Competitive Equilibria in a Comonotone Market

Transcription

1 Competitive Equilibria in a Comonotone Market 1/51 Competitive Equilibria in a Comonotone Market Ruodu Wang wang Department of Statistics and Actuarial Science University of Waterloo The 2nd International Workshop on Optimal (Re)Insurance CUFE Beijing, China July / 51

2 Competitive Equilibria in a Comonotone Market 2/51 Table of contents Introduction Competitive equilibria with dual utilities Competitive equilibria with rank dependent utilities An algorithm for computing competitive equilibria Conclusion Based on joint work with Tim Boonen (Amsterdam) and Fangda Liu (CUFE, Beijing) 2 / 51

3 Competitive Equilibria in a Comonotone Market 3/51 Introduction Introduction Competitive equilibria with dual utilities Competitive equilibria with rank dependent utilities An algorithm for computing competitive equilibria Conclusion 3 / 51

4 Competitive Equilibria in a Comonotone Market 4/51 Introduction Complete market and competitive equilibria The market A one-period exchange market is described by a probability space (Ω, B, P) and a set of bounded random future wealths X. There are n agents and N = {1,..., n}. Each of them is endowed with an endowment ξ i X and uses an objective functional V i : X R to model his preference. The total future wealth is X = n i=1 ξ i, and its range R(X ) R is an interval. The current price of a random wealth Y X is given by E Q [Y ] for some pricing measure Q P, where P is the set of probability measures absolutely continuous w.r.t. P. Q will be an output of the market equilibrium. 4 / 51

5 Competitive Equilibria in a Comonotone Market 5/51 Introduction Complete market and competitive equilibria Competitive Equilibria (Arrow-Debreu Equilibria) In an equilibrium, aggregate supplies will equal aggregate demands for every market state. 5 / 51

6 Competitive Equilibria in a Comonotone Market 5/51 Introduction Complete market and competitive equilibria Competitive Equilibria (Arrow-Debreu Equilibria) In an equilibrium, aggregate supplies will equal aggregate demands for every market state. Definition An allocation (X 1,..., X n ) X n and a pricing measure Q P constitute an (Arrow-Debreu) competitive equilibrium if For i N, X i satisfies the budget constraint: E Q [X i ] E Q [ξ i ] For i N, X i maximizes the agent s objective: V i (Y ) V i (X i ), for all Y X and E Q [Y ] E Q [ξ i ]. The market is cleared: n X i = i=1 n ξ i. i=1 5 / 51

7 Competitive Equilibria in a Comonotone Market 6/51 Introduction Complete market and competitive equilibria In a complete market, the set of admissible allocations is { } n A n (X ) = (X 1,..., X n ) X n : X i = X. Competitive equilibria in a complete market: Early work (expected utility): Arrow-Debreu 54, Borch 62 Cumulative perspective theory: De Gorgi-Hens-Rieger 10 Concave dual utility: Garlier-Dana 08, Dana 11, Boonen 15 Rank dependent utility: Xia-Zhou 16, Jin-Xia-Zhou 18 For objectives other than expected utilities, finding competitive equilibria is a generally very challenging question i=1 6 / 51

8 Competitive Equilibria in a Comonotone Market 7/51 Introduction Complete market and competitive equilibria Comonotonicity Definition A random vector (Y 1,..., Y n ) is comonotonic if (Y 1,..., Y n ) = (f 1 (Y ),..., f n (Y )), holds for some non-decreasing functions f 1,..., f n and a random variable Y. Y can be chosen as n i=1 Y i; e.g. Denneberg 94. (Y 1, Y 2 ) is counter-monotonic if ( Y 1, Y 2 ) is comonotonic. 7 / 51

9 Competitive Equilibria in a Comonotone Market 8/51 Introduction Complete market and competitive equilibria Known results. In a complete market, when agents have the same belief P, under mild conditions, a competitive equilibrium ((X 1,..., X n ), Q) A n (X ) P satisfies i. (X 1,..., X n ) is comonotonic. ii. (X i, η) is counter-monotonic. iii. (X, η) is counter-monotonic, where η is the pricing kernel η = dq dp Obsevation. A complete market leads to comonotonic allocations, which are counter-monotone with the pricing kernel. 8 / 51

