Possibility functions and regular economies

Transcription

1 Possibility functions and regular economies Jean-Marc Bonnisseau 1 CES, Université Paris 1 Panthéon-Sorbonne CNRS Elena L. del Mercato 2 CSEF, Università degli Studi di Salerno Abstract We consider pure exchange economies with additional constraints on the consumption possibility different from the budget constraints. We analyze the case in which the constrained consumption possibility of each household is described in terms of a function, called possibility function, which depends on the individual initial endowment. Under a sufficient condition which is satisfied in many economically meaningful examples, we prove that economies are regular for almost all initial endowments and almost all possibility functions. JEL classification: C62, D11, D50. Key words: General economic equilibrium, consumption possibility, regular economies. This version of the paper was prepared during reciprocal visits of E. L. del Mercato to CES and of J.-M. Bonnisseau to CSEF, University of Salerno. The hospitality of both institutions is gratefully acknowledged. Presented at the Conférence Internationale sur les Mathématiques de l Optimization et de la Décision in Guadeloupe (2006), at the 15th European Workshop on General Equilibrium Theory in Lisbon (2006), it has also benefitted from the comments of these audiences. 1 Centre d Economie de la Sorbonne (CES), Université Paris 1 Panthéon-Sorbonne CNRS, Boulevard de l Hôpital, Paris Cedex 13, France. Jean-Marc.Bonnisseau@univ-paris1.fr. 2 Centre for Studies in Economics and Finance (CSEF), Dipartimento di Scienze Economiche e Statistiche, Università degli Studi di Salerno, Via Ponte don Melillo I, Fisciano (SA), Italy. edmercat@unisa.it. 1 January 2007

2 1 Introduction The global approach to equilibrium analysis of smooth economies is based on the central concept of regular economy. An economy is regular if the number of equilibrium prices is finite and, around each equilibrium price, there exists a local differentiable selection of the equilibrium price set with respect to the fundamental parameters of the economy. In the global analysis of smooth economies à la Arrow Debreu, wealth constraint is the only constraint for household s behavior. We can quote many authors, Debreu (1970, 1976), Balasko (1988), Dierker (1982), Mas-Colell (1985), Smale (1974a). Actually, Debreu (1959) considered for every household h an additional constraint, that is the individual consumption set X h defined as the set of all consumption alternatives which are a priori possible for household h. The author provides as example a survival possibility to work,...the decision for an individual to have during next year as sole input one pound of rice and as output one thousand hours of some type of labor could not be carried out. This idea is formalized and analyzed later in Bonnisseau and Rivera (2005) where consumptions may be restricted to be above minimal levels. Our approach is based on the fundamental intuition of Debreu, that consumption alternatives are limited by additional constraints different from the budget constraint. Indeed, first, our skills and our abilities clearly affect our possibility to make decision. Second, our information and our memory modify the perception of our possibilities (but not necessarily of our preferences). As such, differential and asymmetric information might be modelled as an additional constraint on the agents consumption possibility depending on information. Moreover, since many interactions take place among economic agents, the choices of other people constrain our possibility to make decision in every place and in every situation. This work is the first step to encompass and to study the impact of that richer phenomenons in the global analysis of equilibria. There are two main features in this paper. First, the consumption sets are described in terms of functions (see also Smale, 1974b, and del Mercato, 2004). We call possibility function a function describing a consumption set. The possibility function represents the restricted consumption possibility on commodity markets. Many different authors have considered restrictions on the markets, Cass (1990), Balasko, Cass and Siconolfi (1990), Polemarchakis and Siconolfi (1997), Cass, Siconolfi and Villanacci (2001). But the difference with our approach is that in these papers the restriction is on financial markets. Second, the possibility functions depend on individual initial endowments. 2

3 Indeed, in a consumption-leisure model, the consumption in leisure is bounded below by a bound which depends on the maximal possible workload of the household. This workload is precisely the initial endowment in the commodity labor which is the opposite of the leisure. If some trading rules impose some limitations on the possible net trade of an household on the market, this leads to a consumption set which depends on the initial endowment, since the possible consumption are constrained as the initial endowment plus the possible net trade. Moreover, if one considers the individual initial endowment as an indicator of social status, it affects our consumption possibility. First, our social status clearly affects our knowledge and our information on the quality and on the quantity of available goods. Second, our skills and our abilities depend on our social status and they clearly constrain our possibility to make decision. A further example, the accessibility to the commodity markets is a measure of the development degree of a country which clearly depends on its resources. Besides, as Mas-Colell and Smale have pointed out (see Smale, 1974b), the case in which the consumption set of each household depends on his initial endowment becomes important since...it would make more sense to allow greater latitude for the initial allocation in the definition of economy. In this paper, we consider a pure exchange economy with a finite number of households. Each household is characterized by an initial endowment of commodities, a possibility function and a utility function. The possibility function of each household depends on his initial endowment. Taking prices and initial endowment as given, each household maximizes his utility function in his consumption set under his budget constraint. The definition of competitive equilibrium follows. To get existence of equilibria, the assumptions on the utility functions are standard in smooth economies. In Assumption 2, the assumptions on the possibility functions are adapted from del Mercato (2006) in a natural way. 3 The main results of this paper deal with the well known regularity results of Debreu (1970, 1976), Balasko (1988), Dierker (1982), Mas-Colell (1985), Smale (1974a, 1974b). Their analysis make appear that classical differentiability and regularity results hold whenever all agents are in the interior of their consumption sets at equilibrium. Since nothing prevents the equilibrium allocations to be on the boundary of the consumption sets, we follow the strategy laid out by Cass, Siconolfi and Villanacci (2001) to prove generic differentiability and regularity results. But the dependency of each possibility function with respect to the individual initial endowment leads to technical difficulties. For that rea- 3 del Mercato (2006) proved the existence of equilibria for a pure exchange economy in which the possibility function of each household also depends on the consumption choices of others. 3

4 son, we consider simple perturbations of the possibility functions. This is the key idea for our treatment. Besides, we need an additional assumption on the possibility functions, namely Assumption 4, which covers three relevant cases, that is when the possibility function of at least one household satisfies one of the following conditions: 1. it does not depend on the initial endowment, 2. it depends on the net trade, 3. it is with separable variables. The main result are Theorem 17 and Corollary 18 which embody the regularity results. Theorem 17 states the generic regularity result for perturbed economies, and Corollary 18 provides the regularity result for almost all initial endowments and for almost all possibility functions. The paper is organized as follows. Section 2 is devoted to our basic model and assumptions. In Section 3, the concept of competitive equilibrium is further analyzed. Using Kuhn Tucker conditions, we characterize equilibria in terms of equilibrium function. Then, Theorem 10 states the non-emptiness and the compactness of the equilibrium set. In Section 4, we state the definitions of regular economy and perturbed economy. The main result of the paper is Theorem 17 which states that, generically, the perturbed economies are regular. As a consequence of Theorem 17 we obtain Corollary 18 which states that, in an open and dense subset of the space of initial endowments and possibility functions, the economies are regular. In Subsection 4.1, first we provide the definition of border line case for perturbed economies (i.e., a situation in which, at equilibrium, a consumption is on the boundary of the consumption set and the associated Lagrange multiplier vanishes). Second, Proposition 21 establishes that, generically, border line cases do not occur. In Subsection 4.2, the strategy of the proof of Theorem 17 is detailed. Especially as a first step, we deduce from Proposition 21 that the perturbed equilibrium function is generically differentiable at each equilibrium allocation. Finally, in Subsection 4.3 we prove Corollary 18. All the proofs are gathered in Appendix, except for Theorem 17 and Corollary The model and the assumptions There are C < physical commodities labelled by superscript c {1,..., C}. The commodity space is R C ++. There are H < households labelled by subscript h H := {1,..., H}. Each household h H is characterized by an initial endowment of commodities, a possibility function and a utility function. The possibility function of household h depends on his initial endowment. The notations are summarized below. x c h is the consumption of commodity c by household h; 4

