Proper Welfare Weights for Social Optimization Problems Alexis Anagnostopoulos (Stony Brook University) Eva Cárceles-Poveda (Stony Brook University) Yair Tauman (IDC and Stony Brook University) June 24th of 2011
Motivation Social optimization problems (SOP) are central to economic analysis. ASOP: Maximizes the sum of individual weighted utilities over all feasible allocations that satisfy certain constraints. The allocations are constrained efficient. Every set of individual welfare weights defines a social welfare function (SWF) and hence determines a constrained efficient allocation.
Motivation With complete markets, the welfare theorems imply that one can select the individual weights as the ones that implement competitive equilibria. Under incomplete markets, the welfare theorems do not hold and it is not obvious how to determine these weights. The purpose of this paper is to provide a mechanism that determines proper weights to be applied to SOP.
Determination of proper weights The approach for determining the proper weights is the following: For every SWF, defined by a certain set of weights, we define the contribution of each individual to the total welfare through his individual endowments. We provide an axiomatic approach to the per unit contribution of goods to welfare for every individual. The contribution of an individual to total welfare is the contribution of all the units of his initial endowments. We define proper weights as the ones such that the weighted utility of each individual equals his contribution to total welfare if these weights are used.
Related Literature The axiomatic approach has been used in the literature on cost allocation to determine the per unit costs of different goods when the production cost is non separable and it includes a joint cost component. Main contributions are: Billera/Heath (82), Mirman/Tauman (82), Samet/Tauman (82), Young (85), Neyman and Mondrer (89), Tauman (89). We apply this idea to SOP and we extend the axiomatic approach to the space of SWF. This paper was written when Tauman was a CORE fellow in 1979-1980 and it appeared as a CORE working paper.
Related Literature Macroeconomists use SWF in models with incomplete markets when: Computing optimal taxes: Aiyagari (1995), Conesa et al (2009). Evaluating the impact of tax reforms: Domeij and Heathcote (2007), Abraham and Carceles-Poveda (2010), Anagnostopoulos et al (2010). Studying the constrained efficienct allocations: Davila, Hong and Rios Rull (2007) To evaluate social welfare in these models, utilitarian SWF (equal weighs for everyone) are typically assumed. In this paper, we suggest to use instead proper weights.
Outline The Economy The axiomatic approach to the notion of per unit contribution of goods and individuals determines a family of contribution mechanisms. Analytical and numerical examples with incomplete markets.
The Economy goods and agents indexed by {1 2}: = ³ 1 R + is the vector of initial endowments of. = ³ 1 R + is the vector of consumptions of. ³ is the utility function of A1. is continuously differentiable and concave on R + with (0) = 0forall.
Social Optimization Problem (SOP) Definition 1. A social optimization problem (SOP) maximizes the weighted sum of utilities under constraints, where the individual weights =( ) satisfy 0forall and P ³ =1 =1. The resulting value function 1 is a SWF. ³ 1 max X =1 ³ R +, =1 1 ( ) 0. ( ) 0 s.t. (1)
Social Optimization Problem (SOP) Example of a pure exchange economy with complete markets: ( ) = max ( ) X =1 = X =1 X =1 ³ = R + s.t. (2) The constraint is linear and is a function of only. ( ) is concave and continuously differentiable on R +,with (0) = 0. In other SOPs the constraints might not be linear/convex and the SWF may not be concave.
Proper Welfare Weights Question: What set of welfare weights to use in a SOP? We postulate that welfare should be allocated to individuals according to their contribution and we choose welfare weights that satisfy this principle. We define proper welfare weights (and proper allocations) to be such that the weighted utility of an individual at the constrained efficient allocation equals his contribution to aggregate welfare. Let ( ) be the per unit contribution of good by agent to social welfare given initial endowments R +. ( ) will be determined axiomatically later on.
Proper Welfare Weights The total contribution of agent is ( ) R. Definition 2. Consider a SOP with weights = ( ) and let () R + be a maximizer. and () areproper iff ³ () ³ = () ( ) (3) ( ) Proper weights and proper allocations are such that the weighted utility of relative to must equal the contribution of relative to
Contribution Mechanism Let = be the total number of goods (differentiated by type and owner) Let =( 1 ) R is the vector of initial endowments. F is the set of all functions defined on R + such that (1) is a SWF, (2) is continuously differentiable on R + and (3) (0) = 0. Definition 3. A contribution mechanism is a function ( ) which associates with every good =1, F and R + an element ( ) representing the per unit contribution of the good to the total welfare. We characterize the set of all contribution mechanisms which satisfy the following four axioms.
Axiomatic Approach: Rescaling Axiom 1: Rescaling. Let F.Let R ++ and let F be defined by ( 1 ) ( 1 1 ) Then, for each R + and each =1 ( ) = ( ( 1 1 )) ( ) should not depend on the units of measurement, namely, if you change the scale you get an equivalent change in the contribution.