10 Competitive Equilibria in a Comonotone Market 8/51 Introduction Complete market and competitive equilibria Known results. In a complete market, when agents have the same belief P, under mild conditions, a competitive equilibrium ((X 1,..., X n ), Q) A n (X ) P satisfies i. (X 1,..., X n ) is comonotonic. ii. (X i, η) is counter-monotonic. iii. (X, η) is counter-monotonic, where η is the pricing kernel η = dq dp Obsevation. A complete market leads to comonotonic allocations, which are counter-monotone with the pricing kernel. Question. What happens if we constrain the feasible set of allocations to be comonotonic in the first place? 8 / 51

11 Competitive Equilibria in a Comonotone Market 9/51 Introduction Incomplete comonotone market and competitive equilibria Insurance In an insurance policy, the underlying risk is Y. Y is shared by a policyholder and several insurers. To avoid Moral Hazard, no one should have the incentive to hope for a larger loss. Slow growth property. For the policyholder, the ceded part f (Y ) should be comonotonic with the retained part Y f (Y ), or equivalently 0 f (x) f (y) x y, 0 y x. 9 / 51

12 Competitive Equilibria in a Comonotone Market 10/51 Introduction Incomplete comonotone market and competitive equilibria Comonotone Market Allocations are constrained in the set C(X ) = {Y X : (Y, X Y ) is comonotonic }. Thus Y C(X ) if and only if Y = f (X ) for some f F, where { } F = f : R R f is continuous and a.e. differentiable, 0 f. (z) 1 for z R 10 / 51

13 Competitive Equilibria in a Comonotone Market 10/51 Introduction Incomplete comonotone market and competitive equilibria Comonotone Market Allocations are constrained in the set C(X ) = {Y X : (Y, X Y ) is comonotonic }. Thus Y C(X ) if and only if Y = f (X ) for some f F, where { } F = f : R R f is continuous and a.e. differentiable, 0 f. (z) 1 for z R The set of admissible allocations if { A c n(x ) = (X 1,..., X n ) (C(X )) n : } n X i = X. i=1 10 / 51

14 Competitive Equilibria in a Comonotone Market 10/51 Introduction Incomplete comonotone market and competitive equilibria Comonotone Market Allocations are constrained in the set C(X ) = {Y X : (Y, X Y ) is comonotonic }. Thus Y C(X ) if and only if Y = f (X ) for some f F, where { } F = f : R R f is continuous and a.e. differentiable, 0 f. (z) 1 for z R The set of admissible allocations if { A c n(x ) = (X 1,..., X n ) (C(X )) n : } n X i = X. In the comonotone market, a competitive equilibrium is a pair ((X 1,..., X n ), Q) A c n(x ) P such that E Q [X i ] E Q [ξ i ] and V i (X i ) = max { V i (Y i ) : Y i C(X ), E Q [Y i ] E Q [ξ i ] }, i N. i=1 10 / 51

15 Competitive Equilibria in a Comonotone Market 11/51 Introduction Incomplete comonotone market and competitive equilibria For the same set of agents Comonotone market Complete market 11 / 51

16 Competitive Equilibria in a Comonotone Market 11/51 Introduction Incomplete comonotone market and competitive equilibria For the same set of agents Comonotone market Complete market Constrained CE (CCE) Unconstrained CE (UCE) 11 / 51

17 Competitive Equilibria in a Comonotone Market 11/51 Introduction Incomplete comonotone market and competitive equilibria For the same set of agents Comonotone market Complete market Constrained CE (CCE) Unconstrained CE (UCE) Some results. A UCE is always a CCE (under some mild conditions). 11 / 51

18 Competitive Equilibria in a Comonotone Market 11/51 Introduction Incomplete comonotone market and competitive equilibria For the same set of agents Some results. Comonotone market Complete market Constrained CE (CCE) Unconstrained CE (UCE) A UCE is always a CCE (under some mild conditions). A CCE is not necessarily a UCE. In a CCE ((X 1,..., X n ), Q), X i, i N and X may not be counter-monotonic with η = dq/dp. Thus, a sharp contrast to the case of complete market. 11 / 51