5 x h := (x 1 h,.., x c h,.., x C h ); x := (x h ) h H R CH ++. e c h is the initial endowment of commodity c owned by household h; e h := (e 1 h,.., e c h,.., e C h ) R C ++ ; e := (e h ) h H R CH ++. As in general equilibrium model à la Arrow Debreu, each household h H has to choose a consumption in his consumption set, i.e. in the set of all consumption alternatives which are a priori possible for him. In our paper, the consumption set of household h is described in terms of a function χ h called possibility function. Moreover, the possibility function of household h depends on his initial endowment. That is, given e h R C ++, the consumption set of the household h is the following set, X h (e h ) = { x h R C ++ : χ h (x h, e h ) 0 } where χ h : R C ++ R C ++ R; χ := (χ h ) h H. Each household h H has preferences described by a utility function u h from R C ++ to R, and u h (x h ) R is the utility of household h associated with the consumption x h R C ++; u := (u h ) h H. E := (e, χ, u) is an economy. p c is the price of one unit of commodity c; p := (p 1,.., p c,.., p C ) R C ++. Given w = (w 1,.., w c,.., w C ) R C, we denote w \ := (w 1,.., w c,.., w C 1 ) R C 1 From now on, we make the following assumptions on (χ, u). The assumptions on u h are standard in smooth general equilibrium models. The assumptions on χ h are adapted from del Mercato (2006) in a natural way. Assumption 1 For all h H, (1) u h is a C 2 function. (2) u h is differentiably strictly increasing, i.e. x h R C ++, D xh u h (x h) 0. (3) u h is differentiably strictly quasi-concave, i.e. (x h, y) R C ++ R C \{0}, D xh u h (x h)y = 0 [yd 2 x h u h (x h)]y < 0. (4) For each u Im u h, cl R C{x h R C ++ : u h (x h ) u} R C ++. Assumption 2 For all h H, (1) χ h is a C 2 function. For each e R CH ++, (2) (Convexity of the consumption set) the function χ h (, e h ) is quasi-concave. 4 (3) (Non-empty intersection with the budget set) There exists x h R C ++ such that χ h ( x h, e h ) > 0 and x h e h. (4) (Non-satiation) For each x h R C ++, D xh χ h (x h, e h ) 0 and there exists c {1,..., C} such that D x c h χ h (x h, e h ) 0. 4 Since χ h is C 2, we have that (x h, y) RC ++ R C and D xh χ h (x h, e h)y = 0 [yd 2 x h χ h (x h, e h)]y 0. 5

6 (5) (Global desirability) For each x R CH ++ and for each c {1,..., C} there exists h(c) H such that D x c χ h(c)(x h(c) h(c), e h(c)) 0. From Assumptions 2.1 and 2.2, the usual assumptions on closedness and convexity of the consumption set hold true. From Assumption 2.3, given the initial endowment e h R C ++, the consumption set X h (e h ) = { x h R C ++ : χ h (x h, e h ) 0 } has a non-empty intersection with the budget set. Assumption 2.3 corresponds to the survival assumption. A rough version of survival assumption states that for each household his initial endowment is an interior point of his consumption set. Assumption 2.3 is a survival condition of each household h even if e h is not assumed to belong to the consumption set X h (e h ). The first condition in Assumption 2.4 means that the boundary of the consumption set is smooth. Moreover, the possibility for each household to be locally non-satiated remaining in his consumption set is crucial in order to expect existence of equilibria. Note that the second condition in Assumption 2.4. is weaker than the following one, which seems to be economically reasonable: each household is free for at least one good, i.e. given the initial endowment e h R C ++, there is at least one good c such that D x c h χ h (x h, e h ) = 0 for each x h R C ++. Finally, commodities must be desirable. The idea is the following: a good is globally desirable if for each consumption configuration of the economy there is one household that can locally strictly increase his utility by consuming a larger quantity of this good remaining in his consumption set. According to Assumption 1.2, it is trivially true that each good is desirable if there is at least one super-household, i.e. a household who is not constrained in the consumption possibilities, that is his consumption set is R C ++. Assumption 2.5 is weaker than the existence of a super-household. As a consequence of Assumptions we get the following proposition. It will play a fundamental role in the characterization of household h s maximization problem in terms of Kuhn Tucker conditions and in the result of generic regularity (in particular see the proof of Lemmas 20 and 23). Proposition 3 Let (x h, e h, p) R C ++ R C ++ R C ++ such that χ h (x h, e h ) = 0 and p(x h e h ) = 0. Then, p and D xh χ h (x h, e h ) are linearly independent. The above assumptions are enough to get the existence of competitive equilibria. But, for our purpose, that is the generic regularity of economies, we need an additional assumption on the possibility functions χ. Assumption 4 For each e R CH ++, 6

7 (x, y) R CH ++ R CH and D xh χ h (x h, e h)y h = 0 for all h H, imply that there exists k H such that [y k (D 2 x k χ k (x k, e k) + D 2 e k x k χ k (x k, e k))]y k 0. Observe that if we skip the term D 2 e k x k χ k (x k, e k), then we go back to Assumption 2.2. In the following remark, we provide three relevant cases in which Assumption 4 is satisfied. Remark 5 (1) If one possibility function χ k does not depend on the initial endowment, then Assumption 2.2 implies that Assumption 4 is satisfied. Then, our analysis encompasses the case analyzed by Smale (1974.b) in which all the possibility functions do not depend on initial endowments. (2) If one possibility function χ k depends on the net trade, i.e. χ k (x k, e k ) := χ k (x k e k ) where χ k is a C 2 function from R C to R, then Assumption 4 holds true. (3) If one possibility function χ k is with separable variables, i.e. χ k (x k, e k ) := χ k (x k ) + g k (e k ) where χ k and g k are C 2 functions from R C ++ to R, then Assumption 2.2 implies that Assumption 4 is satisfied. The set of utility functions (u h ) h H which satisfy Assumption 1 is denoted by U. The set of possibility functions (χ h ) h H which satisfy Assumptions 2 and 4 is denoted X. 3 Competitive equilibria First, we provide household h s maximization problem, market clearing conditions and the definition of competitive equilibrium. Second, we characterize in (3) the solution of household h s maximization problem in terms of Kuhn Tucker conditions. Then, in Remark 9 we restate equilibria in terms of solutions of equations, from which we deduce the equilibrium function F defined in (4). Finally, Theorem 10 states the non-emptiness and the compactness of the equilibrium set. Without loss of generality, commodity C is the numeraire good. Then, given p \ R C 1 ++ with innocuous abuse of notation we denote p := (p \, 1) R C ++. Given an economy E and p \ R C 1 ++, household h s maximization problem is max xh R C ++ u h(x h ) subject to χ h (x h, e h ) 0 px h pe h (1) 7

