UNIVERSITY OF 
 
 ILLINOIS LIBRARY 
 
 AT URBANA-CHAMPAIGN 
 
 BOOKSTACKS 
 
DZ 
 
 CENTRAL CIRCULATION BOOKSTACKS 
 
 The person charging this material is re- 
 sponsible for its renewal or its return to 
 the library from which it was borrowed 
 on or before the Latest Date stamped 
 below. You may be charged a minimum 
 fee of $75.00 for each lost book. 
 
 Theft, mutilation, and underlining of books are reasons 
 for disciplinary action and may result in dismissal from 
 the University. 
 
 TO RENEW CALL TELEPHONE CENTER, 333-8400 
 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA CHAMPAIGN 
 
 APR 2 9 1998 
 AUG 3 1998 
 
 When renewing by phone, write new due date below 
 previous due date. L162 
 
330 
 
 3385 
 
 420 COPY 2 
 
 STX 
 
 BEB 
 
 FACULTY WORKING 
 PAPER NO. 1420 
 
 Ioffe's Normal Cone and the Foundations 
 of Welfare Economics: An Example 
 
 M. Ali Khan 
 
 AP! 
 
 1COQ 
 
 College of Commerce and Business Administration 
 Bureau of Economic and Business Research 
 University of Illinois, Urbana-Champaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 1420 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urb ana -Champaign 
 
 December 1987 
 
 Ioffe's Normal Cone and the Foundations 
 of Welfare Economics: An Example 
 
 M. Ali Khan, Professor 
 Department of Economics 
 
Ioffe's Normal Cone and the Foundations 
 of Welfare Economics: An Examplet 
 
 by 
 
 M. Ali Khan* 
 November 1987 
 
 Abstract . We announce an example of an economy with an infinite dimen- 
 sional commodity space for which the extension of the second fundamen- 
 tal theorem of welfare economics is valid if marginal rates of substi- 
 tution are formalized in terms of either the Clarke normal cone or the 
 Ioffe normal cone but in which the former is strictly contained in the 
 latter. This is in direct contradiction to the finite dimensional 
 situation. 
 
 tThis research is supported, in part, by a N. S.F. grant. The 
 author is grateful for the comments and questions of D. Diaraantaras, 
 L. Mckenzie, W. Thompson and the participants of the economic theory 
 seminar at the University of Rochester. Errors are solely his. 
 
 *Department of Economics, University of Illinois, 1206 South Sixth 
 Street, Champaign, Illinois, 61820. 
 
Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/ioffesnormalcone1420khan 
 
1. Introduction 
 
 In a recent paper, the author reported an extension of the so- 
 called second fundamental theorem of welfare economics in which the 
 marginal rates of substitution are formalized through the use of the 
 Ioffe normal cone, see Khan [1987]. In particular, it was shown that 
 under a mild constraint qualification and modulo scalar multiples, the 
 Ioffe normal cones to the production and the "no-worse-than" sets at 
 the respective production and consumption plans have a non-empty and 
 non-zero intersection. If these sets are generated by dif f erentiable 
 functions, the Ioffe normal cones reduce to singletons and the result 
 yields the conventional necessary conditions for a Pareto optimal 
 allocation as in Hicks [1939], Lange [1942], Allais [1943], Samuelson 
 [1947] and Graaf [1957]. If these sets are convex, the Ioffe normal 
 cones reduce to the cones conventional in the sense of convex analysis 
 and the result yields the statement that Pareto optimal allocations 
 can be sustained through profit maximization by producers and expen- 
 diture minimization by consumers as in Arrow [1951], Debreu [1951, 
 1954], Malinvaud [1953] and Koopmans [1957]. Finally, since the Ioffe 
 normal cone is contained in the Clarke normal cone, the result also 
 generalizes recent work of Khan-Vohra [1987], Yun [1984], Quinzii 
 [1986], Cornet [1986], which in turn generalized the earlier work of 
 Guesnerie [1975]. 
 
 The results in Khan [1987] are limited to an economy with a finite 
 dimensional commodity space and were initially motivated by the con- 
 sideration of an example of an economy with a production set without 
 free disposal and for which the marginal rate of substitution at a 
 
-2- 
 
 Pareto optimal production plan, as formalized by the Clarke normal 
 cone at that plan, is the entire dual space. It was shown that the 
 Ioffe normal cone at such a production plan is more in keeping with 
 our intuitive notion of a marginal rate of substitution. In this 
 note, we present an example of an economy with an infinite dimensional 
 commodity space that furnishes an opposite conclusion. In particular, 
 the economy has a production set without free disposal and for which 
 the Ioffe normal cone at the Pareto optimal production plan is the 
 en t i r e dual space, but the Clarke normal cone is a strict subset. 
 Intuition gets somewhat stretched in an infinite dimensional setting 
 but the Clarke normal cone at such a production plan does not seem 
 unreasonable. Our example underscores the unpredictability of the 
 Ioffe normal cone in an infinite dimensional setting and is based on 
 Treiman [1983]. 
 