19 Competitive Equilibria in a Comonotone Market 12/51 Introduction Incomplete comonotone market and competitive equilibria Pareto-optimality Definition (Pareto-optimal allocations) Fix objective functionals V 1,..., V n, total wealth X X and initial endowments ξ 1,..., ξ n X. (i) In the comonotone market, an allocation (X 1,..., X n ) A c n(x ) is Pareto-optimal if for any allocation (Y 1,..., Y n ) A c n(x ), V i (Y i ) V i (X i ) for i N implies V i (Y i ) = V i (X i ) for i N. (ii) In the complete market, an allocation (X 1,..., X n ) A n (X ) is Pareto-optimal if for any allocation (Y 1,..., Y n ) A n (X ), V i (Y i ) V i (X i ) for i N implies V i (Y i ) = V i (X i ) for i N. 12 / 51

20 Competitive Equilibria in a Comonotone Market 13/51 Introduction Dual utilities and rank-dependent utilities Dual utility The set of distortion functions { G = g : [0, 1] [0, 1] g is continuous and increasing, g(0) = 0 and g(1) = 1 }. Definition A dual utility (DU) functional D g with distortion function g G is defined as a Choquet integral, namely, for Y X, 0 D g (Y ) = Y d (g P) := (g(s Y (z)) 1) dz + g(s Y (z))dz. References: Yaari 87, Denneberg 94, Wang-Panjer-Young / 51

21 Competitive Equilibria in a Comonotone Market 14/51 Introduction Dual utilities and rank-dependent utilities Rank dependent utility Definition For an increasing function u : R R { } and a distortion function g G, a rank-dependent utility (RDU) functional R u,g is given by R u,g (Y ) = D g (u(y )) = u(y )d(g P), Y X. R u,g is consistent with strong risk aversion if and only if u is concave and g is convex. The expected utility functional (EU) is a special case of RDU when g(x) = x for x [0, 1]. The DU is a special case of an RDU when u(x) = x for x R. 14 / 51

22 Competitive Equilibria in a Comonotone Market 15/51 Competitive equilibria with dual utilities Introduction Competitive equilibria with dual utilities Competitive equilibria with rank dependent utilities An algorithm for computing competitive equilibria Conclusion 15 / 51

23 Competitive Equilibria in a Comonotone Market 16/51 Competitive equilibria with dual utilities Competitive equilibria with dual utilities In a comonotone market, where agents are equipped with dual utilities, we investigate following issues. Solving the individual optimization. Existence and the close form of a competitive equilibrium. Fundamental theorems of welfare economics. 16 / 51

24 Competitive Equilibria in a Comonotone Market 17/51 Competitive equilibria with dual utilities Individual optimization DU-comonotone market Individual optimization: Each agent is to find Xi which solves max V i(x i ) = D g (X i ) = X i C(X ) s.t. E Q [X i ] E Q [ξ i ]. Ω X i d (g i P), (1) 17 / 51

25 Competitive Equilibria in a Comonotone Market 17/51 Competitive equilibria with dual utilities Individual optimization DU-comonotone market Individual optimization: Each agent is to find Xi which solves max V i(x i ) = D g (X i ) = X i C(X ) s.t. E Q [X i ] E Q [ξ i ]. Ω X i d (g i P), (1) Proposition For a fixed Q and f i F, the random variable Xi = f i (X ) solves (1) if and only if for a.e. z R(X ), f i (z) = { 1, if gi (S X (z)) > Q(X > z), 0, if g i (S X (z)) < Q(X > z). (2) 17 / 51

26 Competitive Equilibria in a Comonotone Market 18/51 Competitive equilibria with dual utilities Existence of a competitive equilibrium in a DU-comonotone market Existence of CCE Theorem (1) In the DU-comonotone market, the following holds: (i) A competitive equilibrium always exists. 18 / 51

27 Competitive Equilibria in a Comonotone Market 18/51 Competitive equilibria with dual utilities Existence of a competitive equilibrium in a DU-comonotone market Existence of CCE Theorem (1) In the DU-comonotone market, the following holds: (i) A competitive equilibrium always exists. (ii) The pair ((X 1,..., X n ), Q) is a competitive equilibrium if and only if (a) g N,2 (S X (z)) Q(X > z) g N,1 (S X (z)) for z R(X ), where g N,1 and g N.2 is the largest and the second largest in {g i, i N}. (b) For i N, Xi = f i (X ) E Q [f i (X )] + E Q [ξ i ] almost surely where f i satisfies (2) with n i=1 f i(x ) = X. 18 / 51