8 We say that x = (x h ) h H satisfies market clearing conditions if x h = e h (2) h H h H That is, if the aggregate consumption is equal to the total resources. Definition 6 (x, p \ ) R CH ++ R C 1 ++ is a competitive equilibrium for E if for all h H, x h solves problem (1) at p \, and x satisfies market clearing conditions (2). Proposition 7 Given E and p \ R C 1 ++, Problem (1) has a unique solution. An element x h is the solution to problem (1) if and only if there exists (λ h, µ h) R ++ R such that (x h, λ h, µ h) is the unique solution of the following system at p \. (h.1) D xh u h (x h ) λ h p + µ h D xh χ h (x h, e h ) = 0 (h.2) p(x h e h ) = 0 (h.3) min {µ h, χ h (x h, e h )} = 0 (3) In the following remark we just observe that at optimum an analogous condition to Smale s Assumption holds true (see NCP Hypothesis, Smale, 1974.b). Remark 8 Let x h be the solution to problem (1) at E and p \ R C From Propositions 3 and 7, and Assumptions 1.2 and 2.4, we get χ h (x h, e h ) = 0 implies that D xh u h (x h) and D xh χ h (x h, e h ) are linearly independent. Define the set of endogenous variables as Ξ := (R C ++ R ++ R) H R C 1 ++, with generic element ξ := (x, λ, µ, p \ ) := ((x h, λ h, µ h ) h H, p \ ). We can now describe extended equilibria using system (3) and market clearing conditions (2). Observe that, from Definition 6 and Proposition 7, the market clearing condition for good C is redundant (see equations (h.2) h H in (3)). Therefore, in the following remark we omit in (2) the condition for good C. Remark 9 ξ Ξ is an extended competitive equilibrium for E if and only if (x h, λ h, µ h) solves system (3) at p \ for all h H, and (x \ h e \ h ) = 0. h H With innocuous abuse of terminology, we call ξ simply an equilibrium. Given an economy E, define the equilibrium function F : Ξ R dim Ξ F (ξ) := ((F h.1 (ξ), F h.2 (ξ), F h.3 (ξ)) h H, F M (ξ)) (4) where F h.1 (ξ) := D xh u h (x h ) λ h p + µ h D xh χ h (x h, e h ), F h.2 (ξ) := p(x h e h ), 8

9 F h.3 (ξ) := min {µ h, χ h (x h, e h )}, and F M (ξ) := h H(x \ h e\ h ). From Remark 9, ξ Ξ is an equilibrium for E if and only if F (ξ ) = 0. The following theorem provides non-emptiness and compactness results for the equilibrium set F 1 (0). Theorem 10 For each economy E R CH ++ X U, the set of equilibria for E is nonempty and compact. 4 Regular economies Let us begin with the definition of a regular economy. Definition 11 E is a regular economy if (1) for each ξ F 1 (0), F is differentiable at ξ, and (2) 0 is a regular value for F, which means that for each ξ F 1 (0), the differential mapping D ξ F (ξ ) is onto. The analysis of Debreu (1970, 1976), Balasko (1988), Dierker (1982), Mas- Colell (1985), show that the differentiability of F holds whenever all agents are in the interior of their consumption sets at equilibrium. 5 Since nothing prevents the equilibrium allocations to be on the boundary of the consumption sets, for each h H the function F h.3 (ξ) = min {µ h, χ h (x h, e h )} is not everywhere differentiable. Then, first of all, it shall be show that, generically, the equilibrium function is differentiable at each equilibrium. To prove generic differentiability and regularity results we follow the strategy laid out by Cass, Siconolfi and Villanacci (2001) on the portfolio restricted participation side. But the dependency of χ h with respect to the initial endowments leads to technical difficulties. For that reason, we consider simple perturbations of the possibility functions. This is the key idea of our main results, namely Theorem 17 and Corollary 18, which embody the regularity results. Theorem 17 states the result of generic regularity for perturbed economies. Corollary 18 provides the regularity result for almost all initial endowments and for almost all possibility functions. Remark 12 In the following definitions and in Theorem 17, we take for fixed an arbitrary (χ, u) X U. 5 Indeed, in their analysis, each consumption set simply coincides with R C ++. 9

10 Definition 13 A perturbed economy E a associated with possibility levels a = (a h ) h H R H is defined by E a = (e, χ a, u) where χ a h(x h, e h ) = χ h (x h, e h ) + a h for all h H. A perturbed economy is parameterized by endowments and possibility levels taken in the following set Λ := (e, a) RCH ++ R h H, x H h R C ++ such that χ h ( x h, e h ) + a h > 0 and x h e h Remark 14 By Assumptions 2.1 and 2.3, we get Λ is an open subset of R CH ++ R H. R CH ++ R H + Λ. In particular, for each e R CH ++, (e, 0) Λ. This means that for each e R CH ++, Λ embodies the economy E = (e, χ, u). It is an easy matter to check that for all h H, the perturbed possibility functions χ h (, e h )+a h satisfy Assumptions 2 and 4 for each (e, a) Λ. Then, first, in (5) we define the equilibrium function for perturbed economies. Second, we present Proposition 15 which states the non-emptiness and the compactness of the equilibrium set for perturbed economies. For each (e, a) Λ, ξ is an equilibrium of the perturbed economy E a if and only if ξ is a zero of the equilibrium function F e,a : ξ Ξ F e,a (ξ) := F (ξ, e, a) R dim Ξ (5) where the function F : Ξ Λ R dim Ξ is defined by F (ξ, e, a) := (( F h.1 (ξ, e, a), F h.2 (ξ, e, a), F h.3 (ξ, e, a)) h H, F M (ξ, e, a)) (6) F h.1 (ξ, e, a) := D xh u h (x h ) λ h p+µ h D xh χ h (x h, e h ), F h.2 (ξ, e, a) := p(x h e h ), F h.3 (ξ, e, a) := min {µ h, χ h (x h, e h ) + a h }, and F M (ξ, e, a) := h H(x \ h e\ h ). Proposition 15 For each (e, a) Λ, F 1 e,a (0) is nonempty and it is compact. As a direct consequence of Remark 14, Assumptions 2.1 and 2.2, we obtain the following proposition. The continuous selection functions given in Proposition 16 will play a fundamental role in the properness result used to show generic differentiability and regularity results (see Lemma 19 and its proof). Proposition 16 For each h H, Λ h denote the projection of Λ on R C ++ R. For each h H, there exists a continuous function x h : Λ h R C ++ such that for each (e h, a h ) Λ h, χ h ( x h (e h, a h ), e h ) + a h > 0 and x h (e h, a h ) e h. 10

