The Cake-Eating problem: Non-linear sharing rules

Transcription

1 The Cake-Eating problem: Non-linear sharing rules Eugenio Peluso 1 and Alain Trannoy 2 Conference In Honor of Louis Eeckhoudt June Department of Economics, University of Verona (Italy) 2 Aix-Marseille Schoolf of Economics, EHESS eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

2 Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

3 Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

4 eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

5 Outline of the talk 1 The model eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

6 Outline of the talk 1 The model 2 The stories eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

7 Outline of the talk 1 The model 2 The stories 3 The results Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

8 The Model A program P with identical utility n max a i v(x i ) x i=1 s.t. p 0 x = y. (1) v : R +! R strictly increasing and concave, satis es "Inada conditions" and is the same for each attribute; The goods are ranked such that the "kernel prices" p i a i are decreasing with i Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

9 The aim of the paper The FOC x i v 0 (xi ) v 0 (xj ) = p i a j p j a i = π ij 8i, j < x j i π ij > 1, i < j (2) Exploring integrability conditions How is the shape of the demand of the least demanded good related to the properties of the utility function? Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

10 Stories Individual wealth sharing: Arrow Debreu securities, Standard Portofolio (tax evasion n=2) eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

11 Stories Individual wealth sharing: Investor who allocates wealth over assets carrying di erent risk Arrow Debreu securities, Standard Portofolio (tax evasion n=2) eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

12 Stories Individual wealth sharing: Investor who allocates wealth over assets carrying di erent risk Arrow Debreu securities, Standard Portofolio (tax evasion n=2) Consumer choosing a consumption plan over n periods eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

13 Stories Individual wealth sharing: Investor who allocates wealth over assets carrying di erent risk Arrow Debreu securities, Standard Portofolio (tax evasion n=2) Consumer choosing a consumption plan over n periods Individual deciding her optimal insurance coverage (n=2) eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

14 Stories Individual wealth sharing: Investor who allocates wealth over assets carrying di erent risk Arrow Debreu securities, Standard Portofolio (tax evasion n=2) Consumer choosing a consumption plan over n periods Individual deciding her optimal insurance coverage (n=2) Group sharing problem(same utility but unequal weights) eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

15 Stories Individual wealth sharing: Investor who allocates wealth over assets carrying di erent risk Arrow Debreu securities, Standard Portofolio (tax evasion n=2) Consumer choosing a consumption plan over n periods Individual deciding her optimal insurance coverage (n=2) Group sharing problem(same utility but unequal weights) Household sharing a given wealth among its members eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

16 Stories Individual wealth sharing: Investor who allocates wealth over assets carrying di erent risk Arrow Debreu securities, Standard Portofolio (tax evasion n=2) Consumer choosing a consumption plan over n periods Individual deciding her optimal insurance coverage (n=2) Group sharing problem(same utility but unequal weights) Household sharing a given wealth among its members Group sharing risks eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

17 Applications to individual decision-making 1- Arrow Debreu contingency claims eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

18 Applications to individual decision-making 1- Arrow Debreu contingency claims states prob demand. price 1 a x 1 p a x 2 p 2 eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

19 Applications to individual decision-making 1- Arrow Debreu contingency claims states prob demand. price 1 a x 1 p a x 2 p 2 y = initial wealth, v = state independent utility eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

20 Applications to individual decision-making 1- Arrow Debreu contingency claims states prob demand. price 1 a x 1 p a x 2 p 2 y = initial wealth, v = state independent utility x 1 (y, p; a) = demand of the contingent claim with "kernel price" p 1 a. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

21 Applications to individual decision-making 1- Arrow Debreu contingency claims states prob demand. price 1 a x 1 p a x 2 p 2 y = initial wealth, v = state independent utility x 1 (y, p; a) = demand of the contingent claim with "kernel price" p 1 a. 2- Intertemporal consumption choice eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

22 Applications to individual decision-making 1- Arrow Debreu contingency claims states prob demand. price 1 a x 1 p a x 2 p 2 y = initial wealth, v = state independent utility x 1 (y, p; a) = demand of the contingent claim with "kernel price" p 1 a. 2- Intertemporal consumption choice Ingredients: Initial wealth y, interest rate r, intertemporal separable utility v(x 1 ) + βv(x 2 ) with discount factor β 1. Then eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

