answered e a s i l y . ADD measure. F i r s t , notice the d e f i n i t i o n (2) of the Suppose that u a i s preferred to u b i n terms of the Paretian p a r t i a l ordering, then S(u ) i s a subset of S ( u ) . a Noting that production b p o s s i b i l i t i e s are fixed, Q(u ) i s a subset of Q(u ) and hence r ( u ) < r ( u ) . a b a b In the case of the HB measure, Pareto inclusiveness d i r e c t l y follows from the nondecreasingness of the expenditure function with respect to u (see Diewert (1982;541)) and i t s d e f i n i t i o n (3). The other question i s more d i f f i c u l t to solve. The ADD measure r = r(u,p) becomes a function of both u and p, so i t cannot be i n d i v i d u a l i s t i c ; i.e., i t i s always affected by the choice of B, which i s not related to i n d i - viduals' welfare. We extend the concept of an ' i n d i v i d u a l i s t i c ' evaluation by saying that r i s o r d i n a l l y i n d i v i d u a l i s t i c i f and only i f the ordering of u t i l i t y vectors induced by r for given p i s not affected by the choice of p. This d e f i n i t i o n i s formalized as follows: (5) r(u ,p ) > r(u ,p ) iff r(u ,p ) > r(u ,p ) a a b a a b for a l l p b a >0 N b and p k The p r o f i t function ir , k = 1 , . . . , K i s defined as (6) i r ( p ) = max {p y k where p > 0 „ . T x : y e S }, k k = 1,...,K, b > 0 . N 7. The regularity properties of the p r o f i t function are summarized i n Diewert (1982;580-1). We assume below that the production p o s s i b i l i t i e s preferences are quasiconcave. sets are convex and Then, (1) i s equivalent to the following dual max min problem: (7) L A D D ( u , p ) = r(u,p) = max m i n r {r(1-p p)+E ^ p x +E ^ ir (p) T p > Q T h h k 1 k 1 - E f m (p,u )}, h h 1 h The proof i s an a p p l i c a t i o n of the Uzawa ( 1958;34)-Karlin (1959;201) Saddle Point Theorem (see Appendix I I ) . If we further assume that r i s twice 7 continuously d i f f e r e n t i a b l e at the relevant values of u and p, then (5) is equivalent to requiring u to be separable^ i n r ( u , p ) ; that i s , r(u,p) satisfies (8) 8(| 7! -)/ap = 0 for a l l i , j = 1 , . . . , H and a l l n = 1 , . . . , N . ou. ou . n r D JL E We assume that the f i r s t order necessary conditions for the max min problem (7) are equalities and define the solution as (r^,p^). Then the well-known envelope theorem implies that or/3u^ = - dm (p^,u^)/9u^, i = 1 , . . . , H . 1 s t i t u t i n g i t into (8) and using the r e l a t i o n : 9 m (p,u^)/9u^3p 2 1 ffl = [ d x ^ ( p , y ) / 3 y ^ ] [ d m ( p ° , u ^ ) / 3 u ^ ] for i = 1 , . . . , H and m = 1 . . . , N , we have 1 r (9) E.I [axi(p yJ)/ay.-8xj(p ,y5)/ay.]Op2/8P ) = 0 0 1 0 f ll for a l l i , j = 1 , . . . , H and a l l n = 1 , . . . , N , Sub- 8. where x^"(p,y.), m i for i = 1 , . . . , H i s the ordinary demand function for the nth good the i t h consumer and y? = m ( p ° , u ^ ) . 1 Conditions (9) are s a t i s f i e d i f Gorman's (1953;73) r e s t r i c t i o n on preferences i s s a t i s f i e d ; either i.e., preferences are quasi-homothetic and t h e i r Engel curves are p a r a l l e l to each other, (since the f i r s t term i n the left-hand side of (9) i s 0 for a l l i , j , m ) or i f the production p o s s i b i l i t i e s i s 0 for a l l m and n). sets are linear (since the second vector (9) has the following meaning: when we increase any one reference good P , then the s c a r c i t y of the nth good increases so that n the system of shadow prices associated with (7) p^ changes, and t h i s change must be orthogonal to the difference of the gradients of the Engel curves for any two consumers at the optimum. This condition does not seem to me to be s a t i s f i e d globally except for the two cases above l i s t e d . We now turn to the HB measure L . By the same token as the ADD HD measure, L „ ( u \ u ^ , p ^ ) n separable i n L U D HD . i s o r d i n a l l y i n d i v i d u a l i s t i c i f and only i f u^ i s Remember that u^ i s one Pareto optimal u t i l i t y a l l o c a t i o n and p^ i s i t s supporting price vector. Therefore p^ is a function of u^ (and 1 other parameters of the general equilibrium) so that s e p a r a b i l i t y of u equivalent to the condition. (10) ( — ^ f / — * H f ) /duf = 0 f o r a l l i , j = 1 , . . . , H and a l l h = 1 , . . . , H . u. 9u. r 3 ^HB i 0 1 1 „ 1 = - 3m (p ,ui)/du{. Substitute t h i s ou. l (10) and we find the following equivalent conditions: Using d e f i n i t i o n (3), into is 9. f o r a l l i , j = 1,...,H and a l l h = 1,...,H. C o n d i t i o n s (11) seem analogous t o (9), except f o r t h e d i f f e r e n c e between dp°/dp i n (9) and d^P/bvP i n (11). m n m n The former i s t h e change o f t h e s u p p o r t p r i c e s o f t h e A l l a i s - D e b r e u - D i e w e r t optimum w i t h r e s p e c t t o an i n c r e a s e o f the n t h good i n t h e r e f e r e n c e bundle, w h i l e t h e l a t t e r i s t h e change o f t h e support p r i c e s o f t h e r e f e r e n c e P a r e t o o p t i m a l a l l o c a t i o n w i t h r e s p e c t t o an i n c r e a s e o f t h e u t i l i t y o f t h e h t h household. T h e r e f o r e , as i n t h e ADD measure, t h e r e does n o t seem t o e x i s t p l a u s i b l e c o n d i t i o n s t o g u a r a n t e e t h e HB measure t o be o r d i n a l l y i n d i v i d u a l i s t i c except f o r t h e two c o n d i t i o n s c i t e d above; i . e . , Gorman's p r e f e r e n c e r e s t r i c t i o n o r l i n e a r production possibilities. Up t o now we have l e a r n e d t h a t b o t h measures a r e P a r e t o i n c l u s i v e b u t not i n d i v i d u a l i s t i c i n g e n e r a l . The c o n d i t i o n s n e c e s s a r y t o make w e l f a r e p r e s c r i p t i o n s by t h e ADD measure o r d i n a l l y i n d i v i d u a l i s t i c a r e as s t r i n g e n t as those needed by t h e HB measure. However, t h e economic i m p l i c a t i o n s o f t h e two measures a r e c o m p l e t e l y d i f f e r e n t . The ADD i n d e x measures pure t e c h n i c a l e f f i c i e n c y i n terms o f t h e r e f e r e n c e bundle o f goods, and t h e HB index measures t h e l o s s o f b o t h e f f i c i e n c y and e q u i t y by i n d i c a t i n g t h e monetary v a l u e of t h e d i f f e r e n c e between t h e s o c i a l optimum and t h e observed equilibrium. Although based on pure e f f i c i e n c y c o n s i d e r a t i o n s , u s i n g t h e ADD measure t o rank s o c i a l s t a t e s means t h a t i m p l i c i t l y i t i s b e i n g used as a measure o f s o c i a l w e l f a r e ( i n s t e a d o f as j u s t an e s t i m a t e o f t h e r e s o u r c e a l l o c a t i o n waste o f one observed e q u i l i b r i u m ) , and as I have shown, t h i s 10. method of valuing s o c i a l states i s affected by the choice of reference bundle of goods. Therefore, to add equity aspects to the ADD measure, we have to choose a reference bundle so that goods which are s o c i a l l y valuable are weighted more heavily. However, i t i s d i f f i c u l t to determine what these goods are, and what weights s h a l l be attached to them. In contrast, the HB measure i s a sum of money-metric scaling u t i l i t y functions and i t has a natural interpretation as a BSSWF, provided a reference price vector fixed. is Another drawback of the ADD measure i s that i t cannot be an appropriate welfare indicator i f there i s technological change ( i . e . , it is not w e l f a r i s t i n the sense that i t depends on technological parameters). HB measure i s free of t h i s defect, (see Section i f the reference price vector i s The fixed 5.2) Let us compare these measures from another viewpoint. Are these measures useful when the shadow price vector does not e x i s t because of nonconvexities or externalities? We w i l l show i n the l a t e r chapters of t h i s essay that the ADD measure i s a very powerful tool to analyze deadweight loss under such market imperfections. It seems that we can also use the HB measure equally well to study deadweight loss i n such circumstances. When we choose a reference Pareto optimal a l l o c a t i o n u^, we find both the optimal shadow prices p^ for priced goods and the optimal demands q^ of external goods or nonpriced goods. A l l we need i s to compare the sum of the negative of the equivalent variations m^(p^,q^,u^) - m (p ,q , u ) , where m (p,q,u ) i s a r e s t r i c t e d expenditure function h (see Diewert (1986;170-6)). Note that the c a l c u l a t i o n of the two measures necessitates global computation of the optimal equilibrium which is very d i f f i c u l t to implement empir- 11. ically. Therefore i n t h i s essay, we concentrate on the study of approximate measures of welfare. In the following section, we compare the approximate ADD measure and HB measure and discuss which one i s more implementable i n empirical research. 1-2. The A p p r o x i m a t i o n Approach t o the Measurement o f Waste This section i s devoted to an introduction to our approximation approach to the measurement of waste. We f i r s t derive a second order approximation to the ADD measure of waste (1). This approximate measure depends on the eco- nomic environment and types of d i s t o r t i o n s . are complete, We assume i n i t i a l l y that markets technologies are convex and that the only source of i s i n d i r e c t taxes levied on consumers. distortions Extensions of these assumptions are a main theme of the l a t e r chapters, so that we only work with the prototype model in t h i s chapter. Given these assumptions, (1) i s equivalent to (6). At t h i s point, we use the concept of the overspending function B which w i l l be f u l l y u t i l i z e d i n t h i s essay which is defined as _ „ . p B(q,P,u) = E H h = 1 h . » r « H T " - h m (q u ) - C ( h h = ( ) q x n K k , » - E _ » (P) • k 0 In Appendix I, B i s restated with i t s economic interpretation and i t s properties are summarized. concisely as (12) Using the d e f i n i t i o n (A.1), (7) may be rewritten follows: r ° = max min {r(1-p p) N T n useful B(p,p,u )}. 1 12. Using the Uzawa (1958) - K a r l i n (1959) Theorem i n reverse, (12) i s also equivalent to: (13) - max {B(p,p,u ) 1 : p p > 1}. T N If (p°,r°) solves (12), p ° solves (13) with r ° being i t s associated Lagrangean m u l t i p l i e r . In order to obtain a second order approximation to r^, we assume: (p^,r^) solves (12); (ii) hold with equalities so that p ° » 0 „ ; ( i i i ) d i f f e r e n t i a b l e at ( p ° , p ° ) ; sufficient the f i r s t order necessary conditions for (iv) Samuelson's (i) (12) B ( q , p , u ) i s twice continuously 1 (1947;361) strong second order conditions hold for (13) when the inequality constraints are replaced by e q u a l i t i e s . Let us consider the following system of equations i n N+1 unknowns p and r which are functions of a scalar variable z, for 0 < z < 1: (14) - V B(p(z) + t z , p(z), u ) (15) 1 - p(z) B = 0. 1 q p + t z , p(z), u ) 1 - r(z)p = 0, T When z = 0, (12) - 7 B(p(z) i f p(0) (14) and (15) coincide with the f i r s t order conditions for = p ° and r(0) = r ° . Suppose p(1) producer prices normalized by (15) i n d i r e c t tax rates t . Setting r(1) = p 1 i s the set of observed i n a tax-distorted equilibrium with = 0, when z = 1 (14) i s then the set of equations characterizing the equality of demand and supply i n the taxdistorted equilibrium. If we assume that appropriate lump-sum transfers 13. from the government to consumers are chosen, then there exist budget constraints for the H consumers compatible with (14) and (15). From these equations, i t i s also the case that s a t i s f a c t i o n of the government budget constraint i s implied. Let us d i f f e r e n t i a t e B (16) 2 qq (14) - (15) t o t a l l y with respect to z. + B , P'U) Z PP B We have t qq z r' (z) z 2 where q = p + tz i s the f i r s t set of arguments for B and B ^ = B(p(z) 1 . . z + t z , p(z), u ) for I,} = q , p , u . Note that B = 0„ „ . Also note that the qp N «N left-hand side matrix of (16) and Woodland (1977)). (iii), is non-singular by assumption (iv) (see Diewert Therefore, using the d i f f e r e n t i a b i l i t y assumptions by the I m p l i c i t Function Theorem there exist once continuously d i f f e r e n t i a b l e functions p(z) and r(z) at z close to 0 that s a t i s f y (15). (17) We show i n Appendix III that the following equation i s - r'(z) = - z t T (14) and satisfied. BJi (p'(z) + t ) . qq We readily have (18) from (17). r'(0) = 0 Using (17), equation follows. i t i s shown i n Appendix IV that the following 14. (19) - r"(0) = - (p'(0) + t ) T B J (p'(0) + t) - p ' ( 0 ) T B 0 p'(0) > 0 where the inequality comes from the concavity of B with respect to q and p (see Appendix I ) . A second order Taylor approximation to the ADD measure i s given by (noting that r(1) = 0), (20) L A D D = r(0) - r(1) - MP'(0) T where we use (18) and (19). = B ° p'(0) + [p'(0) + t ] T B ° [p'(0) + t]} > 0, Equation (16) i s used to compute p ' ( 0 ) . Information we need to evaluate (ii) r(0) - (r(0) + r ' ( 0 ) + J,r'"(0)) = (20) is: (i) the set of i n d i r e c t taxes t, the second order derivatives of the overspending function with respect to prices which equals the producers' aggregate substitution matrix and the consumers' aggregate compensated substitution matrix respectively, evaluated at the optimum equilibrium. The remarkable advantage of t h i s approximation approach i s that i t can be implemented from the derivatives of the overspending function evaluated at the optimum equilibrium, so that we need not know global functional forms for u t i l i t y and production functions. However, as long as we must know the derivatives at the optimum as i n (20), we must actually know the optimal prices so that we must compute the optimum or we must depend on some 'guessing' process about the values at the optimum. Harberger (1964) suggested replacing these (unobservable) derivatives by those which are evaluated at the observed distorted equilibrium, since they can be calculated using data p r e v a i l i n g at the observed equilibrium. This method can be j u s t i f i e d more 15. rigorously by Diewert's ( 1976; 118) Quadratic Approximation Lemma which showed that the approximation (21) L = r(0) - r(1) = r(0) - {r(0) A D D + Jjr'tO) + J j r ' d ) } i s also exact as the approximation (20) when the functional form i s quadratic (see also Diewert (1985(b);238)). Evaluating (16) at z = 1 and using (17), we can show that - r ' ( 1 ) i s i d e n t i c a l with -r"(0) i n (19) except that a l l the relevant functions are evaluated at z = 1; i . e . , -r'(1) and at the observed equilibrium; i s nonnegative due to the semidefiniteness consumer substitution matrices. (22) L -Jftip (1) 1 A D D £ T p'(1) properties of the producer Using also (18), we find + [p'(1) + t ] T B g [p'(1) + t]} 2 0. g This approximation uses only information observable at the prevailing equilibrium as Harberger o r i g i n a l l y required, so that i t i s highly valuable in empirical a n a l y s i s . The next task i s to compute an approximation of the HB measure for the same economy and compare i t with the approximation of the ADD measure. To begin with, we must c l a r i f y which reference optimal equilibrium to pick from a set of Pareto optimal allocations to calculate the HB measure or i t s approximations. According to Negishi's (1960) theorem, every competitive equilibrium i s a solution of the maximum of a linear s o c i a l welfare function H h h T T.^-^ f for some set of weights a = (a^,...,a^) given resource constraints a and production p o s s i b i l i t i e s of the economy, where i t i s assumed that f* , 1 , . . . , H are concave functions. 1 h= In our model, this means that for some vector 16. a, a perfectly competitive equilibrium i s a solution of the following programming problem: (23) Max {E ? a f (x )s.t. x ,y h h k h h h 1 E ^ x N E j ^ y * + E^x* ; 1 y k e S , k k = 1,...,K}. Using the Uzawa-Karlin Saddle-point Theorem using the d e f i n i t i o n (4), (A.1), we can rewrite (23) as follows (6) and (the c a l c u l a t i o n is analogous to the derivation of (7) i n Appendix II): (24) Max Min {a u T B( p,u)}. P( N We assume that (i) (u^,p^) solves (24), (24) hold with equality so that p ° » 0 , N (ii) the f i r s t order conditions for (iii) B i s twice continuously d i f f e r e n t i a b l e at the optimum, and (iv) B ^ + B ^ i s negative d e f i n i t e . qq PP assumptions (i) and ( i i ) , we find the f i r s t order conditions for (24) (25) a = 7 B(p,p,u), (26) - V B(p,p,u) - V B(p,p,u) = 0. From are: Condition (26) i s the equality of demand and supply at the optimum while (25) i s the rule to equate the marginal s o c i a l importance of each person to the inverse of his marginal u t i l i t y of income (see Negishi (1960)).9 Note that the solution depends on a which i s equivalent to picking a reference 17. equilibrium. We have to pick one reference equilibrium from various competitive e q u i l i b r i a corresponding to various a. Varian (1974, 1976) persuasively discussed the welfare significance of the equal d i v i s i o n equilibrium, which i s a perfectly competitive equilibrium obtained from the equal d i v i s i o n of i n i t i a l endowments across i n d i v i d u a l s . Varian (1976), following the approach of Negishi (1960), also examined the r e l a t i o n s h i p between his theory of fairness and more t r a d i t i o n a l welfare economics based on the concept of a s o c i a l welfare function, which we followed i n t h i s section. By Negishi s theorem, the equal d i v i s i o n equilibrium i s 1 also characterized as a solution to a nonlinear programming (23) for some choice of a. By finding this a and associated reference price vector p^, we can find the HB measure. We now compute the second order approximation to the HB measure around the optimal equilibrium i n an analogous way as we computed the approximation to the ADD measure. F i r s t we construct a z-equilibrium: (27) V B(p(z) + t z , p(z), u(z)) (28) - 7 B(p(z) + t z , p(z), u(z)) u g When z = 0, = a + Xz ; - V B(p(z) + t z , p(z),u(z)) = 0. p (27) and (28) coincide with the f i r s t order conditions for the maximum of s o c i a l welfare (25) and (26), s p°. When z = 1, i f we define u(0) = u ° and p(0) (28) i s a set of equations to show the market clearing conditions at the tax-distorted equilibrium, i f u(1) = u^ and p(1) the values p r e v a i l i n g at the observed distorted equilibrium. = p^ are If we assume that the l e v e l of lump-sum transfers from the government to consumers are 18. appropriately chosen, there e x i s t budget constraints for consumers compatible with (27) and (28). (28) and these budget constraints imply the budget balance of the government. When z = 1, (27) quantifies the d i s t o r t i o n s at the observed equilibrium; i . e . , 'equity' -X^ shows the difference between the marginal s o c i a l importance of the hth person and the inverse of his marginal u t i l i t y of income. It must be noted that both a and the marginal u t i l i t y of income are not invariant to a monotone transformation of f^(x^). However, they are adjusted proportionally so that (25) i s v a l i d . We must also adjust X^ proportionally to h's marginal u t i l i t y of income and a so that (27) is valid. Now d i f f e r e n t i a t e 3 , uq uu (29) (27) and (28) with respect to z; B , qu' B Z Z qq + B Z pp u' (z) -X + B t uq P' (z) B t qq z 2 where B ^ = V^B(p(z) + t z , p(z), u(z)) 0 H)<N . Z for i , j = q , p , u . Note that B z u p = Assumptions ( i i i ) and (iv) guarantee, v i a the I m p l i c i t Function Theorem, that once continuously d i f f e r e n t i a b l e functions u(z) and p(z) satisfying (29) e x i s t at z close to 0. sides of (29) and using property ( i i i ) T T Premultiplying [ 0 „ , p(z) ] to both H of the overspending function i n Appendix I, we can derive (30) V m (p(z) + t z , u ( z ) ) u£ (z) h u h = z [ t B ( p ' ( z ) + t) + t T z VJVJ T B z VJ u u'(z)], 1 19. analogously to the derivation of (17) z =0, i n Appendix III. Evaluating (30) at we get (31) h=lV [ h ( p °' h h U ) u ( 0 ) = °" Analogously to the derivation of (19) i n Appendix IV, we next differentiate (30) with respect to z, and evaluate at z = 0 to obtain (32) U'(0) T + E ! V B ° u'(0) h u - P'(0) Premultiplying (29) h { P 1 T °' h U ) U h ( 0 ) = t T B qq ( p ' ( 0 ) + fc) B Ju'(O). toLp'(0) ] and adding the T evaluated at z = 0 by n resulting i d e n t i t y to (32), we have u'(0) B V(0) + E " V m (p°,u°)u^(0) (33) T = - p'(0) B °p'(0) h h u 1 T u - [p'(0) + t ] T p B ° [p (0) + t] 1 g > 0. A second order Taylor approximation to the HB measure (3) at z = 0 i s as follows: ( 3 4 ) L HB 5 - E h=lV °' 2 h h(p E Substituting (31) and (33) u )u (0) " H[u-(0)V\r(0) h-lV °' S> h(p u U h into (34) we have ( 0 ) ^ J + 20. (35) L £ - J i { p ' ( 0 ) V p ' ( 0 ) + [p'(0) + t ] B ° [ p ' ( 0 ) + t]} HB pp qq T > 0. To compute (35), we could again replace B?^ by B ^ i n (35) and (29) since the B^j are observable, again following H a r b e r g e r . 