11 We can now state the main result of the paper about the generic regularity for perturbed economies. Theorem 17 The set Λ r of (e, a) Λ such that E a = (e, χ a, u) is regular is an open and full measure subset of Λ. Moreover, endow the set X with the product topology of the topology of C 2 uniform convergence on compact sets. As a consequence of Theorem 17 we get the following corollary which provides the regularity result in an open and dense subset of the space of initial endowments and possibility functions. Corollary 18 For each u U, the set R u of (e, χ) R CH ++ X such that E = (e, χ, u) is a regular economy is an open and dense subset of R CH ++ X. In Subsection 4.1, first we state the definition of border line cases for F. Second, Proposition 21 establishes that border line cases occur outside an open and full measure subset Θ of the space Λ. The proof of Proposition 21 is built upon Lemmas 19 and 20. In Subsection 4.2, the strategy of the proof for Theorem 17 is detailed. As a first step, we deduce from Proposition 21 that F is differentiable in F 1 (0) (Ξ Θ). Then, we show Theorem 17 using Lemma 23 and Proposition 24. Finally, in Subsection 4.3 we prove Corollary 18 using Lemmas 25, 26 and 27. To show Corollary 18, we follow a similar strategy to the one presented by Villanacci et al. (2002) Border line cases Given (ξ, e, a) F 1 (0), we say that household h is at a border line case if µ h = χ h (x h, e h ) + a h = 0. The main result of this subsection is Proposition 21 stating that border line cases occur outside an open and full measure subset Θ of the space Λ. To construct the set Θ and to prove that Θ is open and full measure subset of Λ we need introduce some preliminary definitions and lemmas. Define B h := { (ξ, e, a) F 1 (0) : µ h = χ h (x h, e h ) + a h = 0 } and B := B h h H B h is closed in F 1 (0) for each h H, then B is closed in F 1 (0). Define also the restriction to F 1 (0) of the projection of Ξ Λ onto Λ, Φ : (ξ, e, a) F 1 (0) Φ(ξ, e, a) := (e, a) Λ and Θ := Λ \ Φ(B) (7) 6 See Chapter 15, Section 5. 11

12 By definition, for each (ξ, e, a) F 1 (0) (Ξ Θ) and for each h H, either µ h > 0 or χ h (x h, e h ) + a h > 0 We have to prove that Φ(B) is closed and of measure zero in Λ. The closedness of Φ(B) follows from the closedness of B in F 1 (0) and from the properness of Φ obtained by the following lemma. Lemma 19 The function Φ is proper. To show that Φ(B) is of measure zero, define P := J = {H 1, H 2, H 3 } H i H, i = 1, 2, 3; H 1 H 2 H 3 = H; H i H j =, i, j = 1, 2, 3, i j; and H 3 Let J = {H 1, H 2, H 3 } P, for each i = 1, 2, 3 denote by H i (J ) the set H i in J, and by H i (J ) the number of element of H i (J ). We define Ξ J := R (C+1)H ++ (R H 1(J ) + H 3 (J ) R H 2(J ) ++ ) R (C 1) ++ (8) Observe that dim Ξ J = dim Ξ. Let the function F J : Ξ J Λ R dim Ξ J F J (ξ, e, a) := (( F h.1 (ξ, e, a), F h.2 (ξ, e, a), h.3 F J (ξ, e, a)) h H, F M (ξ, e, a)) where F J differs from F defined in (6), for the domain and for the component defined below F h.3 J F h.3 J (ξ, e, a) := Moreover, given J P define the set µ h if h H 1 (J ) H 3 (J ), χ h (x h, e h ) + a h if h H 2 (J ) Θ J := {(ξ, e, a) F 1 J (0) : χ h (x h, e h ) + a h = 0, h H 3 (J )} Given an arbitrary (ξ, e, a) B, we can define endogenously J (ξ, e, a) := {H 1 (ξ, e, a), H 2 (ξ, e, a), H 3 (ξ, e, a)} P with H 1 (ξ, e, a) := {h H : µ h = 0 and χ h (x h, e h ) + a h > 0}, H 2 (ξ, e, a) := {h H : µ h > 0 and χ h (x h, e h ) + a h = 0}, H 3 (ξ, e, a) := {h H : µ h = χ h (x h, e h ) + a h = 0}. Then (ξ, e, a) Θ J (ξ,e,a), and we get Φ(B) J P Φ(Θ J ) (9) 12

13 Since the number of sets involved in the above union in finite, to show that Φ(B) is of measure zero it is enough to show that Φ(Θ J ) is of measure zero, for each J P. To show that Φ(Θ J ) is of measure zero for each J P, we need the following definitions and the following key lemma. Given J P, for each h H 3 (J ) define the function F J, h : Ξ J Λ R dim Ξ J +1 F J, h(ξ, e, a) := ( F J (ξ, e, a), h.4 F J (ξ, e, a)) F h.4 J where (ξ, e, a) := χ h(x h, e h) + a h. Moreover, for each (e, a) Λ, define the function F J, h,e,a : ξ Ξ J F J, h,e,a (ξ) := F J, h(ξ, e, a) R dim Ξ J +1. Observe that for each J P, for each h H 3 (J ) and for each (e, a) Λ, F J, h and F J, h,e,a are differentiable on all their domain. Lemma 20 For each J P and for each h H 3 (J ), 0 is a regular value for F J, h. Then, from results of differential topology and Sard s theorem (see Theorems 28 and 30 in Appendix), given J P, for each h H 3 (J ) there exists a full 1 measure subset Ω J, h of Λ such that for each (e, a) Ω J, h, F (0) =. J, h,e,a Given J P, let Ω J := h H 3 (J ) Ω J, h Ω J is a full measure subset of Λ. Let (e, a) Ω J, by definition we have that there exists h 1 H 3 (J ) such that F (0) =. If (e, a) Φ(Θ J, h,e,a J ), 1 by Proposition 15, there exists ξ Ξ J such that ξ F J,h,e,a(0) for each h H 3 (J ), and we get a contradiction. Then, Φ(Θ J ) Λ \ Ω J Since Λ \ Ω J is of measure zero, Φ(Θ J ) is of measure zero as well. By (9), we have that Φ(B) is of measure zero. Then, from (7), Lemmas 19 and 20, we get the following proposition. Proposition 21 There exists an open and full measure subset Θ of Λ such that for each (ξ, e, a) F 1 (0) (Ξ Θ) and for each h H, either µ h > 0 or χ h (x h, e h ) + a h > Generic regularity In this subsection we prove that Theorem 17 holds true. From now on, the set Θ is the open and full measure subset of Λ obtained in Proposition 21. First of all, observe that F is differentiable in F 1 (0) (Ξ Θ). Indeed, given 13