 Section 2 reviews some basic concepts and Section 3 presents the 
 example. Section 4 is devoted to three concluding remarks. 
 
 2. The Basic Concepts 
 
 In this section we present the definitions of the Clarke and Ioffe 
 cones. Since our example is set in an infinite dimensional space, we 
 develop these definitions in the context of an arbitrary Banach space 
 E. However, for concreteness , the reader may choose to think in terms 
 of Euclidean n-space R or, for that matter, in terms of R . 
 
 We begin with a tangential approximant of a set introduced by 
 Bouligand [1932] and termed the contingent cone. It was first applied 
 in economics by Otani-Sicilian [1977]. 
 
-3- 
 
 Definition 2.1 The contingent cone of Y C E at y z Y is the set 
 
 k k 
 
 T (Y,y) = {x e C:_"»a sequence {t } of positive numbers with t ■*■ 
 K 
 
 k k k k 
 
 and a sequence {x } with x ■*■ x such that (y+t x ) e Y for all k}. 
 
 Definition 2.2 The Clarke tangent cone of Y - E at y z Y is the set 
 
 T r (Y,y) = {x z E: For any sequence {t } of positive numbers with 
 
 k k k k 
 
 t + and any sequence {y } with y e Y, y + y, there exists a 
 
 k k k k k 
 
 sequence {x } with x ■*■ x such that (y +t x ) z Y for all k}. 
 
 + 
 Definition 2. 3 For any cone A - E, the polar cone A is the set 
 
 {y z E*: <y,x> _< for all x e A} where E is the topological dual of 
 E. [Note that (R n )* = R n ]. 
 
 We shall denote the polars of T (Y,y) and T (Y,y) by N (Y,y) and 
 
 K. (, K 
 
 N (Y,y) and refer to them as the contingent normal cone and Clarke 
 normal cone respectively. For more details into these definitions, 
 see Clarke [1983] and Khan-Vohra [1987]. 
 
 Definition 2.4 The Ioffe normal cone to Y _ E at y z Y is given by 
 
 k k k 
 the set N (Y,y) = {x e E:^. a sequence {y } with y e Y, y ■*■ y and 
 
 k k k 
 x e N K ( Y >y ) with x * x} . 
 
 For details, see Ioffe [1981, 1984] and Khan [1987]. 
 
 The reader can check his understanding of these basic concepts by 
 referring to the technology depicted as Y in Figure la. The con- 
 tingent cone T (Y,y) is given by the shaded cone in Figure lb, the 
 
 Ioffe normal cone N (Y,y) by the two arrows in Fig. lb and the Clarke 
 
 a 
 
 normal cone N (Y,y) by the cone enclosed by these arrows. The Clarke 
 
-4- 
 
 tangent cone T (Y,y) is the set of all vectors making an obtuse angle 
 with elements of N (Y,y) and the contingent normal cone N (Y,y) is 
 zero. 
 
 We conclude this subsection by mentioning a result due to Cornet 
 that brings out the central position of the contingent cone. 
 
 Theorem : Let Y be a nonempty closed set in R . Then T (Y,y) = 
 
 n k k k 
 
 {x e R : For all sequences {y } with y e Y, y ■*■ y, there exists 
 
 k k k 
 x e T (Y,y ) with x ■*■ x} . 
 
 Proof: See Borwein-Strojwas [1985, Theorem 4.1]. 
 
 3. The Example 
 
 We work in c , the space of sequences of real numbers converging to 
 
 zero and endowed with the supremura norm. This is a Banach space with 
 
 I as its dual (see, for example, Dunf ord-Schwartz [1957] for details 
 
 and elementary properties). For any x,y in c , let x > y denote x. >_ 
 
 y. for all coordinates i in N, the space of positive integers. Let 
 
 c = {xec : x.>0 for all i in N}. £„ has an analogous meaning. 
 o+ o i— 1+ 
 
 The economy we present consists of c as the commodity space and 
 
 two agents, a producer with production set Y «S c and a consumer with 
 
 o 
 
 e c as the endowment, X = {x e c : x„ > 0, x~ > 0} as the consurap- 
 o o 2 — 3 — 
 
 tion set and > as the preference relation on X. The production set Y 
 is given by 
 
 {tw: t > 0}VJ{ U A } 
 
 _ M m 
 
 meN 
 
-5- 
 
 where 
 
 A = {^ z} I " {- z + I A ~) ' {\ x + w}} 
 mm m v -.. n Z n 
 
 neN m 
 
 [a,b) * A = {ta: t e [a,b) _~ R and a e A} 
 
 z = (1, -1, 0, 0, .. . ) e c 
 
 w = (0, 0, 1, -1, 0, ...) e c 
 
 x = - e , . , n e N 
 n n+4 
 
 and e. is a vector in c with all coordinates zero except for the ith 
 l o 
 
 coordinate which is unity. 
 
