Optimal Income Taxation and Public-Goods Provision with Preference and Productivity Shocks

Transcription

1 Optimal Income Taxation and Public-Goods Provision with Preference and Productivity Shocks F. Bierbrauer March 1, 2011

2 Introduction Objective of the paper: Design of Optimal Taxation and Public Good provision with Private Information Unknown distribution of shocks Coalition Proofness

3 Introduction Private Information - Standard Mirrlees Problem Information about productivities are private Need for incentives for revealing information

4 Introduction Private Information - Standard Mirrlees Problem Information about productivities are private Need for incentives for revealing information Unkown Distribution: Productivities and Public good valuation is unknown Agents Beliefs when reporting? Robust Mechanism Design approach: Belief free implementation (Ex-post)

5 Introduction Coalition Proofness Standard Mirrleessian Tax system has clear gains from collusion with similar agents. Interest Groups Agents with similar taste can unite to aect policy choice Relevance of Information Aggregation With group deviations, the problem of aggregating information becomes non trivial.

6 Model Large economy: continuum i [0, 1] of agents; Valuation of public good θ Θ = {θ L, θ H } and productivity w W = {w L, w H } are private information; Preferences given by U ( q, c, y, w i, θ i) = θ i q + u (c) y w i ; Aggregate state is s = (f H, p H, p L ) [0, 1] 3 S where: f H = share of w H (high skill) p H = share of high skill with θ H ; p L = share of low skill with θ H ;

7 Model A social choice function is (q, c, y), where: q : S R + is the public good choice; c : S W Θ R + is the consumption allocation; y : S W Θ R + is the production allocation;

8 Model A social choice function is (q, c, y), where: q : S R + is the public good choice; c : S W Θ R + is the consumption allocation; y : S W Θ R + is the production allocation; A social choice function is feasible if [ ] p f H (y (s, w H, θ H ) c (s, w H, θ H )) H + (1 p H ) (y (s, w H, θ L ) c (s, w H, θ L )) [ ] p + (1 f H ) L (y (s, w L, θ H ) c (s, w L, θ H )) r (q (s)), + (1 p L ) (y (s, w L, θ L ) c (s, w L, θ L )) for all s S. r (.) is the public good cost function.

9 Model A type space is a measurable space (T, T ) together with functions π : T W Θ and β : T ( (T )) where:

10 Model A type space is a measurable space (T, T ) together with functions π : T W Θ and β : T ( (T )) where: π ( t i) = ( w ( t i), θ ( t i)) is an agent's payo type;

11 Model A type space is a measurable space (T, T ) together with functions π : T W Θ and β : T ( (T )) where: π ( t i) = ( w ( t i), θ ( t i)) is an agent's payo type; β ( t i) ( (T )) is the agent's belief over possible distribution of types (other agent's type) Compared with (T i ) in nite case.

12 Model A type space is a measurable space (T, T ) together with functions π : T W Θ and β : T ( (T )) where: π ( t i) = ( w ( t i), θ ( t i)) is an agent's payo type; β ( t i) ( (T )) is the agent's belief over possible distribution of types (other agent's type) Compared with (T i ) in nite case. Every distribution over types δ (T ) implies a distribution over payo types s (δ) (W Θ).

13 Denition A Mechanism is M = [(A, A), Q, C, Y ], where: A is the set of possible messages; Q : (A) R + is public good allocation; (C, Y ) : (A) A R 2 + are the consumption and labor allocations;

14 Denition A Mechanism is M = [(A, A), Q, C, Y ], where: A is the set of possible messages; Q : (A) R + is public good allocation; (C, Y ) : (A) A R 2 + are the consumption and labor allocations; A mechanism induces a (large) game. A strategy is σ : T A; Payos, given action a A, opponent's action χ (A) and type t U M (a, χ; t) = U (Q (χ), C (χ, a), Y (χ, a), w (t), θ (t)) ;

15 Model A Bayes-Nash equilibria is a strategy σ such that for all t T, ( ) ( ) U M σ (t), χ σ ; t dβ (δ t) U M a, χ σ ; t dβ (δ t), δ for all a A. Where χ σ δ = δ σ 1 is the action prole induced by type distribution δ (T ) and strategy σ. δ

16 Model A Bayes-Nash equilibria is a strategy σ such that for all t T, ( ) ( ) U M σ (t), χ σ ; t dβ (δ t) U M a, χ σ ; t dβ (δ t), δ for all a A. Where χ σ δ = δ σ 1 is the action prole induced by type distribution δ (T ) and strategy σ. A social choice function (q, c, y) is implementable in the type space (T, T ) if there is a mechanism M and a Bayes Nash equilibria of the induced game such that ) q (s (δ)) = Q (χ σ δ ; ) for all δ (T ). c (s (δ), w (t), θ (t)) = C y (s (δ), w (t), θ (t)) = Y (χ σ δ δ, σ (t) ; ), (χ σ δ, σ (t)