28 Competitive Equilibria in a Comonotone Market 18/51 Competitive equilibria with dual utilities Existence of a competitive equilibrium in a DU-comonotone market Existence of CCE Theorem (1) In the DU-comonotone market, the following holds: (i) A competitive equilibrium always exists. (ii) The pair ((X 1,..., X n ), Q) is a competitive equilibrium if and only if (a) g N,2 (S X (z)) Q(X > z) g N,1 (S X (z)) for z R(X ), where g N,1 and g N.2 is the largest and the second largest in {g i, i N}. (b) For i N, Xi = f i (X ) E Q [f i (X )] + E Q [ξ i ] almost surely where f i satisfies (2) with n i=1 f i(x ) = X. Sharp contrast I: A competitive equilibrium always exists in a DU-comonotone market; does NOT necessary exist in a DU-complete market where the distortion functions are not convex (e.g. Embrechts-Liu-Wang 18, the case of VaR) 18 / 51

29 Competitive Equilibria in a Comonotone Market 19/51 Competitive equilibria with dual utilities Existence of a competitive equilibrium in a DU-comonotone market Example of equilibrium pricing measure g 1 g 2 g 3 Q(X S 1 X (s)) s (a) Distortion functions and equilibrium price 19 / 51

30 Competitive Equilibria in a Comonotone Market 20/51 Competitive equilibria with dual utilities Existence of a competitive equilibrium in a DU-comonotone market Uniqueness of CCE In the DU-comonotone market, (i) If g N,1 (t) > g N,2 (t) for almost everywhere t [0, 1], then the equilibrium allocation is unique up to constant shifts, and the equilibrium price is not unique. 20 / 51

31 Competitive Equilibria in a Comonotone Market 20/51 Competitive equilibria with dual utilities Existence of a competitive equilibrium in a DU-comonotone market Uniqueness of CCE In the DU-comonotone market, (i) If g N,1 (t) > g N,2 (t) for almost everywhere t [0, 1], then the equilibrium allocation is unique up to constant shifts, and the equilibrium price is not unique. (ii) If g N,1 (t) = g N,2 (t) for almost everywhere t [0, 1], then the equilibrium price is unique, and the equilibrium allocation is not unique. 20 / 51

32 Competitive Equilibria in a Comonotone Market 20/51 Competitive equilibria with dual utilities Existence of a competitive equilibrium in a DU-comonotone market Uniqueness of CCE In the DU-comonotone market, (i) If g N,1 (t) > g N,2 (t) for almost everywhere t [0, 1], then the equilibrium allocation is unique up to constant shifts, and the equilibrium price is not unique. (ii) If g N,1 (t) = g N,2 (t) for almost everywhere t [0, 1], then the equilibrium price is unique, and the equilibrium allocation is not unique. Sharp contrast II: The equilibrium price is unique in a DU-complete market; e.g. Boonen 15 is NOT necessary unique in a DU-comonotone market. 20 / 51

33 Competitive Equilibria in a Comonotone Market 21/51 Competitive equilibria with dual utilities Pareto optimality and fundamental theorems of welfare economics FTWE Theorem (2) In the DU-comonotone market, Without central coordination 21 / 51

34 Competitive Equilibria in a Comonotone Market 21/51 Competitive equilibria with dual utilities Pareto optimality and fundamental theorems of welfare economics FTWE Theorem (2) In the DU-comonotone market, (i) an equilibrium allocation is necessarily Pareto-optimal; Without central coordination 1st FTWE Invisible hand : a competitive market leads to an efficient allocation of resources. 21 / 51

35 Competitive Equilibria in a Comonotone Market 21/51 Competitive equilibria with dual utilities Pareto optimality and fundamental theorems of welfare economics FTWE Theorem (2) In the DU-comonotone market, (i) an equilibrium allocation is necessarily Pareto-optimal; (ii) a Pareto-optimal allocation is necessarily an equilibrium allocation for some choice of endowments. Without central coordination 1st FTWE Invisible hand : a competitive market leads to an efficient allocation of resources. 2nd FTWE Any desired Pareto-efficient allocation can be attained by market competition with transfers. 21 / 51