23 Applications to individual decision-making 1- Arrow Debreu contingency claims states prob demand. price 1 a x 1 p a x 2 p 2 y = initial wealth, v = state independent utility x 1 (y, p; a) = demand of the contingent claim with "kernel price" p 1 a. 2- Intertemporal consumption choice Ingredients: Initial wealth y, interest rate r, intertemporal separable utility v(x 1 ) + βv(x 2 ) with discount factor β 1. Then time weights prices 1 a = 1 1+β a = β 1+β 1 1+r. Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

24 Group decision-making Intra-household allocation: No prices, Samuelson s household welfare function. max av(x 1 ) + (1 a)v(x 2 ) x 1,x 2 s.t. p 1 x 1 + p 2 x 2 = y eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

25 Group decision-making Intra-household allocation: No prices, Samuelson s household welfare function. max av(x 1 ) + (1 a)v(x 2 ) x 1,x 2 s.t. p 1 x 1 + p 2 x 2 = y Risk-sharing: θ 2 Θ states of the world, risk: F : Θ! [0, 1],while v(x) are the identical vnm utility of the two individuals. Z Z max a v(x 1 (θ))df (θ) + (1 a) v(x 2 (θ))df (θ), with a 2 (0, 1 x 1,x 2 2 ] Θ s.t. z 1 (θ) + z 2 (θ) = y(θ) = x 1 (θ) + x 2 (θ), 8θ 2 Θ; x 1 0; x 2 0. Θ eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

26 Group decision-making Intra-household allocation: No prices, Samuelson s household welfare function. max av(x 1 ) + (1 a)v(x 2 ) x 1,x 2 s.t. p 1 x 1 + p 2 x 2 = y Risk-sharing: θ 2 Θ states of the world, risk: F : Θ! [0, 1],while v(x) are the identical vnm utility of the two individuals. Z Z max a v(x 1 (θ))df (θ) + (1 a) v(x 2 (θ))df (θ), with a 2 (0, 1 x 1,x 2 2 ] Θ s.t. z 1 (θ) + z 2 (θ) = y(θ) = x 1 (θ) + x 2 (θ), 8θ 2 Θ; x 1 0; x 2 0. Borch (1960): the consumption in each state of the world only depends on the total wealth in that state. Wealth is not transferable from one state to another. Θ eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

27 Group decision-making Intra-household allocation: No prices, Samuelson s household welfare function. max av(x 1 ) + (1 a)v(x 2 ) x 1,x 2 s.t. p 1 x 1 + p 2 x 2 = y Risk-sharing: θ 2 Θ states of the world, risk: F : Θ! [0, 1],while v(x) are the identical vnm utility of the two individuals. Z Z max a v(x 1 (θ))df (θ) + (1 a) v(x 2 (θ))df (θ), with a 2 (0, 1 x 1,x 2 2 ] Θ s.t. z 1 (θ) + z 2 (θ) = y(θ) = x 1 (θ) + x 2 (θ), 8θ 2 Θ; x 1 0; x 2 0. Borch (1960): the consumption in each state of the world only depends on the total wealth in that state. Wealth is not transferable from one state to another. Solving the risk-sharing problem then reduces to solve the intra-household allocation for any feasible y. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26 Θ

28 Integrability conditions for any n (exposition for n=2). We normalize a = 1/2, p 1 = p > 1 and p 2 = 1.Then x1 (y, p, a) x(y, p). eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

29 Integrability conditions for any n (exposition for n=2). We normalize a = 1/2, p 1 = p > 1 and p 2 = 1.Then x1 (y, p, a) x(y, p). Let h(x, p) be the demand of good 2 as a function of good 1 and p. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

30 Integrability conditions for any n (exposition for n=2). We normalize a = 1/2, p 1 = p > 1 and p 2 = 1.Then x1 (y, p, a) x(y, p). Let h(x, p) be the demand of good 2 as a function of good 1 and p. h(x, p) = g(x, p) px, where g(x, p) is the inverse function of x(y, p) wrt y using the fact that the two goods are normal. Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

31 Integrability conditions Proposition A function x(y, π), strictly increasing with y and decreasing with p is a solution of program P for all y 2 R + and for all p > 1, i there exist a positive function A(x) such that: h x (x, p) = A(x)p (3) h p (x, p) Then A represents the Arrow-Pratt absolute risk aversion coe cient, that is v 0 (x) = exp Z x 0 A(s)ds. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

32 Integrability conditions: examples x1 1 (y, p) = 2p y γ, for γ < 1 does not satisfy the integrability conditions. If h(x, p) = (1 + x) p 1, we get h x p =. Then h is h p (1 + x) ln(1 + x) the solution of P with the log-integral utility function v(x) = Z x 0 1 ln(1+s) ds If h(x, p) = ln(1 + e x p) ln p, we get h x = ex h p 1 + e x p, solution of P under the linex utility function v(x) = x e x. Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

