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- 1. Incentive separability Pawel Doligalski (Bristol & GRAPE) Piotr Dworczak (Northwestern & GRAPE) Joanna Krysta (Standford) Filip Tokarski (Stanford GSB) This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Starting grant IMD-101040122) 1
- 2. Efficiency-equity trade-off: distortions are useful for redistribution • central concept in Public Finance 2
- 3. Efficiency-equity trade-off: distortions are useful for redistribution • central concept in Public Finance However, sometimes the efficiency-equity trade-off can be avoided: • Diamond-Mirrlees ’71: distortionary taxes on firms are redundant • Atkinson-Stiglitz ’76: differentiated consumption taxes are redundant with a non-linear income tax • Sadka ’76, Seade ’77: there should be “no distortion at the top” We study the general logic behind these results using Mechanism Design 2
- 4. What we do We consider a general framework allowing for complex incentive constraints - private information, moral hazard, voluntary participation etc We introduce a new notion of incentive separability - A set of decisions is incentive-separable if perturbing these decisions along agents’ indifference curves preserves all incentive constraints. 3
- 5. What we do We consider a general framework allowing for complex incentive constraints - private information, moral hazard, voluntary participation etc We introduce a new notion of incentive separability - A set of decisions is incentive-separable if perturbing these decisions along agents’ indifference curves preserves all incentive constraints. Main theorem: It is optimal to allow agents to make unrestricted choices over incentive-separable decisions, given some prices and budgets. 3
- 6. What we do We consider a general framework allowing for complex incentive constraints - private information, moral hazard, voluntary participation etc We introduce a new notion of incentive separability - A set of decisions is incentive-separable if perturbing these decisions along agents’ indifference curves preserves all incentive constraints. Main theorem: It is optimal to allow agents to make unrestricted choices over incentive-separable decisions, given some prices and budgets. The theorem allows us to • Extend and unify Atkinson-Stiglitz and Diamond-Mirrlees theorems - optimal to jointly remove consumption and production distortions - harmonizing consumption taxes with distorted production—a bad idea • Propose a novel argument for the optimality of food vouchers 3
- 7. Literature review • Redundancy (or not) of commodity taxes with nonlinear income tax Atkinson & Stiglitz (1976); Christiansen (1981); Cremer, Gahvari & Ladoux (1998); Laroque (2005); Kaplow (2006); Gauthier & Laroque (2009); Cremer & Gahvari (1995); Cremer, Pestieau & Rochet (2001); Saez (2002); da Costa & Werning (2002); Golosov, Kocherlakota & Tsyvinski (2003) • Optimal in-kind redistribution Nichols & Zeckhauser (1982); Currie & Gahvari (2008); Condorelli (2012); Akbarpour r ○ Dworczak r ○ Kominers (2023) • Optimality of production efficiency Diamond & Mirrlees (1971); Stiglitz & Dasgupta (1971), Naito (1999), Hammond (2000) 4
- 8. Outline of the talk 1. Framework 2. General results 3. Applications 5
- 9. Framework
- 10. • A unit mass of agents with types θ ∈ [0, 1] ≡ Θ, uniformly distributed • Agents’ utility over a vector of K decisions x ∈ RK + U (x, θ) - x contains e.g. consumption of goods, labor supply, effort, ... - U continuous in x, measurable in θ - we make no ”single crossing” assumptions 6
- 11. • A unit mass of agents with types θ ∈ [0, 1] ≡ Θ, uniformly distributed • Agents’ utility over a vector of K decisions x ∈ RK + U (x, θ) - x contains e.g. consumption of goods, labor supply, effort, ... - U continuous in x, measurable in θ - we make no ”single crossing” assumptions • An allocation rule x : Θ → RK + • associated aggregate allocation: x = ´ x(θ) dθ • associated utility profile Ux , where Ux (θ) = U(x(θ), θ) 6
- 12. • Social planner with objective function W (Ux , x), - W continuous and weakly decreasing in x - dependence on Ux : redistributive (welfarist) objective - dependence on x: preferences over allocations beyond agents utilities - e.g. (opportunity) cost of resources, preference for tax revenue, aggregate constraints 7