10 I t i s interesting to com- pare (35) with the second order approximation to the gain i n s o c i a l welfare using the linear welfare function i n moving to the optimum from the distorted equilibrium, E " a [ f ( x ) - f ( x h h ( 3 6 ) l L = HB £ h h 0 h 1 + J h 1 ) ] = L ^ . We find that 4»'(0) B Su'(0) > L T u H B where the t i l d e shows i t i s an approximation of the o r i g i n a l measure and the inequality comes from the positive semidefiniteness of B ^ , which i s implied by the concavity of the u t i l i t y functions. According to Varian (1976;257), the l i n e a r u t i l i t y function does not count the problem of equity. Therefore, when moving from the equitable equilibrium to market distorted equilibrium, only measures efficiency loss and does not evaluate i t s equity l o s s . t h i s sense, L may be taken as a lower bound of the welfare r change. In 11 Li However, (36) shows that L H R i s even smaller than L ^ . This i s because, with diminishing marginal u t i l i t y of income, increasing the inequality i n terms of u t i l i t y (or r e a l income) holding the (weighted) sum of u t i l i t y constant tends to increase the aggregate expenditure necessary to a t t a i n the reference u t i l i t y allocation. This problem of inequity i n the HB measure may not a r i s e i f we adopt money metric u t i l i t y scaling so that u 1,...,H. 2 1 H, = L H B = m (p^,u ), n h h h= i f t h i s i s the case, v B ( p ° , u ) = 1„ and B = 0 „ „ so that u H uu H«H and = L ^ . With t h i s assumption we can regard the HB measure as 0 B 21 summing the change of u t i l i t i e s of i n d i v i d u a l s ; i . e . , i t is a u t i l i t a r i a n measure of welfare. We now compare the empirical implementability of L and L H B i n (35). A Q D i n (20) and (22) Though (20) and (35) look i d e n t i c a l , t h e i r meanings are completely d i f f e r e n t . First, the substitution matrices are evaluated at distorted l e v e l of u t i l i t i e s i n (20) while they are evaluated at optimal l e v e l of u t i l i t i e s i n (35). of equations, (16) and (29). was already stressed, Second, p'(0) i s calculated from d i f f e r e n t sets The f i r s t difference i s i n e s s e n t i a l , since, as we replace these matrices with matrices evaluated at an observed distorted equilibrium. However, the second difference matters 1 1 the substitution matrices, B , B , tax rates t and pp qq reference bundles 8 are a l l information required to compute p'(1) and hence crucially. (22). In (22), In (35), we need both the substitution matrices B \ B ^ and income PP qq effect matrices B ^, tax rates t and the d i s t r i b u t i o n a l d i s t o r t i o n parameters X so that the informational requirements are much higher. Though i t is 1 1 possible to calculate B and B from l o c a l information on ordinary demand pp qq curves and supply curves at the distorted equilibrium, we have to know the 1 1 1 1 marginal u t i l i t y of income V B(p +t,p ,u ) to compute X from (27) or B from ordinary demand curves. Even i f we adopt the money metric scaling convention using the optimal p r i c e s , this does not give information on the marginal u t i l i t y of income at the observed equilibrium, and t h i s i s what we r e a l l y require. p 1 + t, If we adopt money metric scaling at the observed distorted prices 1 . then B i s easy to calculate since v B(p qu u case we also have B UU = 0„ „ . n*H 0 anymore. In t h i s However, we s t i l l cannot compute X from (27) since now we do not have 1„ = v B ( p ° , p ° , u ) ; M 1 1 1 + t , p ,u ) = 1 „ . H i.e., a i s not a vector of ones U McKenzie (1983, chapter 3) studied the methodology for c a l c u l a t i n g 22. the money metrics, and he c o r r e c t l y pointed out that the marginal u t i l i t y of income i s not an operational concept without knowing the u t i l i t y function. His approach i s based on normalizing the marginal u t i l i t y of income at one price vector, but i n our case, we have to know i t at two sets of prices p and p + t, and we cannot normalize twice. Diewert (1984;36) already pointed out that his approximate HB measure depends on the hypothetical income vector at the optimum which i s d i f f i c u l t to obtain. Though we adopted a different method of approximation, the same problem seems to occur by the measurement of marginal u t i l i t y of income (more rigorously, the difference between the marginal s o c i a l importance and the inverse of the marginal u t i l i t y of income), instead of the measurement of hypothetical income. these observations, In l i g h t of we must conclude that the approximate HB measure lacks empirical o p e r a t i o n a l l y without a knowledge of the o r i g i n a l u t i l i t y functions whereas the ADD measure i s free from this problem. Note that this c r i t i c i s m w i l l also apply even i f we compute the waste using the Bergson-Samuelsonian s o c i a l welfare function. It i s c h i e f l y for t h i s reason that we adopt the ADD measure as our welfare c r i t e r i o n . Needless to say, however, the informational advantage of using the ADD measure does not mean that i t i s a superior measure to either the HB measure or the BSSWF. As long as we can measure the difference between the weight of a linear BSSWF and the inverse of the marginal u t i l i t y of income, the same type of analysis as is presented i n Chapters 2 and 4 for the ADD measure can be carried out using the HB measure or a BSSWF. We have compared the informational requirements for the approximations of the ADD and the HB measures to be empirically computable, and i n this context we have found a remarkable property of the ADD measure: i t i s comput- able from l o c a l information on supply curves and ordinary demand curves at the observed equilibrium. A natural defect of our approximation approach i s that the approximation might deviate from i t s true value considerably when the 'gap' between two e q u i l i b r i a i s large. The numerical general equilibrium approach by Shoven and Whalley (1972, 1973, 1977) chooses an alternative way to compute e q u i l i b r i a d i r e c t l y corresponding to various tax and expenditure policies so that a more exact welfare evaluation seems a v a i l a b l e . However, an obvious drawback of the numerical general equilibrium approach i s that we must have information on global functional forms of u t i l i t y and production functions. In contrast, our approximation approach requires only second order derivatives of these functions evaluated at the observed equilibrium. As an important c o r o l l a r y of t h i s fact, derived from any set of f l e x i b l e the observed equilibrium. our approximate measure can be functional forms using information based on On the contrary, i n the numerical general e q u i l i - brium approach, very r e s t r i c t i v e functional forms are adopted to make global computation possible, and these r e s t r i c t i o n s are easily rejected i n econo- metric tests using more general functional forms (see Jorgenson (1984;140)). Moreover, the approximation approach does not involve any numerical computations that are more complicated than a single matrix inversion, whereas there are often substantial numerical d i f f i c u l t i e s equilibria. involved i n computing general Therefore, these two competing programs have their own pros and cons so that i t would be d i f f i c u l t to judge which one i s u n i v e r s a l l y superior to the other. 1 3 The measurement of waste i s prominently a p r a c t i c a l subject. As i s pointed out by Harberger (1964;58), the comparison of welfare measures i s only constructive way to give a policy prescription under the 'nth best' the 24. situation,1* i . e . , feasible by comparing the amount of waste corresponding to various p o l i c i e s we can give a ranking among them even i f there are various other d i s t o r t i o n s . However, as long as we use approximations, we cannot avoid approximation errors which might cause erroneous policy assessment. For example, Green and Sheshinski (1979) pointed out that Harberger's t r i a n g l e approximation may change considerably by changing the choice of approximation point. In t h i s context, they c r i t i c i z e d Feldstein (1978) who measured the net benefit of c a p i t a l income tax reform by comparing Harberger's (1964) measure at two taxed e q u i l i b r i a . noted that there e x i s t differences Green and Sheshinski between Feldstein's Harberger measure and a second order approximation of income gain evaluated at the i n i t i a l tax equilibrium. A s i m i l a r c r i t i c i s m also applies to Turunen (1986) who applied the approximate ADD measure for the numerical assessment of gains from t a r i f f reform. It would be possible to derive Green-Sheshinski l i k e approximate gains formula for tax reform which i s a second order approximation to the change of the ADD measure evaluated at an i n i t i a l tax e q u i l i b r i u m . 1 5 However, due to the complexity of the r e s u l t i n g formula, we have omitted derivation. this Therefore, t h i s approximation error may lead to reversals i n the true ranking of p o l i c i e s based on the exact amount of waste. We have to admit a dilemma that we cannot get an exact welfare measure for various sets of p o l i c i e s either by approximation or by equilibrium computation while we have to reach some decision on the choice or reform of economic p o l i c i e s . In the second best theory approach originated by Lipsey and Lancaster (1956), recommendations for p o l i c i e s or t h e i r p a r t i a l reforms are given using the programming method under the constraint that some of the optimality conditions are not met, or some of the instruments to a t t a i n the f i r s t best i s r e s t r i c t e d . This approach has successfully derived many i n t e r - esting r e s u l t s i n optimum taxation theory, piecemeal p o l i c y recommendations and cost-benefit analysis.16 drawbacks of this approach. However, we have to note at least two basic First, i n contrast to the f i r s t best s o l u t i o n , general second best solutions cannot be decentralized i n a simple p r i n c i p l e (see Guesnerie (1979)) so that the p o s s i b i l i t y of meaningful policy recommendations i s quite r e s t r i c t e d except under rather s i m p l i f i e d second best situations as i n an optimal taxation economy . Second, since most of the second best r e s u l t s depend on l o c a l necessary conditions for optimality, they suffer from t h e o r e t i c a l c r i t i c i s m s from the viewpoint of general equilibrium theory. As i s shown by Foster and Sonnenschein (1970) and Hatta (1977), multiple e q u i l i b r i a and i n s t a b i l i t y can e a s i l y occur i n a well-behaved economy with t a x - d i s t o r t i o n s . Harris (1977) pointed out that the sufficiency of the necessary conditions for second best optimality depends on the t h i r d order derivatives so that the i n t e r p r e t a t i o n of these sufficiency is not easy. In contrast, conditions tax reform approach due o r i g i n a l l y to Meade (1955) avoids the problem by r e s t r i c t i n g i t s attention to the l o c a l area around the observed d i s t o r t e d equilibrium. Various authors, represented by D i x i t (1975) and Hatta (1977), derived sufficiency conditions for welfare improvement by some p o l i c y changes. Unfortunately, these conditions depend on many r e s t r i c t i v e assumptions. E s p e c i a l l y , the assumption that the policy maker can change the set of taxes incrementally i s often i r r e l e v a n t , since reform a l t e r n a t i v e s are discrete changes of taxation. its By the same token, it i s often the case that the reform alternatives are i n s t i t u t i o n a l l y r e s t r i c t e d to the ones which are short of f u l l y s a t i s f y i n g the s u f f i c i e n t conditions. In these cases, t h i s approach cannot t e l l anything about the ranking of 26. policies, but our approach can. Furthermore, the sequence of l o c a l improvements may not converge to global optimum, but may stay on a l o c a l optimum or some stationary point. These problems seem to give l i m i t a t i o n s on the use of l o c a l optimality or improvement conditions for policy recommendations. Considering these defects, we seem to be obliged to conform to a convent i o n a l view on second best; i . e . , i f conditions on propositions are met, implement the prescribed p o l i c y . If the actual economic situations do not coincide with the conditions, or we do not have enough information to judge whether i t i s actually the case, we cannot t e l l anything from the second best theory. P a r t i c u l a r l y , even i f conditions are not met for positive best propositions, second this does not j u s t i f y the status quo i n any way, since even i n this case, the deadweight loss of the economy could be too large to neglect. Following Harberger (1964), "The Economics of nth Best," to measure the deadweight loss associated with the economy's being i n any given nonoptimal position i s of high p r a c t i c a l importance when we cannot know how to make the best of a bad s i t u a t i o n . 1-3. A Diagraamatic Exposition In t h i s section we i l l u s t r a t e diagrammatically the ADD measure and the HB measure and their approximations using a simple model i n order to c l a r i f y the i n t u i t i v e content of the discussions i n the previous section. We assume that there i s one good and one production factor (labour). One aggregate firm produces the good y using labour v according to the production function y <. g(v). We also assume that there i s a single consumer who enjoys u t i l i t y u from the consumption of the good x and l e i s u r e L by 27. means of the u t i l i t y function f ( x , L ) . The i n i t i a l endowment of labour i s v and there i s a zero endowment of the good. We f i r s t specify the tax-distorted observed equilibrium. labour as numeraire so that i t s p r i c e , w = 1 . We choose We assume that there i s a s p e c i f i c tax t on the good levied for consumption so that i t s producer price i s p whereas i t s consumer price i s p + t . It i s also assumed that the s p e c i f i c tax revenue i s transferred to the consumer as a lump-sum transfer. Then, using the p r o f i t function w(1,p) dual to y < g(v) and the expenditure function dual to f ( x , L ) , the observed equilibrium i s characterized by the market c l e a r i n g conditions for the good and labour; (37) V TT(1,P) - v m ( 1 , p+t, (38) V TT(1,P) - v m(1,p+t,u) + v = 0. W u) = 0 w 1 1 We assume that (p ,u ) solves (37) and (38) uniquely. From the homogeneity properties of TT and m, we can deduce (39) i.e., v + i r ( 1 , p ) + t V m , ( 1 , p + t , u ) = m( 1 , p + t , u ) ; 1 1 1 1 1 p the budget constraint of the representative i n d i v i d u a l i s satisfied. Now we define the ADD measure of waste i n t h i s simple model. We assume that the surplus of the economy i s measured by the numeraire good, labour. Therefore, the general primal programming problem ( 1 ) and i t s dual (7) are s i m p l i f i e d respectively i n t h i s model as follows: 28 ADD ( 4 0 ) L (41) 5 M a X = Min y,v,L ^ " { ~ L V : y 1 {TT(1,P) - m(1,p,u ) 1 p > Q ' x y 1 g ( v ) ' f ( x ' L ) 1 " } + v}. A We assume that p = p > 0 i s a unique solution of the f i r s t order condition: (42) V TT(1, ) - 7 m ( 1 , p \ u ) A 1 P tr Therefore, = 0. tr (40) and (41) can be rewritten as (43) v + 7 w(1, ) - V m ( 1 , p \ u ) = L (44) v + ii(1,p ) A w A Note that (43) 1 P w = L and (44) properties of TT and m. + m(1,p ,u ). A & n n A D D 1 are equivalent by using (42) and the homogeneity We can i l l u s t r a t e the ADD measure of waste diagrammatically i n F i g . 3. The program (40) b o i l s down to searching for a point where the horizontal length of the lens-shaped area formed by the 1 . production p o s s i b i l i t i e s maximal. set and the indifference curve with u = u This maximum i s characterized by an equal slope 1/p is of the two curves. In t h i s simple example, we can also express the ADD measure of waste as a more familiar Hotelling-Harberger-like c u r v i l i n e a r t r i a n g l e ABC i n F i g . 4. This can be proven as follows. The area ABC i s defined from F i g . 4 as (45) ABC = ; From t h i s we have ABC = m(1,p +t,u )-mn,p ,u )+Tr(1,p )-ir(1,p )-tV m( 1,p +t,u ), P 1 1 A = v - m( 1, p , u ) A = L 1 1 A 1 + ir(1,p ) 1 (from A 1 (39)) (from (44)). ADD In F i g . 4, we have also drawn two triangles ABC and ABC". ABC i s a linear approximation to ABC using the slopes of the Hicksian demand curve and the supply curve at the optimum point whereas ABC" i s a l i n e a r approximation to ABC using the slopes of the two curves at the distorted equilibrium. These two triangles of waste (20) correspond to the two approximations of the ADD measure and (22) i n t h i s simple example. To show t h i s , l e t us first construct a z-equilibrium as i n the previous section for t h i s simple model as follows: (46) V i(1,p(z)) - v m(1,p(z)+tz,u ) = 0, (47) V ir(1,p(z)) - V m(1,p(z)+tz,u ) + v = r ( z ) , w w 1 p 1 where 0 < z < 1 and p(0) z = 0, (46) and (47) = p , p(1) A = p , r(0) correspond to (42) 1 = L and (43), A m ) and r(1) = 0. When and when z = 1, they 30. correspond to (37) and (38). T o t a l l y d i f f e r e n t i a t i n g (46) and (47) with respect to z we can compute r ' ( z ) . From t h i s r ' ( 0 ) , r"(0) and r'(1) are also computable so that we can calculate the two approximations to the ADD measure of waste (20) and (21) as follows (see Appendix V ) : th P °P (48) 2 ^ oPoP " t E 2 (49) 2(E 1 pp 1 PP s S 0 PP O PP 1 PP - s 1 PP ) where E = V m( 1 ,p(z)+tz,u ) and S = V ir(1,p(z)) for z = 0,1 pp pp pp pp irv Z 2 1 z 2 As AB = t and the height of the t r i a n g l e ABC i s 0 (tE PP (48) equals the area ABC' while the height of ABC" i s S (tE 1 PP so that (49) equals the area ABC". (See Appendix V.) - S ° ), ° ) / ( E ° PP PP S 1 ) / ( E PP - S 1 PP PP 1 ) PP Note that the slope of A C equals the slope of the demand curve at C while the slope of B C equals the slope of the supply curve at the point C. Therefore, i n t h i s simple model, our triangular expression of the deadweight loss corresponds to that by Harberger (1964) except that we allowed for nonlinear production possibilities set. We next turn to a diagrammatic interpretation of the HB measure of welfare and i t s approximations. For t h i s purpose, we f i r s t have to find a price vector which supports the s o c i a l optimum. In a single consumer economy, i t may be defined as the price vector which corresponds to the 31 . u t i l i t y maximum given resource and technology constraints. Therefore, i t is B B the p r i c e solution (p ,w ) to the concave programming problem below: (50) Max {f (x,L) (51) = Max M i n u : y ) x, v ) v H , p > 0 w > ( ) y < g(v)} { u - m(w,p,u) + n(w,p) + wv}. B B B We assume that an i n t e r i o r optimum point (u ,p ,w ) solves (51) uniquely with B p B > 0 and w > 0. conditions for It i s a solution to the following f i r s t order necessary (51): (52) v m(w ,p ,u ) = 1, (53) V i(w ,p ) (54) v + V ir(w ,p ) u B - v m ( w , p u ) = 0, B B p 1 B p B B w B ( - V m ( w , p , u ) = 0. B B B w As (53) and (54) are unchanged by a proportional change of w and p, we set w =1. For t h i s normalization, we can assume that (52) i s always met by choosing a money-metric normalization of the u t i l i t y function at the reference p r i c e (1,p ). Therefore we can delete (52) from the system and B assume that (53) and (54) determine u normalization, B and p B from w = 1 . (53) and (54) imply the following budget constraint of the representative consumer for the optimum price vector (1,p (55) Using t h i s v + ir(1,p ) D = m(1,p ,u ). D D ): 32. Now the HB measure of welfare (3) i n t h i s simple model may be defined as follows: (56) L Fig. = m(1,p ,u ) B H B - m(1,p ,u ) B B 1 5. i l l u s t r a t e s the HB measure for t h i s simple model. This i s nothing but a Hicksian compensating v a r i a t i o n when moving from a tax-distorted equilibrium to a s o c i a l optimum. We also i l l u s t r a t e L _ using a Hotelling-Harberger-like expression i n u rib Fig. 6. This figure i s the same as F i g . 4 for the Hicksian compensated demand curve for the good V m(1,p,u ) using the tax-distorted u t i l i t y l e v e l P u^ and the supply curve v i ( 1 , p ) . We also include the compensated demand P 1 curve for the good for the s o c i a l l y optimum u t i l i t y l e v e l , Fig. Vpm(1,p,u ). 6 corresponds to the case where the good i s normal so that Vpin(1,p,u ) i s above V m(1,p.u**). If the good i s i n f e r i o r , the former curve i s below the l a t t e r curve and i f the good changes from a normal to an i n f e r i o r good then the two curves i n t e r s e c t . above. Using F i g . Our results below apply to a l l cases l i s t e d 6, the HB measure can be shown to be equal to the sum of two c u r v i l i n e a r t r i a n g l e s AFE and FBD. To show t h i s , f i r s t note that L „ can D Ho be decomposed as follows: (57) L = {m(1,p ,u ) - m ( 1 , + t , u ) } + {m(1,p +t,u ) - m ( 1 , p , u ) } . B H B B 1 1 1 1 B 1 P Substituting (39) and (55) into the f i r s t term on the right-hand side of (57), L H B may be further rewritten as 33. (58) However, the sum of the areas AFE and FBD, denoted as AFBCDE, is (59) P 1 p 1 Performing the integration i n (59) yields the expression i n (58). For this simple model, the triangles ABG and ABC" drawn i n F i g . 