33 Integrability without prices The "group" case Group decision-making set-up: prices are xed (eventually equal to 1) and weights are xed eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

34 Integrability without prices The "group" case Group decision-making set-up: prices are xed (eventually equal to 1) and weights are xed Does identical utility impose more restrictions on the class of non-linear sharing functions generated by P beyond x 1 < x 2 for all y? eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

35 Integrability without prices The "group" case Group decision-making set-up: prices are xed (eventually equal to 1) and weights are xed Does identical utility impose more restrictions on the class of non-linear sharing functions generated by P beyond x 1 < x 2 for all y? Answer: No for n = 2, eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

36 Integrability without prices The "group" case Group decision-making set-up: prices are xed (eventually equal to 1) and weights are xed Does identical utility impose more restrictions on the class of non-linear sharing functions generated by P beyond x 1 < x 2 for all y? Answer: No for n = 2, Answer: Yes for n > 2 but?. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

37 Integrability without prices The "group" case Group decision-making set-up: prices are xed (eventually equal to 1) and weights are xed Does identical utility impose more restrictions on the class of non-linear sharing functions generated by P beyond x 1 < x 2 for all y? Answer: No for n = 2, Answer: Yes for n > 2 but?. Proposition eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

38 Integrability without prices The "group" case Group decision-making set-up: prices are xed (eventually equal to 1) and weights are xed Does identical utility impose more restrictions on the class of non-linear sharing functions generated by P beyond x 1 < x 2 for all y? Answer: No for n = 2, Answer: Yes for n > 2 but?. Proposition For all f (y) and a 2 (0, 1/2), there exists a continuous di erentiable utility function v such that, for all y 2 R +, from Program (1) we get x1 (y; a)) = f (y). eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

39 The sharing function A sharing function f maps wealth y into the quantity consumed or invested in one good x 1 = f (y) eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

40 The sharing function A sharing function f maps wealth y into the quantity consumed or invested in one good x 1 = f (y) From p 1 x 1 + p 2 x 2 = y we know x 1 = x 2 =) x 1 = y p 1 +p 2 y p 1 + p 2 f(y) y eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

41 Three classes of diverging sharing functions Type 1: Class M, or "Moving Away" sharing functions f y p 1 + p 2 f5 M y Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

42 Type 2: Class P, or "progressive" sharing functions f y p 1 + p 2 f5 P y Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

43 Type 3: Class C, or "concave" f y p 1 + p 2 f5 C y Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

44 2.d Remark The classes are nested C P M f y p 1 + p 2 f y p 1 + p 2 f y p 1 + p 2 f5 M f5 P f5 C y y a b c y Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

45 A characterization result for the rst good Proposition Suppose that x1 (y; ) is twice continuously di erentiable. Then: i) v 2 DARA () x1 (y; ) 2 M for all π 1 ii) v 2 DRRA () x1 (y; ) 2 P for all π 1. iii) v 2 CT () x1 (y; ) 2 C for all π 1. Where eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

46 A characterization result for the rst good Proposition Suppose that x1 (y; ) is twice continuously di erentiable. Then: i) v 2 DARA () x1 (y; ) 2 M for all π 1 ii) v 2 DRRA () x1 (y; ) 2 P for all π 1. iii) v 2 CT () x1 (y; ) 2 C for all π 1. Where DARA = Decreasing Absolute Risk Aversion eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

47 A characterization result for the rst good Proposition Suppose that x1 (y; ) is twice continuously di erentiable. Then: i) v 2 DARA () x1 (y; ) 2 M for all π 1 ii) v 2 DRRA () x1 (y; ) 2 P for all π 1. iii) v 2 CT () x1 (y; ) 2 C for all π 1. Where DARA = Decreasing Absolute Risk Aversion DRRA = Decreasing Relative Risk Aversion Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

48 A characterization result for the rst good Proposition Suppose that x1 (y; ) is twice continuously di erentiable. Then: i) v 2 DARA () x1 (y; ) 2 M for all π 1 ii) v 2 DRRA () x1 (y; ) 2 P for all π 1. iii) v 2 CT () x1 (y; ) 2 C for all π 1. Where DARA = Decreasing Absolute Risk Aversion DRRA = Decreasing Relative Risk Aversion CT = Convex Tolerance Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