- 13. • Social planner with objective function W (Ux , x), - W continuous and weakly decreasing in x - dependence on Ux : redistributive (welfarist) objective - dependence on x: preferences over allocations beyond agents utilities - e.g. (opportunity) cost of resources, preference for tax revenue, aggregate constraints • Planner chooses allocation x : Θ → RK + subject to incentive constraints x ∈ I ⊆ (RK +)Θ - no a priori restrictions on I - I can represent IC constraints due to private info, moral hazard, etc. • An allocation x ∈ I is called feasible 7
- 14. • Social planner with objective function W (Ux , x), - W continuous and weakly decreasing in x - dependence on Ux : redistributive (welfarist) objective - dependence on x: preferences over allocations beyond agents utilities - e.g. (opportunity) cost of resources, preference for tax revenue, aggregate constraints • Planner chooses allocation x : Θ → RK + subject to incentive constraints x ∈ I ⊆ (RK +)Θ - no a priori restrictions on I - I can represent IC constraints due to private info, moral hazard, etc. • An allocation x ∈ I is called feasible • Notation: For a subset of decisions S ⊆ {1, ..., K} : xS = (xi )i∈S x−S = (xi )i̸∈S x = (xS , x−S ) 7
- 15. Incentive separability Def: Decisions S ⊆ {1, ..., K} are incentive-separable (at feasible alloc x0) if {(xS , x−S 0 ) : U(xS (θ), x−S 0 (θ), θ) = U(x0(θ), θ), ∀θ ∈ Θ} ⊆ I. Decisions S are incentive-separable (IS) if all perturbations of xS that keep all types indifferent satisfy incentive constraints 8
- 16. Incentive separability Def: Decisions S ⊆ {1, ..., K} are incentive-separable (at feasible alloc x0) if {(xS , x−S 0 ) : U(xS (θ), x−S 0 (θ), θ) = U(x0(θ), θ), ∀θ ∈ Θ} ⊆ I. Decisions S are incentive-separable (IS) if all perturbations of xS that keep all types indifferent satisfy incentive constraints IS is a joint restriction on preferences and incentives. Examples: 1. Voluntary participation: all decisions are IS 2. Private information: decisions S are weakly-separable ⇒ S are IS - S are weakly-separable when U(x, θ) = Ũ(v(xS ), x−S , θ) 3. Moral hazard: hidden action a affects the distribution of observed state ω - U(x, θ) = −c(a) + P ω P(ω|a)uω(yω), yω ∈ RL + - Decisions within each state ω (namely, yω) are IS 4. Easy to extend to combinations of frictions, dynamic private info, aggregate states, ... 8
- 17. Preliminaries Consider a feasible allocation x0. We will keep x−S 0 and Ux0 fixed, and re-optimize over the allocation of incentive-separable goods xS : Uθ(xS (θ)) := U(xS (θ), x−S 0 (θ), θ) R(xS ) := W (Ux0 , (xS , x−S 0 )) Assumptions: 1. Uθ is locally nonsatiated, for all θ 2. U(xS 0 (θ)) ≥ U(0), for all θ 3. For any x0, there exists integrable x̄(θ) such that Uθ(y) = Uθ(xS 0 (θ)) =⇒ y ≤ x̄(θ) 9
- 18. General results
- 19. Suppose that decisions S ⊆ {1, ..., K} are incentive-separable Def: A feasible allocation x0 is S-undistorted if xS 0 solves max xS R(xS ) s.t. Uθ(xS (θ)) = Uθ(xS 0 (θ)), ∀θ ∈ Θ 10
- 20. Suppose that decisions S ⊆ {1, ..., K} are incentive-separable Def: A feasible allocation x0 is S-undistorted if xS 0 solves max xS R(xS ) s.t. Uθ(xS (θ)) = Uθ(xS 0 (θ)), ∀θ ∈ Θ Lemma 1 (Optimality) Any feasible x0 that is not S-undistorted can be improved upon. 10
- 21. Suppose that decisions S ⊆ {1, ..., K} are incentive-separable Def: A feasible allocation x0 is S-undistorted if xS 0 solves max xS R(xS ) s.t. Uθ(xS (θ)) = Uθ(xS 0 (θ)), ∀θ ∈ Θ Lemma 1 (Optimality) Any feasible x0 that is not S-undistorted can be improved upon. Proof: Since x0 is not S-undistorted, there exists allocation x∗ = (xS ∗ , x−S 0 ) s.t. 1. Utility levels are unaffected: Uθ(xS ∗ (θ)) = Uθ(xS 0 (θ)), 2. Planner’s objective strictly improves under x∗ relative to x0. Since goods S are incentive-separable, 1. implies that x∗ is also feasible. □ 10
- 22. Simplifying Assumption: R(xS ) is linear in xS with coefficients −λ ≪ 0 - Interpretation: constant marginal costs of producing xS - Too strong, we will relax it in a moment 11
- 23. Simplifying Assumption: R(xS ) is linear in xS with coefficients −λ ≪ 0 - Interpretation: constant marginal costs of producing xS - Too strong, we will relax it in a moment Lemma 2 (Decentralization) Any feasible S−undistorted allocation x0 can be decentralized with prices λ. That is, there exists a budget assignment m : Θ → R+ such that, for all θ ∈ Θ, xS 0 (θ) solves max xS ∈R |S| + Uθ(xS ) subject to λ · xS ≤ m(θ). 11