17 Robust Implementation A social choice function is robustly implementable if for any type space it is implementable.

18 Robust Implementation Theorem A social choice function is Robustly Implementable i for any s S, every (w, θ) W Θ and every (w, θ ) W Θ, y (s, w, θ) θq (s) + u (c (s, w, θ)) w θq (s) + u (c (s, w, θ )) y (s, w, θ ) w

19 Robust Implementation Consequences of the Theorem: Consumption and labor allocation cannot depend on θ: c (s, w, θ) = c (s, w) and y (s, w, θ) = y (s, w).

20 Robust Implementation Consequences of the Theorem: Consumption and labor allocation cannot depend on θ: c (s, w, θ) = c (s, w) and y (s, w, θ) = y (s, w). So IC becomes u (c (s, w)) y (s, w) w for all w, w W and for all s S. u (c (s, w )) y (s, w ), w

21 Robust Implementation Consequences of the Theorem: Consumption and labor allocation cannot depend on θ: c (s, w, θ) = c (s, w) and y (s, w, θ) = y (s, w). So IC becomes u (c (s, w)) y (s, w) w for all w, w W and for all s S. Feasibility constraint becomes u (c (s, w )) y (s, w ), w f H [y (s, w H ) c (s, w H )]+(1 f H ) [y (s, w L ) c (s, w L )] r (q (s)), for all s S.

22 Mirrlees Problem For each s S, we dene the utilitarian welfare measure ( W (s) = θq(s) (s) + f H u (c (s, w H )) y (s, w ) H) ( + (1 f H ) u (c (s, w L )) y (s, w ) L) w L Where θ (s) = (f H p H + (1 f H ) p L ) θ H + (f H (1 p H ) + (1 f H ) (1 p L )) θ L. We can solve a standard Mirrlees problem pointwise in s S. w H

23 Mirrlees Problem Theorem: For each s S, the solution to the welfare maximization problem is characterized by 1. Consumption: u (c (s, w H )) = 1 w H and u (c (s, w L )) = 1 w 1 f H w L H w H w w L 1 f H w L ; H w L

24 Mirrlees Problem Theorem: For each s S, the solution to the welfare maximization problem is characterized by 1. Consumption: u (c (s, w H )) = 1 w H 2. Implicit marginal taxes: τ (s, w H ) = 0 and τ (s, w L ) = 1 and u (c (s, w L )) = 1 w 1 f H w L H w H w w L 1 f H w L ; H w L 1 w L u (c (s, w L )) > 0;

25 Mirrlees Problem Theorem: For each s S, the solution to the welfare maximization problem is characterized by 1. Consumption: u (c (s, w H )) = 1 w H 2. Implicit marginal taxes: τ (s, w H ) = 0 and τ (s, w L ) = 1 and u (c (s, w L )) = 1 w 1 f H w L H w H w w L 1 f H w L ; H w L 1 w L u (c (s, w L )) > 0; 3. Samuelson Rule public good provision: where λ (s) = f H wh + (1 f H) w L θ (s) = λ (s) r (q (s)), is the marginal cost of funds.

26 Mirrlees Protable deviation Dene V (s, w, θ) θq (s) + u (c (s, w, θ)) y(s,w,θ) w. ( V (s, w, θ) = θ 1 ) q p k w r (s) (q (s)) p k = 1 ( θw θ ) (s) q (s), w λ (s) p k In case θ (s) = θ H we have V (s, w L, θ H ) p k = θ H w ( w L 1 ) q (s) < 0. λ (s) p k (w L, θ H ) agents should understate their preference for public good. Societal marginal cost of funds is lower than their own.

27 Coalition Proofness Denition Given a mechanism M and a type space T, a strategy σ is a Coalition Proof Bayes Nash equilibrium if it is a Bayes Nash equilibrium and There is no set of types T T and a strategy σ T such that: ) The strategy prole (σ T \T, σ T is a Bayes-Nash equilibrium; Deviators are better o, i.e., for all t T U M (( σ T \T, σ T ), σ T (t), t ) > U M (σ, σ (t), t) ;

28 Coalition Proofness Denition The coalition T is subcoalition proof: There exists no strict subcoalition T T and strategy σ T such ( ) σ T \T, σ T \T, σ T is a Bayes-Nash equilibrium and that for all t T, (( ) ) U M σ T \T, σ T \T, σ T, σ T (t), t U M (( σ T \T, σ T ), σ T (t), t )