14 (ξ, e, a ) F 1 (0) (Ξ Θ), define H 1 (ξ, e, a ) := H 1 := {h H : µ h = 0 and χ h (x h, e h) + a h > 0} H 2 (ξ, e, a ) := H 2 := {h H : µ h > 0 and χ h (x h, e h) + a h = 0} (10) By Proposition 21, we get H 1 H 2 = H and H 1 H 2 =. Since linear functions and possibility functions are continuous, there is an open neighborhood I of (ξ, e, a ) in Ξ Θ such that for each (ξ, e, a) I, µ F h.3 h if h H1 (ξ, e, a) = χ h (x h, e h ) + a h if h H2 Remark 22 From now on, the domain of F will be Ξ Θ instead of Ξ Λ. The main result of this subsection is Proposition 24. To prove Proposition 24, we need the following definitions and the following key lemma. Observe that there is a slight difference between the below definitions and the ones given in Subsection 4.1. In Subsection 4.1, we considered the set P of appropriate {H 1, H 2, H 3 } with H 3. In this subsection, we are interested to describe the case in which H 3 =. Then, we define A := {I := {H 1, H 2 } : H i H, i = 1, 2; H 1 H 2 = H and H 1 H 2 = } Let I = {H 1, H 2 } A, for each i = 1, 2 denote by H i (I) the set H i in I, and by H i (I) the number of element of H i (I). We define Ξ I := R (C+1)H ++ (R H 1(I) R H 2(I) ++ ) R (C 1) ++ (11) and we observe that dim Ξ I = dim Ξ. Let the function F I : Ξ I Θ R dim Ξ I F I (ξ, e, a) := (( F h.1 (ξ, e, a), F h.2 (ξ, e, a), F h.3 (ξ, e, a)) h H, F M (ξ, e, a)) where F I differs from F defined in (6), for the domain and for the component defined below F h.3 I F h.3 I (ξ, e, a) := µ h if h H 1 (I), χ h (x h, e h ) + a h if h H 2 (I) All the above definitions allows us to conclude that for each I A, F I differentiable on all its domain. is Lemma 23 For each I A and for each (ξ, e, a ) matrix D F (ξ,e,a) I (ξ, e, a ) has full row rank. 1 F I (0), the Jacobian 14

15 Observe that by (10) and Remark 22, for each (ξ, e, a ) F 1 (0) we get (1) I := {H1, H2} A, (2) (ξ, e, a 1 ) F I (0), and (3) D F (ξ,e,a) (ξ, e, a ) = D F (ξ,e,a) I (ξ, e, a ). Then, from Lemma 23 we can state the following result. Proposition 24 0 is a regular value for F. From the above proposition and Sard s theorem (see Theorem 30 in Appendix), there is a full measure subset Θ of Θ such that for each (e, a) Θ, 0 is a regular value for F e,a. Since Θ is a full measure subset of Λ, Θ is a full measure subset of Λ. Since the following set Λ r := {(e, a) Λ : E a = (e, χ a, u) is regular} contains Θ, Λ r is a full measure subset of Λ. Moreover, from Lemma 19 and Corollary 31 in the Appendix, it follows that Λ r is an open subset of Λ. Therefore, Theorem 17 holds true. 4.3 Regularity in an open and dense subset In this subsection we prove Corollary 18. From now on, we take for fixed an arbitrary u U. By the following lemma we deduce that the set R of (e, χ) such that E := (e, χ, u) is a regular economy is a dense subset of R CH ++ X. Lemma 25 There exists a dense subset D of R CH ++ X such that for each (e, χ) D the economy E := (e, χ, u) is regular. To prove that R is open we need introduce some preliminary definitions and lemmas. With innocuous abuse of notation, define the following function F : Ξ R CH ++ X R dim Ξ F (ξ, e, χ) := ((F h.1 (ξ, e, χ), F h.2 (ξ, e, χ), F h.3 (ξ, e, χ)) h H, F M (ξ, e, χ)) (12) F h.1 (ξ, e, χ) := D xh u h (x h ) λ h p+µ h D xh χ h (x h, e h ), F h.2 (ξ, e, χ) := p(x h e h ), F h.3 (ξ, e, χ) := min {µ h, χ h (x h, e h )}, and F M (ξ, e, χ) := h H(x \ h e\ h ). F is a continuous function, since the convergence of a sequence (χ v h) v N means uniform convergence on compact sets of (χ v h) v N and of (Dχ v h) v N. Moreover, for each (e, χ) R CH ++ X define the function F e,χ : ξ Ξ F e,χ (ξ) := F (ξ, e, χ) R dim Ξ 15

16 and note that it corresponds to the equilibrium function defined in (4). Define also the restriction to F 1 (0) of the projection of Ξ R CH ++ X onto R CH ++ X, Π : (ξ, e, χ) F 1 (0) Π(ξ, e, χ) := (e, χ) R CH ++ X By Theorem 10 and Definition 11, we get R = (R CH ++ X ) \ Π(C 1 C 2 ) where C 1 := {(ξ, e, χ) F 1 (0) : F e,χ is not differentiable in ξ } C 2 := {(ξ, e, χ) F 1 (0) : rank D ξ F e,χ (ξ ) < dim Ξ} The closedness of Π(C 1 C 2 ) in R CH ++ X follows from the closedness of C 1 C 2 in F 1 (0) and the properness of Π obtained by the following lemmas. Then, R is open. Lemma 26 The set C 1 C 2 is closed in F 1 (0). Lemma 27 The function Π is proper. 5 Appendix Proof of Proposition 3. Otherwise, suppose that D xh χ h (x h, e h ) = βp with β 0. Since p 0 and χ h (x h, e h ) = 0, by Assumption 2.4 we get β > 0. Then, D xh χ h (x h, e h ) 0. By Assumption 2.3, x h R C ++ satisfies χ h ( x h, e h ) > 0 and x h e h. From Assumptions 2.1 and 2.2, χ h is C 1 and quasi-concave, then χ h ( x h, e h ) χ h (x h, e h ) > 0 D xh χ h (x h, e h )( x h x h) 0. Therefore, D xh χ h (x h, e h )(e h x h) > 0 since D xh χ h (x h, e h ) 0 and x h e h. That is βp(e h x h) > 0, contradicting p(x h e h ) = 0. Proof of Proposition 7. The existence of a solution to problem (1) follows from Assumptions 1.1, 1.4, 2.1 and 2.3, and the uniqueness from Assumptions 1.3 and 2.2. By the well known Kuhn Tucker theorem for non-linear programming, the sufficiency of the Kuhn Tucker conditions follows from Assumptions 1.3 and 2.2. The necessity of the Kuhn Tucker conditions follows from Propositions 3, and by Assumptions 1.2 and 2.4 we get λ h R ++. Proof of Theorem 10. Theorem 10 is a particular case of existence and compactness results obtained by del Mercato (2006) using homotopy arguments. Then, we just provide the homotopy H adapted to our model and we return the reader to del Mercato (2006) for the detailed proof. Take for fixed E = (e h, χ h, u h ) h H and define the homotopy H : Ξ [0, 1] R dim Ξ, Ψ(ξ, 1 2t) if 0 t 1 2 H(ξ, t) := Γ(ξ, 2 2t) if 1 < t