Axiomatic Approach: Consistency Axiom 2: Consistency. Suppose that F and F 1. If for every R + () = X =1 then for each =1 and for each R + ( ) = X =1 If two (or more) goods are the same (they play the same role in the SWF), they should have the same per unit contribution.
Axiomatic Approach: Additivity Axiom 3: Additivity. Suppose that, and belong to F.Iffor every R + then for every R + () = ()+ () ( ) = ( )+ ( ) If the welfare function canbebrokenintothesumoftwocomponents and, then the per unit contribution of a good is the sum of the per unit contributions of that good from and.
Axiomatic Approach: Positivity Axiom 4: Positivity. Let F and R ++. Suppose that is non decreasing for all 0 ( means that for all =1). Then ( ) ( ) If increasing initial endowments results in a larger increase in welfare in economy compared to, then the per unit contribution in the former is larger than in the latter. Given additivity, positivity guarantees that for R + ( )is continuous in the C 1 -norm.
Contribution Mechanisms Theorem 1. ( ) is a contribution mechanism on F = =1 F which satisfies Axioms 1-4 iff there exists a nonnegative measure on ([0 1] ß) (ß is the set of all Borel subsets of [0 1]) such that for each, F and R + 6= 0 ( ) = Z 1 0 () (), =1 (4) Moreover, (4) defines a one to one mapping from all the nonnegative measures on ([0 1] ß) onto the set of all contribution mechanisms satisfying the four axioms.
Contribution Mechanisms Assume that grows from 0 to R + homogeneously. At every point with [0 1] and for every good, compute the per unit increase in welfare resulting from an infinitesimal change of. The contribution of a good to the welfare function is the average of the marginal effects of that good on welfare weighted by the measure. The axiomatic approach does not specify what measure should be used. But two modifications can uniquely determine this.
Unique Contribution Mechanisms Axiom 5. Welfare Allocation. For every F and R +, P =1 ( ) = (). If we add axiom 5, requiring that contributions have to add up to the total welfare, the unique measure satisying A1-A5 is the Lebesgue measure on [0 1]: ( ) = Z 1 0 (), =1 This averages the marginal effects on welfare uniformly along the diagonal [0].
Unique Contribution Mechanisms The second modification strengthens positivity. Axiom 4. Strong Positivity. Let F and R ++.If is non decreasing at each in a neighborhood of then ( ) 0, for 1. The only measure that satisfies Axioms 1-3 and 4 is the atomic probability measure whose whole mass is concentrated at the point =1: ( ) = (), =1 Accordingly, the contribution of the th good is simply equal to its marginal effect on welfare at.
Complete Markets Proposition 1. Consider a pure exchange economy with complete markets. Then an allocation is proper for the contribution mechanism satisfying Axioms 1-3 and 4 iff it is a Walrasian (competitive) equilibrium with respect to the initial endowments. Proposition 2. Consider a pure exchange economy with complete markets. Suppose that the utilities ( ) are all homogeneous of degree for some 0. Then an allocation is proper for any contribution mechanism satisfying Axioms 1-4 iff it is a Walrasian (competitive) equilibrium with respect to the initial endowments.
Application: Optimal Taxation The economy is populated by two agents =1 2 and a government. The government chooses a set of weights ( 1 ) and a lump sum tax of for each agent so that 1 + 2 =0. In the first period, agents have an initial endowment ( )andthey decide how much to consume ( ) and how much to save ( ). In the second period, agents receive a return of ( ) from their investments. As an example, we assume that ( )= Agents have the following utility: =( ) 1 2 + ( ( )) 1 2