36 Competitive Equilibria in a Comonotone Market 22/51 Competitive equilibria with rank dependent utilities Introduction Competitive equilibria with dual utilities Competitive equilibria with rank dependent utilities An algorithm for computing competitive equilibria Conclusion 22 / 51

37 Competitive Equilibria in a Comonotone Market 23/51 Competitive equilibria with rank dependent utilities Competitive equilibria with rank-dependent utilities In a comonotone market, where agents are equipped with rank-dependent utilities, we investigate following issues. Existence of a competitive equilibrium. First fundamental theorems of welfare economics. EU market approach. 23 / 51

38 Competitive Equilibria in a Comonotone Market 24/51 Competitive equilibria with rank dependent utilities General results in a RDU-comonotone market RDU-comonotone market Individual optimization: or X i max V i(x i ) = R ui,g i (X i ) = X i C(X ) s.t. E Q [X i ] E Q [ξ i ] Ω u i (X i )d(g i P) = Y i E Q [Y i ] + E Q [ξ i ], where { ( Y i arg max Vi Y E Q [Y ] + E Q [ξ i ] )}. Y C(X ) 24 / 51

39 Competitive Equilibria in a Comonotone Market 24/51 Competitive equilibria with rank dependent utilities General results in a RDU-comonotone market RDU-comonotone market Individual optimization: or X i max V i(x i ) = R ui,g i (X i ) = X i C(X ) s.t. E Q [X i ] E Q [ξ i ] Ω u i (X i )d(g i P) = Y i E Q [Y i ] + E Q [ξ i ], where { ( Y i arg max Vi Y E Q [Y ] + E Q [ξ i ] )}. Y C(X ) Recall: Constrained competitive equilibrium (CCE) ((Xi,..., Xn ), Q) A c n P is a CCE if E Q [Xi ] = E Q [ξ i ] and V i (X i ) = max { V i (Y i ) : Y i C(X ), E Q [Y i ] E Q [ξ i ] }. 24 / 51

40 Competitive Equilibria in a Comonotone Market 25/51 Competitive equilibria with rank dependent utilities General results in a RDU-comonotone market Existence & FTWE Assumptions. In a RDU-comonotone market or an RDU-complete market with given ξ 1,..., ξ n, X X, u 1,..., u n are strictly increasing, strictly concave and continuously differentiable functions and u i > on (d i, ), i N. g 1,..., g n G are continuously differentiable. 25 / 51

41 Competitive Equilibria in a Comonotone Market 25/51 Competitive equilibria with rank dependent utilities General results in a RDU-comonotone market Existence & FTWE Assumptions. In a RDU-comonotone market or an RDU-complete market with given ξ 1,..., ξ n, X X, u 1,..., u n are strictly increasing, strictly concave and continuously differentiable functions and u i > on (d i, ), i N. g 1,..., g n G are continuously differentiable. Theorem (3) Consider the RDU-comonotone market. 1. (Existence.) If ξ i d i and ξ i is a continuos function of X, i N, then a competitive equilibrium exists. 2. (1st FTWE.) An equilibrium allocation satisfying V i (X i ) > for all i N is necessarily Pareto-optimal. 25 / 51

42 Competitive Equilibria in a Comonotone Market 26/51 Competitive equilibria with rank dependent utilities Expected utility approach for the competitive equilibria Expected-utility with heterogeneous beliefs Observation. V i (Y ) = R gi,u i (Y ) = E Q i [u i (Y )], for all Y C(X ), where Q i P, i N such that Q i (X > t) = g i P(X > t) for all t R. 26 / 51

43 Competitive Equilibria in a Comonotone Market 26/51 Competitive equilibria with rank dependent utilities Expected utility approach for the competitive equilibria Expected-utility with heterogeneous beliefs Observation. V i (Y ) = R gi,u i (Y ) = E Q i [u i (Y )], for all Y C(X ), where Q i P, i N such that Q i (X > t) = g i P(X > t) for all t R. In a comonotone market, the individual RDU optimization problem translates to a EU problem with heterogeneous beliefs Q i, i N: max X i C(X ) EQ i [u i (X i )] s.t. E Q [X i ] E Q [ξ i ]. 26 / 51