17 where Ψ, Γ : Ξ [0, 1] R dim Ξ are defined by Ψ(ξ, τ) := ((Ψ h.1 (ξ, τ), Ψ h.2 (ξ, τ), Ψ h.3 (ξ, τ)) h H, Ψ M (ξ, τ)) Ψ h.1 (ξ, τ) = D xh u h (x h ) λ h p, Ψ h.2 (ξ, τ) = p(x h e τ h), Ψ h.3 (ξ, τ) = min {µ h, χ h ( x h, e h )} and Ψ M (ξ, τ) = x \ h e τ\ h, h H h H and Γ(ξ, τ) := ((Γ h.1 (ξ, τ), Γ h.2 (ξ, τ), Γ h.3 (ξ, τ)) h H, Γ M (ξ, τ)) Γ h.1 (ξ, τ) = D xh u h (x h ) λ h p + µ h (1 τ)d xh χ h ((1 τ)x h + τ x h, e h ), Γ h.2 (ξ, τ) = p(x h e h ), Γ h.3 (ξ, τ) = min {µ h, χ h ((1 τ)x h + τ x h, e h )} and Γ M (ξ, τ) = x \ h h H h He \ h. For each h H, x h is given by Assumptions 2.3, and for each τ [0, 1], e τ h := (1 τ)e h +τx h with (x h ) h H Pareto optimal allocation of the exchange economy Ẽ := (X h, u h, e h ) h H à la Arrow Debreu, where X h := R C ++ for each h H. Proof of Proposition 15. It is enough to adapt the proof of Theorem 10. Take for fixed (e, a) Λ, the homotopy H : Ξ [0, 1] R dim Ξ does not change. The homotopies Ψ, Γ : Ξ [0, 1] R dim Ξ adapted to this case differ from the ones used in the proof of Theorem 10 since Ψ h.3 (ξ, τ) = min {µ h, χ h ( x h, e h ) + a h } and Γ h.3 (ξ, τ) = min {µ h, χ h ((1 τ)x h + τ x h, e h ) + a h }. For each h H, x h is given by the definition of Λ. Proof of Proposition 16. Let h H. By Remark 14, for each (e h, a h ) Λ h the set { x h R C ++ : χ h (x h, e h ) + a h > 0 and x h e h } is not empty, and by Assumption 2.2 it is a convex set. Define the correspondence φ h : Λ h R C φ h : (e h, a h ) φ h (e h, a h ) := { x h R C ++ : χ h (x h, e h ) + a h > 0 and x h e h } From Assumption 2.1, for each x h R C the following set φ 1 h (x h) := {(e h, a h ) Λ h : χ h (x h, e h ) + a h > 0 and x h e h } is open in Λ h. Moreover, Λ h equipped with the metric induced by the Euclidean distance is metrizable, thus paracompact. Then, we have the desired result since the correspondence φ h satisfies the assumptions of Michael s Selection theorem (see Proposition in Florenzano, 2003). Proof of Lemma 19. We have to show that any sequence (ξ v, e v, a v ) v N F 1 (0), up to a subsequence, converges to an element of F 1 (0), knowing that 17

18 (e v, a v ) v N Λ, up to a subsequence, converges to (e, a ) Λ. (x v ) v N, up to a subsequence, converges to x R CH ++. (x v ) v N R CH ++, and from F M (ξ v, e v, a v ) = 0 and F k.2 (ξ v, e v, a v ) = 0 in (6), x v k = e v h x v h h H h k h He v h for each k H. Since (e v h) v N converges to e h R C ++ for each h H, (x v ) v N is bounded from above by an element of R C ++. Then, (x v ) v N, up to a subsequence, converges to x 0. Now, we prove that x h 0 for each h H. By F h.1 (ξ v, e v, a v ) = 0 and F h.2 (ξ v, e v, a v ) = 0, u h (x v h) u h ( x h (e v h, a v h)) for every v N, where x h is the continuous selection function given by Proposition 16. Define 1 := (1,..., 1) R C ++, from Assumption 1.2 we have that for each ε > 0, u h (x v h + ε1) u h ( x h (e v h, a v h)) for every v N. So taking the limit on v, since (e v h, a v h) v N converges to (e h, a h) Λ h, and u h and x h are continuous, for each ε > 0 we get u h (x h + ε1) u h ( x h (e h, a h)) := u h. By Assumption 1.4, x h R C ++ since x h belongs to the set cl R C{x h R C ++ : u h (x h ) u h }. (λ v, µ v ) v N, up to a subsequence, converges to (λ, µ ) R H + R H +. It is enough to show that (λ v hp v, µ v h) v N is bounded for each h H. Then, (λ v hp v, µ v h) v N R C ++ R +, up to a subsequence, converges to (πh, µ h) R C + R +, and λ h = πh C since p vc = 1 for each v N. Suppose otherwise that there is a subsequence of (λ v hp v, µ v h) v N (that without loss of generality we continue to denote ( with (λ v hp v, µ v h) v N ) such that (λ v hp v, µ v λ h) +. Consider the sequence v h pv,µ v h (λ v h pv,µ v ) in the sphere, h ( ) )v N λ a compact set. Then, up to a subsequence v h pv,µ v h (λ v h pv,µ v ) (π h, µ h ) 0. h Since µ v h 0 and λ v hp v 0 for each v N, we get π h 0 and µ h 0. By F h.1 (ξ v, e v, a v ) = 0 in (6), λ v hp v = D xh u h (x v h)+µ v hd xh χ h (x v h, e v h) for each v N. Now, divide both sides by (λ v hp v, µ v h) and take the limits. From Assumptions 1.1 and 2.1, we get π h = µ h D xh χ h (x h, e h) µ h > 0, otherwise we get (π h, µ h ) = 0. By Assumption 2.4, D xh χ h (x h, e h) 0. Then, π h 0. From Kuhn-Tucker necessary and sufficient conditions, we get π h x h = min xh R C ++ π hx h subject to χ h (x h, e h) 0 (13) By F h.2 (ξ v, e v, a v ) = 0 in (6), we get λ v hp v x v h = λ v hp v e v h for each v N. Now, divide both sides by (λ v hp v, µ v h) and take the limits. We get π h x h = π h e h. By Assumption 2.3, χ h ( x h, e h) > 0 and π h x h < π h e h = π h x h contradict (13). 18

