Combining Equity and Efficiency. in Health Care. John Hooker. Joint work with H. P. Williams, LSE. Imperial College, November 2010

Transcription

1 Combining Equity and Efficiency in Health Care John Hooker Carnegie Mellon University Joint work with H. P. Williams, LSE Imperial College, November

2 Just Distribution The problem: How to distribute resources This image cannot currently be displayed. with a focus on health care. Health care Education Tax breaks Government Salaries benefits 2

3 Justice and Optimization Two classical criteria for distributive justice: This image cannot currently be displayed. Utilitarianism Difference principle of John Rawls 3

4 Justice and Optimization Two classical criteria for distributive justice: This image cannot currently be displayed. Utilitarianism Difference principle of John Rawls Both can be viewed as mathematical optimization problems. 4

5 Justice and Optimization Utilitarianism seeks allocation of resources to individuals that maximizes total utility. This image cannot currently be displayed. 5

6 Justice and Optimization Utilitarianism seeks allocation of resources to individuals that maximizes total utility. This image cannot currently be displayed. The Rawlsian difference principle calls for maximizing the minimum individual utility. 6

7 Justice and Optimization Utilitarianism seeks allocation of resources to individuals that maximizes total utility. This image cannot currently be displayed. The Rawlsian difference principle calls for maximizing the minimum individual utility. The two principles can also be combined. 7

8 Outline This image cannot currently be displayed. Utilitarian principle Optimality analysis Difference principle A combined principle Key application: Health care Mixed integer model & example 8

9 Utilitarian Principle 9

10 Utilitarian Principle A just distribution of resources is one that maximizes total expected utility. This image cannot currently be displayed. 10

11 Utilitarian Principle A just distribution of resources is one that maximizes total expected utility. This image cannot currently be displayed. Let x i = resources initially allocated to person i u i (x i ) = utility that results from the allocation 11

12 Utilitarian Principle A just distribution of resources is one that maximizes total expected utility. This image cannot currently be displayed. Let x i = resources initially allocated to person i u i (x i ) = utility that results from the allocation Resources may be more productive when allocated to some people than to others. For example, some illnesses may be easier to treat. We call u i a production function. 12

13 Utilitarian Principle A typical production function u i 13

14 Utilitarian Model The utility maximization problem: max u ( x ) n i = 1 i i n 1 = 1 i i x 0, all i i x = Total budget 14

15 Utilitarian Model Elementary KKT analysis yields the optimal solution: u ( x ) = = u ( x ) 1 1 n n Marginal productivity Distribute resources to equalize marginal productivity. 15

16 Utilitarian Model Elementary KKT analysis yields the optimal solution: u ( x ) = = u ( x ) 1 1 n n Marginal productivity Distribute resources to equalize marginal productivity. If we index individuals in order of productivity u i ( ) u i + 1 ( ), all i then less productive investments receive fewer resources. 16

17 Utilitarian Model Elementary KKT analysis yields the optimal solution: u ( x ) = = u ( x ) 1 1 n n Marginal productivity Distribute resources to equalize marginal productivity. If we index individuals in order of productivity u i ( ) u i + 1 ( ), all i then less productive investments receive fewer resources. For convenience assume u i (x i ) = c i x i p 17

18 p = 0.5 Utility maximizing distribution Production functions u i (x i ) for 5 individuals Resource allocation 18 u5 u4 u3 u2 u1 Production

19 p = 0.5 Utility maximizing allocation Resource allocation 19 u5 u4 u3 u2 u1 Production

20 Utilitarian Model Classical utilitarian argument: concave utility functions tend to make the utilitarian solution more egalitarian. 20

21 Utilitarian Model Classical utilitarian argument: concave utility functions tend to make the utilitarian solution more egalitarian. A completely egalitarian allocation x 1 = = x n is optimal only when u (1/ n ) = = u (1/ n ) 1 n So, equality is optimal only when everyone has the same marginal productivity in an egalitarian allocation. 21

22 Utilitarian Model u ( x ) = c x Recall that where p 0 i i i i p The optimal wealth allocation is x = c c 1 n 1 1 p 1 p i i j j = 1 1 When p < 1: Allocation is completely egalitarian only if c 1 = = c n Otherwise the most egalitarian allocation occurs when p 0: x i = j c i c j 22

23 Utilitarian Model The most egalitarian optimal allocation: people receive wealth in proportion to productivity c i. And this occurs only when productivity very insensitive to investment (p 0). 23

24 Utility maximizing resource alllocation Productivity coefficient ci p = 0.9 p = 0.8 p = 0.7 p = 0.5 p = 0 24 Resources received

25 Utilitarianism But consider this distribution p = 0.5 Utility Maximizing Distribution Wealth allocation u5 u4 u3 u2 u1 25 Production

26 Utility Loss Due to Equality p 26 Utility with equality/max utility

27 Utility Loss Due to Equality When output is proportional to investment, equality has high cost (cuts utility in half) p 27 Utility with equality/max utility