- 24. Simplifying Assumption: R(xS ) is linear in xS with coefficients −λ ≪ 0 - Interpretation: constant marginal costs of producing xS - Too strong, we will relax it in a moment Lemma 2 (Decentralization) Any feasible S−undistorted allocation x0 can be decentralized with prices λ. That is, there exists a budget assignment m : Θ → R+ such that, for all θ ∈ Θ, xS 0 (θ) solves max xS ∈R |S| + Uθ(xS ) subject to λ · xS ≤ m(θ). Proof: x0 is S-undistorted if xS 0 solves max xS −λ · xS s.t. Uθ(xS (θ)) = Uθ(xS 0 (θ)), ∀θ ∈ Θ It is a collection of expenditure minimization problems. The claim follows from consumer duality (Mas-Colell et al. 1995, Prop 3.E.1). □ 11
- 25. Main theorem (simplified) Combining Lemmas 1 and 2, we arrive at: Theorem Starting at any feasible allocation x0, the planner’s objective can be (weakly) improved by allowing agents to purchase incentive-separable goods at prices λ subject to type-dependent budgets. • There should be no distortions between IS goods • Agents can trade IS goods freely given prices and budgets • With production, prices of IS goods should be equal to marginal costs • Result silent about non-IS goods 12
- 26. Main theorem (general) Assumption (R has bounded marginals) There exist scalars c̄ > c > 0 such that, for all y ∈ RK , k ∈ {1, ..., K}, ϵ > 0 c̄ ≥ |R(y + ϵek ) − R(y)| ϵ ≥ c. Theorem 1 Starting at any feasible allocation x0, the planner’s objective can be (weakly) improved by allowing agents to purchase incentive-separable goods at undistorted prices subject to type-dependent budgets. • Undistorted prices: prices that decentralize an S-undistorted allocation • R has bounded marginals → the planner will always select an allocation implying strictly positive prices 13
- 27. Main theorem (general) Assumption (R has bounded marginals) There exist scalars c̄ > c > 0 such that, for all y ∈ RK , k ∈ {1, ..., K}, ϵ > 0 c̄ ≥ |R(y + ϵek ) − R(y)| ϵ ≥ c. Theorem 1 Starting at any feasible allocation x0, the planner’s objective can be (weakly) improved by allowing agents to purchase incentive-separable goods at undistorted prices subject to type-dependent budgets. • Undistorted prices: prices that decentralize an S-undistorted allocation • R has bounded marginals → the planner will always select an allocation implying strictly positive prices The results can be easily extended to allow for: • the set of IS goods to depend on type θ and initial allocation x0 • additional constraints faced by the planner: x ∈ F 13
- 28. Applications
- 29. APPLICATION 1: ATKINSON-STIGLITZ Atkinson-Stiglitz model: - θ: privately observed productivity - c ∈ RK−1 + : vector of consumption of goods - produced with const marginal costs λ - L: labor in efficiency units - consumption is weakly separable: U(v(c), L, θ) → consumption goods c are incentive-separable 14
- 30. APPLICATION 1: ATKINSON-STIGLITZ Atkinson-Stiglitz model: - θ: privately observed productivity - c ∈ RK−1 + : vector of consumption of goods - produced with const marginal costs λ - L: labor in efficiency units - consumption is weakly separable: U(v(c), L, θ) → consumption goods c are incentive-separable - Planner’s objective is (α is the marginal value of public funds): W (Ux , x) = V (Ux ) + α(L − K−1 X k=1 λk ck ). 14
- 31. APPLICATION 1: ATKINSON-STIGLITZ Atkinson-Stiglitz model: - θ: privately observed productivity - c ∈ RK−1 + : vector of consumption of goods - produced with const marginal costs λ - L: labor in efficiency units - consumption is weakly separable: U(v(c), L, θ) → consumption goods c are incentive-separable - Planner’s objective is (α is the marginal value of public funds): W (Ux , x) = V (Ux ) + α(L − K−1 X k=1 λk ck ). By Theorem 1, it is welfare-improving to let agents trade goods c freely given prices proportional to marginal costs and type-dependent budgets Theorem 1 implies the Atkinson-Stiglitz theorem: - prices proportional to marginal costs → no distortionary taxes on c - type-dependent budgets → implemented with non-linear income tax 14