6 correspond to the approximation to the HB measure where ABG corresponds to (35) and ABC" i s i t s variant where observed information is used. shown as follows. (60) First, v T(1,P(Z)) (61) construct a z-equilibrium: - V m(1,p(z) + t z , tr v + V »(1,p(z)) w u(z)) = 0, tr - v m(1,p(z) + t z , u(z)) w where 0 <. z < 1 and p(0) = p , p(1) = p , u(0) B z = 0, This may be 1 = 0, = u B and u(1) = u . g 1 When (60) and (61) correspond to (53) and (54) with w = 1 and when z = 1 they correspond to (37) and (38). respect to z, we can derive u ' ( z ) . T o t a l l y d i f f e r e n t i a t i n g (60) and (61) with From this u'(0) and u"(0) can also be computed so that we can calculate the second order approximation to the HB measure (34) as (see Appendix V I ) . (62) 34. where E * = V J m ( 1 , p ( z ) PP PP + t z , u ( z ) ) and S = 7 ir(1,p(z)) for z = PP 0,1. 2 z PP Note t h a t E ^ and S ^ a r e d i f f e r e n t from t h e analogous e x p r e s s i o n i n (48) s i n c e , i n ( 6 2 ) , t h e d e r i v a t i v e s a r e e v a l u a t e d a t the s o c i a l l y optimum p o i n t B B (1,p ,u ). As AB = t and t h e h e i g h t of t h e t r i a n g l e ABG i s (tE J S ?)/([ J - S J), (62) e q u a l s t h e a r e a ABG. I f (62) i s f u r t h e r approx- imated by r e p l a c i n g t h e d e r i v a t i v e s E ° and S by t h o s e o b s e r v a b l e d e r i v a PP PP 1 •) tives E and S , t h e n t h i s a p p r o x i m a t i o n i s i d e n t i c a l t o (49) which i s a 0 trkr trtr suggested a p p r o x i m a t i o n of t h e ADD 6 as ABC", measure. (49) i s i l l u s t r a t e d i n F i g . which i s a l s o shown i n F i g . 4. I n t h i s s i m p l e model, t h e ADD measure and t h e HB measure c o i n c i d e i f t h e two p o i n t s C and D c o i n c i d e i n F i g . 6; i . e . , the ADD optimum c o i n c i d e . a r e second (48) and (62) (or A B C optimum and t h e i n F i g . 4 and ABG o r d e r a p p r o x i m a t i o n s of the ADD social i n F i g . 6), which measure, and the HB measure c o i n c i d e i f t h e c u r v a t u r e s of the compensated demand f u n c t i o n s and the s u p p l y f u n c t i o n a t p o i n t s C and D a r e the same. t o t h e s e f u n c t i o n s depending However, the f u r t h e r approximations on the d e r i v a t i v e s of t h e s u p p l y and compensated demand f u n c t i o n s a t t h e observed e q u i l i b r i u m c o i n c i d e f o r t h i s s i m p l e model as the t r i a n g l e ABC". I t i s , however, c l e a r from t h e d i s c u s s i o n of t h e p r e v i o u s s e c t i o n t h a t t h i s i d e n t i t y cannot go through f o r a g e n e r a l many-consumer model. F i n a l l y , a l l of t h e s e a p p r o x i m a t i o n s c o i n c i d e 1 B i f V m(1,p,u ) = 7 m(1,p,u ) f o r a l l p. P T h i s i s t h e case where t h e r e i s no P income e f f e c t f o r t h e good and, i n t h i s case, t h e M a r s h a l l i a n consumer's s u r p l u s c o i n c i d e s w i t h t h e ADD and the HB measures (see H i c k s (1946;38-41)). 35. FOOTNOTES 1 FOR CHAPTER 1 These two measures were examined comparatively by Diewert (1981, 1984, 1985a). 2 x » 0 means that each element of the vector x i s s t r i c t l y p o s i t i v e , N x > 0„ means that each element of x i s nonnegative, and x > 0„ means x > 0., — N N but x T* 0 3 N> N — A superscript T means transpose. Note also that the solution to (1) may not correspond to a Pareto optimal point. This does not, however, contradict the Pareto inclusiveness of the ADD measure which i s discussed i n this section. * See Diewert (1982;554) for the r e g u l a r i t y properties that must be s a t i s f i e d by the functions m . h 5 Most welfare evaluation methods cannot even generate orderings. example, the Kaldor (1939)-Hicks (1939)-Scitovsky complete nor t r a n s i t i v e (see Gorman (1955)). For (1941-2(a)) test i s neither Aggregate Hicksian (1941-2) compensating and equivalent variations cannot be t r a n s i t i v e i f the base price i s not fixed (see, for example, Mohring (1971;365-7) or Blackorby and Donaldson (1985;256-7)) . 6 A widely adopted welfare measure by Diamond and McFadden (1974) can be shown to be an equivalent v a r i a t i o n where tax-distorted prices are base prices. Therefore, i t cannot give a consistent ranking of u t i l i t i e s various tax schemes even i n a single-consumer economy, i . e . , inclusive. 7 across not Pareto See Kay (1980) and Pazner and Sadka (1980). We also assume that the Slater constraint q u a l i f i c a t i o n condition applies i n t h i s economy; i . e . , we require that a feasible solution for (1) exists that s a t i s f i e s the f i r s t N inequality constraints s t r i c t l y . The d e f i n i t i o n s of various concepts of separability and t h e i r economic 8 applications are surveyed i n Geary and Morishima (1973) and Blackorby, Primont and Russell (1978). Pages 52-61 of the l a t t e r book are important for our a n a l y s i s . With appropriate lump-sum transfers across households, 9 the budget constraints of individuals are s a t i s f i e d and the government budget constraint i s implied by them and (26). Combined with Negishi's theorem, the program (23) and i t s interpretation may be regarded as a restatement of the second fundamental theorem of welfare economics, due o r i g i n a l l y to Arrow (1951). It i s d i f f i c u l t i n t h i s case to interpret t h i s further approximation 1 0 by adopting the Quadratic Approximation Lemma in the same manner as with the ADD measure. However, using the money-metric u t i l i t y scaling adopted l a t e r , we can show that - ^ { p ( 1 ) B p ' ( 1 ) + [p'(1)+t] B [p'(1)+t]} + Jau'(1) B ^ 1 T T pp u'(1) T qq u i s also accurate for quadratic functions as (35) by this Lemma. If we assume that there exists a concave Bergson-Samuelsonian s o c i a l 1 1 welfare function which i s maximized at the equal d i v i s i o n equilibrium, we can show that the second order approximation to the difference of the BSSWF, evaluated at an optimum or distorted equilibrium L inequality L 1 2 R S R S , satisfies 2L L The term money metric u t i l i t y was introduced into economics by Samuelson (1974), but the concept dates back to McKenzie (1957). that m (p^,u^) h i s s t r i c t l y increasing i n u . given by Weymark (1985). x 1 the h We assumed Its s u f f i c i e n t condition was We also assume that m ( p ^ , f ( x ) ) i s concave i n h h h for the reference price p^, but t h i s i s not guaranteed i n general. Blackorby and Donaldson (1986). See Applications of money metrics to applied welfare economics are given by King (1983) and McKenzie (1983). Most computable general equilibrium models adopt n e o - c l a s s i c a l 1 3 market assumptions. perfect However, Pigott and Whalley (1982) incorporated public goods and Harris (1984) introduced increasing returns to scale by allowing fixed costs i n numerical general equilibrium models. 1 * In contributions collected i n Harberger (1974), he applied his methodology i n various policy assessments. Many studies use the HB measure or Marshallian consumer surpluses for the same purpose (see Currie-MurphySchmitz (1971)). 1 5 Needless to say, both second order approximations as well as a mean value of the two f i r s t order derivatives are exact approximations for quadratic functions. 1 6 We do not survey these studies i n this paper. Excellent surveys were provided by Auerbach (1985;86-118), Mirrlees (1986) and Dreze and Stern ( 1986). 38. Appendices for Chapter 1 Appendix I : The Properties of the Overspending Function. An overspending function, introduced into economics by Bhagwati, Brecher and Hatta (1983;608) summarizes the general equilibrium relations of an economy within one equation. It may be interpreted as the aggregate net expenditure of consumers facing prices q minus the aggregate p r o f i t s of firms facing prices p. It i n h e r i t s many useful properties of expenditure functions and p r o f i t functions which are exhibited i n Diewert (1982). We c o l l e c t several important properties for l a t e r use. An overspending function i s defined by (A.1) B(q,p,u) = E " m ( q , u ) - E " q x h h 1 T h h h 0 - E^ iT (p) . k 0 I t has the following properties. (i) (ii) B i s concave with respect to p and q. If B(q,p,u) i s once continuously d i f f e r e n t i a b l e with respect to q and p at ( q , p , u ) , then V B(q,p,u) i s the aggregate net consumption q vector and -V B(q,p,u) i s the aggregate net production vector, p (iii) The following i d e n t i t i e s are v a l i d for any (q,p,u) i f B i s twice continuously d i f f e r e n t i a b l e at T (A.2) (A.3) q \ u (q,p,u): T ° ( , u B | T * <3» (<!.u,>/3u l 8n (q,u )/3u ), H 1 H H 3 9 . (A.4) where PP . = V,j.B(q,p,u) for i , j = q , p , u . Property (i) follows from the fact that an expenditure function i s concave with respect to prices and a p r o f i t function i s convex with respect to p r i c e s . Property ( i i ) i s a straightforward consequence of H o t e l l i n g ' s (1932;594) lemma and the Hicks (1946;331)-Shephard (iii) (1953;11) lemma. Property i s a consequence of the l i n e a r homogeneity of an expenditure function and a p r o f i t function with respect to p r i c e s . Appendix I I In t h i s Appendix, we show that (1) and (7) are equivalent given quasiconcave u t i l i t y functions and convex production sets, provided the Slater constraint q u a l i f i c a t i o n holds. In (1), convex from the quasi-concavity of f ( x ) , h and the i n e q u a l i t i e s are l i n e a r . S h k the set {x : f (x ) 2 u ) n i s also assumed to be convex Therefore, (1) i s a concave programming problem and the Uzawa-Karlin Saddle Point Theorem i s applicable. Rewrite (1) as: (A.5) r 0 = max {r + mm P>0, N T P [E f (x ) 2 uj h is h f y k=1 H k + E H ih - E h=1 h = 1,...,H; y k H h x - pr] h=1 e S , k = 1,...,K}, k 40. (A.6) = max min {r(1-p B) + ~ N T r p>0 E ^ p V +£,,!!,[max + E^^max p y T = max min r ^ { r ( 1 - p 8) + E ^ P ^ T p20 1 1 :y k k - p x T h : f (x ) > h h e S ]} + E ^ir (p) k k k - E " m (P,uJ)} h h 1 using the d e f i n i t i o n s (4) and (6). Appendix III In t h i s Appendix, we derive (17). Premultiply both sides of (16) by T [p(z) ,0]. (A.7) Using (15), we have: p(z) B q(p"(z) + t) + p ( z ) B p ' ( z ) = - r ' ( z ) . T T q pp 1 From (A.2) and (A.4) evaluated at (q,p,u) = (p(z) + t z , p ( z ) , u ) , we have T z T z T z T p(z) B = - zt B and p(z) B = 0 „ . Substituting these equations into qq qq pp N (A.7), we have (17). Appendix IV In t h i s Appendix, we derive (19). Differentiate (17) with respect to z and evaluate at z = 0, and we have (A.8) - r"(0) = - t B g ( P ' ( 0 ) T q + t). Next premultiply both sides of (16) evaluated at z = 0, by We obtain [p*(0) ,r'(0)]. T UjJ 41 . (A.9) - p'lOlVtpMO) qq + t) - p'(0) B T Adding (A.8) and (A.9), we have V ( O ) = 0. pp (19). Appendix V Total d i f f e r e n t i a t i o n of (46), o P'(z) S - E wp wp , -1 r'(z) s (A.10) 2 , - E PP 2 PP 2 L 2 z 2 1 where E - = V .m(1,p(z)+tz,u ) and (47) gives the following: PP E z i Premultiplying both sides by [ p ( z ) , 1 ] , S z wp + p(z)S z PP E = 0, 2 2 .ir( 1 ,p(z)) for i , j , = p,w. = using the + (p(z)+tz)E 2 wp t wp 2 = 0, ' pp identities ' we have (A.11) r ' ( z ) = ztE *(p'(z)+t). srtr Inverting the right-hand side matrix of (A.10), we have P'(z) = ( E 2 p t)/(S z p -E z p ) Substituting i t into (A.11), we have 42. (A.12) r'(z) = (zt S 2 E )/(S - E ) . PP PP PP PP Z Z Z Z Using (A.12), we can compute the two approximations (20) and (21) which correspond to (48) and (49) respectively. Now draw a perpendicular l i n e from point C to AB and define the cross point with AB, H. Then the height of the t r i a n g l e i s C H . - E ^-AH = S ^-(t-AH). PP PP From these two equations, ( t E °S ° ) / ( E ° - S ° ) . PP PP PP PP We have C H = we can solve C H = The proof of ABC" i s perfectly analogous. Appendix VI Totally differentiating -T S PU' pp -E ,S - E wu wp wp z Z E z pp (60) and (61) with respect to z, we have u' (z) E t PP P'(z) E t wp Z (A.13) Z Z 2 z 2 where E— = V .m( 1, p(z)+tz, u(z)) , z i We compute u'(z) z 2 = V^irt 1 ,p(z)) for i , j = w,p,u. by inverting the left-hand side matrix of (A.13). the determinant of the matrix D i s (A. 14) D = E z wu (S Z - E pp z ) pp - E (S - E * ) . pu wp wp Z Z Using the l i n e a r homogeneity properties of m and TT, First, 43. E w u (P + ( z ) + t z ) E P u V = { 1 , p ( z ) + t z , u ( z ) ' ) E +(p(z)+tz)E = 0, S +p(z)S = 0, wp pp wp ^ pp Z Z Z we can rewrite (A.14) as (A.15) D = - 7 m(1,p(z)+tz,u(z))(E -S ) - z t E S . u PP PP p u pp Z Z Z Z v The numerator of u ' ( z ) , defined as N, i s given by (A.16) N = {S - E > E t - (S I - E ) E t wp pp pp pp wp z Z wp z z z = {-p(z)S + ( p ( z ) + t z ) E } E t + (S - E ) ( p ( z ) + t z ) E PP pp pp pp pp Z z = zt s 2 Z Z z Z t pp E pp P P Z 2 From (A.15) and (A.16) we have (A.17) u'(z) = -{zt S 2 2 p E Z p }/{V m(1,p(z) tz,u(z))(E + u 2 p -S z p + ztE ) Z p S 2 p }. From (A.17) we have, u'(0> = 0 and u-'(0) = - { t S ° E ° > / { V m ( 1 , ° , u ° ) ( E ° - S ° ) > Substituting them into (34), 2 p we have p u P p (62). We can show analogously as Appendix V that (62) ABG. p coincides with the area 44. pr i s the reference bundle AB i s the ADD measure Fig. 1 The ADD Measure of Waste 45 . X 2 Scitovsky set 1 S(u ) Scitovsky set 0 S(u ) AB i s the HB measure Fig. 2 The HB Measure of Welfare Fig. 3 The ADD Measure: A One-Consumer Two-Goods Economy 47 . V m(1,p u) A n l = 7 n(1,p ) Xr A P x ' y Fig. 4 The ADD Measure and i t s Approximations: A One-Consumer Two-Goods Economy Fig. The HB Measure: 5 A One-Consumer Two-Goods Economy Fig. 6 The HB Measure and Its Approximations: A One-Consumer Two-Goods Economy 50. CHAPTER 2 THE MEASUREMENT OF WASTE IN A PUBLIC GOODS ECONOMY 2-1. Introduction In the long history of the study on the measurement of deadweight loss in applied welfare economics, the waste due to i n d i r e c t taxation has been the main concern of t h i s l i t e r a t u r e . This section proposes a methodology for measuring the waste due to an externality, which seems to be an alternative and equally important s i t u a t i o n involving a market f a i l u r e . methodology i s applicable to other e x t e r n a l i t i e s , Though our here we focus on the problem of public goods. Consider a government which c o l l e c t s revenue from both lump-sum and i n d i r e c t taxation and provides public goods. This economy exhibits the waste due to a price d i s t o r t i o n and to an incomplete market at the same time. As was already suggested by Harberger (1964;73), the deadweight loss of the whole economy depends on the difference between the s o c i a l benefit and s o c i a l cost of public goods i n addition to the set of i n d i r e c t taxes (or mark-up rates of noncompetitive firms). We derive approximations to the A l l a i s - Debreu-Diewert measure of waste of t h i s public good economy, and we show that the approximate deadweight loss can be expressed i n terms of the derivatives of r e s t r i c t e d expenditure functions and r e s t r i c t e d p r o f i t functions evaluated at the observed equilibrium as long as we know the marginal benefits of public goods for consumers. In deriving the approximate waste, we need not assume l o c a l l i n e a r i t y of the production p o s s i b i l i t i e s set as i n Harberger (1964) and we need not assume r e s t r i c t i v e functional forms for u t i l i t y and production function as i n the numerical or applied general equilibrium l i t e r ature. The waste to be studied i s due to the simultaneous existence of 51 . distortionary taxes and the nonoptimal provision of public goods. Needless to say, a simple sum of these two types of waste cannot even approximate the simultaneous loss measured i n this section. The next section i s devoted to the description of our model of a public goods economy, while section 3 defines the Allais-Debreu-Diewert measure of waste i n t h i s economy. In section 4, we compute second order approximations to the ADD measure to gain more insight about the nature of the waste. We also i n t e r p r e t the empirical significance of the approximate ADD measure. In section 5, some drawbacks to our approximate ADD measures are discussed and a diagrammatic exposition of our analysis i s presented. The Model 2-2. Our model i s similar to the one used i n Section 1-1 and 1-2 except that we now introduce public goods into the model. There are N private goods . . T which are traded at positive prices p = ( p ^ , . . . , p ^ ) and I public goods which affect both consumers' u t i l i t i e s and the production p o s s i b i l i t i e s sets of T firms. A quantity vector of public goods i s denoted as G = ( G , . . . , G ) 1 I > Oj. There are K p r o f i t maximizing private firms which produce goods and services by u t i l i z i n g both private and public inputs using the production k k k p o s s i b i l i t i e s set S for k = 1 , . . . , K , i . e . , i f (y , - G) e S , then the k k kT vector of net outputs y = ( y ^ , . . . , y ) i s producible by sector k using the k vector of public goods G. The sector k r e s t r i c t e d p r o f i t function ir , which k i s dual to the production p o s s i b i l i t i e s set S , i s : N (1) ir (p,G) = max {p y : (y, - G) e S K T y k }, k = 1 K, 52. where p > 0^. We assume that either G or an entrepreneurial factor i s a l i m i t i n g factor of production, so that S scale when G i s fixed. exhibits decreasing returns to (See Meade (1952) for the d e f i n i t i o n of an 'unpaid factor" public input.) The vector of public goods i s produced by the government, k = 0, which has the production p o s s i b i l i t i e s set S^. produces G using the input vector y. If (y,G) e S^, the government If some component of y i s p o s i t i v e , government i s j o i n t l y producing the corresponding private good with G. government r e s t r i c t e d p r o f i t function TT^, which i s dual to S^, (2) the The is: 0 T O TT (p,G) = max {p y : (y,G) e S }, y where p > 0 . 1 N M Let us now look at the consumer side of our model. are H i n d i v i d u a l s , h = 1 , . . . , H , i n the economy. We assume that there The preference of individual h can be represented by a quasi-concave u t i l i t y function f* 1 defined over a translated orthant i n R * , N + Q* . 1 Define the i n d i v i d u a l h h h r e s t r i c t e d expenditure function m , which i s dual to f , for h = 1 , . . . , H , by: (3) where p > 0 (x,G) e Q } h N and U r e Range f . We suppose that each i n d i v i d u a l h possesses nonnegative endowment vector of private goods, x h > 0 „ , for h = 1 , . . . , H . N also allow the government, which i s h = 0, to have an i n i t i a l endowment vector x^ -> 0N „. We As i n section 1-2, the government raises revenue by the set of i n d i r e c t T taxes t = ( t . | , . . . , t ) to provide the public goods. N make a net transfer g to i n d i v i d u a l h. h lump-sum tax c o l l e c t e d from person h. consumers face p + t > 0 The government can also If g^ < 0, -g^ i s the amount of Producers face prices p > 0 N whereas at the observed distorted equilibrium. N We use the overspending function defined by (4) B(q,p,G,u) = E " m ( q , G , u ) h h 1 h h=o q x to characterize our general equilibrium system. " k=0 E f ( P , G ) Diewert ( 1986;131-155, 170-176) showed that the properties of a p r o f i t function and an expenditure function are v a l i d i n t h e i r r e s t r i c t e d functional form. properties of an overspending function (i) Chapter 1 are v a l i d for (4). - (iii) This means that the l i s t e d i n Appendix I to Diewert (1986) also showed that: (iv) a r e s t r i c t e d p r o f i t function i s concave with respect to G i f the production possibilities set i s convex and a r e s t r i c t e d expenditure function i s convex with respect to G, so that B i s convex with respect to G, (v) -V_m^(q,G,u, ), o n for h = 1,...,H, i s the marginal benefit vector of consumer h for the v public goods; V v (p,G) for k = 1 , . . . , K i s the marginal benefit vector of firm k for the public goods, and -V_ir^(p,G) i s a marginal cost vector for the o n public goods, public goods. so that -V^B(q,p,G,u) shows the aggregate net benefit vector of From the l i n e a r homogeneity of B with respect to p r i c e s , identity (5) qT V T + P T B P G = ( V G B ) holds i n addition to (1.A.2) - (1.A.4). the The system of equations characterizing the observed equilibrium i s now hk stated i n a f a i r l y simple manner where a i s defined as the f r a c t i o n of a hk firm k held by i n d i v i d u a l h, with 0 < a < 1 for h = 1 , . . . , H and k = 1 , . . . , K and EjJ^a* * = 1 for k = 1 , . . . , K . 1 m (p+t G,u ) (7) V B(p+t,p,G,u) + V B(p+t,p,G,u) (8) -V B(p+t,p,G,u) Here (6) shows the budget constraints h ( h = g q + (p+t) x + E ^a Tr (p,G), h = 1,...,H, (6) T h h h K p V = d. G for the H individuals and (7) equality of demand and supply for goods 1 , . . . , N . s t r a i n t i s implied by (6) and (7). k k The government budget con- From the property (v), the net marginal benefit vector of the public goods. s i s t e n t with the well-known Samuelson (1954)-Kaizuka d i n (8) If d = 0^, (8) public goods are not supplied optimally. i s con- (1965) conditions We assume that the for distortions parameter d arises because of the limited a b i l i t y of the government We regard (6) defines Therefore, d f 0j means that the the optimal provision of public goods. provide public goods shows the to efficiently. - (8) as a general equilibrium system which determines P n « - - - P « r d , u and one component of t and g given the remaining components of t and g, with p^ = 1 as numeraire and G fixed. 