44 Competitive Equilibria in a Comonotone Market 26/51 Competitive equilibria with rank dependent utilities Expected utility approach for the competitive equilibria Expected-utility with heterogeneous beliefs Observation. V i (Y ) = R gi,u i (Y ) = E Q i [u i (Y )], for all Y C(X ), where Q i P, i N such that Q i (X > t) = g i P(X > t) for all t R. In a comonotone market, the individual RDU optimization problem translates to a EU problem with heterogeneous beliefs Q i, i N: max X i C(X ) EQ i [u i (X i )] s.t. E Q [X i ] E Q [ξ i ]. V i (Y ) = E Q i [u i (Y )] relies on the fact that (Y, X ) is comonotonic, and it does not necessarily hold on X. 26 / 51

45 Competitive Equilibria in a Comonotone Market 27/51 Competitive equilibria with rank dependent utilities Expected utility approach for the competitive equilibria RDU markets & EU markets RDU-comonotone market CCE EU-comonotone market CCE (generally) (if comonotonic) RDU-complete market UCE Individual objectives: EU-comonotone market = = EU-complete market UCE (relatively well studied) EU-complete market max E Q i [u i (X i )] s.t. E Q [X i ] E Q [ξ i ]. X i C(X ) max X i X EQ i [u i (X i )] s.t. E Q [X i ] E Q [ξ i ]. 27 / 51

46 Competitive Equilibria in a Comonotone Market 28/51 Competitive equilibria with rank dependent utilities Expected utility approach for the competitive equilibria Competitive equilibria in an EU-complete market Optimization problems in the EU-complete market with heterogeneous beliefs Q i, i N: max X i X EQ i [u i (X i )] s.t. E Q [X i ] E Q [ξ i ], i N. Individual optimization has a unique solution (e.g. Föllmer-Schied 16) ( ) dq X i = (u i ) 1 λ i, E Q [X i ] = E Q [ξ i ] dq i The market clearing condition n i=1 ( ) dq (u i ) 1 λ i = X, dq i 28 / 51

47 Competitive Equilibria in a Comonotone Market 29/51 Competitive equilibria with rank dependent utilities Expected utility approach for the competitive equilibria Theorem (4) Suppose that ((X 1,..., X n ), Q) is an UCE in the EU-complete market. If ( dq,..., dq ) is comonotonic, dq 1 dq n then it is a CCE in the RDU-comonotone market. 29 / 51

48 Competitive Equilibria in a Comonotone Market 29/51 Competitive equilibria with rank dependent utilities Expected utility approach for the competitive equilibria Theorem (4) Suppose that ((X 1,..., X n ), Q) is an UCE in the EU-complete market. If ( dq,..., dq ) is comonotonic, dq 1 dq n then it is a CCE in the RDU-comonotone market. Sharp contrast III: In a CCE, the pricing kernel η = dq dp = dq dq i g i (S X (X )) is not necessarily a decreasing function of X when g i is not convex. Our model could accommodate the pricing kernel puzzle that pricing kernel is not necessarily counter-monotonic with X by empirical observations (e.g. Hens-Reichlin 13). 29 / 51

49 Competitive Equilibria in a Comonotone Market 30/51 Competitive equilibria with rank dependent utilities RDU-exponential-comonotone market Exponential utilities For i N, assume u i (x) = e x θ i, x R, where θ 1,..., θ n > 0 are parameters representing risk tolerance. Proposition With exponential utilities, if the following condition holds, inf inf x R j=1,...,n dq dp = exp { θ 1 + q j(x) q j (x) n i=1 dx { ( n ( 1 dqi θ i ln θ dp i=1 ( n ( dqi X θ i ln dq j i=1 θ i q i (x) } 0, θ q i (x) where θ = n i=1 θ i and q i (x) = dq i (X x), then a CCE is given by ) )} + c X, X j = θ j θ ) c ) + c j. 30 / 51

50 Competitive Equilibria in a Comonotone Market 31/51 Competitive equilibria with rank dependent utilities RDU-exponential-comonotone market (b) Equilibrium pricing kernal 31 / 51