- 32. APPLICATION 1: ATKINSON-STIGLITZ Theorem 1 is much more general than the Atkinson-Stiglitz theorem: 1. We allow for more complex settings and incentives - only a subset of commodities may be incentive separable - there may be moral hazard (or other incentive problems) - combine redistributive (Mirrlees ’71) and social-insurance (Varian ’80) strands of optimal income tax literature 2. We rule out any distortionary mechanisms - e.g. quotas, public provision,... 3. What is essential is the ability to implement budgets for S goods, rather than the availability of nonlinear income tax - Theorem 1 applies even if earnings are unobserved (e.g. due to tax evasion) but total expenditure on IS goods is observed 15
- 33. APPLICATION 2: FOOD VOUCHERS Let E ⊆ {1, ..., K} denote consumption of various food items, −E : other commodities and decisions Jensen & Miller (2010): idiosyncratic food tastes more pronounced when food consumption is high. 16
- 34. APPLICATION 2: FOOD VOUCHERS Let E ⊆ {1, ..., K} denote consumption of various food items, −E : other commodities and decisions Jensen & Miller (2010): idiosyncratic food tastes more pronounced when food consumption is high. • Let v(xE ) denote the nutritional value of food (common to all agents) • Utility function U(x, θ) = UL(v(xE ), x−E , θ) if v(xE ) ≤ v (only nutrition matters) UH (x, θ) if v(xE ) > v (nutrition and tastes matter) - v: nutritional threshold above which idiosync. food tastes become active 16
- 35. APPLICATION 2: FOOD VOUCHERS Let E ⊆ {1, ..., K} denote consumption of various food items, −E : other commodities and decisions Jensen & Miller (2010): idiosyncratic food tastes more pronounced when food consumption is high. • Let v(xE ) denote the nutritional value of food (common to all agents) • Utility function U(x, θ) = UL(v(xE ), x−E , θ) if v(xE ) ≤ v (only nutrition matters) UH (x, θ) if v(xE ) > v (nutrition and tastes matter) - v: nutritional threshold above which idiosync. food tastes become active • θ is privately observed → food items are IS for types θ s.t. v(xE (θ)) ≤ v • Food items produced with const marginal cost 16
- 36. APPLICATION 2: FOOD VOUCHERS Conflicting motives for taxes on food: - food choices of the poor are IS → suboptimal to distort - the rich exhibit heterogeneous tastes for food → optimal to distort their food choices (Saez, 2002) 17
- 37. APPLICATION 2: FOOD VOUCHERS Conflicting motives for taxes on food: - food choices of the poor are IS → suboptimal to distort - the rich exhibit heterogeneous tastes for food → optimal to distort their food choices (Saez, 2002) Solution? A food vouchers program: Corollary 1 Consider a feasible allocation x0. The planner’s objective can be improved by assigning budgets to all agents with θ such that v(xE (θ)) ≤ v, and letting them spend these budgets on food items E (but no other goods) priced at marginal cost. - means-tested: allocated only to ”poor” types {θ : v(xE (θ)) ≤ v} - allows to buy food at producer prices: isolates from tax distortions 17
- 38. APPLICATION 2: FOOD VOUCHERS Conflicting motives for taxes on food: - food choices of the poor are IS → suboptimal to distort - the rich exhibit heterogeneous tastes for food → optimal to distort their food choices (Saez, 2002) Solution? A food vouchers program: Corollary 1 Consider a feasible allocation x0. The planner’s objective can be improved by assigning budgets to all agents with θ such that v(xE (θ)) ≤ v, and letting them spend these budgets on food items E (but no other goods) priced at marginal cost. - means-tested: allocated only to ”poor” types {θ : v(xE (θ)) ≤ v} - allows to buy food at producer prices: isolates from tax distortions Compare to the U.S. food stamps program (SNAP) - means-tested, purchases exempt from commodity taxes (state and local) - however, restricted eligibility of unemployed—suboptimal in our setting 17
- 39. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES x: allocation of consumption goods and other households decisions 18
- 40. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES x: allocation of consumption goods and other households decisions Production sector consists of J firms. For a firm j ∈ {1, ..., J}: • zj ∈ RK is the production vector (negative entries represent inputs) • Zj is the set of possible production vectors z = (z1, ..., zJ ) is a production plan, z = PJ j=1 zj is aggregate production vector Z is the Minkowski sum of Zj over j 18
