Optimal Risk Sharing with Optimistic and Pessimistic Decision Makers

Transcription

1 Optimal Risk Sharing with Optimistic and Pessimistic Decision Makers A. Araujo 1,2, J.M. Bonnisseau 3, A. Chateauneuf 3 and R. Novinski 1 1 Instituto Nacional de Matematica Pura e Aplicada 2 Fundacao Getulio Vargas 3 Paris School of Economics, Université Paris 1 Panthéon-Sorbonne Centre d Economie de la Sorbonne, UMR CNRS 8174 Economic Theory Conference University of Kansas December 9-10, 2011 Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 1

2 Outline 1 Introduction Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 2

3 Outline 1 Introduction Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 3

4 Introduction Main motivation: we aim at proving that under mild conditions individually Pareto rational optima will exist even in presence of non-convex preferences. We essentially consider preferences on l + hence accommodating the behavior of DM in front of a countable flow x of payoffs, or else choosing among financial assets x whose outcomes depend on the realization of a countable set of states of the world. Therefore, our conditions for the existence of P.O. can be interpreted as myopia in the first context and as pessimism or reasonable optimism in the second context. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 4

5 Introduction (continued) Replacing the uncertain context by a risky one, we then derive some necessary properties of risk sharing. For strong risk averters these conditions are exactly the usual ones of pairwise comonotonicity, for strong risk lovers exclusively corner conditions or pairwise comonotonicity can appear. A crucial result for our work is the proof that the closed unit ball B(0, 1) for the normed dual l of the normed space l 1 is compact in the Mackey topology τ(l, l 1 ). Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 5

6 Outline 1 Introduction Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 6

7 Theorem Let B(S) be the normed space of bounded real mappings on S endowed with the sup-norm. Then: Corollary 1) B(S) is the norm dual of the space rca(s) of all regular and bounded Borel measures for S endowed with the discrete topology. 2) The closed unit ball of B(S) is compact for the Mackey topology τ(b(s), rca(s)). The closed unit ball B(0, 1) for the normed dual l of the normed space l 1 is compact in the Mackey topology τ(l, l 1 ). Proof. Taking S = N, clearly l = B(S). Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 7

8 Outline 1 Introduction Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 8

9 Existence of individually rational Pareto efficient allocations under Mackey upper-semicontinuous preferences Assumption (1) For every i = 1,..., m, i is a transitive reflexive preorder on l +. (ii) For every i and every y i l +, {x i l + x i i y i } is τ-closed where τ is the Mackey topology τ(l, l 1 ). Theorem Under Assumptions (1) and (2), individually rational Pareto efficient allocations exist. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 9

10 Examples of Mackey upper-semicontinuous preferences Definition Let (S, A) be a measurable space and B (A) be the space of bounded A-measurable mappings from S to R. v is a capacity on A if v( ) = 0, v(s) = 1, A, B A, A B v(a) v(b). x B (A), the Choquet integral of x with respect to v, denoted xdv is defined by 0 (v(x t) 1)dt v(x t)dt. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 10

11 >From Araujo et al. General Equilibrium, Wariness and efficient bubbles", JET forthcoming, we know that preferences on l represented by a Choquet integral is Mackey usc if and only if is strongly myopic where: Definition A preference relation on l, is strongly myopic if (x, y) l l x y, z l +, implies that there exists n large enough such that x y + z En where z En (m) = 0 if m < n and z(m) if m n. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 11

12 Pessimistic Mackey upper-semicontinuous preferences Definition A Choquet DM will be said pessimistic if v is convex, i.e., A, B A, v(a B) + v(a B) v(a) + v(b). Recall that if v is convex, then xdv = min{e P (x) P additive probability measure v} Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 12

13 Proposition For a pessimistic decision maker, the following assertions are equivalent: (1) is Mackey upper-semicontinuous; (2) v is outer continuous, i.e., A n, A A = P(N), A n A v(a n ) v(a). Example I(x) = inf N {x(n)} or equally I(x) = xdv where v(a) = 0 A N, v(n) = 1 Remark that I is not Mackey lower semicontinuous. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 13

14 Optimistic Mackey upper-semicontinuous preferences Definition A Choquet DM will be said optimistic if v is concave, i.e., A, B A, v(a B) + v(a B) v(a) + v(b). Recall that if v is concave, xdv = max{e P (x) P additive probability measure v} or else x, y B (A), x y, α ]0, 1[ αx + (1 α)y y. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 14

15 Proposition For an optimistic decision maker the following assertions are equivalent: (1) is Mackey upper-semicontinuous; (2) is Mackey lower-semicontinuous; (3) v is outer continuous at, i.e., A n P(N), A n v(a n ) 0. Remark. I(x) = sup N {x(n)}, i.e., I(x) = xdv with v(a) = 1 A, v( ) = 0 is not Mackey upper-semicontinuous. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 15

16 Example where no existence of nontrivial individually rational Pareto optima if such a too optimistic" DM is present Consumption space X = l +, p s = 1 2. s DM1 U 1 (x) = s N p sx s, initial endowments ω 1 = (2 1 s+1 ) s N DM2 U 2 (x) = sup s N x s, ω 2 = ω 1. The unique Pareto optimal allocation is ( x 1 = ω 1 + ω 2, x 2 = 0). No individually rational Pareto optimal allocation. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 16

17 Outline 1 Introduction Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 17

18 for risk lovers and risk averse decision makers Interpret l as the set of all bounded A measurable mappings x from N to R where A = P(N) and assume that a σ-additive probability P is given on (N, P(N)). So any x can now be interpreted as a random variable. Definition x is less risky than y for the second order stochastic dominance denoted x SSD y if t P(x u)du t P(y u)du for all t R. x is strictly less risky than y denoted x SSD y if one of the previous inequality is strict. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 18

19 Definition Let i be the preference relation of a DM. i is a strict (strong) risk averter if x SSD y x i y and x SSD y x i y. i is a strict (strong) risk lover if x SSD y x i y and x SSD y x i y. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 19

20 Examples of Mackey upper-semicontinuous strict risk lover of strict risk averter Consider a Choquet decision maker where v = f P, i.e., v is a distortion of a probability P, i.e., f : [0, 1] [0, 1], f (0) = 0, f (1) = 1, f strictly increasing. If f is strictly concave and continuous, then x l I(x) = xdv f with v f = f P defines a Mackey upper-semicontinuous strict risk lover. If f is strictly convex and continuous, then x l I(x) = xdv f with v f = f P defines a Mackey upper-semicontinuous strict risk averter. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 20

21 The main result Theorem Consider m strict risk averters i = 1,..., m with Mackey upper-semicontinuous preferences i and n strict risk lovers j = 1,..., n with Mackey upper-semicontinuous preferences j with initial endowments ω i l ++, ω j l ++. Then individual rational Pareto efficient allocations (in (l + ) m+n ) exist. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 21

22 Theorem continued Theorem For such PO (x i, i = 1,..., m; y j, j = 1,..., n) we have: 1) The allocations of risk averters are pairwise comonotonic, i.e., (x i1 (s) x i1 (t))(x i2 (s) x i2 (t)) 0 (s, t) N 2, (i 1, i 2 ) {1,..., m} 2. 2) For two risk lovers j 1, j 2 : (y j1 (s) y j1 (t))(y j2 (s) y j2 (t)) < 0 (y j1 (s) y j1 (t))(y j2 (s) y j2 (t)) = 0 (i.e. in at least one state, one of these two risk lovers gets 0). 3) If all allocations of risk lovers are strictly positive, then these allocations are pairwise comonotonic. Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 22