1 1 1 distorted equilibrium (u ,p ,G , t , d , g ) exists. We assume that an observed 55. 2-3. An Allais-Debreu-Diewert Measure of Waste An Allais-Debreu-Diewert measure of waste that was defined and discussed i n Chapter 1 i s now u t i l i z e d to measure the waste due to the public good externalities. T Pick a nonnegative reference vector of private goods p = (P^,...,P^) T . > 0 and consider the following primal programming problem: N (9) r ° = max r,x {r : (i) E x h=1 H ,y , G . (ii) (iii) where u = (u.,...,u„) + pr < E v k=0 K k + E i h=0 H f ( x , G ) > u^ ; (x ,G) e Q , h = 1 , . . . , H ; h h h (y ,-G) e S , k = 1,...K; k k h (y°,G) observed d i s t o r t e d equilibrium defined i n the previous section. interpretation of L ^ repeat i t here. ; h e S° } i s the u t i l i t y vector which corresponds to the n 1 h The = r^ i s discussed i n Chapter 1 so that we w i l l not D D For s i m p l i c i t y of computation, our reference bundle does not include public goods. numeraire good, i . e . , A l l a i s proposed to measure the waste i n terms of a T . • i n our context p = ( 1 , 0 , . . . , 0 ) . Debreu's c o e f f i c i e n t of resource u t i l i z a t i o n model (which assumed that p was proportional to the economy's t o t a l endowment vector) is also consistent with our present model since we assumed that there were no endowments of public goods. Given the l e v e l of G, (9) i s a concave programming problem so that we can derive i t s dual equivalent problem-. (10) r ° = max [max m i n G r {r(1-p p) T p > 0 - B(p,p,G,u)}].2 (The process of derivation i s analogous to that i n Appendix 1-II.) 56. If G°, r ° and p ° solve (10), then p r ° i s a measure of the resources that can be extracted from the economy while maintaining households at t h e i r d i s t o r t e d equilibrium u t i l i t y levels and i s a corresponding "optimal" level of public goods and p ° i s a vector of private goods prices which supports the efficient equilibrium. Note that i n t h i s "optimal" equilibrium, not only are public goods being provided e f f i c i e n t l y , but also a l l commodity tax d i s t o r t i o n s have been removed. Given the l e v e l of G, (10) may be rewritten by using the Uzawa-Karlin Saddle Point Theorem i n reverse as (11) r ° = - max {B(p,p,G,u ) : p p > 1} N 1 n T where B i s the overspending function defined by (4). solves (10), If G°, r ° and p ° then p^ solves (11) and r^ i s the associated Lagrangean m u l t i p l i e r for the constraint i n (11). and p^ solve (10), It i s also the case that i f G°, r ° then G^ i s the solution to the following unconstrained maximization problem: (12) max G {r°(1-p p) 0 T - B(p°,p ,G,u )} 0 Our expressions for the ADD measure, 1 (10) and (11), approach to the measurement of deadweight loss. present our basic However, these abstract expressions do not indicate how the magnitude of the loss depends on the size of d i s t o r t i o n parameters t and d. Furthermore, the global computation of 57. (10) i s very d i f f i c u l t as was discussed i n Chapter 1. Therefore, we turn to the computation of second order approximations to the ADD measure. 2-4. Second Order Approximations To obtain a second order approximation to the loss measure, we require some stronger assumptions. Suppose that: (i) B i s twice continuously d i f f e r e n t i a b l e with respect to q, p and G at the optimum of (10); Oj, p^ » 0 ( i i ) G° » so that the f i r s t order necessary conditions for the max min N problem (10) hold with equality; ( i i i ) Samuelson's (1947;361) strong second order s u f f i c i e n t conditions hold for (11) when the inequality constraints are replaced by e q u a l i t i e s , and these conditions also hold for (12). Consider the following system of equations i n the N + I + 1 unknowns, p, G and r , regarded as functions of a scalar parameter z defined for 0 <. z <. 1: (13) V B(p(z)+tz,p(z),G(z),u )+V B(p(z)+tz,p(z),G(z),u )+pr(z) (14) V B ( p ( z ) + t z , p ( z ) , G ( z ) , u ) = - zd, (15) 1 - p(z) fi = 0. 1 1 q p = 0 , N 1 G T When z = 0, define p(0) = p ° , G(0) = G° and r(0) = r ° . Then (13) - become the f i r s t order conditions for the max min problem (10). t i v e l y , when z = 1, define p(1) = p \ G(1) = G and r(1) = 0. 1 the reference waste bundle 8 s a t i s f i e s the normalization (15) AlternaSuppose that 58 .1T (16) p = 1 by choosing the scale of p appropriately, which seems quite innocuous. (13) - (15) coincide with (7), (8) and (16). Therefore, (13) t e r i z e s the observed distorted equilibrium when z = 1. and (8) are s a t i s f i e d , Then, - (15) charac- Note that when (7) (6) i s also s a t i s f i e d for the observed choice of g, . n Therefore, we can safely conclude that (13) - (15) maps the Allais-DebreuDiewert reference equilibrium into the observed distorted equilibrium as z i s adjusted from 0 to 1. D i f f e r e n t i a t i n g the system (13) - (15) with respect to z and evaluating at z = 0, we obtain ° B + B , B ° + B qq pp qG pG 0 (17) B Gq 0 + B r , p Gp ' GG B P'(0) B ° t qq G' (0) B ° t + d Gq r' (0) , o where the second order derivatives of the overspending function B^?, i , j = q, p, G are evaluated at the optimum z = 0. follows: The meaning of the B^9 are as B q q i s an aggregate consumers' compensated substitution matrix whereas - B p p i s an aggregate producers' substitution matrix evaluated at the optimum; B ^ shows the change of aggregate compensated demands with respect qo to an increase of public goods and B - p G shows the change of aggregate net supply of goods for firms with respect to an increase i n the public good supply. 59. Now regard (17) assumptions (iii) as an i d e n t i t y i n z, v a l i d for z close to 0. Our introduced at the outset of t h i s section imply that an inverse exists for the matrix on the left-hand side of Diewert-Woodland (1977, Appendix I ) ) . (13) - (15) Premultiply both sides of (See Hence, by the i m p l i c i t Function Theorem, there e x i s t once continuously d i f f e r e n t i a b l e and r(z) which s a t i s f y (17). functions p(z), G(z) i n a neighbourhood of z = 0. (17) evaluated at z close to 0 by [p(z) (1.A.2), (1,A.4), and (5) evaluated at the T , T 0 , 0]. Using i d e n t i t i e s , z-equilibrium, and then using (14) and (15) we get (18) r'(z).= z[t B T ( p ' ( z ) + t) + t B ! G ' ( Z ) + d G ' ( z ) ] . qq qG Z T (The process for deriving (18) From (18) (19) T i s s i m i l a r to the one i n Appendix 1 - I I I . ) . we readily have r'(0) = 0. Now d i f f e r e n t i a t e (18) with respect to z, evaluate at z = 0, and adding the T i d e n t i t y derived by premultiplying [p'(0) (17), (20) T , - G'(0) ,0] to both sides of we find - r"(0) = G'(0) B ° G ' ( 0 ) uu T - p'(0)\V(0) - [p (0)+t] B 1 pp (The d e r i v a t i o n i s analogous to the one i n Appendix 1-IV.) T °[p'(0)+t] qq 60. Note that the l a s t two terms i n the right-hand side of (20) are nonnegative because of the concavity of B with respect to p r i c e s . We also assume that B ^ i s p o s i t i v e semidefinite; t h i s assumption i s s a t i s f i e d i f the production p o s s i b i l i t i e s sets are a l l convex, but i t i s much milder than assuming global convexity i n production. I n t u i t i v e l y , i t means that the concavity of the u t i l i t y functions outweighs any nonconvexity i n aggregate production with respect to public goods i n the neighbourhood of the optimum. Given t h i s assumption, -r"(0) > 0 i s implied. The Allais-Debreu-Diewert measure of waste r(0) may be written as ( 2 1 ) ADD L = I since r(1) = 0 . ( 0 " ) r ( 1 ) A second order approximation to L^p i s obtained by using a D Taylor series expansion evaluated at z = 0, (22) L A [ ) D * - [r'(0) + Jjr"(0)] = - Jir"(0) Using (19) we therefore have the following theorem. Theorem 1 ADD * " ^ " L r ( 0 ) 1 0 where the inequality i s v a l i d from (20) and i t s following discussion. i s quadratic, (22) provides an exact expression for I « use the expression for r"(0) given i n (20). A n n - If r To compute (22), The vectors of derivatives 61 . p'(0) and G'(0) i n (20) can be calculated by inverting the matrix on the left-hand side of (17). Therefore, the information required to calculate the approximate ADD measure i s the reference bundle P, the d i s t o r t i o n parameters (t, d) and the second order derivatives of the overspending function evalu- ated at the optimum. Let us s c r u t i n i z e the informational requirements for computing (22) more carefully. The vectors t and p are d i r e c t l y observable. d, we must know the consumers' marginal benefits at the observed consumer p r i c e s . To know the vector from public goods evaluated This means that we must overcome the w e l l - known preference revelation problem for public goods. Furthermore, to estimate the matrix B„^, we need to know the derivatives of the net marginal oo benefits for public goods for both consumers and producers. To calculate the other second order derivatives of the overspending function, we need to know the f i r s t order derivatives of the net supply functions of firms for private goods and the compensated demand functions of consumers, which depend on both prices and public goods. the second set i s not. elasticities Though the f i r s t set of functions i s observable, It i s well-known, however, that the compensated price can be computed from data on the ordinary demand functions using the derivatives with respect to both prices and income i n the Slutsky equation. (See, for example, Diewert (1982;572).) S i m i l a r l y , the derivatives of the compensated demand functions with respect to public goods can also be computed from market demand functions using ' S l u t s k y - l i k e ' equations Wildasin (1984;230)). (see The fact that we need information on the second order derivatives of the overspending function evaluated at the optimum considerably decreases the usefulness of (22), since these values are not observable 62. (at the market d i s t o r t e d equilibrium) and, i n general, are d i f f e r e n t from the values observed i n the distorted equilibrium. An a l t e r n a t i v e approach to approximating L f t D D Diewert's (1976;118) Quadratic Approximation Lemma. can be developed using This Lemma demonstrates that r(0) - r(1) can be appproximated by - ( 1 / 2 ) ( r ' ( 0 ) + r ' ( 1 ) ) , with the approximation being exact i f r i s quadratic. Note that t h i s approximation formula does not employ second-order derivatives of r . Suppose that (17) i s v a l i d for z close to 1 (instead of our previous assumption that i t i s v a l i d for z close to 0). obtain that r'(1) Setting z = 1 i n (18), we i s equal to the right-hand side of (20) evaluated at z = 1 instead of at z = 0. Using (19) and Diewert's Quadratic Approximation Lemma, we have the following c o r o l l a r y : Corollary (23) 1.1 L A D D * -(1/2) r'(1) > 0. A desirable a t t r i b u t e of t h i s approximation i s that i t only u t i l i z e s l o c a l information at the observed equilibrium. We thus see that both of our approximations to the deadweight loss measure r^ can be calculated from the derivatives up to second order of the overspending function evaluated at the reference equilibrium i n the case of (22) and evaluated at the observed equilibrium i n the case of (23). In p a r t i c u l a r , i t i s not necessary to make any assumptions concerning the functional form of B or place any r e s t r i c t i o n s on the values of observed economic v a r i a b l e s , other than the general r e s t r i c t i o n s used i n describing our model. On the contrary, to calculate r , as opposed to an approximation to r ° , i t would be necessary to adopt s p e c i f i c (and possibly restrictive) functional forms i n order to solve the max-min problem (10) g l o b a l l y . 2-5. Conclusion This chapter has discussed the measurement of waste and i t s local approximations for an economy facing distortions due to i n d i r e c t taxation and nonoptimal levels of public good production. Use has been made of the ADD measure defined i n Chapter 1 and two l o c a l approximations to the exact measure were calculated. tion on an overspending These approximations only required l o c a l informafunction. Figs. 7 - 9 i l l u s t r a t e the diagrammatic interpretation of the ADD measure of waste i n a public goods economy and i t s approximations. that there i s one private good and one public good. an aggregate production p o s s i b i l i t i e s Suppose In F i g . 7, we have drawn set that transforms the private good into the public good and the indifference curve of the representative consumer corresponding to the u t i l i t y l e v e l received at the observed distorted equilibrium. Though we cannot introduce a distortionary taxation in this one private good economy, the observed equilibrium i s not optimal because of the distortionary provision of the public good, and i t is expressed by the discrepancy of the marginal rate of substitution and the marginal rate of transformation at the equilibrium. By choosing the r e f e r - ence bundle to consist only of the private good, the ADD measure, as shown i n F i g . 7, i s a maximum surplus of the private good with holding the u t i l i t y level of the consumer and satisfying the production p o s s i b i l i t i e s set. The point where the surplus good i s maximized i s characterized by the equality of the m a r g i n a l r a t e o f s u b s t i t u t i o n and the m a r g i n a l r a t e of t r a n s f o r m a t i o n . The ADD measure can be r e i n t e r p r e t e d i n a H o t e l l i n g - H a r b e r g e r way s i m p l e model as i n F i g . 8. in this The m a r g i n a l b e n e f i t o f t h e p u b l i c good i s t h e m a r g i n a l r a t e of s u b s t i t u t i o n a t u = u 1 as a f u n c t i o n of the amount of the p u b l i c good, and t h e m a r g i n a l c o s t of t h e p u b l i c good i s t h e m a r g i n a l r a t e of t r a n s f o r m a t i o n as a f u n c t i o n of the amount of the p u b l i c good. mum At the o p t i - t h e y c o i n c i d e , but t h e former i s h i g h e r than t h e l a t t e r a t t h e d i s t o r t e d e q u i l i b r i u m , and t h e i r d i s c r e p a n c y i s denoted as d. We measure of waste i s shown as a c u r v i l i n e a r t r i a n g l e ABC approximations (22) and can show t h a t the ADD and t h a t i t s two (23) c o i n c i d e w i t h the t r i a n g l e s ABC and A B C ' . The d e r i v a t i o n i s analogous t o Appendix V of c h a p t e r 1 f o r the i n t e r p r e t a t i o n of the t a x l o s s as shown i n F i g . 4. In F i g . 8, the a p p r o x i m a t i o n s are rather a c c u r a t e i n comparison t o t h e t r u e amount of waste, but i t i s d i f f i c u l t t o t e l l how general. w e l l the a p p r o x i m a t i o n s can approximate the t r u e amount of waste i n I n F i g . 9, we show an example of one consumer economy w i t h l i n e a r p r o d u c t i o n p o s s i b i l i t i e s s e t where two a p p r o x i m a t i o n s even i n t h i s s i m p l e model. Tsuneki can be q u i t e i n a c c u r a t e (1987a) g i v e s a more e x t e n s i v e d i s c u s s i o n on t h i s n u m e r i c a l example and concludes t h a t the approximations can g i v e a t l e a s t an o r d e r of magnitude e s t i m a t e of t h e t r u e amount of waste and the a p p r o x i m a t i o n s can work q u i t e w e l l as l o n g as the optimum and the d i s t o r t e d e q u i l i b r i u m a r e not f a r a p a r t . To c o n c l u d e t h i s c h a p t e r : we can i n c o r p o r a t e t h e c h o i c e of p u b l i c goods by governments (which a r e used both by consumers and p r o d u c e r s ) t i o n a l general e q u i l i b r i u m Harberger-type measurement of deadweight l o s s framework by a d o p t i n g t h e A l l a i s - D e b r e u - D i e w e r t approach. more g e n e r a l than Harberger's in a tradi- Our approach i s a n a l y s i s i n t h e sense t h a t i t a l l o w s f o r ( i ) 65. the choice of f l e x i b l e functional forms (instead of linear ones as i n Harberger or CES-type ones i n the numerical general equilibrium l i t e r a t u r e ) for the production sectors and ( i i ) the loss due to i n d i r e c t taxation and the nonoptimal provision of public goods i s evaluated simultaneously. FOOTNOTES FOR CHAPTER 2 1 Assuming that there i s a single government production sector involves no loss of generality. 2 Formula (10) See Tsuneki (1987a) for more d e t a i l s . follows using d e f i n i t i o n s (1958)-Karlin (1959) Saddle Point Theorem. constraint q u a l i f i c a t i o n condition applies. (1) - (3) and the Uzawa We assume that S l a t e r ' s 67 . Fig. 7 The ADD Measure i n a Public Goods Economy 68. Fig. 8 The ADD Measure and i t s Approximations i n a Public Goods Economy 69 . Marginal Benefit of P u b l i c Good Marginal Cost o f P u b l i c Good 0 1—f Optimum Distorted Equilibrium Public Good Fig. 9 An Example where Approximations o f the ADD Measure a r e I n a c c u r a t e 70. CHAPTER 3 PROJECT EVALUATION RULES FOR THE PROVISION OF PUBLIC GOODS 3-1. Introduction The tion, t h e o r y of the p r o v i s i o n of a p u b l i c good w i t h d i s t o r t i o n a r y first s e t f o r t h by Pigou r u l e t o equate the sum (1947), m a i n t a i n s taxa- t h a t the Samuelsonian (1954) of m a r g i n a l b e n e f i t s t o i t s marginal c o s t cannot be an a p p r o p r i a t e r u l e f o r the maximum of s o c i a l w e l f a r e . A main o b j e c t i v e of the p r e s e n t c h a p t e r i s t o f o r m u l a t e some c o s t - b e n e f i t r u l e s f o r the p r o v i s i o n of a p u b l i c good which d e f i n i t e l y improves the w e l f a r e of a l l the i n d i v i d u a l s w i t h i n the economy. T h i s means t h a t our approach c o n s i d e r s s u f f i c i e n t c o n d i t i o n s f o r the e x i s t e n c e of a Pareto improvement when the p u b l i c good i s p r o v i d e d i n a d i s t o r t i o n a r y f a s h i o n , and transfers, ( i ) i n d i r e c t tax r a t e s , (iii) ( i i ) i n d i r e c t taxes r a t e s and lump-sum t r a n s f e r s , are a l l o w e d t o vary w i t h the lump-sum provision of p u b l i c good. In case (i), we suggest a G e n e r a l i z e d P i g o v i a n Rule which i s a many- person g e n e r a l i z a t i o n of the second-best good p r o v i s i o n which i s due and (iii) we t o A t k i n s o n and public S t e r n (1974), w h i l e i n cases ( i i ) suggest a G e n e r a l i z e d Samuelsonian Rule and a M o d i f i e d Harberger-Bruce-Harris Samuelsonian o p t i m a l i t y c o n d i t i o n f o r the Rule, where a l l of them more or l e s s d i f f e r from the rule. S i n c e our r u l e s are v a l i d when the e q u i l i b r i u m i s away from the second- b e s t optimum, our approach c o n t r a s t s w i t h the p r e v i o u s l i t e r a t u r e on project e v a l u a t i o n r u l e s f o r p u b l i c goods by S t i g l i t z and Dasgupta (1971), Atkinson and King S t e r n (1974), Diamond (1975), A t k i n s o n and Stiglitz (1980) and 71 (1986). They analyze the f i r s t order necessary conditions for ( i n t e r i o r ) second best s o c i a l welfare optima. Another objective of t h i s chapter i s , however, to reconcile these two apparently different approaches and to together strands of previous discussions within our framework. the cost-benefit 1 tie We show that rules i n this chapter are v a l i d both as necessary and s u f f i c i e n t conditions i f the manipulable taxation scheme i s optimized. After describing our model i n the next section, 3-3 studies an economy where distortionary commodity taxes are used to finance the provision of public goods; lump-sum transfers are not a v a i l a b l e . (1974) cost-benefit Atkinson and Stern's rule for public goods provision, which generalized the r e s u l t in Pigou's (1947) pioneering study, i s extended to a heterogeneousconsumers' economy i n t h i s section. In Atkinson and Stern's (1974) model, the marginal u t i l i t y of income does not equal the marginal s o c i a l cost of r a i s i n g one d o l l a r by i n d i r e c t taxation; t h i s difference arises because there i s a welfare cost due to i n d i r e c t taxation and there i s an income effect due to taxation on tax revenue. The f i r s t d i s t o r t i o n i s emphasized by Pigou, but the second one i s neglected by him. When we extend the Atkinson and Stern r e s u l t to a heterogeneous-consumers' economy, two differences a r i s e . First, the income effect of taxation is the sum of individual income effects with the hth weight being the share of tax revenue paid by the hth i n d i v i d u a l . Second, the change i n the income d i s t r i b u t i o n that results from increased taxation affects the s o c i a l cost of taxation; e . g . , i f the tax i s levied on people with high s o c i a l importance, the s o c i a l cost of taxation w i l l be higher. Section 3-4 discusses the cases where lump-sum transfers are available to finance an increased supply of public goods. If we can perturb both 72. i n d i r e c t tax rates and lump-sum transfers at the same time, a generalization of a t r a d i t i o n a l Samuelsonian r u l e , generalized Samuelsonian rule applies for the project evaluation. However, when there exists unchangeable i n d i r e c t tax d i s t o r t i o n s , we derive a Modified Harberger-Bruce-Harris rule for evaluating the public good. This approach proceeds by using the lump-sum tranfers to keep everyone on t h e i r i n i t i a l indifference curves when the supply of a public good i s increased. The induced change i n the net supply of private goods i s then evaluated using Harberger's generalized weighted-average shadow prices for fixed i n d i r e c t tax d i s t o r t i o n s . Since we adopted the approach of searching for s u f f i c i e n t conditions for a Pareto improvement, our cost-benefit rules can be implemented with know- ledge of the i n i t i a l demand and supply vectors and of the derivatives of the aggregate demand and supply functions evaluated at the observed equilibrium value, as long as preferences for public goods can be determined. Our approach may be contrasted with an alternative approach which searches for necessary conditions for an i n t e r i o r welfare optimum. cost-benefit In this approach, the rules depend on the derivatives of the aggregate demand and supply functions evaluated at the optimum point. 