Application: Optimal Taxation Given ( ), agents maximize utility by choosing. order conditions imply: The first 1 ( ) 1 2 = Ã!1 2 Solving for the optimal effort, we obtain: = 2 ( ) 1+ 2 If we plug in the optimal effort, we obtain the welfare of agent, : =( ) 1 2 + ( )1 2 =( ) 1 ³ 2 1 2 1+ 2
Application: Optimal Taxation We now determine the optimal tax ( 1 =, 2 = ). For a given set of welfare weights ( 1 ), we definethesocialwelfarefunction: 1 +(1 ) 2 = ( 1 ) 1 2 ³ 1+1 2 12 +(1 )( 2 + ) 1 2 ³ 1+2 2 1 2 The first order condition for the optimal tax = 0 implies the following optimal tax: = (1 ³ )2 1 1+2 2 2 ³ 2 1+1 2 2 ³ 1+ 1 2 +(1 ) 2 ³ 1+ 2 2
Application: Optimal Taxation Plugging the taxes back in we can obtain the Ramsey allocations: 1 = 1 2 ( 1 ) 1+ 1 2 = 2 = 2 2 ( 2 + ) 1+ 2 2 = 1 = 1 1 = 2 = 2 2 + = 2 1 2 ( 1 + 2 ) 2 ³ 1+ 1 2 +(1 ) 2 ³ 1+ 2 2 (1 ) 2 2 2 ( 1 + 2 ) 2 ³ 1+ 1 2 +(1 ) 2 ³ 1+ 2 2 2 ( 1 + 2 ) 2 ³ 1+ 1 2 +(1 ) 2 ³ 1+ 2 2 (1 ) 2 ( 1 + 2 ) 2 ³ 1+ 1 2 +(1 ) 2 ³ 1+ 2 2
Application: Optimal Taxation To find the contributions, we can plug the allocations back in to get the maximized social welfare as a function of : = ( 1 ) 1 2 ³ 1+1 2 12 +(1 )( 2 + ) 1 2 ³ 1+2 2 1 2 = ³ 2 ³ 1+ 1 2 +(1 ) 2 ³ 1+ 2 2 12 ( 1 + 2 ) 1 2 Let = 2 +(1 ) 2 and = 2 ³ 2 1 +(1 ) 2 2. The per unit contributions for =1 2 with respect to initial endowments are given by: = Z 1 0 ³ 2 ³ 1+ 1 2 +(1 ) 2 ³ 2 1 2 1+ 2 2( 1 + 2 ) 1 = 2 Ã + 1 + 2!1 2
Application: Optimal Taxation Theproperlamdaisthesolutionof: 1 = 1 1 1 The proper lamda is equal to: = ³³ 1+2 2 1 1 2 ³³ 1+1 2 2 1 2 + ³³ 1+ 2 2 1 1 2 and the proper tax is =0forany 1 2 1, 2.
Application: Optimal Taxation Substituting = and =0weobtain: 1 = 2 = 1 2 1 ³ 1+1 2 2 2 2 ³ 1+2 2 1 = 2 = 1 ³ 1+1 2 2 ³ 1+2 2
Application: Subcases Suppose that 1 = 2 = and 1 = 2 =. Then, = 1 2. Suppose that 1 = and 2 =. Then the proper weight of agent 1is: ³ 1+ 2 1 2 = ³ 2 12 1 1+ 2 + ³ 2 12 1+ 1 2 1 1as That is, for any 1 and 2,thehigher is the return on investment of agent 2 relative to 1 the lower isherproperweight.
Application: Subcases Suppose that 1 =, 2 =. Then the proper weight of agent 1 is: ³ 1+2 2 1 2 = ³ 1+2 2 12 + 1 2 ³ 1+1 2 1 2 0as This means that the higher is the initial endowment of agent 2 relative to 1 the higher is her weight.
Application: Extension We now assume that the contribution is a function of both the initial endowments and the productivities, namely, = ( 1 2 1 2 ). Let = 2 +(1 ) 2 = 2 ³ 2 1 +(1 ) 2 2 = 1 2 2( 1 + 2 ) 1 2 Theaggregatewelfarefunctionis: =[ + ]1 2 ( 1 + 2 ) 1 2
Let 1 = 2 2 ( 1 + 2 ) 1 2 2 1 2 Application: Extension and 2 = (1 )2 2 ( 1 + 2 ) 1 2 Then, 2 1 2 ( 1 2 1 2 ) = ( 1 2 1 2 ) = + + 1 2 1 2 The contributions Z 1 = 0 + and 1 2 are equal to: and = Z 1 0 + 1 2
Application: Extension The contributions and can be rewritten as: 1 = µ +1 2 ln + µ 1 2 ln +1 2 1 1 = µ +1 2 ln + µ 1 2 ln +1 2 +1 The proper lambda now solves 1 = 1 1 1 + 1 1 1.
Numerical Results 1 2 1 2 1 2 1 2 1 2 1 1 1 2 004 053 042 064 052 039 131 165 1 1 1 20 014 072 038 107 047 006 124 443 2 1 1 1 0 058 089 044 110 055 190 134 5 1 1 1 0 069 223 044 276 055 300 134 5 1 1 2 006 072 220 066 272 040 298 167 2 1 2 1 005 055 127 042 078 052 232 130
Application: Optimal Taxation An economy with two periods, one good, a continuum of households represented by the unit interval [0 1], a continuum of identical firms and a government. There are two types of households that are indexed by {1 2}. A proportion 1 of them are each initially endowed with 1 units of the good and the rest of households (a proportion 1 1 ) are initially endowed with 2 units of the good. At = 1 the government announces a policy defined by a pair (), where is the government expenditure and is a proportional capital income tax that is used to finance.