51 Competitive Equilibria in a Comonotone Market 32/51 An algorithm for computing competitive equilibria Introduction Competitive equilibria with dual utilities Competitive equilibria with rank dependent utilities An algorithm for computing competitive equilibria Conclusion 32 / 51

52 Competitive Equilibria in a Comonotone Market 33/51 An algorithm for computing competitive equilibria Algorithm Idea of the algorithm Discretization. Take ˆX = {x 1,..., x m } such that ε = x i+1 x i is small enough. The initial wealth of agent i is ψ0 i = m k=1 δi 0,k I(x x k). The initial guess of the price is q 0,k = ˆQ 0 ( ˆX x k ), k = 1,..., m. Initial input. ˆX, ˆQ 0 = ˆP, ψ0 i = ˆξ i if ξ i C(X ), otherwise ψ0 i = E ˆQ 0 [ ˆξ i ] E ˆQ 0 [ ˆX. ˆX ] Updating process. In each step, we update δ i 0,k [0, ε] and q 0,k consequently such that each ε is optimal allocated and the market is cleared. 33 / 51

53 Competitive Equilibria in a Comonotone Market 34/51 An algorithm for computing competitive equilibria Algorithm The initial wealth. ψ i 0 δ i 0,1 ε 2ε 3ε 4ε X 34 / 51

54 Competitive Equilibria in a Comonotone Market 34/51 An algorithm for computing competitive equilibria Algorithm The initial wealth. ψ i 0 δ i 0,1 ε 2ε 3ε 4ε X After the first step: ψ i 1 ψ i 1 δ1,1 i = ε δ1,1 i = 0 ε 2ε 3ε 4ε X ε 2ε 3ε 4ε X 34 / 51

55 Competitive Equilibria in a Comonotone Market 35/51 An algorithm for computing competitive equilibria Examples Examples Simple setup. N = {1, 2, 3} X U[0, 10], ε = 0.01, m = 1000 ξ i = X /3, i = 1, 2, 3 35 / 51

56 Competitive Equilibria in a Comonotone Market 36/51 An algorithm for computing competitive equilibria Examples Example 1 - Dual utility Assumptions. Distortion functions, for s [0, 1] γ = 0.4 γ = 0.6 γ = 0.8 g(s; γ i ) = s γ i (s γ i + (1 s) γ i ) 1/γ i, where γ 1 = 0.4, γ 2 = 0.6, and γ 3 = 0.8 (Tversky-Kahneman 92). g(s;γ) s (c) Distortion functions 36 / 51

57 Competitive Equilibria in a Comonotone Market 37/51 An algorithm for computing competitive equilibria Examples Example 1 - Dual utility For the inverse-s shape distortion functions, UCE may not exist, but CCE exists. Certainty equivalents (CEQ). For i N, let CEQ prior i be constants s.t. V i (CEQ prior i ) = V i (ξ i ) and V i (CEQ post i ) = V i (X i ) and CEQ post i CEQ prior i CEQ post i (theoretical/algorithm) % increase Agent / Agent / Agent / / 51

58 Competitive Equilibria in a Comonotone Market 38/51 An algorithm for computing competitive equilibria Examples Example 1 - Dual utility g1 g2 g3 Q(X S 1 X (s)) 0.8 g1 g2 g3 Q(X S 1 X (s)) s (d) Distortion functions and equilibrium price (exact) s (e) Distortion functions and equilibrium price (algorithm) 38 / 51

59 Competitive Equilibria in a Comonotone Market 39/51 An algorithm for computing competitive equilibria Examples Example 1 - Dual utility X1 X2 X3 4 3 X1 X2 X3 2 2 Xi 1 0 Xi X (f) Equilibrium allocation (exact) X (g) Equilibrium allocation (algorithm) 39 / 51

60 Competitive Equilibria in a Comonotone Market 40/51 An algorithm for computing competitive equilibria Examples Example 2 - RDU with explicit solutions Assumptions. Distortion functions, for s [0, 1] ( ) s g i (s) = ag 1 + 2δ ; γ i + b, where γ 1 = 0.55, γ 2 = 0.6, and γ 3 = Exponential utilities u i (x) = e x/θ i with θ 1 = 2, θ 2 = 1.5 and θ 3 = 1. gi(s) 1 g1 g2 0.8 g s (h) Distortion functions 40 / 51