29 Robust and Coalition-Proof Implementation Denition For a given type space, SCF (q, c, y) is implementable as a coalition proof BNE, if there is a mechanism M and a strategy σ such that 1. σ is a coalition-prof Bayes-Nash equilibrium 2. For all δ and t we have, σ (t)-almost surely, Q (α(δ, σ )) = q (s(δ)) C (α(δ, σ ), a(t)) = c (s(δ), w(t), θ(t)) Y (α(δ, σ ), a(t)) = y (s(δ), w(t), θ(t)) A SCF is robustly implementable and coalition-proof if, given (T, T ) and π, there is a mechanism M and a strategy σ such that requirements 1. and 2. are satised, for every belief system β.

30 Necessary Conditions For a given SCF (q, c, y), let V (s, w, θ) = θq(s) + u (c(s, w)) y(s,w) w. Proposition If (q,c,y) is robust and coalition-proof, then it must be true that 1. For any (f H, p L ), V (s, w H, θ L ) is a non-increasing function of p H and V (s, w H, θ H ) is a non-decreasing function of p H. 2. For any (f H, p H ), V (s, w L, θ L ) is a non-increasing function of p L and V (s, w L, θ H ) is a non-decreasing function of p L. Sketch of the Proof.

31 Necessary Conditions Sketch of the Proof of Proposition 3 Sketch of the proof of the monotonicity constraint V (s, w L, θ H ) p L 0 If p L > p L s.t. V ((f H, p L, p H ), w L, θ H ) > V ((f H, p L, p H), w L, θ H ) Consider a type space with a belief system such that β ({ δ : s(δ) = (f H, p H, p L) } t ) = 1 t Deviation σ T for individuals in T = {t : (w(t), θ(t)) = (w L, θ H )}: Play according to σ (t ) with probability p L /p L Play according to σ (ˆt) with probability (1 p L /p ), where L ˆt ˆT = {t : (w(t), θ(t)) = (w L, θ L )} Then α ( δ(δ), σ ) = α ( δ, (σ, T \T σ )) T with s ( δ(δ) ) = (f H, p H, p L ). On this type space, the deviation makes the deviators better o, is subcoalition-proof, and (σ T \T, σ T ) is a Bayes-Nash equilibrium.

32 Necessary Conditions Mirrleesian Social Choice Function (w L, θ H )-guys do not benet from understating θ if V (s,w L,θ H ) p L 0. There is no state s such that the Mirrleesian SCF is coalition-proof.

33 Sucient Conditions Let Ω(ε) be the set of social choice functions which satisfy 1. For all s, the necessary conditions above are satised, and at most one of them is binding 2. For all s, the resource constraint holds 3. For all (s, w, ŵ), u (c(s, w)) y(s, w) w u (c(s, ŵ)) y(s, ŵ) w + ε Proposition A SCF in Ω(ε) is robustly implementable as a coalition-proof BNE Sketch of the Proof.

34 Sucient Conditions Sketch of the Proof of Proposition 4 Given (T, T ) and π, and given a SCF (q, c, y) Ω(ε), we construct a direct mechanism M = [(T, T ), Q, C, Y ] s.t. δ M(T ), t T, Q(δ) = q (s(δ)), C(δ, t) = c (s(δ), w(t)), Y (δ, t) = y (s(δ), w(t)) The TT strategy is a BNE of the game induced by this mechanism, for every belief system β, and is coalition-proof on every type space. There is no deviation that involves lies about skills w since for all δ, θ(t)q ( s ( δ(δ) )) + u ( c ( s ( δ(δ) )) ) y ( s ( δ(δ) ) ), w(t), w(t) w(t) > θ(t)q ( s ( δ(δ) )) ( ( )) ) y + u c s ( δ(δ) ( s ( δ(δ) ) ), ŵ, ŵ w(t)

35 Sucient Conditions Sketch of the Proof of Proposition 4 (ct'd) Suppose all the deviators have the same payo type, e.g. (w H, θ L ) Then δ, f H (δ) = f H ( δ(δ) ), p H (δ) < p H ( δ(δ) ), p L (δ) = p L ( δ(δ) ) But V (s, w H, θ L )/ p H 0, hence the deviators are not better o Suppose both (w H, θ L ) and (w H, θ H ) lie with positive probability If (w H, θ H ) reduce their prob. of lying, then ) δ the communicated distribution of types, ˆδ(δ), is s.t. p H (ˆδ(δ) > p H ( δ(δ) ) Since s, V (s, w H, θ H )/ p H 0, they are weakly better o If there is a subset D with β (D (w H, θ H )) > 0 s.t. the monotonicity constraint is a strict inequality, then they are strictly better o Otherwise, since for all s at most one monotonicity constraint is binding, (w H, θ L ) benet from reducing the probability of a lie