28 As p 0, optimal utility requires highly unequal allocation, but equal allocation is only slightly suboptimal Utility Loss Due to Equality p 28 Utility with equality/max utility

29 Difference Principle 29

30 Problem with Utilitarianism A utility maximizing allocation may be unjust. This image cannot currently be displayed. Seriously ill people may be neglected even though they can be treated. 30

31 Rawlsian Difference Principle Difference principle: A just distribution maximizes the welfare of the worst off. This image cannot currently be displayed. Also known as the maximin principle. Another formulation: inequality is permissible only to extent that it is necessary to improve the welfare of those worst off. 31

32 Rawlsian Difference Principle Social contract argument This image cannot currently be displayed. The rationality of my policy should not depend on who I am. 32

33 Rawlsian Difference Principle Social contract argument This image cannot currently be displayed. The rationality of my policy should not depend on who I am. I should make decisions (formulate a social contract) in an original position, behind a veil of ignorance as to who I am. 33

34 Rawlsian Difference Principle Social contract argument This image cannot currently be displayed. The rationality of my policy should not depend on who I am. I should make decisions (formulate a social contract) in an original position, behind a veil of ignorance as to who I am. I must find the decision acceptable after I learn who I am. 34

35 Rawlsian Difference Principle Social contract argument This image cannot currently be displayed. The rationality of my policy should not depend on who I am. I should make decisions (formulate a social contract) in an original position, behind a veil of ignorance as to who I am. I must find the decision acceptable after I learn who I am. I cannot rationally assent to a policy that puts me on the bottom, unless I would have been even worse off under alternative policies. So the policy must maximize the welfare of the worst off. 35

36 Rawlsian Difference Principle Applies only to basic goods. Things that people want, no matter what else they want. Salaries, tax burden, medical benefits, etc. For example, salary differentials may satisfy the principle if necessary to make the poorest better off. Applies to smallest groups for which outcome is predictable. A lottery passes the test even though it doesn t maximize welfare of worst off the loser is unpredictable. unless the lottery participants as a whole are worst off. 36

37 Rawlsian Difference Principle The optimization problem is: max z z u ( x ), all i i i i i = 1 x 0, all i i x 37

38 Rawlsian Difference Principle The optimization problem is: max z z u ( x ), all i i i i i = 1 x 0, all i i x For this simple constraint set, the solution equalizes all utilities. Given the functions u i defined earlier, x i = j c i c 1 p j 1 p 38

39 Utilitarianism + Equity 39

40 A Composite Model Utilitarian and Rawlsian distributions seem too extreme This image cannot currently be displayed. in practice. How to combine them? 40

41 A Composite Model Utilitarian and Rawlsian distributions seem too extreme This image cannot currently be displayed. in practice. How to combine them? Focus on health care Allocation of resources Combined model Maximize welfare of most seriously ill (Rawlsian)... until this requires undue sacrifice from others 41

42 A Composite Model Proposal: This image cannot currently be displayed. Switch from Rawlsian to utilitarian when inequality exceeds. 42

43 A Composite Model Proposal: This image cannot currently be displayed. Switch from Rawlsian to utilitarian when inequality exceeds. Let u i = utility allocated to person i For 2 persons: Maximize min i {u 1, u 2 } (Rawlsian) when u 1 u 2 Maximize u 1 + u 2 (utilitarian) when u 1 u 2 > 43

44 A Composite Model Contours of social welfare function for 2 persons. u 2 u 1 44

45 A Composite Model Contours of social welfare function for 2 persons. u 2 Rawlsian region { } min u 1, u 2 u 1 45

46 A Composite Model Contours of social welfare function for 2 persons. u 2 Utilitarian region u 1 + u 2 Rawlsian region { } min u 1, u 2 u 1 46

47 Person 1 is harder to treat. But maximizing person 1 s health requires too much sacrifice from person 2. Feasible set u 2 Optimal allocation Suboptimal u 1 47

48 A Composite Model Advantage: Only one parameter Focus for debate. has intuitive meaning (unlike weights) Examine consequences of different settings for Find least objectionable setting Results in a consistent policy 48

49 A Composite Model We want continuous contours u 2 u 1 49

50 A Composite Model We want continuous contours u 2 u 1 + u 2 2min { u } 1, u 2 + So we use affine transform of Rawlsian criterion u 1 50

51 A Composite Model The social welfare problem becomes max z z { } 2min u, u +, if u u u 1 + u 2, otherwise constraints on feasible set 51

52 Epigraph is union of 2 polyhedra. MILP Model u 2 u 2 u 1 u 1 52

53 MILP Model Epigraph is union of 2 polyhedra. Because they have different recession cones, there is no MILP model. u 2 u 2 Recession directions (u 1,u 2,z) (0,1,0) (1,1,2) u 1 (1,0,0) (0,1,1) (1,0,1) u 1 53