49 An extension to a "sequential" setting Proposition (3bis) Let P represent an intertemporal consumption choice, with n = T periods and initial wealth y. Let us consider the associated dynamic programming problem where at time t the consumer chooses the optimal consumption pattern c t, c t+1,..., c T of the remaining T t periods as a function of the current wealth y t. Then the conditions of the previous proposition apply to the sharing function linking the current consumption c t to the current wealth y t for each period t = 1...T 1. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

50 Among CT utility functions, an interesting and general family: linhara utility functions, obtained by adding a linear term to HARA utility functions. Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

51 Among CT utility functions, an interesting and general family: linhara utility functions, obtained by adding a linear term to HARA utility functions. The linex v(x) = αx e βx is well known in the risk literature Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

52 Among CT utility functions, an interesting and general family: linhara utility functions, obtained by adding a linear term to HARA utility functions. The linex v(x) = αx e βx is well known in the risk literature linpower v(x) = k 1 a x1 a + bx, with parameters a > 1, b and k > 0 1 pk a (the corresponding h(x, p) = x is bounded k (λ 1)bx a 1 a x < ) k (p 1)b Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

53 Among CT utility functions, an interesting and general family: linhara utility functions, obtained by adding a linear term to HARA utility functions. The linex v(x) = αx e βx is well known in the risk literature linpower v(x) = k 1 a x1 a + bx, with parameters a > 1, b and k > 0 1 pk a (the corresponding h(x, p) = x is bounded k (λ 1)bx a 1 a x < ) k (p 1)b linlog utility function v(x) = αx + β log x. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

54 Applications: Individual Choice 1- Arrow Debreu contingency claims states prob demand. price 1 a x 1 p a x 2 p 2 y = initial wealth, v = state independent utility eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

55 Applications: Individual Choice 1- Arrow Debreu contingency claims states prob demand. price 1 a x 1 p a x 2 p 2 y = initial wealth, v = state independent utility x1 (y, p; a) = demand for the contingent claim with the highest "kernel price" p 1 a. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

56 Applications: Individual Choice 1- Arrow Debreu contingency claims states prob demand. price 1 a x 1 p a x 2 p 2 y = initial wealth, v = state independent utility x1 (y, p; a) = demand for the contingent claim with the highest "kernel price" p 1 a. Results: eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

57 Applications: Individual Choice 1- Arrow Debreu contingency claims states prob demand. price 1 a x 1 p a x 2 p 2 y = initial wealth, v = state independent utility x1 (y, p; a) = demand for the contingent claim with the highest "kernel price" p 1 a. Results: v 2 DARA () x2 x1 is increasing with y eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

58 Applications: Individual Choice 1- Arrow Debreu contingency claims states prob demand. price 1 a x 1 p a x 2 p 2 y = initial wealth, v = state independent utility x1 (y, p; a) = demand for the contingent claim with the highest "kernel price" p 1 a. Results: v 2 DARA () x2 x1 is increasing with y v 2 DRRA () p 1x 1 y is decreasing with y eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

59 Applications: Individual Choice 1- Arrow Debreu contingency claims states prob demand. price 1 a x 1 p a x 2 p 2 y = initial wealth, v = state independent utility x1 (y, p; a) = demand for the contingent claim with the highest "kernel price" p 1 a. Results: v 2 DARA () x2 x1 is increasing with y v 2 DRRA () p 1x 1 y is decreasing with y v 2 CT () x1 is concave in y (the marginal share of the less demanded attribute decreases with wealth) eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

60 Individual choice Insurance Initial wealth Y ; risk of a loss X in state 1 with probability a. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

61 Individual choice Insurance Initial wealth Y ; risk of a loss X in state 1 with probability a. Insurance contract where 0 C X. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

62 Individual choice Insurance Initial wealth Y ; risk of a loss X in state 1 with probability a. Insurance contract where 0 C X. The premium βc is proportional to the coverage, with β < 1 eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

63 Individual choice Insurance Initial wealth Y ; risk of a loss X in state 1 with probability a. Insurance contract where 0 C X. The premium βc is proportional to the coverage, with β < 1 states prob. nal wealth 1 a x 1 = Y X + (1 β)c 2 1 a x 2 = Y βc eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

64 Individual choice Insurance Initial wealth Y ; risk of a loss X in state 1 with probability a. Insurance contract where 0 C X. The premium βc is proportional to the coverage, with β < 1 states prob. nal wealth 1 a x 1 = Y X + (1 β)c 2 1 a x 2 = Y βc Uninsured loss z 1 = x 2 x 1 eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