36 The Optimal Social Choice Function We now characterize the optimal robust and coalition-proof SCF 1. Choose {q(s), c(s, w), y(s, w)} s,w to maximize E [W (s)] s.t. k {L, H}, V (s, w k, θ L )/ p k 0, V (s, w k, θ H )/ p k 0, and incentive compatibility and resource constraints are satised. 2. Verify that for all s, at most one of the monotonicity constraints holds as an equality, i.e. the SCF belongs to Ω(0). 3. Thus the SCF is approximately coalition-proof, because If ε > 0 s.t. 0 ε ε Ω(ε) is compact, then ε > 0, (q, c, y ) robustly implementable as a coalition-proof BNE s.t. (s, θ, w), U (q (s), c (s, w), y (s, w), w, θ) U (q(s), c(s, w), y(s, w), w, θ) ε

37 The Optimal Social Choice Function We solve the optimization problem by decomposing it into subproblems Problems [ P L (p H, f H ): Choose c(s, w k ), y(s, w k ), q(s) to maximize E W (s) ] θ(p H,p L ) λ θ H w L, f H, p H s.t. IC, RC, and V (s,w L,θ H ) p L 0. Problems P H (p L, f H ): Prevent (w H, θ L )-guys to overstate their θ.

38 The Optimal Social Choice Function Preventing the Low-Skilled from Understating their Preferences For every s, let c (s, w k ), y (s, w k ), q (s) be the solution to the relevant subproblem, and vk (s) = u (c (s, w k )) y (s,w k ). w k Proposition (Solution to Problem P L (p H, f H )) Let f H < 1/2. There is ˆp L s.t., in comparison to the Mirrleesian SCF, 1. For p L < ˆp L, redistribution and public-goods provision are distorted downwards and the implicit marginal tax rates are lower: v L (s) < v L (s), v H (s) > vh(s), q (s) < q (s), τ (s, w k ) τ (s, w k ) 2. For p L = ˆp L, the allocation is undistorted 3. For p L > ˆp L, we have: vl (s) > v L (s), vh (s) < v H (s), q (s) > q (s), τ (s, w L ) < τ (s, w L ), τ (s, w H ) = τ (s, w H ) Sketch of the Proof

39 The Optimal Social Choice Function Sketch of the Proof of Proposition 5 Optimization Problem: First Step: Given p L, and so q(p L ) and v L (p L ), the utility of the high skilled is chosen optimally, v H (p L ) = V H (v L (p L ), r (q(p L ))), where V H (v L, ρ) max u(c H ) y H w H s.t. u(c j ) y j w j u(c k ) y k w j, f H (y H c H ) + (1 f H )(y L c L ) = ρ, u(c L ) y L w L = v L Second Step: Determine the optimal q(p L ), v L (p L ), i.e. maximize 1 { θ(pl )q(p L ) + f H V H (v L (p L ), r (q(p L ))) + (1 f H )v L (p L ) } dp L κ(p H ) s.t. θ H q (p L ) + v L(p L ) 0

40 The Optimal Social Choice Function Sketch of the Proof of Proposition 5 (ct'd) Optimality Conditions: The marginal social benet from increasing public-goods provision is proportional to that from increased redistribution: 1 θ H ( θ + fh V H2 r (q) ) = f H V H1 + 1 f H Implies that public goods provision and redistribution are always distorted in the same direction. (Intuition.) The average distortion" has to be zero, i.e. 1 { θ + fh V H2 r (q) } dp L = 1 κ(p H ) κ(p H ) {f H V H1 + 1 f H } dp L Implies that it is optimal to have the upward distortions of public goods supply concentrated in the region where it contributes most to welfare, i.e. where θ, hence p L, is high. (Intuition.)

41 The Optimal Social Choice Function Preventing the High-Skilled from Overstating their Preferences Proposition (Solution to Problem P H (p L, f H )) Let f H < 1/2. There is ˆp H s.t., in comparison to the Mirrleesian SCF: For p H < ˆp H (resp., p H = ˆp H, p H > ˆp H ), redistribution and public-goods provision are distorted downwards (resp., undistorted, distorted upwards) and the implicit marginal tax rates are lower (resp., same, higher).

42 Conclusion and Comments Positive correlation between public goods provision and redistribution Absent in standard Mirrlees Normative or positive? Dierent from standard Mirrlees Technical critiques Continuum of types Compare with Hellwig and Bierbauer (2011) Sub n (coalitions) Too simple for policy prescriptions Non-separability, Linearity disutility Why coalition proofness?