 Note that Y is a technology capable of producing two outputs, the 
 
 first and the third, and through the use of a countable number of 
 
 inputs. More specifically, it consists of the technique w which can 
 
 be operated at any nonzero level and the technique z which can be 
 
 operated at any level — , m a positive integer. There is also the 
 
 ra 
 
 option of joint production of the first and third commodities but in 
 this case "small" units of the third commodity require specialized 
 inputs given by the elements of x . 
 
 Lemma 3.1 Y is a non-empty, closed subset of c such that (i) 
 
 Y ,c - {0}, (ii) Y \(-Y) = {0}, (iii) T (Y, -z) = {0}, (iv) 
 1 o+ K m 
 
 T_(Y,0) = {tw: t > 0}, (v) N (Y,0) = I. . 
 u — a 1 
 
 Note that (i) formalizes the impossibility of producing something 
 from nothing and (ii) the property of "irreversibility." Furthermore, 
 
 Y is not convex and does not satisfy the property of "free disposal," 
 i.e., for any y e Y, y - (c ) d Y. For details of these properties, 
 see the standard reference, Debreu [1959]. 
 
-6- 
 
 Proof of Lemma 3.1 Nonemptiness of Y is is trivial and properties (i) 
 and (ii) require routine computations. The shortest, though not 
 necessarily the most transparent, proof of closedness, properties 
 (iii) and (iv) can be had by the observation that Y satisfies all the 
 assumptions required for Counterexample 3.1 in Treiman [1983]. (v) 
 
 follows from Definition 2.4 and (iii). 
 
 II 
 
 Now observe that (0,0) e (c x c ) is a Pareto optimal allocation 
 
 o o 
 
 of our two agent economy. If not, there exists y e Y, y * 0, such that 
 
 y _> 0. But this contradicts Lemma 3.1(i). Now the "no worse than" 
 
 set at is given by c . Furthermore, N (c , 0) = -I. , . Since 
 
 o+ C o+ 1+ 
 
 N c (Y,0) = (T C (Y,0)) + = {p e li <p,w> < 0}, we obtain 
 
 = N_(Y,0) , (-N (c ,0)) = {pe£ : p. > (ieN) and p_ [p.} 
 L L o+ 11 — J — 4 
 
 On the other hand, 
 
 " = N (Y,0) , (-N (c .0)) = {pel : p. • (ieN)}. 
 a a o+ 1- i — 
 
 , '- v C 
 The fact that -_ s , _ *J" is forbidden by the principal result in 
 
 the finite dimensional setup in Khan [1987]. 
 
 4. Concluding Remarks 
 
 Ioffe [1981] presents an alternative definition of the so-called 
 
 Ioffe normal cone that may lead to a different object in infinite 
 
 dimensional spaces from the one considered here. For any set Y . E and 
 
 y e Y, denote such a cone by N (Y,y) where 
 
 a 
 
-7- 
 
 I k k k 
 
 N (Y,y) = i{x e E: ZZ a sequence {y } with y ■*■ y, y e Y 
 
 a Fe „ 
 
 k k k k 
 
 and x e N (Y , (y +F),y ) with x ■»• x} , 
 
 and ..is the family of finite dimensional subspaces of E. However, it 
 is easy to check that for the example presented in Section 3, 
 
 N^Y.O) - I.. 
 a i 
 
 Borwein-Strojwas [1985] introduce the notion of compactly epi- 
 Lipschitzian sets that includes the class of epi-Lipschitzian sets 
 introduced in the context of the second welfare theorem by Khan-Vohra 
 [1985]. It is easy to check that Y in the example in Section 3 is not 
 compactly epi-Lipschitzian at the Pareto-optimal production plan con- 
 sisting of zero. One simply needs the characterization of compact 
 
 sets in c as given in Dunford-Schwar£-[1957 , IV. 13. 9]. 
 o 
 
 Finally, it is easy to manufacture examples of economies with an 
 infinite dimensional commodity space in which the extension of the 
 second welfare theorem fails in the sense that the normal cones, 
 Clarke's or Ioffe's, at the Pareto optimal production and consumption 
 plans have an intersection consisting solely of the zero vector. For 
 one such example, simply take the negative of set presented in Klee 
 [1963] as the production set plus initial endowment and the coordinate- 
 wise ordering as the preference relation. The example presented in 
 Section 3 is very different in spirit. Here the problem has to do 
 with the Ioffe normal cone being "too large" rather than "too small." 
 