61 Competitive Equilibria in a Comonotone Market 41/51 An algorithm for computing competitive equilibria Examples Example 2 - RDU with explicit solutions The certainty equivalents before and after risk sharing CEQ prior i CEQ post i (theoretical/algorithm) % increase Agent / Agent / Agent / / 51

62 Competitive Equilibria in a Comonotone Market 42/51 An algorithm for computing competitive equilibria Examples Example 2 - RDU with explicit solutions Q(X z) Q(X z) z z (i) Equilibrium price (exact) (j) Equilibrium price (algorithm) 42 / 51

63 Competitive Equilibria in a Comonotone Market 43/51 An algorithm for computing competitive equilibria Examples Example 2 - RDU with explicit solutions X1 X2 4 X1 X2 3 X3 3 X3 2 2 Xi 1 Xi X (k) Equilibrium allocation (exact) X (l) Equilibrium allocation (algorithm) 43 / 51

64 Competitive Equilibria in a Comonotone Market 44/51 An algorithm for computing competitive equilibria Examples Example 3 - RDU without explicit solutions Assumptions. The three agents use distortion functions g(s; γ i ) = s γ i (s γ i + (1 s) γ i ) 1/γ i, s [0, 1], where γ 1 = 0.4, γ 2 = 0.6, and γ 3 = 0.8. Exponential utility u i (x) = e x/θ i with risk tolerant θ 1 = 3, θ 2 = 2 and θ 3 = / 51

65 Competitive Equilibria in a Comonotone Market 45/51 An algorithm for computing competitive equilibria Examples Example 3 - RDU without explicit solutions The equilibrium is most attractive for the agent with the most distorted probability measure (Agent 1) and for the most risk averse agent (Agent 3). CEQ prior i CEQ post i (algorithm) % increase Agent Agent Agent / 51

66 Competitive Equilibria in a Comonotone Market 46/51 An algorithm for computing competitive equilibria Examples Example 3 - RDU without explicit solutions Q(X z) Xi z (m) Equilibrium price (algorithm) X (n) Equilibrium allocation (algorithm) X1 X2 X3 46 / 51

67 Competitive Equilibria in a Comonotone Market 47/51 Conclusion Summary of our work Introducing the comonotome market. Solving competitive equilibrium problem in a DU-comonotone market. Existence and closed form. Fundamental theorems of welfare economics. Partially solving competitive equilibrium problem in a RDU-comonotone market by a EU approach. Existence. Obtaining CCE under exponential utilities (and power utilities). Proposing an algorithm on determining CCE in general cases. 47 / 51

68 Competitive Equilibria in a Comonotone Market 48/51 Conclusion Open questions in RDU-comonotone market Existence of competitive equilibrium under more general assumptions. Uniqueness of competitive equilibrium. The second fundamental theorem of welfare economics for the RDU market. If Q 1 = = Q n, that is a EU market with the same belief. Then a CCE in EU market is also a CCE in RDU market. Is it the only equilibrium for the comonotone market? Whether the EU-comonotone market and the EU-complete market have the same equilibria? More / 51

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70 Competitive Equilibria in a Comonotone Market 50/51 Conclusion Reference P. Embrechts, H. Liu and R. Wang (2018) Quantile-based risk sharing. Forthcoming in Operations Research. T. Hens and C. Reichlin (2013). Three solutions to the pricing kernel puzzle. Review of Finance 17: H. Jin, J. Xia and X. Zhou (2018) Arrow-Debreu equilib. Forthcoming in Mathematical Finance. A. Tversky and D. Kahneman (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty 5(4): S. Wang, V.R. Young and H.H. Panjer (1997). Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics 21(2): J. Xia and X. Zhou (2016) Arrow-Debreu equilibria for rank-dependent utilities, Mathematical Finance 26(3): M.E. Yaari (1987) The dual theory of choice under risk, Econometrica 55(1): / 51

71 Competitive Equilibria in a Comonotone Market 51/51 Conclusion Thank you Thank you for your kind attention 51 / 51