3-2. The Model The model we u t i l i z e i n t h i s chapter i s i d e n t i c a l with the one we used in the previous chapter to characterize the observed distorted equilibrium, (2.6) - (2.8). We assume for s i m p l i c i t y that p r o f i t income i s taxed away, following Diamond and Mirrlees (1971). completely This assumption can be relaxed by assuming that the entrepreneurial factors are additional commodit i e s (see Diewert (1978) and D i x i t (1979)). restate (2.6) and (2.7) as follows: With these assumptions, we can 73. h = 1, . . . , H (1) (2) 0, t 1 endogenously given p - 1 = 1, t T T = (u-|,...,u ) and T T = ( t , . . . , t ) , g = (g.,,...,g ) and G = We assume that (1) and (2) determine p = 2 (P2F--- PN) ( ». u N H H T (G^,...,G ) N . Note again that by Walras' law (1) and (2) imply the budget constraint of the government i s s a t i s f i e d . When the equality i n (2) replaced by the inequality (<.) by assuming free disposal, we c a l l i t inequality version of is "the (2)." The Pigovian cost-benefit comparative s t a t i c s exercise problem we study i n t h i s chapter i s simply a in which at the i n i t i a l observed equilibrium we perturb G and some of the available tax variables. We assume that G i s a scalar (or a l t e r n a t i v e l y , we assume that only the production of the f i r s t public good i s varied while the other public goods are held f i x e d ) . Three a l t e r n a t i v e rules are derived depending on which taxation instruments we can change. (3) D i f f e r e n t i a t i n g (1) and (2) t o t a l l y , assuming that h = which i s implied by money metric u t i l i t y scaling (see Samuelson (1974)), we obtain: 74. — — -X -X du = dp + B qu -B qq h -B ~ -B PP qq 1 + -B qq 1 w dg + o dG " qG ' B _ N«H_ B P G where the net demand matrix of consumers X = dt is: ] (H«N matrix, with X H « 1 and X ,H«(N-1)) V where the hth row shows the net demand vector of the hth consumer and W = (W ...,W ) 1 f T H = (-V m (p+t,G,u ),..., 1 G 1 -V m (p+t,G,u )) H G T H i s a vector of the marginal benefits of the public good for the consumers. The scalar M C= " k=oV ' E k ( p c ) i s the net aggregate marginal cost of the public good, L i s an H»H unit n matrix and 0 „ „ i s an N«H matrix consisting of zeros. A l l the derivatives of the overspending function B are evaluated at the observed equilibrium point (p+t,p,G,u). 75. Throughout the chapter, we assume that 2 (5) PP = V-.. B(p+t,p,G,u) i s negative PP definite. We express (3) in a d i f f e r e n t way for later use: (6) Adu = B.jdp + B dt 2 + B dt 1 + B dg + B^dG. 3 4 When we refer to "the inequality version of (6)" we mean that the H + 1 , . . . , H+Nth equalities i n (6) are replaced by i n e q u a l i t i e s (<.). This case u t i l i z e s the assumption that an excess supply of goods can be freely disposed. We assume that [ A , - B ^ , - B ] 2 1 exists, so that we can l o c a l l y solve for u, p, and t^ as functions of the exogenous variables, using the I m p l i c i t Function Theorem. This a n a l y t i c a l technique closely follows Diewert (1983b). F i n a l l y , we have to define our welfare c r i t e r i a . In a many-consumer economy, we have to d i s t i n g u i s h between two c r i t e r i a for a welfare improvement. The f i r s t c r i t e r i o n i s the s t r i c t Pareto c r i t e r i o n . improvement occurs i f each person's u t i l i t y i s increased. A s t r i c t Pareto The second T H T c r i t e r i o n makes e x p l i c i t use of the s o c i a l welfare function 8 u = E. „B,u. h=1 h h T where 8 > 0 „ . The l i n e a r function 8 u can be thought of as a l o c a l l i n e a r n r p approximation to a general quasiconcave s o c i a l welfare function evaluated at the i n i t i a l u t i l i t y vector u. In this chapter, we consider a d i f f e r e n t i a l effect of the various sets of tax-expenditure instruments with respect to s o c i a l welfare. satisfying If a set of available instruments i s f u l l y perturbed with (4) and du » 0 occurred, then we define i t as a d i f f e r e n t i a l l y s t r i c t Pareto improvement. If available tools are f u l l y perturbed with T satisfying (4) and p du > 0 occurred, then we define i t as a d i f f e r e n t i a l l y s t r i c t welfare improvement. (These d e f i n i t i o n s follow Diewert (1983b).) Obviously, a d i f f e r e n t i a l l y s t r i c t Pareto improvement ( i . e . , du 2> 0 „ ) i s a H d i f f e r e n t i a l l y s t r i c t welfare improvement for any nonnegative, u t i l i t y weight vector p. for but nonzero, Therefore, i f we can find a s u f f i c i e n t condition the existence of a d i f f e r e n t i a l l y s t r i c t Pareto improvement, then there exists a d i f f e r e n t i a l l y s t r i c t welfare improvement as well. Note also that T T P du > 0 implies the improvement of s o c i a l welfare p u i n a l o c a l sense but the opposite i s not true i n general, since i t i s possible that there exists T an i n f l e x i o n point of p u with respect to the set of instruments so that the T improvement of s o c i a l welfare occurs even i f p du = 0. The same argument applies for the change of individual u t i l i t y . We also define p-optimality . . . . T with respect to some set of instruments as an equilibrium i n which p u i s maximized with respect to the instruments. 3-3. Pigovian Rules Reconsidered Most papers on cost-benefit rules for public goods provision follow the Pigovian t r a d i t i o n and suppose that the government can vary i n d i r e c t tax rates t simultaneously with changes i n the production of the public good dG 2 0; however, lump-sum transfers g are fixed. Atkinson and Stern (1974) gave the most elegant formula for such a cost-benefit consumers have i d e n t i c a l preferences and wealth. rule by assuming that a l l The purpose of t h i s section i s to extend t h e i r formula, which we c a l l a Generalized Pigovian Rule to a heterogeneous-consumers' economy and to compare the economic implications of t h i s new rule with that of Atkinson and Stern. this section as follows: We state our main theorem i n Theorem 3.1 Suppose that public good production i s i r r e v e r s i b l e so that dG > 0 and 2 the government can perturb t a r b i t r a r i l y . Suppose also that (7) and there i s no solution a to a u T T > 0 „ and a X = 0 „ H u N u T that the i n d i r e c t tax revenue R = t v B i s nonzero. q T Then i f a l l for -y > 0„ for which no d i f f e r e n t i a l l y s t r i c t improvement of ^ u H with respect to i n d i r e c t tax rates e x i s t s , (8) <rW(1 + T "j^B h ^f - ~C l ^ q u F ^ l ^ i r 3 i s s a t i s f i e d where R* = t x 1 hth O) T b h 1 > M C " E ! t (ax (p+t G I )/aG) T f t h 1 l f i s the amount of i n d i r e c t tax revenue paid by person and s || = a x ( p + t , G , u ) / a u h h = ax (p+t,G,i )/ai , h h h h h = I,...,H i s a vector of income effects for the hth i n d i v i d u a l , then there exists a d i f f e r e n t i a l l y s t r i c t Pareto improvement du » 0 „ . h 78. (ii) If the pre-project equilibrium i s 8-optimal with respect to the of t, and i f no) fro *^- hls\»h<Ah > MC - is satisfied, choice E ^ t 1 (9x (p+t,G,I )/r3G) h h then (10) i s a necessary condition for a d i f f e r e n t i a l l y strict T increase i n s o c i a l welfare B du > 0. PROOF: (i) (11) A Pareto improvement with dG 2 0 exists i f and only i f there e x i s t dG > 0, du, dt such that du » 0 version of (6) i s s a t i s f i e d T there does not exist an a [a a ] T r and the inequality with dg = 0^. Applying Motzkin's Theorem (see Appendix I ) , (12) H this i s equivalent T -T = [ a , a ^ , a ] such that u > o j , a [ B , B , B ] = 0 _*, a A > o j , T T 1 2 3 to 2N and a B T If a Pareto improving i n d i r e c t tax perturbatin i s possible, always s a t i s f i e d with dG = 0 and the problem i s vacuous. assume that such an improvement does not e x i s t . 5 < 0. then (11) i s Therefore, we Then there exists an a T = 79. [ a J , a a ] such that [ a a ] > 0^, a [ B B , B ] = 0 T T 1 ( T v i ; 2 3 _ , a A > 0^. T 2 N T For any a • • T T that s a t i s f i e s t h i s condition, we define t = a A. We would l i k e to show that for such a, (12) i s s a t i s f i e d . Suppose (8) holds, and also suppose, T contrary to the theorem, a solution to (12) e x i s t s . T T Subtracting a B = 0 ^ 3 T from a B = using the i d e n t i t y (1.A.4) and using the supposition (5), 1 we have (see Appendix II) (13) a = a^. Using (13), the i d e n t i t y (2.5), and the definitions of W and MC, we get (14) a B = (aj + a T 5 1 f*)W - a ^ C + a ^ B . T Suppose that a^ = 0. Then, a = 0 from (13). Therefore, a [ B B ] T T T T = 0 „ implies a X = 0 „ . Furthermore, since a A > 0., , we have a > 0 „ . This N u N ' H u H N 1 1 ( contradicts the supposition (7), so that (12) i s s a t i s f i e d . > 0. 2 Suppose that a^ T T T Now postmultiply t^ and t to a B = 0 and a B = 0 _^ respectively, add 2 3 N them together and using (13) and (1.A.2) we have (see Appendix III) H (15) a n L / \ h=1 a 1 T K 7 R = 1 1 t/R. qq T T We also have a A = t = ( , 1 H y,...,'r)so show that (see Appendix III) using (13) and (1.A.3) and (3), we can 80. h h • u T h — = — - 1 + tS , a a qu 3 (16) h=1,...,H. n x 1 1 Substituting (16) "1 into (15), we find t B t Now substituting (16) U and (17) T U V . R W^ into (14) and using the S l u t s k y - l i k e equation by Wildasin (1984;230),3 we get T (18) t \ . a ^ W . U 1 - E J A u ^ ?"C/«R ) - MC + E " t ( 9 x ( p + t , G , I ) / 3 G ) } . T h Therefore, (ii) (8) T implies a B 1 c h h > 0 and we have a contradiction. If t i s chosen optimally at the pre-project equilibrium then i t i s a solution to the problem: (19) T max ~ {B u : (1) and the inequality version of u,p,t satisfied}. The f i r s t order Kuhn-Tucker conditions for (19) are: (2) are 81 (20) there e x i s t [a.,,a ] > OjJ, such that a A = 8 , a [ B B , B ] = 0 _^ T T T T r 2 2N 3 since the Mangasarian-Fromovitz constraint q u a l i f i c a t i o n conditions are implied by the existence of [ A , - B - B ] 1 f -1 2 (see Mangasarian (1969; 172-3)). A d i f f e r e n t i a l l y s t r i c t improvement i n s o c i a l welfare exists i f and only if (11) is satisfied with du » 0 H T replaced by p du > 0. i s given by replacing a A > 0^ i n (12) T by a A = p . T T Its dual condition This dual condition, and T (20) imply a B establish 5 > 0, which i s equivalent to (10) using the argument to (8) from (12). Q. E. D. We now have to consider the economic implications of Theorem 3.1. The assumption that i n d i r e c t tax revenue i s nonzero i s standard i n the optimal tax l i t e r a t u r e . Assumption (7) i s more subtle, but i t may well be j u s t i f i e d , since i t i s implied by the existence of a Diamond and Mirrlees' good (1971 ;23). More generally, (7) i s the condition for the existence of Pareto improving price changes ignoring production constraints, and equivalently there exists a Hicksian composite good i n net demand (or net supply) by a l l consumers. Then, lowering (raising) consumers better off call (8) and (10) the price of the Hicksian good makes a l l (see Weymark (1979)). With these assumptions, the Generalized Pigovian Rules (GPR hereafter) many-person Pigovian rules for the provision of public goods. we may or the There are several i n t e r e s t i n g interpretations of these two formulae. Let us f i r s t consider the r e l a t i o n between (8) and (10). Obviously, the only difference between the two formulae i s that we must consider any semipositive u t i l i t y weight vector for which a s o c i a l welfare improving tax perturbation does not exist i n the former, while we specify the weight p i n the l a t t e r . This may be explained as follows. We f i r s t assume that i n d i r e c t 82. taxes are set so that we cannot make a d i f f e r e n t i a l l y s t r i c t Pareto improvement with dG = 0. Otherwise, the problem i s t r i v i a l . However, once the i n d i r e c t taxes are set i n this manner, there exists at least one weight vector f (and probably many) so that the i n d i r e c t taxes are set such that T increasing i u i s impossible (see D i x i t (1979, 152)). Therefore we can use T the f i r s t order necessary conditions of the maximal s o c i a l welfare f u with respect to i n d i r e c t taxes, and hence the rest of the problem i s an extension of Atkinson and Stern's (1974) r e s u l t on s o c i a l l y optimal provision of public good with optimal taxes to a many-consumer economy. Furthermore, i f we specify t = 6 assuming that the economy i s at the 8-optimum, then we can get (10). We now discuss how to extend the Atkinson and Stern's cost-benefit rules to a many-consumer economy. i.e., at a 8 optimum, _ (1974, 122) With taxes set optimally, (14) has the following interpretation. At the p-optimum, a|j, h = 1 , . . . , H , from the programming (19). and a^ are the Lagrange m u l t i p l i e r s As d ( 8 u ) / d g T = a , ajj i s a net benefit of b h giving hth person one unit of numeraire good by r a i s i n g the i n d i r e c t taxes. T -1 It i s also the case that d(8 u)/dx^ = a^, a.^ i s the s o c i a l gam of the society to have one more unit of the numeraire good (so that i n d i r e c t taxes are reduced). Therefore, (a y + a^)/a^ i s a gross benefit i n terms of s o c i a l value of the numeraire of giving hth person one unit of numeraire good, and hence i t i s Diamond's (1975;341) s o c i a l marginal u t i l i t y of income a^, h = 1, T . . . . ...,H. Therefore, we can rewrite a B > 0 using the d e f i n i t i o n of a, , h = 1, c . . . , H i t i s equivalent to D h 83 . ( 2 1 ) E h=1 h h > a W M - ^ q C C where the left-hand side i s the s o c i a l value of the public good while the right-hand side i s the net s o c i a l cost of the public good both measured i n terms of the s o c i a l value of numeraire. a (22) h T T can also be rewritten from a A = B using (13) a = h + t S , qu' T a h = 1, . . . ,H h 1 as which coincides with Diamond's (1975;341) o r i g i n a l formula. (22) (23) into (21), By substituting we have I l A ^+ t S T h=1 a^ v )W qu' h h > MC - t B qG T Using Wildasin's (1984;231) S l u t s k y - l i k e equation for public goods footnote 3), (24) (23) may be further rewritten as (E " B W )/a h h 1 (see h > MC - t ( 9 E " x ( p + t , G , I ) / a G ) . T 1 The left-hand side of (24) h h 1 h i s the weighted sum of the marginal willingness to pay for public goods discounted by the shadow cost of r a i s i n g one d o l l a r by i n d i r e c t taxation. The right-hand side i s the marginal cost of the public good minus the complementarity effect of public goods provision which means the effect of public good provision on tax revenue due to the complementarity 84. between public and private goods. Therefore, (24) extends the formula (3) of Atkinson and Stern (1974; 122) to a many consumer context. What was emphasized by Atkinson and Stern was that 1/a^ may not necessarily smaller than unity, i n spite of Pigou's be seen from our formula (17) for 1/a.j. between our formula and t h e i r s . L (1947;34) conjecture. be This may also There are two main differences First, the revenue effect of taxation H T h h T h .t S R /R i s a weighted sum of the i n d i v i d u a l revenue effect t S h=1 qu ' * . qu where the hth weight i s the share of t o t a l taxes paid by the hth i n d i v i d u a l . T When there i s only one person this expression i s simply t S and Stern (1974;123)). q u (see Atkinson Second, i n a many consumer context one also has d i s t r i b u t i o n a l effects to consider. Raising one d o l l a r by taxation involves changing the d i s t r i b u t i o n of income proportionately to the tax shares of individuals. This d i s t r i b u t i o n a l effect i s reflected in the term H h h h E _ - | 8 (R /R) • If the tax i s levied on people with high s o c i a l importance 8 , n then t h i s expression increases as does a^; i . e . , the s o c i a l cost of r a i s i n g one d o l l a r i s higher because of the increase of s o c i a l inequity. concerns are summarized i n the GPR (10). These To see the d i s t r i b u t i v e concern i n (10) more f u l l y , we define the covariance term following Feldstein (1972); (25) H <p_ = E * G (26) H _ 1 <p = E A R n _ S W ^ * - / H, 8W n ^ 1 h ?^ / H. 8 R where p, R and W are defined as p = E f P / H , h h 1 R = R/H and W = E f W / H . h h 1 the c o r r e l a t i o n between the s o c i a l importance and the d i s t r i b u t i o n of As 85 . marginal willingnesses to pay or of tax burdens increases, increase. Substituting (25) and (26) (27) into (10) - Eh^Sgu n>_ and tp yields: > MC - t Ox (p+t,G,I )/3G). T h h This formula e x p l i c i t l y shows the importance of d i s t r i b u t i o n a l concern i n a <P many-person GPR by the term — . If the d i s t r i b u t i o n of the public goods R G benefits are regressive or the d i s t r i b u t i o n of the tax burden i s progressive, the s o c i a l welfare of the public good must be valued higher than the simple sum of the marginal willingnesses to pay. Before closing t h i s section, we should mention the r e l a t i o n between our model and the recent work by King (1986). Our formula (21) with the h interpretation of a by (22) i s obviously indentical with his formula (31) King (1986;281) so that i t i s possible to interpret (21) i n his way. in His result i s more general than ours i n the sense that he i s not assuming the Pareto e f f i c i e n t i n d i r e c t taxation, but our approach i s more complete than his i n the sense that he i s not deriving the e x p l i c i t formula and i n t e r p r e t a tion of the shadow price of government revenue l i k e (17) of ours, for i t u t t e r l y depends on the a r b i t r a r y structure of i n d i r e c t taxation i n his model. 3-4. Cases Where Lump-sum Transfers Are Available In contrast to the previous section where lump-sum transfers cannot be changed, the conventional Samuelsonian project evaluation rule which equates the sum of the marginal willingnesses to pay with the marginal cost of the 86 . public good has a strong i n t u i t i v e appeal when lump-sum taxes are available for financing the public good. We show i n this section that the Samuelson rule i s appropriate with some generalizations i f both i n d i r e c t taxes and lump-sum transfers are v a r i a b l e , whereas i t i s not appropriate i f there exits unchangeable d i s t o r t i o n s due to i n d i r e c t taxation. Let us f i r s t consider the case where we can change i n d i r e c t taxes and lump-sum tranfers at the same time. Theorem 3.2 (i) i.e., Suppose that the government can change t dG > 0. If, E!W + t (28) h and g when G i s increased, 1 h T B q G > MC, then there exists a s t r i c t Pareto improvement du » 0 „ . n (ii) Suppose that t a B-optimum. and g are chosen so that the pre-project equilibrium i s Then (28) i s also necessary for the existence of a d i f f e r e n T t i a l l y s t r i c t increase of s o c i a l welfare 8 du > 0. PROOF: (i) A s u f f i c i e n t condition for the existence of a Pareto improvement with dG > 0 i s : 87 . (29) there exists dG > 0, dt, dg, such that the inequality version of (6) i s s a t i s f i e d with du > 0 . H By Motzkin's Theorem, this i s equivalent T (30) to: T ~T there does not exist an a = [ a , a^, a ] such that: [a a ] 2 oJ,a [B B ,B ,B ] = 0 _ * , a A > ojj, and a B u T T v T 1 ( 2 3 4 T 2 N + H The argument used to show the equivalence of (29) and (30) that used to show the equivalence of (11) and (12) 5 < 0. i s s i m i l a r to in Appendix I, so i s omitted. Suppose (28) holds, but also suppose, contrary to the theorem, a . . T T . solution to (30) e x i s t s . The conditions a B, = 0 implies a =0... In the 4 H u H T T TJ proof of Theorem 3.1, i t i s shown that a [B^B^] = 0 ^_ 2 T can rewrite a B by using a = 0 and (13) 5 u H c (31) If a a B T 1 5 TI = a/(-B T > 0, then a B 5 p G -B q(J h + N We t B T q G - MC) . . . a contradiction. T If a^ = 0, a = 0 _^ from (13). implies (13). as ) = a, ( E ^ W > 0 by (28), 2 T Therefore, a A = 0^ and again we have a contradiction. (ii) If t and g are optimally chosen at the pre-project equilibrium, then they are a solution to the problem: (32) T max ~ . {8 u : (1) and the inequality version of (2) are u,p,t,g satisfied}. 