19 (p v\ ) v N, up to a subsequence, converges to p \ R C By Assumptions 1.2 and 2.5, λ k = D xk u k (x k) + µ kd xk χ k (x k, e k) > 0, for some k = h(c) H. From the previous step, (λ v kp v\ ) v N admits a subsequence converging to π \ k 0. Then, (p v\ ) v N, up to a subsequence, converges to p \ 0, since λ k > 0. Now, suppose that there is c C, such that p c = 0. By Assumptions 1.2 and 2.5, for some k = h(c) H we get 0 < D x c u k (x k k ) + µ k D x c χ k (x k k e k ) = λ k p c = 0 that is a contradiction. λ R H ++. Otherwise, suppose that λ h = 0 for some h H. By F h.1 (ξ v, e v, a v ) = 0 in (6), λ v hp v = D xh u h (x v h) + µ v hd xh χ h (x v h, e v h) for each v N. Taking the limit, from Assumptions 1.1 and 2.1 we get 0 = λ hp = D xh u h (x h) + µ hd xh χ h (x h, e h). By Assumptions 1.2 and 2.4, for some good c = 1,.., C we get 0 < D x c h u h (x h) + µ hd x c h χ h (x h, e h) = λ hp c = 0 that is a contradiction. Proof of Lemma 20. We have to show that for each (ξ, e, a ) the Jacobian matrix D (ξ,e,a) F J, h(ξ, e, a ) has full row rank. F 1 J, h (0), Let := (( x h, λ h, µ h ) h H, p \, v) R (C+2)H R C 1 R. It is enough to show that D (ξ,e,a) F J, h(ξ, e, a ) = 0 = 0. To prove it, we consider the computation of the partial Jacobian matrix with respect to the variables ((x h, λ h, µ h ) h H, e h, p \, a h) The partial system D (ξ,e,a) F J, h(ξ, e, a ) = 0 is written in detail below. x h Dx 2 h u h (x h) λ h p + p \ [I C 1 0] = 0 if h H 1 (J ) (H 3 (J ) \ { h}) x hd x hu h(x 2 h) λ hp + p \ [I C 1 0] + vd x hχ h(x h, e h) = 0 [ x h D 2 u xh h (x h ) + µ h D2 χ xh h (x h, e h )] λ h p + µ h D xh χ h (x h, e h ) + p\ [I C 1 0] = 0 if h H 2 (J ) x h p = 0 for each h H x h D xh χ h (x h, e h) + µ h = 0 if h H 1 (J ) H 3 (J ) x h D xh χ h (x h, e h ) = 0 if h H 2 (J ) λ hp p \ [I C 1 0] + vd e hχ h(x h, e h) = 0 h Hλ h x \ h + λ h (x \ h e \ h ) = 0 h H v = 0 (14) 19

20 Since v = 0 and p C = 1, we get λ h = 0 and p \ = 0. Then, from the above system, we get and [ x h D 2 x h u h (x h)] x h = 0 if h H 1 (J ) (H 3 (J ) \ { h}) (15) [ x h D 2 x h u h (x h )] x h = µ h [ x h D2 x h χ h (x h, e h )] x h if h H 2 (J ) Assumption 2.2 and µ h > 0 for each h H 2 (J ) imply that [ x h D 2 x h u h (x h )] x h 0 if h H 2 (J ) (16) Observe that from F J, h(ξ, e, a ) = 0 and system (14), we get D xh u h (x h) x h = λ hp x h = 0 if h H 1 (J ) H 3 (J ), and D xh u h (x h ) x h = λ h p x h µ h D x h χ h (x h, e h ) x h = 0 if h H 2 (J ) Then, (15), (16) and Assumption 1.3 imply that x h = 0 for each h h. Therefore, the relevant equations of system (14) become λ h p = 0 if h H 1 (J ) (H 3 (J ) \ { h}) x hd 2 x hu h(x h) = 0 λ h p µ h D xh χ h (x h, e h ) = 0 if h H 2 (J ) x hp = 0 µ h = 0 if h H 1 (J )(H 3 (J ) \ { h}) x hd x hχ h(x h, e h) + µ h = 0 λ h x \ h + λ h (x \ h h h e \ h ) = 0 Since p 0, λ h = 0 for each h H 1 (J ) (H 3 (J ) \ { h}). Moreover, since F J, h(ξ, e, a ) = 0, Proposition 3 imply that λ h = µ h = 0 for each h H 2 (J ). Then x h = 0 since λ h > 0 and x hp = 0. Finally, µ h = 0. Therefore, = 0. Proof of Lemma 23. Let := (( x h, λ h, µ h ) h H, p \ ) R (C+2)H R C 1. It is enough to show that D (ξ,e,a) F I (ξ, e, a ) = 0 = 0. We consider two cases: 1. H 1 (I), and 2. H 1 (I) =. Case 1. H 1 (I). Without loosing of generality we suppose 1 H 1 (I). In this case, we consider the computation of the partial Jacobian matrix with respect to the variables ((x h, λ h, µ h ) h H, e 1 ) 20

21 Then, the correspondent partial system D F (ξ,e,a) I (ξ, e, a ) = 0 is written in detail below. x h Dx 2 h u h (x h) λ h p + p \ [I C 1 0] = 0 if h H 1 (I) [ x h D 2 u xh h (x h ) + µ h D2 χ xh h (x h, e h )] λ h p + µ h D xh χ h (x h, e h ) + p\ [I C 1 0] = 0 if h H 2 (I) x h p = 0 for each h H x h D xh χ h (x h, e h) + µ h = 0 if h H 1 (I) x h D xh χ h (x h, e h ) = 0 if h H 2 (I) λ 1 p p \ [I C 1 0] = 0 Since p C = 1, we get λ 1 = 0 and p \ = 0. From the above system, we get [ x h D 2 x h u h (x h)] x h = 0 if h H 1 (I), and (17) [ x h D 2 x h u h (x h )] x h = µ h [ x h D2 x h χ h (x h, e h )] x h if h H 2 (I) Using similar arguments as in the proof of Lemma 20, for each h H we get x h = 0. Therefore, the relevant equations of system (17) become λ h p = 0 if h H 1 (I) λ h p µ h D xh χ h (x h, e h ) = 0 if h H 2 (I) µ h = 0 if h H 1 (I) Since p 0, λ h = 0 for each h H 1 (I). Moreover, since F I (ξ, e, a ) = 0, Proposition 3 imply that λ h = µ h = 0 for each h H 2 (I). Therefore, = 0. Case 2. H 1 (I) =. Then, H 2 (I) = H. The computation of the partial Jacobian matrix with respect to the variables is described below. (x h, λ h, µ h, e h, a h ) h H x h λ h µ h e h a h F (h.1) D2 x h u h (x h )+µ h D2 x h χ h (x h,e h ) p T D xh χ h (x h,e h )T µ h D2 e h x h χ h (x h,e h ) F (h.2) p p F (h.3) I D xh χ h (x h,e h ) De h χ h(x h,e h ) 1 F M [I C 1 0] [I C 1 0] 21