54 Impose constraints u 1 u 2 M MILP Model u 2 M M u 1 54

55 This equalizes recession cones. MILP Model u 2 u 2 u 1 Recession directions (u 1,u 2,z) (1,1,2) (1,1,2) u 1 55

56 MILP Model We have the model max z z 2 u + + ( M ) δ, i = 1,2 z u + u + (1 δ ) 1 2 u u M, u u M u, u δ i {0,1} constraints on feasible set u 1 56

57 MILP Model We have the model max z z 2 u + + ( M ) δ, i = 1,2 z u + u + (1 δ ) 1 2 u u M, u u M u, u δ i {0,1} constraints on feasible set This is a convex hull formulation. u 1 57

58 n-person Model Rewrite the 2-person social welfare function as { } min u 1, u u min + ( u 1 u min ) + ( u 2 u min ) { } max 0, + α = α u 1 58

59 n-person Model Rewrite the 2-person social welfare function as { } min u 1, u u min + ( u 1 u min ) + ( u 2 u min ) This can be generalized to n persons: ( ) ( n 1) + nu + u j u n min min j = 1 + { } max 0, + α = α 59

60 n-person Model Rewrite the 2-person social welfare function as { } min u 1, u u min + ( u 1 u min ) + ( u 2 u min ) { } max 0, + α = α This can be generalized to n persons: ( ) ( n 1) + nu + u j u n min min j = 1 + Epigraph is a union of n! polyhedra with same recession direction (u,z) = (1,,1,n) if we require u i u j M So there is an MILP model 60

61 n-person MILP Model To avoid n! 0-1 variables, add auxiliary variables w ij max z z u + w, all i i ij j i w + u + δ ( M ), all i, j with i j ij i ij w u + (1 δ ), all i, j with i j ij j ij u u M, all i, j i j u 0, all i δ i ij {0,1}, all i, j with i j 61

62 n-person MILP Model To avoid n! 0-1 variables, add auxiliary variables w ij max z z u + w, all i i ij j i w + u + δ ( M ), all i, j with i j ij i ij w u + (1 δ ), all i, j with i j ij j ij u u M, all i, j i j u 0, all i δ i ij {0,1}, all i, j with i j Theorem. The model is correct (not easy to prove). u 1 62

63 n-person MILP Model To avoid n! 0-1 variables, add auxiliary variables w ij max z z u + w, all i i ij j i w + u + δ ( M ), all i, j with i j ij i ij w u + (1 δ ), all i, j with i j ij j ij u u M, all i, j i j u 0, all i δ i ij {0,1}, all i, j with i j u 1 Theorem. The model is correct (not easy to prove). Theorem. This is a convex hull formulation (not easy to prove). 63

64 n-group Model In practice, funds may be allocated to groups of different sizes For example, disease/treatment categories. u i Let = average utility gained by a person in group i n i = size of group i 64

65 n-group Model 2-person case with n 1 < n 2. Contours have slope n 1 /n 2 u 2 M M u 1 65

66 n-group MILP Model Again add auxiliary variables w ij max z z ( n 1) + nu + w, all i w n ( u + ) + δ n ( M ), all i, j with i j w u + (1 δ ) n, all i, j with i j δ i i i ij j i ij j i ij j ij j ij j u u M, all i, j i j u 0, all i i ij {0,1}, all i, j with i j Theorem. The model is correct. Theorem. This is a convex hull formulation. u 1 66

67 Health Example Measure utility in QALYs (quality-adjusted life years). QALY and cost data based on Briggs & Gray, (2000) etc. Each group is a disease/treatment pair. Treatments are discrete, so group funding is all-or-nothing. Divide groups into relatively homogeneous subgroups. 67

68 Health Example Add constraints to define feasible set max z ( n 1) + nu + w, all i w n ( u + ) + δ n ( M ), all i, j with i j w u + (1 δ ) n, all i, j with i j δ i i i ij j i u u M, all i, j u 0, all i {0,1}, all i, j with i j u = qy + ij j i ij j ij j ij j i j i ij α i i i i i i i i y {0,1}, all i i z ncy budget y i indicates whether group i is funded u 1 68

69 QALY & cost data Part 1 69

70 QALY & cost data Part 2 70

71 Results Total budget 3 million 71

72 Utilitarian solution Results 72

73 Rawlsian solution Results 73

74 Fund for all Results 74

75 Results More dialysis with larger, beginning with longer life span 75

76 Results Abrupt change at =

77 Results Come and go together 77

78 Results In-out-in 78

79 Results Most rapid change. Possible range for politically acceptable compromise 79

80 Results 32 groups, 1089 integer variables Solution time (CPLEX 12.2) is < 0.5 sec for each 80

81 Solution time vs. = No. of groups 81

82 Future Work Generalize Rawlsian criterion to lexmax. Find principled justification for choice of. 82