88. The f i r s t order Kuhn-Tucker conditions for (32) (33) ~T, there exists [ a . , a ] a Suppose (33) , -T , ,. ' T, „T > Q , such that a A = 6 , [B B ,B B 3 T r is satisfied are: 2 3 ( but (28) 4 = 0 i s not. T 2N+H-1" The argument following (30) then T establishes that a Bj < 0, so (30) i s not s a t i s f i e d . Consequently, (28) i s also necessary for the existence of a d i f f e r e n t i a l l y s t r i c t increase of s o c i a l welfare 8 du > 0 at a 8-optimum for t and g. Q. E. D. To understand the implications of the Generalized Samuelsonian Rule (GSR hereafter) (28), l e t us assume that i n d i r e c t taxes and transfers are Pareto e f f i c i e n t l y , set so that we cannot make a d i f f e r e n t i a l l y s t r i c t Pareto improvement with dG = 0. Pareto e f f i c i e n t that the economy i s i n f i r s t best. i n d i r e c t taxes and transfers imply It i s well-known that the proportional commodity tax rates t = 8(p+t) for some r e a l number 8 i s f i r s t best with some appropriate lump-sum transfers. Substituting this r e l a t i o n into (28) and using (p+t) B equals T (34) (1-8) = - E f » , h E ° W h 1 h 1 h (28) > MC. This means that the sum of marginal willingnesses to pay for the public good deflated by 8 (which i s a r a t i o between producer and consumer prices) must be compared with the marginal cost. Needless to say, i f no i n d i r e c t taxes, then the Samuelsonian rule applies. 8 = 0 so that there are 89 . Though proportional i n d i r e c t taxes are always f i r s t best, there may e x i s t some other f i r s t best taxes depending on the structure of the economy. For example, i f there i s no room for technological s u b s t i t u t a b i l i t y among private goods so that B = 0, , then any i n d i r e c t taxes can be p-optimal N«N PP I with apropriate lump-sum transfers (see Diewert (1978)). It i s obvious i n t h i s case that the use of the simple Samuelsonian rule is erroneous and we have to use the GSR (28). We now move to an alternative case where we can perturb g and G while holding the commodity tax d i s t o r t i o n s t fixed. We c a l l the r e s u l t i n g rule within the following proposition, a Modified Harberger-Bruce-Harris Rule (MHBHR hereafter), since i t i s an a p p l i c a t i o n of Harberger (1971) and Bruce-Harris (1982) to a project evaluation approach to the production of a public good (see also Diewert (1983b)). Theorem 3.3 (i) Suppose that the government can change only the transfer vector g when G i s increased; i . e . , Rule^ i s (35) where (36) T e ~T = [CM ] = 1 dG > 0. Then, the Modified Harberger-Bruce-Harris 90. Condition (35) du » 0 (ii) fl is sufficient for the existence of a Pareto improvement . If i n the tax-distorted pre-project economy, g was chosen optimally, then (35) is also a necessary condition for a small increase i n public good production to lead to a d i f f e r e n t i a l l y s t r i c t welfare improvement. PROOF: (i) (37) A sufficient condition for a Pareto improvement i s : there exists dg and dG _> 0 such that the inequality version of (6) i s s a t i s f i e d and du > 0 „ . n Condition (37) i s equivalent to the following Motzkin dual condition: T (38) ~T T there does not exist [a , a a ] = a oj, 1 f such that 0 J, oj. aA > a B < 0, a [ B B , B ] = N+ [a aj] > Suppose (35) holds, but also suppose that, contrary to the theorem, a T T solution to (38) e x i s t s . The conditions a B^ = 0^ imply a = 0^. Hence we T T can rewrite a B. = 0„ . as: 1 N-1 T T T 5 V 2 4 v y (39) a = a ^ p + e) using (1.A.2) and (1.A.4) (see Appendix IV). Using a = 0 , (36) and (39), 91 . a B T = a, (P + ) [ - B - B T 5 E p G q G ]. T If a 1 (39) . (ii) > 0, (35) > 0, a contradiction. If a = 0, a = 0 from T T With a„ = 0„ we have a A = 0 „ , and again we have a contradiction. U implies a B 5 1 n N 1 H If g i s optimally chosen at the pre-project equilibrium, g i s a solution to the problem. (40) max T - . {B u ; (1) and the inequality version of (2) are u,p,t ,g 1 satisfied}. The f i r s t order Kuhn-Tucker conditions for (40) (41) there exists [a^,a ] _> 0 , N Suppose (41) is satisfied T establishes that a B c are: such that a A = 8 ,a [B^,B2,B ] = 0 but (35) i s not. 4 N + H > The argument following (38) . . . < 0, so (38) i s not s a t i s f i e d . Consequently, (35) then is also necessary for a d i f f e r e n t i a l l y s t r i c t increase of s o c i a l welfare at a B-optimum for g. Q. E . D. The economic i n t u i t i o n behind the two Propositions i n this section i s as follows. Given the pre-project l e v e l s of u t i l i t y , increasing the provision of the public good permits a reduction i n the consumption of private goods but requires additional inputs for the increased public good production. appropriately offsetting the marginal benefits of the public good ( i . e . , By the 92 . externality) by changing lump-sum transfers to keep consumers at t h e i r o r i g i n a l u t i l i t y l e v e l s , i t i s only necessary to evaluate the r e s u l t i n g change i n the quantities of the private goods by appropriate shadow p r i c e s . If the vector of tax rates t i s v a r i a b l e , the production price vector i s appropriate shadow price vector. See (31) i s a version of the production efficiency (1971). behind a GSR (28) . the This r e s u l t theorem i n Diamond and Mirrlees If t i s fixed, a MHBHR (35) must be adopted which uses a Harberger- Bruce-Harris shadow price vector. 3-5. Conclusion Our present chapter has derived project evaluation formulae for the provision of public goods i n various second-best situations. three cases. (1) the case where i n d i r e c t tax rates can be varied; (2) case where both lump-sum transfers and i n d i r e c t tax-rates the case where lump-sum transfers are varied. the can be varied (3) We showed that the the use of a GPR, a GSR and a MHBHR are suggested for cases (1), tively. We considered (2) and (3) respec- Our basic point i s that project evaluation rules must vary depending on what instruments we can change when we a l t e r the supply of public goods. We have to note that there are severe limitations i n u t i l i z i n g our costbenefit rules; i . e . , we have ignored the preference revelation problem for public goods i n measuring the marginal willingnesses to pay W for consumers. Once t h i s d i f f i c u l t y i s overcome, our rules can be implemented by using only l o c a l information observable at the pre-project equilibrium, that i s , l e v e l of taxes, public goods, the prices, incomes, and the f i r s t order d e r i v a - t i v e s of the ordinary demand functions and the net supply functions for 93. private goods (which depend on both prices and public goods). and B 2.4. Note that B can be computed from ordinary demand functions as we pointed out i n Note further that information on W i s necessary to use the MHBHR (35) as we need to compute B - from data on the ordinary demand functions. go There- fore, this rule i s also vulnerable to the f r e e - r i d e r problem. We have shown that i t i s f a i r l y easy to obtain s u f f i c i e n t conditions for the existence of a small Pareto improvement corresponding to an increase i n public goods production, given that various taxation instruments are a v a i l able. It seems that t h i s approach i s more useful compared to the t r a d i t i o n a l approach which derives the f i r s t order conditions for an i n t e r i o r second-best welfare optimum. Our results also show that conventional cost-benefit for the provision of public goods are not always correct. rules 94. FOOTNOTES FOR CHAPTER 3 1 Our approach draws on the methodology found i n the tax reform literature, e.g., Guesnerie (1977), Diewert (1978), D i x i t (1979) and Weymark (1979), and the project evaluation study by Diewert (1983b). Wildasin (1984) also worked with a framework similar to ours, but his paper has various r e s t r i c t i v e assumptions; e . g . , all only one commodity tax rate i s variable and other goods are untaxed. 2 If we evaluate a possible reduction i n the production of the public good dG <. 0, a l l we need i s to reverse the d i r e c t i o n of the i n e q u a l i t i e s the cost-benefit omitted. rule. in The proof is straightforward and hence may be The same comment applies to a l l cost-benefit formulae i n t h i s chapter. 3 It i s given by \S^. 9 x ( p + t , G , I ) / 3 G = {3x (p+t,G,u )/9G} + n h h h T Premultiplymg by t E ^t (9x ( T and i n summation over h, we have h h P + t,G,I )/9G) which i s used to derive * If t = 0 N h = t B T E ^t W S T q G + h h b q , (18). so that there are no pre-existing tax d i s t o r t i o n s , then the Modified Harberger-Bruce-Harris Rule i s i d e n t i c a l with the t r a d i t i o n a l Samuelsonian r u l e . The proof i s straightforward. APPENDICES FOR CHAPTER 3 Appendix I : The derivation of (12). Motzkin's Theorem i s as follows: Either Ex » 0, Fx > 0, Gx = 0 has a solution x where E i s a nonvacu1T 2T 3T ous matrix, F and G are matrices and x i s a vector or v E + v F + v G = T 1 2 1 2 3 0 , v > 0, v 2 0 has a solution where v , v and v are vectors, but not both. See Mangasarian (1969). We now apply i t to rewrite (11). = 0 . U Decompose A, B. (i=1,2,3,5) H B^dg can be dropped from (6), for dg between A*, B?, which are the top H rows I and A**, B?*, which are the bottom N rows. ' l x = [du, dp , d t E F = [ I 1 Define dt , dG] , r H ' °H«(2N+H) ' ] -A** A o** n** n** 1 ' 1 ' °2 ' 3 ' B B n** ti 5 '2N+H where e 2 N + H i s a unit vector with unity i n 2N+Hth row, and G = [-A*,B*,B*,B*,B*]. 96. Then, the primal condition of the Motzkin's Theorem i s i d e n t i c a l with (11). ^ "j* "* T T T Defining v . j d - H row vector), v = [ a ^ a ,v] where v i s a scalar, v = a , the 2 3 dual condition i s : there i s no solution v^, a^, a, v, a v v > 0, a 1 such that 2 0, a 2 0, v 2 0, 1 - a A = 0, a [ B B , B ] = 0 T y T T 1 ( 2 which i s i n turn i d e n t i c a l with 3 2 N _ T , a B T 5 + v = 0, (12). Appendix I I : The derivation of (13). T T T Subtracting a B = 0 _^ from a B^ = 3 (A.1) - a.B 1 N T " e have - - a B~~ = 0„ f. PP N-1 T From (1.A.4), (A. 2) where p (A.3) B - + p B-~ = 0„ « pp N-1 T p^p 1 = 1. ( a / Substituting (A.2) into (A.1), we have - a ) By assumption (5), T B p p = 0^. B«-~ i s nonsingular, which implies (13). u 97 Appendix III: The derivation of (15) and (16) T T Postmultiply t^ and t to a Bj = 0 and a B 3 T = 0^_y and adding them together, we have (A.4) a*Xt + [ a a ] B t T 1 f Substituting (A.5) q q = 0. (13) and rewriting the f i r s t term of E A a V h=i u + a , p B t = 0. 1 qq T 1 Substituting the i d e n t i t y (1.A.2) rearranging terms, we get (15). (A.6) = aj + [ a a ] B T 1 ( Substituting (A.7) a m (13) q u into the second term of (A.5), and T T We can rewrite a A = f as . into (A.6), we have m m + a„p B = -y • u r qu 1 However, from (1.A.3) and (3) we get (A.S) = (p + t ) B T Substituting <»•» q u t J• (A.8) into (A.7) we have «J - 7 T (A.4), we get - a, l „ • a , t \ . u 98 From the d e f i n i t i o n of B and S i n (9), qu qu Appendix IV: T a B (A.10) 1 The derivation of N -a.[B U (A.11) (39). T = 0 _ can be rewritten as 1 using a (A.9) i s i d e n t i c a l with = 0„. n B 1 - + B -] - a [B-~ + B~~] = 0 „ « q«q PiP qq PP N-1 T From (1.A.4) and (1.A.2), we get - + p B~~ = 0 „ 1 T P-,P N-1 PP and (A. 12) t B - + B -+ p B~~ = 0 qq q^ qq N-1' T T M respectively. (A. 13) ^ Therefore, adding up (A.11) and (A.12), we have B - + B q-iq ~ = - t B - - p B~~ - p B ~ qq qq PP T Substituting (A.13) into (A. 14) T T (A.10), a . [ t B - + p B ~ + p B-~] = a [ B ~ + B~~] 1 qq qq pp qq PP T T T T 1 J Inverting the matrix [B~~ + B - ~ ] , and using d e f i n i t i o n (36), qq pp follows. (39) (16) 99 . CHAPTER 4 INCREASING RETURNS, IMPERFECT COMPETITION AND THE MEASUREMENT OF WASTE 4-1 Introduction In the presence of increasing returns to scale i n production, i t is well-known that Pareto optimal e q u i l i b r i a may not be decentralized through perfect competition and moreover, imperfect competition prevails frequently. Therefore, both positive and normative analysis of resource a l l o c a t i o n with increasing returns to scale becomes an important topic i n applied welfare economics. The normative problem of developing mechanisms to support Pareto optima i n the presence of increasing returns to scale has been discussed by many authors, including Arrow and Hurwicz (1960), Guesnerie (1975) and Brown and Heal (1980). The second best p r i c i n g problem of public u t i l i t i e s facing a revenue constraint i s discussed by the optimal p r i c i n g and taxation l i t e r a ture beginning with Boiteux (1956). There have been numerous positive analy- ses of o l i g o p o l i s t i c markets in the vast l i t e r a t u r e on strategic interactions among incumbent firms or among incumbent firms and potential entrants. More- over, there i s a large l i t e r a t u r e on Chamberlinian (1962) monopolistic competition. In contrast, the measurement of waste due to imperfect competition with increasing returns to scale i s a r e l a t i v e l y less developed area, although the important seminal paper by Hotelling (1938) dealt with this topic. The aim of this chapter i s to consider this measurement of waste problem. Let us f i r s t review the problem discussed by Hotelling (1938) and l i s t the points which seem to c a l l for extensions. First, Hotelling claimed that f i r s t best optimality i s characterized by the marginal cost p r i n c i p l e , i.e., 100. the price of the product should equal i t s marginal cost, for increasing returns to scale firms. However, i t was pointed out by Arrow and Hurwicz (1960) that this solution is not necessarily optimal with a general nonconvex technology, and Silberberg (1980) pointed out that Hotelling (1938) i s actually not proving the optimality of marginal cost p r i c i n g . Therefore, in the l i t e r a t u r e on the measuremnt of deadweight l o s s , which includes Debreu (1954), Harberger (1964) and Diewert (1981, 1983(a), 1985(a)) i n order to avoid t h i s d i f f i c u l t y i t i s assumed that a l l firms have a convex technology. Therefore, i n order to compute the deadweight loss, we f i r s t characterize the optimality i n nonconvex economy rigorously. Second, H o t e l l i n g ' s (1938) measure of waste does not seem to be correct i n a general equilibrium sense, and furthermore, requires the computation of an optimum equilibrium which necessitates global information on consumer preferences and technology, so that we would l i k e to derive a measure of waste which can be evaluated using only l o c a l information on preferences and technology, so that the measure is more useful i n empirical research on the measurement of waste. In this chapter, we show that these problems can be solved i n a s a t i s factory way, at least i n our s i m p l i f i e d model. Our findings i n this chapter may be summarized as follows. We can derive a Hotelling-Harberger type general equilibrium approximate deadweight loss measure due to imperfect competition allowing for quite general d i f f e r entiable functional forms for production and u t i l i t y functions, including production functions that exhibit increasing returns to scale. This approxi- mate measure can be implemented from l o c a l information up to the second order obtained at an observed distorted equilibrium. There are different waste measures depending on the types of increasing returns to scale, since the 101 characterization of the optimum depends on these types of increasing returns to scale. In the next section, we construct a model employing the assumptions that production functions are quasiconcave, factor markets are competitive, and the number of firms i n one production sector i s fixed. We characterize the imperfectly competitive general equilibrium by a system of equations. In 4-3, we derive an Allais-Debreu-Diewert measure of waste with increasing returns to scale and show that the corresponding optimum equilibrium is characterized by the marginal cost p r i n c i p l e . In 4-4, we compute a second order approximation to the ADD loss measure, discuss i t s informational requirements, and show how our measure generalizes H o t e l l i n g ' s o r i g i n a l approach and other works on deadweight loss which assume technologies are convex. We also discuss various relaxations of our assumptions, and l i m i t a t i o n s on applying our approach to empirical studies of various market imperfections. Section 4-5 concludes with a diagrammatic interpretation of our approximate measures. 4-2. The Model We assume that there are N goods i n the economy, where the corresponding T price vector i s p E ( p ^ , - . . , P ^ ) » 0 , N and that only sector n produces the nth good for n = 1 , . . . , N by combining the other goods and M nonproducible factors. This vector of primary factors has the vector of factor prices w = (w ...,w ) r M T » 0 . M Each production unit i s assumed to have a quasi-concave production function f ( x , . . . , x , n 1 N v 1 ( . . . , v ) ; that i s , M for a given l e v e l of output y , marginal rates of technical substitution between inputs are d i m i n i s h i n g . This assumption i s weaker than global convexity i n production; the 1 102 . p o s s i b i l i t y of increasing returns to scale i s allowed for when we change the l e v e l of output i n t h i s c h a r a c t e r i z a t i o n . 2 We define the sector n cost function C ( p , w , y ) as n n (1) C (p,w,y ) = rain {p x + w v : f ( x , v ) > y }, n = 1 , . . . , N . n x>0. ,v>0„ ' - n' ' ' ' —N M n T v A C l i T n r 2 T C n is i d e n t i c a l to the expenditure function m* defined by (1.4), except that 1 the u t i l i t y l e v e l i s replaced by the production l e v e l . We assume that the regularity conditions l i s t e d i n Diewert (1982;554) are s a t i s f i e d . There are H households i n t h i s economy and t h e i r demands are characterized i n terms of the expenditure functions h T T h m (p,w,u. ) = min ,{p a+w b : f (a,b) n a, D (2) h where P. i s a (translated) h > u, ,(a,b)eQ }, n h = 1,...,H, N+M orthant of R defined as i n (1.4). We assume _ that the hth household holds the vector of i n i t i a l endowments Y = ,-hT (a rhT.T ,b ) . To characterize the general equilibrium, we u t i l i z e the overspending function B defined by: (3) B(y,p,w,u) = E ^ { m ( p , w , u ) - p a h h T 1 h h - wb} T h - [ ,{p y - C (p,w,y )}, n=1 n n n N where y = ( y , . . . , y ) 1 N T and u = n (u ,...,u ) . T 1 H Compared with the overspending functions i n previous chapters, consumers and producers are facing the same prices i n (3) so that we no longer have two 103 . D e f i n i t i o n ( 3 ) may be s i m p l i f i e d by defining sets of prices as arguments. T TT Q = (p ,w ) as follows: B(y Q,u) (4) ( = E " {m (Q,u ) h h 1 h - Q Y} - ^ { p ^ T h - C (Q,y )}. n n In the same manner as we derived the properties of an overspending function in Appendix I of chapter 1, we can e a s i l y derive the following properties for the new overspending function: (i) B i s concave with respect to Q ; ( i i ) i s once continuously d i f f e r e n t i a b l e with respect to prices, the vector of excess demands; ( i i i ) prices. From t h i s , ( 5 ) Q T B V^Bty.QjU) if B equals B i s l i n e a r l y homogeneous with respect the r e s u l t i n g i d e n t i t i e s are satisfied: Q Q = °N M' + and (6) Q B T where B Q y = V B(y,Q,u) = V B(y,Q,u) 2 Q Q T y Q and B = V B(y,Q,u). 2 Q y Q Note that - V B ( y , Q , u ) y is a vector whose i t h component i s the difference between the price and marginal cost of the i t h good. Now using the above r e l a t i o n s , we characterize the general equilibrium 1 1 1 1 1T 1T T (y , Q ,u ) where Q = (p ,w ) as (7) follows: h, 1 „ 1 . ^IT^h , N h n 1 1 „ n , - 1 1., m (Q , u ) = Q Y + E a { P y - C (Q ,y )} ft r h f n = 1 n n n , + g , h , . h = 1,...,H, to 104 . (8) (9) N+M where a hn i s the share of the nth firm held by the hth i n d i v i d u a l and H hn Ej^a = 1, for n = 1 , . . . , N . The number g^, h = 1 , . . . , H , lump-sum transfer given to the hth i n d i v i d u a l and t = (t^ shows the net T ,...,t^) where t R i s the monopolistic mark-up imposed by firm n on his sales. We can show that (7) and (9) imply that the sum of the transfers g^, h = 1,...,H, equals zero. the H i n d i v i d u a l s . The equations i n (7) are the budget constraints of The equations i n (8) state that the difference between the price of the i t h good and i t s marginal cost i s equal to the mark-up t^. For perfectly competitive firms t = 0^, but with imperfect competition we expect t » 0^. With increasing returns to scale, firms must charge prices larger than t h e i r marginal costs i n order to a t t a i n nonnegative profits. This does not necessarily mean that the monopolistic markup i s fixed for monopolists. We just define t ex-post at the equilibrium as the between consumer prices and marginal costs. difference Noting that v^B equals the vector of excess demands, the equations i n (9) are the market clearing conditions for the equilibrium. Therefore, (7) to (9) characterize an imperfectly competitive general equilibrium, as elaborated by Negishi (1960-1), Arrow and Hahn (1971, Ch. 6) and Roberts and Sonnenschein (1977). 