-8- 
 
 References 
 
 Allais, M. , 1943, A la recherche d'une discipline economique, vol. 1 
 (Ateliers Industria, Paris). 
 
 Arrow, K. J., 1951, An extension of the basic theorem of classical 
 welfare economies, Proceedings of the Second Berkeley Symposium 
 (University of California Press, Berkeley). 
 
 Borwein, J. and H. M. Strojwas, 1985, Tangential Approximations, 
 Nonlinear Analysis 9, 1347-1366. 
 
 Bouligand, G. , 1932, Introducton a la geometrie inf initesimale directe 
 (Vuibert, Paris). 
 
 Clarke, F. H. , 1983, Optimization and nonsmooth analysis (John Wiley, 
 New York). 
 
 Cornet, B. , 1986, The second welfare theorem in nonconvex economies, 
 CORE Discussion Paper No. 8630. 
 
 Debreu, G. , 1951, The coefficient of resource utilization, 
 Econometrica 19, 273-292. 
 
 Debreu, G. , 1954, Valuation equilibrium and Pareto optimum, Proceedings 
 of the National Academy of Sciences 40, 588-592. 
 
 Debreu, G. , 1959, Theory of value (John Wiley, New York). 
 
 Dunford, N. and J. T. Schwartz, 1967, Linear operators, Part I 
 (Interscience, New York). 
 
 Graaf, J. de Van, 1957, Theoretical welfare economies (Cambridge 
 University Press, Cambridge). 
 
 Hicks, J. R. , 1939, The foundations of welfare economics, Economic 
 Journal 49, 696-712. Also Prefatory Note to reprint in 1984, 
 Wealth and welfare: Collected essays on economic theory, vol. 1 
 (Basil Blackwell, Oxford). 
 
 Guesnerie, R. , 1975, Pareto optimality in non-convex economies, 
 Econometrica 43, 1-29. 
 
 Ioffe, A. D. , 1981, Sous-dif f erentielles approchees de fonctions 
 
 numerique, Coraptes Rendus Acad. Sci. Paris Ser. A-B, 292, 675-678. 
 
 Ioffe, A. D. , 1984, Approximate subdif ferentials and applications I: 
 the finite dimensional theory, Transactions of the American 
 Mathematical Society 281, 389-416. 
 
-9- 
 
 Khan, M. Ali, 1987, The Ioffe normal cone and the foundations of 
 
 welfare economics, B. E.B.R. Working Paper No. 1388, University of 
 Illinois. 
 
 Khan, M. Ali and R. Vohra, 1985, Pareto optimal allocations of non- 
 convex economies in locally convex spaces, B. E.B.R. Working Paper 
 No. 1373, forthcoming in Nonlinear Analysis. 
 
 Khan, M. Ali and R. Vohra, 1987, An extension of the second welfare 
 theorem to economies with non-convexities and public goods, 
 Quarterly Journal of Economics 102, 223-241. Circulated in 1984. 
 
 Klee, V., 1963, On a question of Bishop and Phelps, American Journal 
 of Mathematics 85, 95-98. 
 
 Koopmans, T. , 1957, Three essays on the state of economic science 
 (McGraw Hill, New York). 
 
 Lange, 0., 1942, The foundations of welfare economics, Econoraetrica 
 10, 215-228. 
 
 Malinvaud, E. , 1953, Capital accumulation and efficient allocation of 
 resources, Econoraetrica 21, 233-268. 
 
 Otani, Y. and J. Sicilian, 1977, Externalities and problems of non- 
 convexity and overhead costs in welfare economics, Journal of 
 Economic Theory 14, 239-251. 
 
 Quinzii, M. , 1986, Renderaents croissants et equilibre generale, Ph.D. 
 dissertation, Universite de Paris II. 
 
 Samuelson, P. A., 1947, Foundations of economic analysis (Harvard 
 University Press, Cambridge). 
 
 Treiraan, J. S. , 1983, Characterization of Clarke's tangent and normal 
 cones in finite and infinite dimensions, Nonlinear Analysis 7, 
 771-783. 
 
 Yun, K. K. , 1984, Pareto optimality in non-convex economies and mar- 
 ginal cost pricing equilibria, Proceedings of the First National 
 Conference of Korean Economists. 
 
 D/464 
 
Ns 
 
 * .* , 
 
 Fig. la 
 
 N (Y,y) 
 a 
 
 T K (Y,y) 
 
 Fig lb 
 
HECKMAN LJ 
 BINDERY INC. |s| 
 
 
 JUN95 
 
 R,™„d To Picas? N. MANCHESTER, 
 
 Bound -To -lkasc- | ND|ANA 46 g 62