- 41. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES x: allocation of consumption goods and other households decisions Production sector consists of J firms. For a firm j ∈ {1, ..., J}: • zj ∈ RK is the production vector (negative entries represent inputs) • Zj is the set of possible production vectors z = (z1, ..., zJ ) is a production plan, z = PJ j=1 zj is aggregate production vector Z is the Minkowski sum of Zj over j Allocation (x, z) is feasible if x ∈ I, z ∈ Z and z ≥ x The planner maximizes W (Ux , z − x), non-decreasing in z − x 18
- 42. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES x: allocation of consumption goods and other households decisions Production sector consists of J firms. For a firm j ∈ {1, ..., J}: • zj ∈ RK is the production vector (negative entries represent inputs) • Zj is the set of possible production vectors z = (z1, ..., zJ ) is a production plan, z = PJ j=1 zj is aggregate production vector Z is the Minkowski sum of Zj over j Allocation (x, z) is feasible if x ∈ I, z ∈ Z and z ≥ x The planner maximizes W (Ux , z − x), non-decreasing in z − x A feasible production plan z0 is efficient if there does not exist z ∈ Z such that z ≥ z0 and z ̸= z0 18
- 43. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES Corollary 2 For any feasible allocation, the planner’s objective can be (weakly) improved by choosing an S-undistorted allocation of incentive-separable goods and an efficient production plan. Key insight: production decisions are (trivially) incentive-separable. 19
- 44. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES Corollary 2 For any feasible allocation, the planner’s objective can be (weakly) improved by choosing an S-undistorted allocation of incentive-separable goods and an efficient production plan. Key insight: production decisions are (trivially) incentive-separable. To ensure positive decentralizing prices, we need additional assumptions on feasible production plans: Assumption The aggregate production set Z is closed, bounded from above and convex. Furthermore, for any z0 ∈ Z and any nonempty proper subset A ⊂ {1, ..., K}, there exists z ∈ Z such that zA ≤ zA 0 , z−A ≥ z−A 0 and z−A ̸= z−A 0 . 19
- 45. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES Theorem 2 For any feasible allocation, there exists a price vector λ ∈ RK ++ such that the planner’s objective can be (weakly) improved by simultaneously: (i) allowing agents to purchase incentive-separable goods at prices λ subject to type-dependent budgets; (ii) allowing firms to maximize profits and trade all goods taking prices λ as given, and taxing their profits lump-sum. 1. Beneficial to jointly remove distortions in production and IS-consumption 2. Achieved by allowing hhs and firms to trade goods at common prices However, harmonizing consumption taxes w/out addressing production distortions could reduce welfare: • with a production distortions, the producer prices are distorted 20
- 46. CONCLUSIONS • Incentive separability: a useful notion to study optimality and decentralization in complex incentive problems • We focused on applications within Public Finance - Atkinson-Stiglitz theorem - food stamps - Atkinson-Stiglitz meet Diamond-Mirrlees • Potential applications in other contexts - employee compensation / principal-agent problems within organizations - monopoly pricing with multiple goods / attributes - ... 21
- 48. Examples of incentive-separable decisions IS decisions: {(xS , x−S 0 ) : U(xS (θ), x−S 0 (θ), θ) = U(x0(θ), θ), ∀θ ∈ Θ} ⊆ I. 1. Voluntary participation: I = { x : U(x(θ), θ) ≥ U(θ), ∀θ ∈ Θ } ; All decisions are incentive-separable. 2. Private information: I = x : U(x(θ), θ) ≥ max θ′∈Θ U(x(θ′ ), θ), ∀θ ∈ Θ ; Weak separability of decisions S, U(x(θ), θ) = e U (v(xS (θ)), x−S (θ), θ), ∀θ ∈ Θ, implies incentive separability of S. 22
- 49. Examples of incentive-separable decisions 3. Moral hazard: Hidden action a ∈ A affects the distribution of observed state ω: U((a, y), (τ, ω)) = X ω u(y; ω)P(ω|a, τ) − c(a; τ), y ∈ RL +; Set I: Agent with type τ reports τ truthfully, takes recommended action a(τ), and consumes y(τ, ω). With θ = (τ, ω) and x = (a, y), decisions y are incentive-separable. 4. Easy to extend to combinations of incentive constraints, dynamic private information, verifiable information, labels, ... 23