4-3. The Allais-Debreu-Diewert Measure of Waste Let us f i r s t take an N + M dimensional nonnegative reference bundle of goods and factors A = ( a , 8 ) T T T >0 and each consumer's u t i l i t y l e v e l u}, 105. h = 1,...,H, i n the imperfectly competitive equilibrium, and consider the following primal planning problem: 0 nn\ ( 1 0 ) r , H r a a x h , h n n ... { r : ( 1 a ,b ,y .x ,v ) E - H h=1 h , - N a + E n=1 X n , + a r ,r,H-h . * y r - h=1 + E h=1 ' a n h=1 E E n=1 V P ' E ( i i i ) f ( a , b ) 2 UjJ, (a ,b )eQ h = 1 , . . . , H , h h h h h h (iv) f ( x , v ) 2 y , n = 1,...,N}. n n n n The solution to (10) defines the ADD measure of waste L , ^ ^ = ADD . Problem (10) may be interpreted as maximizing the number of multiples r of the given reference bundle X that can be obtained while maintaining consumers' u t i l i t i e s at u^, h = 1 , . . . , H , technology constraints. and satisfying the materials balance and We assume that a f i n i t e maximum exists for We can also derive a dual expression to (10) as follows. (10). First, let us T f i x y = (y.|, . . . ,Yjj) . From the d e f i n i t i o n of quasi-concavity, the sets f ( x , v ) 2 Y ( n = 1 , . . . , N ) are convex sets belonging to R ^ n n n n +M . Then, the remaining programming problem becomes a concave programming so that we can rewrite (10) using the Uzawa (1958)-Karlin (1959) Saddle Point Theorem as 3 (11) r 0 = max [max min N using d e f i n i t i o n s (1) N+M (2) and (4), T 1 (r(1-Q X ) - B(y,Q,u )}] where Q i s the vector of Lagrangean m u l t i p l i e r s associated with the resource constraints, (i) and ( i i ) . The max-min problem within the squared bracket of (11) can be rewritten using the Uzawa-Karlin Theorem i n reverse as 106 . (12) - m a x n >n W Y I Q ' " ) **.t. Q X > 1} 1 T N+M For the given l e v e l of y, the solution of the max-min problem within becomes r(y) and Q(y) which are functions of y. Then (11) (11) can also be written as 13) r ° = max {r(y)(1-Q(y) X) T - B(y,Q(y),u )} 1 N The global programming problem (10) and (11) define the ADD measure of 1 0 waste when the observed u t i l i t i e s are u , but i t i s d i f f i c u l t to compute r using this approach since we need global information on preferences and technologies. To get more insight about the amount of waste i n r e l a t i o n to the degree of monopoly, and bridge the gap between conventional deadweight loss measures and our ADD measure, we derive a second order approximation to the ADD measure of waste. assumptions as (i) For this purpose, we have to strengthen our follows: (y°,r ,Q ) solves (11) with y ° » 0 , N conditions for (11) hold with e q u a l i t y ; 0 0 M 4 h = 1,...,H, n M M are twice continuously d i f f e r e n t i a b l e with respect to Q at 0 1 (Q ' ^ ; ( i i i ) u Q° » 0 ^ so that the f i r s t order N+M ( i i ) the expenditure functions m*\ n the cost functions C , n = 1 , . . . , N , are twice continuously d i f f e r e n t i a b l e at (Q^,y^); (iv) Samuelson's (1947) strong second order conditions hold for the two problems (12) and (13) when the inequality constraint i n (12) i s replaced by an equality. The regularity condition (i) implies that there are no free goods and a l l firms are useful. Conditions ( i i ) and ( i i i ) are d i f f e r e n t i a b i l i t y 107 . assumptions, (iv) which are natural for a l o c a l analysis such as ours. Condition i s an assumption which guarantees that the maximum of the planning problem (10) is l o c a l l y unique. Our regularity conditions on (12) imply the bordered Hessian -B° -x QQ' * (14) T -A where = V^B(y® A 0 definite 0 , ,Q® are evaluated at z = 0. (15) i s positive ) and the superscript 0 means that the derivatives By defining , -B yy 0 and (16) B°=[-B °,0 ] y N where the superscript 0 means that B ^ and B ^ are evaluated at the optimum, yy yQ our condition i n (iv) i s equivalent to the following condition: (17) 0 0 0 -1 OT A - B (C ) B is negative The condition (17) definite. i s much weaker than assuming marginal costs are increasing, which requires A to be negative d e f i n i t e , for C° i s definite we are admitting the 0 by (14). By merely requiring condition ( i v ) , positive 108 . p o s s i b i l i t y of a downward sloping marginal cost curve, which follows the s p i r i t of Hotelling (1938;255-6) .5 It follows from assumption (i) that an i n t e r i o r solution exists to (11). The f i r s t order conditions are given by: 18) -v B(y°,Q°,u ) = 0 , 1 y N (19) -Ar° - V B ( y ° , 0 ° , u ) = 0 (20) 1 - Q A = 0, 1 Q N + M , 0T where (18) i s a marginal cost p r i c i n g p r i n c i p l e for monopolistic firms, (19) are resource balance equations for goods and factors with Ar"* > 0^ being the vector of surplus goods and factors, and (20) i s a normalization rule for the optimal p r i c e s . 4-4. Second Order Approximations Now comparing the market equilibrium conditions and the f i r s t order conditions for the optimum, we construct a z-equilibrium which depends on a scalar parameter z (0 <. z <. 1); (21) -v B(y(z),Q(z),u ) (22) - V B ( y ( z ) , Q ( z ) , u ) - Ar(z) = 0 (23) 1 - Q(z) A = 0. 1 y = tz, 1 Q T N + M , 109 . If we define (y(0), Q(0), r(0)) = (y ,Q ,i ), then (21) - (23) coincide with the optimality conditions (18) - (20) when z = 0. (yd), Q ( D , r(1)) In contrast, i f we define = ( y , Q , 0 ) , then (21) and (22) coincide with (8) and (9) 1 respectively when z = 1. 1 In this case, (7) i s also s a t i s f i e d appropriate choice of transfers g^, h = 1 , . . . , H . for an From condition (23) at z = 1, we also assume that the market prices satisfy the normalization, (24) 1 = Q X, 1 T by choosing the scale of X appropriately. z-equilibrium (21) Thus we can conclude that the - (23) maps the optimal equilibrium into the imperfectly competitive equilibrium as z i s adjusted from zero to one. Equation (21) that maps the marginal cost p r i c i n g condition (18) into the monopolistic markup equilibrium condition (8) may seem unnatural because markups are decreasing l i n e a r l y , but the change i n t may be nonlinear depending on the behaviour of monopolists. tions, Even i n the case of t a x - d i s t o r - however, i t is possible to choose some nonlinear path of the change of tax rates as the equilibrium i s adjusted and the r e s u l t i n g magnitude of waste depends on this choice of path. (This problem i s also related to the t r a d i t i o n a l problem of path independence i n consumers' surplus analysis.) As i t i s d i f f i c u l t to overcome this a r b i t r a r i n e s s within our framework, we have just assumed that there i s a uniform reduction of monopoly d i s t o r t i o n s . The main theorem i n t h i s section i s as follows: 1 10. Theorem 1: A second order approximation to the ADD measure of waste (10) is given by V(A (25) PROOF: 0 - B°(C°)- B°V t 1 1 > 0. Differentiate (21) - (23) with respect to z and we have A , B Z t y' (z) Z (26) = Q'iz) B Z T , C °N+M r' (z) Z 0 where A , B , C are the matrices A , B z z z and C defined by (15), (16) and (14) evaluated at z, rather than 0. T T Premultiplying (26) by [0 ,Q(z) ,0], we have N (27) - Q ( z ) B y ' ( z ) - Q ( z ) B Q ' ( z ) - Q ( z ) A r ' ( z ) = 0. T Z T Q Substituting (5), (28) Z T Q (6), and (23), and then (21) into (27), we obtain r'(z) = z t y ' ( z ) . T Noting that r(1) = 0, by using a Taylor series expansion the ADD measure of waste L n A n (29) = r< ^ = r° - r r 1 (0) ~ d) r can be approximated by = r ° - { r ° + r'(0) However, from (28), r'(0) + hr"(0)} = - r'(0) = 0 and r"(0) = t y ' ( 0 ) . T - Jjr"(0) Therefore, we have 111. (30) r° - r 1 .= - J^tV (0). Evaluating (26) at z = 0 and inverting the left-hand side matrix yields y'(0) = (A^ - B^(C^) ^B^ ) ^t. T result (25) follows. Substituting t h i s expression into (30), The inequality i n (25) follows from (17). the Q. E . D. The formula (25) gives a general formula of deadweight loss applicable to either a convex or nonconvex economy. This formula i s i d e n t i c a l to the Debreu (1954)-Diewert (1985a) approximate deadweight loss formula when the technologies are convex. However, the converse i s not true, since the optimal shadow price (or i n t r i n s i c price to use Debreu's (1951, 1954) may not exist with increasing returns to scale. This problem i s overcome by our two-stage optimization procedure (11) term) for the characterization of the optimum, an approach which was suggested by Arrow and Hurwicz (1960) and Guesnerie (1975). Our r e s u l t i n g approximate loss formula (25) not from the derivatives of supply functions, is calculated but from the derivatives of r e s t r i c t e d factor demand functions and marginal cost functions evaluated at the optimal l e v e l of output. As our work i s preceded by Hotelling (1938), i t i s important to his work in r e l a t i o n to ours. discuss Hotelling's contributions i n t h i s paper are known to be that (i) he showed the optimality of the marginal cost p r i c i n g p r i n c i p l e of the regulated firms, and that ( i i ) he derived the approximate deadweight loss formula deviating from the optimality above. For the f i r s t point, Silberberg (1980) pointed out that H o t e l l i n g ' s proof i s not a v a l i d one. In section 3 we gave a rigorous proof based on programming that the marginal cost p r i c i n g p r i n c i p l e i s necessary for o p t i - 112. mality i f technologies are quasi-concave. As Arrow and Hurwicz (1960) showed, t h i s condition i s not necessary for general nonconvex technology. Suppose that we a l t e r n a t i v e l y consider Guesnerie's (1975) type 3 firm; i s a firm's technology i s convex i f some input i s given. that By applying results i n Guesnerie ( 1975;12-13), i t i s straightforward to show that optimality i s characterized by the competitive maximization of ' r e s t r i c t e d ' p r o f i t given the level of the input which causes the increasing returns to scale, the equality of the marginal value product with the factor p r i c e . Aoki (1971).)6 and by (See also For the second point, H o t e l l i n g (1938;254) derived a similar deadweight loss measure to our formula (30). Similar to his f i r s t point, however, the derivation of his loss measure lacks true general equilibrium considerations and cannot be v a l i d despite his own conjecture. (1987b)). (See Tsuneki Furthermore, even i f we interpret his measure as i n (30), not useful without knowing how to compute y'(0) Function Theorem. it is using (26) v i a the Implicit We must also note that our approach for the measurement of waste can be applied to an economy including type-3 firms which was not considered by Hotelling (1938). As we have seen, we can derive f i r s t order necessary conditions for the optimality. Then, comparing the optimum with a market equilibrium which includes mark-up rates i n either product or factor markets, we can derive a deadweight loss measure using the methodology employed above. (25). However, the r e s u l t i n g approximate measure i s different from What matters now are derivatives of r e s t r i c t e d p r o f i t functions, given the input that causes the nonconvexity, instead of the derivatives of cost functions. The drawback of our approach i s that these derivatives are not observable at the distorted observed equilibrium. following c o r o l l a r y : It i s somewhat overcome by the 113. Corollary 1.1: T (31) The approximate ADD measure 1 1 -1 1 -Jjt ( A - B (C ) 1 1T B -1 ) t i s also accurate for quadratic functions as PROOF: According to Diewert's (25). (1976;118) Quadratic Approximation Lemma, both - ( r ' ( 0 ) + Jjr'tO)) and -^(r'(0) + r'(1)) give the exact value of r(0) - r(1) if r i s quadratic. The former was adopted to derive (25). Now using the l a t t e r approximation and using (28) we have 0 (32) r 1 - r T = -^ty'(1). Evaluating (26) at z = 1, computing y'(1) matrix and substituting i t into (32), The remarkable property of (31) by inverting the left-hand side we get (31). Q. E. D. i s that we can compute the deadweight loss of the economy from the l o c a l derivatives of demand and supply (cost) functions evaluated at the observed equilibrium. of this observation, i s that (31) One important consequence can be computed using f l e x i b l e forms for u t i l i t y and production functions, functional so that we need not assume r e s t r i c t i v e functional forms to calculate the global optimum point, as is usual i n the numerical general equilibrium l i t e r a t u r e . To derive our approximate loss formulae (25) and (31), several r e s t r i c t i v e assumptions. we maintained The assumption of competitive factor markets can be dropped by introducing mark-up rates on factor prices, even though the resulting formulae become more complicated. The assumption that 114. the production functions must be quasi-concave was required to guarantee the optimality of the marginal cost p r i n c i p l e , and we already discussed how to extend our approach when we dropped the assumption. We have assumed that each industry i s monopolized. the result to the case of an o l i g o p o l i s t i c It i s easy to extend industry i f we know the mark-up rates of firms and that the number of firms within one industry i s fixed for a l l industries. However, i t i s d i f f i c u l t to introduce entry-exit behaviour, since the f i r s t order s o c i a l optimality and market equilibrium conditions incumbents and entrants are characterized by inequalities ities, for the number of firms changes discontinuously for rather than equal- as e q u i l i b r i a are adjusted from the observed equilibrium to the optimum as is shown i n the l i m i t pricing literature. Therefore, i t i s d i f f i c u l t to apply our approach based on the Implicit Function Theorem.1 The only case with entry that we can deal with within our framework i s a Chamberlinian (1962) monopolistic competition with each product produced by homogeneous producers with respect to market shares, product q u a l i l t y and technology. number of firms i s continuous. Suppose also that the Then, the long-run equilibrium i s characterized by the z e r o - p r o f i t conditions of firms, i.e. e q u a l i t i e s where the number of firms i s also endogenous, and Chamberlinian excess capacities cause deadweight l o s s . The optimality conditions are characterized by the marginal cost p r i c i n g p r i n c i p l e and the optimum number of firms i s at the point where the marginal cost equals average cost. However, determined this model may be incomplete as a monopolistic competition model, since product d i v e r s i t y i s exogenous i n our model. with much more s i m p l i f i e d models, Stiglitz (1977). To make i t endogenous, we must work as adopted i n Spence (1976), and D i x i t and 115. 4-5. Conclusion This chapter has reconsidered the methodology for the measurement of waste due to imperfect competition i n the presence of increasing returns to scale. We noted H o t e l l i n g ' s (1938) confusion about the optimality of the marginal cost p r i n c i p l e and the derivation of his deadweight loss formula and rederived his formula as (30). The drawback to H o t e l l i n g ' s measure (30) that i t cannot be computed without finding the optimum beforehand. is This drawback was corrected by our measure (25) and (31) where we required only l o c a l information i n order to measure the deadweight loss. for the loss measure defined by (31), In p a r t i c u l a r , only information observable at the distorted equilibrium i s required to measure the dead loss. Fig. 10 shows a single-consumer economy with one good y and one non- producible production factor v, labour for example. i t y set OA exhibits increasing returns to scale, equilibrium cannot e x i s t . M = (y^' [Y|) v c a n The production p o s s i b i l - so that a competitive However, an imperfectly competitive equilibrium exist where the marginal rate of substitution between the good and labour i n consumption i s different from the rate of substitution i n production. The ADD optimum point D = ( Y Q I Q ) i s V a point where surplus labour is maximized given the u t i l i t y level at the observed distorted equilibrium where the reference bundle 6 consists only of labour. The point D i s characterized by the equality of the marginal rates of substitution i n consumption (or marginal benefit of the good) and the marginal rates of substitution i n production (or marginal cost of the good). these two curves as MB and MC. F i g . 11 shows The true amount of deadweight loss i s shown by the c u r v i l i n e a r t r i a n g l e ABC while the approximate measure (25) by the t r i a n g l e ABC and (31) i s shown by A B C ' . i s shown The proof that ABC, A B C , 116. ABC" r e a l l y correspond to (11), (25), (31) for t h i s simple economy i s analogous to the derivation and construction of (1.45), (1.48), (1.49) i n Chapter 1. Given the l i m i t a t i o n s and assumptions l i s t e d within the chapter, we can apply our generalized H o t e l l i n g ' s measure to various models of imperfect competition and to p u b l i c l y regulated markets when increasing returns to scale are present. We hope that the theoretical foundation provided here for Hotelling's measure w i l l stimulate future empirical research and policy evaluation using i t . 117. FOOTNOTES FOR CHAPTER 4 1 For the production function f ( x ^ , . . . , x , v , . . . , v ) , n N 3f /3x = 0. ' n n 1 M Therefore, the cost function C dual to f ' n n we assume that has the derivative 3C /9p = 0. ' n n r 2 For example, the increasing returns to scale technology obtained by combining a convex production p o s s i b i l i t i e s be dealt with within our framework. set with a large fixed cost can (See Negishi (1962).) Aoki (1971) also used a similar technological assumption to the one adopted here. 3 To apply the theorem, we need to assume that the Slater constraint q u a l i f i c a t i o n condition holds; that i s , for (10) we assume that a feasible exists that s t r i c t l y s a t i s f i e s the f i r s t N+M inequality * With increasing returns to scale, 0 0 0> . constraints. a l o c a l optimum that s a t i s f i e s f i r s t order conditions may not be globally optimal. , 0 solution the We assume that . . . (r ,y ,p ,w ) i s a global optimum. 5 Increasing returns to scale i s usually defined as a more than proportionate increase of output when a l l the inputs are proportionately increased. Baumol, Panzar and W i l l i g (1982;18-21) propose a weaker notion of increasing returns to scale, i.e., decreasing average cost, and showed that i t i s implied by decreasing marginal cost. 6 Arrow and Hurwicz (1960) and Arnott and Harris (1976) gave examples where cost minimization and the marginal cost p r i n c i p l e r e s u l t i n productive i n e f f i c i e n c y i n a type-3 economy. 7 According to the recent study of contestable markets by Baumol, Panzar and W i l l i g (1982), these strategic aspects are immaterial when the fixed cost i s not sunk. Since a natural monopoly must set the price equal to its 118. average cost for a sustainable equilibrium, the mark-up rates t the difference between the average cost and marginal cost, approach i s applicable. equal so that our 119. F i g . 10 The ADD Measure with Increasing Returns to Scale 120. Marginal Benefit of y U V M yo Fig. y 11 The ADD Measure and i t s Approximations with Increasing Returns to Scale 121 CHAPTER 5 PROJECT EVALUATION RULES FOR IMPERFECTLY COMPETITIVE ECONOMIES 5-1 Introduction In t h i s chapter, we are interested i n evaluating the net benefit of introducing a new technology i n the presence of pre-existing d i s t o r t i o n s . This problem, we c a l l project evaluation, may be defined as follows. Given a pre-project general equilibrium where consumers and firms follow some behavioural rules and demand and supply are equal, consider introducing a net output vector, c a l l e d a project. Both consumers and firms adjust to t h i s change and the economy moves to a post-project equilibrium. Project evalua- tion means to determine whether the project increased or decreased s o c i a l welfare. The evaluation of a small project when there i s perfect competition with tax d i s t o r t i o n s was surveyed by Diewert (1983b) and we applied his approach to evaluate the benefit of public goods when there are tax d i s t o r tions i n Chapter 3. Therefore, a natural way to proceed seems to be to extend t h i s approach to the evaluation of a small project i n an imperfectly competitive economy. looks at f i r s t . However, this approach may not be as promising as it Commenting on Davis and Whinston's (1965) use of a perceived demand curve i n the second best theory of imperfect competition, Negishi (1967) pointed out that the second best policy of a public firm i s indeterminate unless the perceived demand curves of the imperfect competitors are known. Therefore, we have to follow a different avenue. In project evaluation, i t i s often the case that a new project has effects which are too large to be approximated by d i f f e r e n t i a l changes so that a shadow-pricing approach must be given up. Project evaluation rules 122 . for large projects have been studied by Negishi (1962) and Harris (1978) for the case of perfect competition with an increasing returns to scale technology due to a large fixed cost. Negishi (1962) studied the welfare i m p l i - cations of the entry of a new firm which i s either a perfect competitor but has a large fixed cost technology or i s the only firm which deviates perfect competition. from Some of Negishi's results were extended and some new rules were developed by Harris (1978), who also considered economies with distortionary taxation and public goods. Negishi's However, Harris (1978) kept (1962) assumptions about perfect competition and a convex technol- ogy with a fixed cost. The purpose of this chapter i s to extend the Negishi and Harris results to an imperfect market economy. This extension to an imperfectly competitive economy may be important considering the above mentioned indeterminacy of the optimum policy when there i s imperfect competition. Our results in this chapter may be summarized as follows. First, the Harris and Negishi results hold even i f the assumption of a convex technology with a large fixed cost i s replaced by general nonconvex technology, provided i t i s assumed that pre and post-project e q u i l i b r i a e x i s t . the extensions of Negishi's Second, some of (1962) results by Harris (1978) depend on an i m p l i c i t weakening of the c r i t e r i o n for welfare improvement made by Harris compared with Negishi's o r i g i n a l welfare c r i t e r i o n . T h i r d l y , but most importantly, most of their rules can be applied i n imperfectly competitive economies generally, again as long as pre and post-project e q u i l i b r i a are assumed to e x i s t . In the next section, we discuss welfare c r i t e r i a for cost-benefit analysis. We discuss some confusion which exists concerning the use of the 123 . compensation p r i n c i p l e and show that the c r i t e r i o n adopted by Negishi (1962) for the acceptance of a project i s more s t r i c t than that by Harris (1978). We cannot judge which c r i t e r i o n i s superior to the other. However, when we develop project evaluation rules, we simply have to be e x p l i c i t on which c r i t e r i o n each rule i s based. After presenting the model i n section 3, i n section 4 we reconsider the rules l i s t e d by Harris (1978), which include Negishi's (1962) o r i g i n a l rules, and show that most of them are applicable in an imperfectly competitive economy. Economic implications and informational requirements for extending project evaluation rules to imperfectly competitive environments are also discussed. 