22 The partial system D F (ξ,e,a) I (ξ, e, a ) = 0 is written in detail below. [ x h D 2 xh u h (x h) + µ hdx 2 h χ h (x h, e h) ] λ h p + µ h D xh χ h (x h, e h) + p \ [I C 1 0] = 0 for each h H x h p = 0 for each h H x h D xh χ h (x h, e h) = 0 for each h H (18) x h µ hde 2 h x h χ h (x h, e h) + λ h p + µ h D eh χ h (x h, e h) p \ [I C 1 0] = 0 for each h H µ h = 0 for each h H Observe that, since F I (ξ, e, a ) = 0, from the above system we get D xh u h (x h) x h = λ hp x h µ hd xh χ h (x h, e h) x h = 0, h H (19) From system (18), for each h H we get also [ x h D 2 x h u h (x h)] x h = µ h[ x h (D 2 x h χ h (x h, e h) + D 2 e h x h χ h (x h, e h))] x h By Assumption 4, [ x k (D 2 x k χ k (x k, e k) + D 2 e k x k χ k (x k, e k))] x k 0 for some k H. Since µ k > 0, (19) and Assumption 1.3 imply that x k = 0. Then, since p C = 1, we get λ k = 0 and p \ = 0. Therefore, the relevant equations of system (18) become [ x h D 2 xh u h (x h) + µ hdx 2 h χ h (x h, e h) ] λ h p = 0 for each h k x h p = 0 for each h k x h D xh χ h (x h, e h) = 0 for each h k From the above system, we get [ x h D 2 x h u h (x h)] x h = µ h[ x h D 2 x h χ h (x h, e h)] x h for each h k From (19), using similar arguments as in the proof of Lemma 20, we get x h = 0 for each h k. Finally, we get λ h = 0 for each h k, and then = 0. Proof of Lemma 25. Let u U. First, observe that for each χ X there is a dense subset Λ χ of R CH ++ R H + such that E a = (e, χ a, u) is regular each (e, a) Λ χ. Indeed, as a consequence of Theorem 17 we have that for each χ X, the set Λ r χ of (e, a) Λ χ such that E a = (e, χ a, u) is regular is an open and dense subset of Λ χ. By Remark 14, for each χ X, R CH ++ R H + Λ χ. Then, Λ χ := Λ r χ (R CH ++ R H +) is a dense subset of R CH ++ R H +. 22

23 Now, suppose otherwise that there are (ē, χ) R CH ++ X, an open neighborhood I of ē in R CH ++ and an open neighborhood N of χ in X such that every (e, χ) I N is not regular. Without loosing of generality we suppose that N = {χ X : h H, d(χ h, χ h ) < ε h } where for each h H, 0 < ε h < 1. It is easy to check that χ + a N, for each a = (a h ) h H with 0 a h < ε h. Then, we can conclude that there is an open subset A of R CH ++ R H + such that E a = (e, χ a, u) is not regular for every (e, a) A. This is a contradiction since Λ χ is a dense subset of R CH ++ R H +. Proof of Lemma 26. C 1 is sequentially closed in F 1 (0). Let (ξ v, e v, χ v ) v N C 1 be a sequence converging to ( ξ, ē, χ) F 1 (0). We want to prove that ( ξ, ē, χ) C 1. Otherwise, suppose that Fē, χ is differentiable in ξ. Then, for each h H, either µ h > 0 or χ h ( x h, ē h ) > 0 Define H 1 := {h H : µ h > 0} and H 2 := {h H : χ h ( x h, ē h ) > 0}. For each h H 1 there is v h such that µ v h > 0 for each v v h, and for each h H 2, for given 0 < ε h < χ h ( x h, ē h ), there is n h such that χ h (x v h, e v h) > ε h for each v n h. Since (χ v ) v N converges uniformly on compact sets and the set {(x v h, e v h) v N } {( x h, ē h )} is compact, we have that for each h H 2 there is m h such that for each m m h and for each v N, χ m h (x v h, e v h) > χ h (x v h, e v h) ε h. Now, take v = max h H {v h, n h, m h }, we have that for each v v µ v h > 0, h H 1 and χ v h(x v h, e v h) > 0, h H 2 That is, F e v,χ v is differentiable in ξv for each v v that is a contradiction. C 2 is closed in F 1 (0). For each (ξ, e, χ) C 2, the determinant of all the square submatrices of D ξ F e,χ (ξ ) of dimension dim Ξ is equal to zero. Since the determinant is a continuous function, C 2 is closed in F 1 (0). Proof of Lemma 27. We have to show that any sequence (ξ v, e v, χ v ) v N F 1 (0), up to a subsequence, converges to an element of F 1 (0), knowing that (e v, χ v ) v N, up to a subsequence, converges to (ē, χ) R CH ++ X. As in the proof of Lemma 19, (x v ) v N, up to a subsequence, converges to x 0. The main difficulty is to prove that for each h H, x h 0. Using similar arguments as in the proof of Proposition 16, it easy to show that there exists a continuous selection function x h : R C ++ R C ++ such that for each e h R C ++, χ h ( x h (e h ), e h ) > 0 and x h (e h ) e h Then, in particular χ h ( x h (ē h ), ē h ) > 0 and x h (ē h ) ē h, and χ h ( x h (e v h), e v h) > 0 and x h (e v h) e v h for each v N. Now, we are going to show that there is v 23

24 such that for each v v χ v h( x h (e v h), e v h) > 0 (20) Since x h is continuous, for given 0 < ε h < χ h ( x h (ē h ), ē h ), there is n h such that χ h ( x h (e v h), e v h) > ε h for each v n h. Since (χ v ) v N converges uniformly on compact sets and the set {( x h (e v h), e v h)) v N } {( x h (ē h ), ē h )} is compact, we have that there is m h such that for each m m h and for each v N, χ m h ( x h (e v h), e v h) > χ h ( x h (ē h ), ē h ) ε h. Now, take v = max{n h, m h }, we have that for each v v, (20) holds true. Then, by F h.1 (ξ v, e v, χ v ) = 0 and F h.2 (ξ v, e v, χ v ) = 0 in (12), u h (x v h) u h ( x h (e v h)) for every v v. Define 1 := (1,..., 1) R C ++, from Assumption 1.2 we have that for each ε > 0, u h (x v h + ε1) u h ( x h (e v h)) for every v v. So taking the limit on v, since u h and x h are continuous, for each ε > 0 we get u h ( x h + ε1) u h ( x h (ē h )) := u h. By Assumption 1.4, x h R C ++ since x h belongs to the set cl R C{x h R C ++ : u h (x h ) u h }. The remaining part of the proof follows the same steps as in the proof of Lemma 19. Theorem 28 (Regular value theorem) Let M, N be C r manifolds of dimensions m and n, respectively. Let f : M N be a C r function. Assume r > max{m n, 0}. If y N is a regular value for f, then (1) if m < n, f 1 (y) =, (2) if m n, either f 1 (y) =, or f 1 (y) is an (m n)-dimensional submanifold of M. Corollary 29 Let M, N be C r manifolds of the same dimension. Let f : M N be a C r function. Assume r 1. Let y N a regular value for f such that f 1 (y) is non-empty and compact. Then, f 1 (y) is a finite subset of M. The following results is a consequence of the Sard Theorem for manifolds. See, for example Villanacci et al. (2002). Theorem 30 Let M, Ω and N be C r manifolds of dimensions m, p and n, respectively. Let f : M Ω N be a C r function. Assume r > max{m n, 0}. If y N is a regular value for f, then there exists a full measure subset Ω of Ω such that for any ω Ω, y N is a regular value for f ω, where f ω : ξ M f ω (ξ) := f(ξ, ω) N Corollary 31 Let M, Ω and N be C r manifolds of dimensions m, p and n, respectively. Let f : M Ω N be a C r function. Assume r > max{m n, 0}. Let Γ be a full measure subset of Ω such that for any ω Γ, y N is a regular 24