65 Individual choice Insurance Initial wealth Y ; risk of a loss X in state 1 with probability a. Insurance contract where 0 C X. The premium βc is proportional to the coverage, with β < 1 states prob. nal wealth 1 a x 1 = Y X + (1 β)c 2 1 a x 2 = Y βc Uninsured loss z 1 = x 2 x 1 Results: Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

66 Individual choice Insurance Initial wealth Y ; risk of a loss X in state 1 with probability a. Insurance contract where 0 C X. The premium βc is proportional to the coverage, with β < 1 states prob. nal wealth 1 a x 1 = Y X + (1 β)c 2 1 a x 2 = Y βc Uninsured loss z 1 = x 2 x 1 Results: v 2 DARA () z 1 is increasing with y Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

67 Individual choice Insurance Initial wealth Y ; risk of a loss X in state 1 with probability a. Insurance contract where 0 C X. The premium βc is proportional to the coverage, with β < 1 states prob. nal wealth 1 a x 1 = Y X + (1 β)c 2 1 a x 2 = Y βc Uninsured loss z 1 = x 2 x 1 Results: v 2 DARA () z1 is increasing with y v 2 DRRA () proportion of uninsured wealth is increasing with y Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

68 Individual choice Insurance Initial wealth Y ; risk of a loss X in state 1 with probability a. Insurance contract where 0 C X. The premium βc is proportional to the coverage, with β < 1 states prob. nal wealth 1 a x 1 = Y X + (1 β)c 2 1 a x 2 = Y βc Uninsured loss z 1 = x 2 x 1 Results: v 2 DARA () z1 is increasing with y v 2 DRRA () proportion of uninsured wealth is increasing with y v 2 CT () uinsured wealth is concave with y eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

69 3.4 Individual Choice 4- Intertemporal Consumption Given the model time wheights prices 1 a = 1 1+β a = β 1+β 1 1+r eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

70 3.4 Individual Choice 4- Intertemporal Consumption Given the model time wheights prices 1 a = 1 1+β a = β 1+β 1 1+r The initial condition λ = p (1 a) 1 p 2 a 1 becomes β 1 1+r. The marginal opportunity cost of saving is lower than the intertemporal MRS =) lower consumption in the rst period. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

71 3.4 Individual Choice 4- Intertemporal Consumption Given the model time wheights prices 1 a = 1 1+β a = β 1+β 1 1+r The initial condition λ = p (1 a) 1 p 2 a 1 becomes β 1 1+r. The marginal opportunity cost of saving is lower than the intertemporal MRS =) lower consumption in the rst period. Results: Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

72 3.4 Individual Choice 4- Intertemporal Consumption Given the model time wheights prices 1 a = 1 1+β a = β 1+β 1 1+r The initial condition λ = p (1 a) 1 p 2 a 1 becomes β 1 1+r. The marginal opportunity cost of saving is lower than the intertemporal MRS =) lower consumption in the rst period. Results: v 2 DARA () saving increasing with y Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

73 3.4 Individual Choice 4- Intertemporal Consumption Given the model time wheights prices 1 a = 1 1+β a = β 1+β 1 1+r The initial condition λ = p (1 a) 1 p 2 a 1 becomes β 1 1+r. The marginal opportunity cost of saving is lower than the intertemporal MRS =) lower consumption in the rst period. Results: v 2 DARA () saving increasing with y v 2 DRRA () decreasing average propensity to consume with wealth (Keynes) Peluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

74 3.4 Individual Choice 4- Intertemporal Consumption Given the model time wheights prices 1 a = 1 1+β a = β 1+β 1 1+r The initial condition λ = p (1 a) 1 p 2 a 1 becomes β 1 1+r. The marginal opportunity cost of saving is lower than the intertemporal MRS =) lower consumption in the rst period. Results: v 2 DARA () saving increasing with y v 2 DRRA () decreasing average propensity to consume with wealth (Keynes) v 2 CT () x1 is concave with y eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

75 Group choice Intra-household allocation Samuelson s household welfare function, with balance of power among the members given by a. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

76 Group choice Intra-household allocation Samuelson s household welfare function, with balance of power among the members given by a. If individual 1 is the "weaker" individual (a 1 2 ) then x 1 (y, a) 1 2 y. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26

77 Group choice Intra-household allocation Samuelson s household welfare function, with balance of power among the members given by a. If individual 1 is the "weaker" individual (a 1 2 ) then x 1 (y, a) 1 2 y. Immediate interpretation of the Proposition 1, for the risk-sharing too. eluso & Trannoy (Conference In Honor of Louis Eeckhoudt)Sharing Rules June / 26