Application: Optimal Taxation At =1,householdsobserve() and decide how much to consume 1 and save,where = 1 +, =1 2. At = 2, households work and consume. Consumption 2 equals income (measured in units of the good) from two sources: capital invested in the first period and labor. Capitalincomeis,where is the gross rate of return. Labor income is,where isthewagerateand is the endowment of labor. The two factor prices and are determined competitively.
Application: Optimal Taxation The endowment of labor is stochastic: = ( with prob. 2 with prob. 1 2 ) The aggregate capital supply at =1is 1 1 +(1 1 ) 2 and the aggregate labor supply at =2is 2 +(1 2 ). The production function is = ( ) and it exhibits constant returns to scale. In a perfectly competitive environment: = ( ) = ( )
Application: Optimal Taxation Households derive utility from the consumption of and from government expenditures. Let (0 1) be the discount factor and let = 1 2 ( ) = (1 ) + for { } The utility of a household of type is given by: ( 1 )= ( 1 )+ h 2 ( 2 ( ))+(1 2 ) ( 2 ( )) i
Application: Optimal Taxation In equilibrium, every household maximizes ( 1 )over 1 given () and given the factor prices and. The government correctly anticipates the behavior of households and chooses () to balance the budget: = Given the budget constraint of the government, in equilibrium, the decisions of households 1 (and therefore ), the aggregate capital and the factor prices and are all functions of only. In what follows, we determine the optimal tax rate and the proper weights and allocations for the households.
Application: Optimal Taxation The objective of the government is to maximize social welfare, given by a weighted average of the individual utilities. The question under consideration is what are the proper weights for the different types of households and what is the resulting proper optimal tax. Let be the density weight of a household of type and define e 1 = 1 1 and e 2 =(1 1 ) 2 to be the weight of each type. We normalize 1 and 2 so that e 1 + e 2 =1.
Application: Optimal Taxation Let () be the equilibrium government expenditure if the government chooses the tax rate. The government maximizes: e ( 1 2 )=max {} 2X =1 e [ ( 1 ())] The maximizer is a function of 1 and 2 as well as 1.Weabuse notation and write the equilibrium utility level (at the optimal tax) as ( 1 2 ).
Application: Optimal Taxation Aproperweight 1 with respect to the Lebesgue measure (the only mechanism that satisfies Axioms 1-5) is the solution of 1 1 [ 1 ( 1 2 )] = 1 1 ³ e ( 1 2 ) (1 1 ) 2 [ 2 ( 1 2 )] = 2 2 ³ e ( 1 2 ) where ( 1 2 ) is the equilibrium utility level of type at the optimal tax given ( 1 2 )and ³ e ( 1 2 ) = Z 1 0 e ( 1 2 )
Application: Optimal Taxation We assume the following functional forms: ( ) = + () = 1 1 ( ) = 1 1 + 1 1 Case 1: Households differ in only. The other parameters are: 2 1 05 15 1 05 05 04 16 05 099
0.5 0.45 0.4 0.35 Capital tax 0.3 0.25 0.2 0.15 0.1 0 0.2 0.4 0.6 0.8 1 Normalized welfare weight of type 1 Case 1: The types only differ in their initial endowments ( 1 =1and 2 =9) The optimal tax increases in the weight 1 of low income type. The higher is 1, the more the planner redistributes towards them. The low income type gets a lower proper weight and the tax decreases from the utilitarian tax of 0.23 to the proper tax of 0.12.
0.45 0.4 0.35 Capital tax 0.3 0.25 0.2 0.15 0.1 0 0.2 0.4 0.6 0.8 1 Normalized welfare weight of type 1 Case 2: The two types differ only in the discount factor ( 1 =04, 2 =099, and 1 = 2 =10) increases in 1, the weight of the impatient households. If they get more weight, the planner chooses a higher tax to increase their first period consumption. The proper weight of the impatient type is higher than his utilitarian weight and the proper tax is bigger than the utilitarian tax.
0.32 0.3 0.28 Capital tax 0.26 0.24 0.22 0.2 0.18 0 0.2 0.4 0.6 0.8 1 Normalized welfare weight of type 1 Case 3: The types only differ in their attitude towards risk ( 1 =09, 2 =05, 1 = 2 =099 and 1 = 2 =5). The optimal tax increases in the weight 1 of the high risk averse households. The higher is their weight, the higher is the tax rate. The high risk averse type gets a lower proper weight and the proper tax is also lower (0.21 versus 0.265) than the utilitarian tax.
Conclusion We apply ideas from cost allocation literature to determine the set of individual weights for social optimization problems. We assign individual weights that are proportional to the contribution of each individual to total welfare. The mechanism used to compute the contribution to welfare satisfies four axioms. We apply our methodology to a problem of optimal taxation and show that the optimal tax could potentially be very different with proper or utilitarian weights.