5-2. Section 5 concludes. Compensation C r i t e r i a for Cost-Benefit Analysis Reconsidered In chapter 1, we analyzed the properties of the ADD measure and the HB measure as a s o c i a l welfare function. It was shown that they are consistent with the Pareto quasiordering, but they are not t y p i c a l l y w e l f a r i s t . We also suggested the use of deadweight loss measures for policy evaluation; a l t e r n a t i v e p o l i c i e s are ranked by the associated level of deadweight loss. The c r u c i a l assumption for using t h i s procedure was that production possibilities sets remain unchanged by these p o l i c i e s . Therefore, this approach works for changes i n tax or regulation p o l i c i e s with a fixed technology. We should note, however, that i t i s impossible to compare the values of these measures when production p o s s i b i l i t i e s the introduction of new projects, w e l f a r i s t assumptions. sets are changed by since these measures do not satisfy For example, the d e f i n i t i o n of Pareto optimal allocations takes as given the production p o s s i b i l i t i e s Pareto optimal a l l o c a t i o n a. sets. Consider a Suppose there i s a change i n technology which 124. permits the attainment of a new a l l o c a t i o n a which i s preferred by a l l consumers to a. Using the ADD measure, a i s always measured to exhibit at least as much deadweight loss as a, as a l l Pareto optimal a l l o c a t i o n s have ADD measures of zero. As a consequence, i t i s inappropriate to use this deadweight loss measure when the technology i s not fixed. The optimal reference equilibrium on which the HB measure i s based depends on technology so i f the introduction of a new project changes the technology, then a unique reference price cannot be determined to calculate the HB measure. Therefore, the topic of this chapter, the evaluation of a new project, necessitates an a l t e r n a t i v e c r i t e r i o n for s o c i a l welfare and especially the problem of u t i l i t y comparison. We recommend two c r i t e r i a ; the f i r s t i s Bergson Samuelson s o c i a l welfare function and the other i s Hicks-Kaldor compensation principle.1 In either c r i t e r i o n , we show i n this chapter that our project evaluation c r i t e r i a can be related to aggregate quantities of i n d i v i d u a l consumption bundles. One natural way to proceed i s to suppose that there exists a Paretoi n c l u s i v e social welfare function by assuming that either there exists an omniscient planner who d i s t r i b u t e s income optimally at any point (see Samuelson (1956)) or consumers' preferences s a t i s f y Gorman's (1953) r e s t r i c t i o n of quasi-homotheticity. It i s obvious that a Bergson-Samuelsonian s o c i a l welfare function can serve as a welfare indicator to evaluate states corresponding to d i f f e r e n t technologies Unfortunately, the i n a consistent manner. i t i s d i f f i c u l t to come to a consensus as to what an appro- priate functional form for the s o c i a l welfare function i s . Also the assump- tion of quasihomothetic preferences i s empirically r e s t r i c t i v e . An alternative method t r a d i t i o n a l l y adopted for the evaluation of projects i s the Kaldor (1939)-Hicks (1939, 1940) compensation p r i n c i p l e , which 125 . states that a move from one state to another should be made i f a potential Pareto improvement can be made.2 However, there are several versions of the compensation p r i n c i p l e and we have to be careful i n d i s t i n g u i s h i n g t h e i r d i f ferent meanings. 1 We suppose that two states of the economy (z,Y^,x^) and 1 (z,Y ,x ) are compared where z i s a vector of i n i t i a l resources which i s fixed, x 1 Y i s an aggregate production p o s s i b i l i t i e s set i n state i = 0,1, and 1 i s an aggregate consumption bundle i n state i = 0,1. T u t i l i t y levels for the H households u = ( u ^ , . . . , u ) We also define the and the Scitovsky sets H SCu ) for i = 0,1 corresponding to the u t i l i t y functions f ; 1 h the u t i l i t y functions are assumed to be continuous from above, quasiconcave, and nonsatiated. {x : E f n 1 x The Scitovsky set for period i i s defined as S(u ) = x 1 x, f ( x ) h h h ) uj, h = 1,...,H}, where u 1 = (u*,...,u^) . assumptions ensure that S ( u ) i s convex (see Scitovsky (1941-2(b)). 1 T Our Now we can define the four types of compensation t e s t . 1 £ p 0 (read state 1 i s preferred to state 0 by the Kaldor strong R S (V principle) i f f x^ e S(u^). 1 R (2) 0 (read state 1 i s preferred to state 0 by the Kaldor weak 1 . 0 p r i n c i p l e ) i f f z + Y intersects with S(u ). 1 R (3) K w p UCT , 0 (read state 1 i s preferred to state 0 by the Hicks strong nor p r i n c i p l e ) i f f x ° f. S ( u ) . 1 1 R p 0 (read state 1 i s preferred to state 0 by the Hicks weak HW (4) p r i n c i p l e ) i f f z + Y° and S(u ) are d i s j o i n t . 1 126 . What i s called Scitovsky's (1941-2(a)) double c r i t e r i o n i s that 0 i s prefer- red to 1 i f f both the Hicks and Kaldor c r i t e r i a are met i n either weak or strong form. The following two propositions are obvious. PROPOSITION 1: If 1 R K g p 0, then 1 R ^ 0, but not vice versa. PROPOSITION 2: If 1 R R W p 0, then 1 R Negishi (1962;88) wrote that i f 0 Rjj^p 1 ( H S p (z+Y ) and S ( u ° ) are d i s j o i n t , 1 then state 1 i s not recommended. and S(u ) are d i s j o i n t , i . e . , 1 1 R H w p 0, but not vice versa. i.e., In page 89, he wrote i f z + Y^ 0 then state 1 i s preferred to 0. Therefore, we may conclude that Negishi adopted the Hicks weak compensation c r i t e r i o n (4) as his project acceptance c r i t e r i o n . (1978;412) suggested that i f x p. 414, 1 In contrast, Harris £ S ( u ° ) then 0 i s preferred to 1 and that on i f x^ i S(u^) then 1 i s preferred to 0. strong compensation c r i t e r i o n (3). Therefore, he u t i l i z e d Hicks By Proposition 2, we deduce that i f project 1 i s accepted by Negishi's c r i t e r i o n , 1 i s also accepted by Harris' criterion, but not vice versa. In economic terms, state 1 meets Harris' acceptance c r i t e r i o n i f the pre-project aggregate consumption bundle x^ cannot be redistributed so as to make everyone as well off as u \ whereas state 1 meets Negishi's acceptance c r i t e r i o n i f everyone cannot be made as well off as u^ even when the best production plans and income d i s t r i b u t i o n policy are executed using the i n i t i a l endowment z and technology Y^. These two project rules are equivalent under the following condition. 127 . PROPOSITION 3: We define perfect competition as an equilibrium where there exits a price vector that equilibrates the markets and consumers are maximizing u t i l i t i e s and producers are maximizing p r o f i t s given the prices. If consumers' preferences are quasiconcave and quasihomothetic i s perfectly competitive, then 1 R„ , cr are 0 implies 1 R and state 0 3 0, i . e . , m 7 n HSP and HWP equivalent. PROOF: From d e f i n i t i o n (3), competitive, there exists p and p 0 T x° 2 P° T x ° £ S(u ). If state 0 i s 1 2 0^ such that p y for a l l y e Y° + z. x 2 P x perfectly for a l l x E S(u ) (See Debreu (1951, 1959)). . . 0 1 . 1 Since the Gorman aggregation conditions are met, x f. S(u ) implies x e S ( u ° ) and S(u ) i s a subset of S ( u ° ) , 1 Therefore,p x > p 0T 0 T x° but x 1 i s not on the boundary of S ( u ° ) . > p y for a l l y e Y° + z and for a l l x e S ( u ) . 0 T 1 Therefore, S(u ) and Y° + z are d i s j o i n t and from d e f i n i t i o n (4) 1 follows. Therefore, 1 R H W p 0 Q. E . D. for perfectly competitive economies i n which the Gorman aggregation conditions hold, the two Hicksian c r i t e r i a are equivalent. If these two conditions are not met, Negishi's c r i t e r i o n i s stronger than that of Harris, so that we have to c l a r i f y whether a project acceptance rule i s based on the Negishi or Harris c r i t e r i o n . 5-3. The Model We now sketch the model of Harris (1978). and H consumers, h = 1 , . . . , H . u t i l i t y functions f (x ) h There are N goods n = 1 , . . . N , Consumers' preferences are represented by where x e Q , a transformed orthant R . h + As for 128 . production, an aggregate closed production set and a net output vector belonging to i t for the private firm sector are denoted by Y and y. A new firm introduced by the government has an operating technology and a net output vector G and g, respectively. properties of the technologies We do not assume anything about the Y and G except closedness (which i s harmless from an empirical point of view). We assume the existence of e q u i l i b r i a as follows. N A before-project equilibrium i s defined as an H + 2 tuple i n R , r o 0 , 0 0,, . . (P ,Y , ( x , . . . , x ) } such that 1 (5) H f (x£) 2 h f h ( x n ) f o r a 1 1 x e n (budget constraint for h under p }, 0 for h = 1 , . . . , H , (6) y° e Y (7) 0 H 0 0^ x =L x = y + z. r h = 1 h In the same way, we define an a f t e r - p r o j e c t equilibrium to be an H + 3 tuple i n R , {p , g , y , (x!j, . . . ,x^)} such that N (8) f h 1 1 1 ^ h^ - ^( ) x f x f n for h = 1 , . . . , H , (9) y 1 eY (10) g 1 e G 1 1 o r a 1 1 X h e {k u d 9 e t constraint for h under p } 1 129 These two definitions seem to incorporate the minimum requirements for an imperfectly competitive e q u i l i b r i a studied by Negishi (1961-2), Arrow and Hahn (1971, Ch. 6), and Roberts and Sonnenschein (1977); i . e . , taking behaviour of consumers, and ( i i i ) (ii) (i) price f e a s i b i l i t y of equilibrium production, equality of demand and supply. The main r e s u l t i n t h i s chapter i s that the existence of before and after-project e q u i l i b r i a i s s u f f i c i e n t for the v a l i d i t y of most of the project evaluation rules developed by Negishi and Harris. 5-4. Project Evaluation Rules Using the Hicksian strong compensation c r i t e r i o n , Harris' two main project evaluation rules (using his numbering of the rules) can be restated within our framework as Rule 2: A sufficient follows. condition to reject a proposed project i s that the project have a net value at before-project prices which i s less than the change i n the p r o f i t s on a l l other production a c t i v i t i e s before-project prices; i . e . , Rule 4: profits A sufficient evaluated at OT 1 OT 0 1 the rejection c r i t e r i o n i s p g < p (y - y ). condition to accept a project i s that minus the (or minus the net value) of the project at post-project prices be less than the change i n p r o f i t s i n the rest of the economy at a f t e r - p r o j e c t prices; i . e . , the acceptance c r i t e r i o n i s -p IT g 1 < p 1T (y 1 0 - y ). 130. PROOF: Substituting the resources constraints (7) and (11) into Rule 2 OT 0 OT 1 and Rule 4, we find that Rule 2 i s equivalent to p x > p x , and Rule 4 i s 1T 1 1T 0 0 1 equivalent to p x > p x . As the Scitovsky sets S(u ) and S(u ) are convex sets, and x^ and x^ belong to the boundary of S(u^) and S(u^), respectively from (5) and (8), Rule 2 implies x 0 1 x i S(u ). 1 £ S ( u ° ) and Rule 4 implies . . . . . Now applying the Hicks strong compensation p r i n c i p l e (3), Rule 2 gives a s u f f i c i e n t condition for state 0 to be preferred to 1, and Rule 4 gives a s u f f i c i e n t condition for state 1 to be preferred to 0. Q. E . D. We proved H a r r i s ' s two main rules without making any assumptions concerning market structure. consumers are price takers. We assumed only that markets clear and In p a r t i c u l a r , we did not assume that either the private production sector or the government optimizes. Therefore, Harris' rules have a very broad a p p l i c a b i l i t y . From Proposition 3, Rule 2 i s v a l i d for Hicks' weak compensation p r i n c i p l e i f the Gorman aggregation conditions for consumers' preferences are met and the a f t e r - p r o j e c t equilibrium i s perfectly competitive. Similarly, Rule 4 i s v a l i d for Hicks' weak compensation p r i n c i p l e i f the aggregation of consumers' preferences conditions are met and the before-project equilibrium i s perfectly competitive.'' This i s the reason why Negishi (1962; 91) assumed that Gorman's preference r e s t r i c t i o n s were met i n his demonstration of the v a l i d i t y of Rule 4. Harris (1978) restated Negishi's 3 i n Harris' numbering, as follows: principle (1962) main two r u l e s , Rule 1 and Rule Referring to Hicks weak compensating 131 . Rule 1: A s u f f i c i e n t condition to r e j e c t a proposed project i s that i t is impossible for the project to show a nonnegative net value at before-project . . equilibrium prices; i . e . , Rule 3. . . OT 1 the r e j e c t i o n c r i t e r i o n i s p g < 0. A s u f f i c i e n t condition to accept a project i s that the project show positive p r o f i t s at post-project prices; i . e . , is p g the acceptance c r i t e r i o n > 0. In general, unless the economy i s competitive, except for the new firm introduced by the government, Rule 1 and Rule 3 are i n v a l i d (see Negishi (1962)).5 However, i f the competition assumption i s met, then Rule 1 implies Rule 2 and Rule 3 implies Rule 4 as the price-taking assumptions for firms OT 0 OT 1 1T 1 1T 0 imply p y 2 p y and p y 2 p y . be v a l i d i n general. Obviously, other implications cannot This means that the Harris rules are more complete than Negishi's for competitive economies (Harris (1978;413)). Stated another way, some project accepted by Harris' Rule 4 may not be accepted by Negishi's Rule 3 and some project rejected by Harris' Rule 2 may not be rejected by Negishi's Rule 1. This indeterminacy of Rules 1 and 3 comes partly from the fact that Negishi adopted Hicks' weak p r i n c i p l e as his welfare c r i t e r i o n , which i s more indeterminate than the Hicks strong p r i n c i p l e adopted by H a r r i s , but c h i e f l y i t i s because the p r o f i t a b i l i t y c r i t e r i o n i s a less exact estimate of the s o c i a l welfare change than the index number approach used i n Rule 2 and Rule 4. The drawback of Rule 2 and Rule 4 seems to be t h e i r more demanding informational requirements, i . e . , as long as the economy is competitive, the informational requirements for implementing Rules 1 and 3 seem less onerous 132. than for Rules 2 and 4. In Rule 1, we only need to know the production p o s s i b i l i t i e s set of the public agency. be somehow predicted. In Rule 3, a f t e r - p r o j e c t prices must However, i n Rules 2 and 4, a f t e r - p r o j e c t output l e v e l s of the rest of the economy are also required, and t h i s i s d i f f i c u l t to obtain ex ante. In summary, Rules 1 and 3 show the f i r s t - b e s t significance of a new technology i n terms of i t s p r o f i t a b i l i t y . They are almost always less exact than Rule 2 and Rule 4, and cannot t e l l us anything i n second best conditions when we do not have perfect competition. Rule 2 and Rule 4 are v a l i d i n both f i r s t best and second best conditions, and i n a second best, i t evaluates the improvement of technical efficiency and market e f f i c i e n c y at the same time. 5-5. Conclusion Harris discussed thirteen project evaluation rules i n his paper. Excluding Rule 1 and Rule 3, which we have discussed, and Rule 8, which i s analogous to Rule 1 i n the tax-distorted economy, a l l of his rules are effec- t i v e for non-competitive market structures, because the proofs of a l l of them are s i m i l a r to the proofs of Rule 2 and Rule 4, or they are contrapositives of other r u l e s . In p a r t i c u l a r , the s a t i s f a c t i o n of Samuelsonian conditions i s not necessary to prove Harris' Rules 11, 12 and 13, which give f i t rules for supplying a public input. cost-bene- Although p r o f i t a b i l i t y of a new project has a normative meaning only i n f i r s t - b e s t situations where the usual marginal conditions hold, the application of index number theorems due to Hicks (1940, 1941-2) and Samuelson (1950) are f r u i t f u l even i n a second best economy. 133 . FOOTNOTES FOR CHAPTER 5 1 Note also that, i n general, the relationship between the compensation p r i n c i p l e and the sum of equivalent or compensating variations i s ambiguous. See Boadway (1974), Smith and Stevens (1975), Foster (1976) and Boadway (1976). 2 In chapter 3, we considered a s u f f i c i e n t condition for the existence of a Pareto improvement. principle. 3 This discussion may be related to the compensation See Bruce and Harris (1982). Quasi-homotheticity i s s a t i s f i e d i f Engel curves are straight and they are p a r a l l e l for a l l consumers. See Gorman (1953). lines Alternatively, we can think of the case where income d i s t r i b u t i o n i s always optimized with respect to a Bergson-Samuelsonian s o c i a l welfare function. In this case, Bergson's s o c i a l indifference surfaces do not intersect and convex to the o r i g i n i f u t i l i t y functions are concave and a s o c i a l welfare function i s quasi-concave (see Gorman (1959) and Negishi (1963)). Replacing the Scitovsky set with the better set of Bergson's indifference surface, the of the discussion goes through. preferences, rest When we mention Gorman's r e s t r i c t i o n s on we can also allow for this alternative case. * Harris (1978;410) pointed out that his welfare c r i t e r i o n i s consistent with an ordering based on the Bergson-Samuelsonian s o c i a l welfare function, if i t exists. 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Essays on the measurement of waste and project evaluation Tsuneki, Atsushi 1987
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Title | Essays on the measurement of waste and project evaluation |
Creator |
Tsuneki, Atsushi |
Publisher | University of British Columbia |
Date Issued | 1987 |
Description | Harberger's methodology for the measurement of deadweight loss is reformulated in a general equilibrium context with adopting the Allais-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 Allais, Debreu, Diewert and Hicks, Boiteux in relation to Bergson-Samuelsonian social welfare function. We first 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 utilities of income of households, which is unavailable with ordinal utility theory. Finally, we give a diagrammatic exposition of the two measures and their approximations to give an intuitive 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 provision 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 interior social optimum. When indirect taxes are fully flexible but lump-sum transfers 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 flexible, 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 discuss various limitations of our approach and the relation between our work and that of Hotelling (1938). In Chapter 5, we consider cost-benefit rules of a large project applicable in the presence of imperfect competition. We show that the index number approach due to Negishi (1962) and Harris (1978) can be extended to handle situations with imperfect competition. |
Subject |
Welfare economics Waste (Economics) Economics -- Mathematical models |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-08-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0097517 |
URI | http://hdl.handle.net/2429/27554 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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