Dynamic Mechanism Design: Robustness and Endogenous Types

Transcription

1 Dynamic Mechanism Design: Robustness and Endogenous Types Alessandro Pavan Northwestern University November 2015

2 Mechanism Design Mechanism Design: auctions, regulation, taxation, political economy, etc...

3 Mechanism Design Mechanism Design: auctions, regulation, taxation, political economy, etc... Standard model: one-time information, one-time decisions

4 Mechanism Design Mechanism Design: auctions, regulation, taxation, political economy, etc... Standard model: one-time information, one-time decisions Many settings - information arrives over time (serially correlated, possibly endogenous) - sequence of decisions

5 Long-Term Contracting Long-term contracting

6 Long-Term Contracting Long-term contracting Trade

7 Long-Term Contracting Long-term contracting Trade Employment

8 Long-Term Contracting Long-term contracting Trade Employment Taxation

9 Long-Term Contracting Long-term contracting Trade Employment Taxation Regulation

10 Long-Term Contracting Long-term contracting Trade Employment Taxation Regulation Financing

11 Long-Term Contracting Long-term contracting Trade Employment Taxation Regulation Financing etc.

12 Long-Term Contracting Value of relationship changes over time

13 Long-Term Contracting Value of relationship changes over time Shocks to:

14 Long-Term Contracting Value of relationship changes over time Shocks to: valuations

15 Long-Term Contracting Value of relationship changes over time Shocks to: valuations costs

16 Long-Term Contracting Value of relationship changes over time Shocks to: valuations costs productivity

17 Long-Term Contracting Value of relationship changes over time Shocks to: valuations costs productivity outside options

18 Long-Term Contracting Value of relationship changes over time Shocks to: valuations costs productivity outside options etc.

19 Long-Term Contracting Value of relationship changes over time Shocks to: valuations costs productivity outside options etc. Changes often anticipated albeit not necessarily commonly observed

20 Questions Structure of optimal LT contract in changing environments

21 Questions Structure of optimal LT contract in changing environments Dynamics of distortions convergence to FB?

22 Dynamic Mechanism Design Applications: - revenue management (Courty and Li, 2000, Battaglini 2005, Boleslavsky and Said, 2013, Ely, Garrett and Hinnosaar, 2014, Board and Skrzypacz, 2015, Akan, Ata, and Dana, 2015,...) - disclosure in auctions (Eso and Szentes, 2007, Bergemann and Wambach (2015), Li and Shi (2015)...) - experimentation (Bergemann and Välimäki, 2010, Pavan, Segal, and Toikka, 2014, Fershtman and Pavan, ) - taxation (Farhi and Werning, 2012, Kapicka, 2013, Stantcheva, 2014, Makris and Pavan,2015,...) - managerial compensation (Garrett and Pavan, 2012, 2014,...) - insurance (Hendel and Lizzeri, 2003, Handel, Hendel, Whinston, 2015,...)

23 Dynamic Mechanism Design Most work: - Myersonian/first-order approach ("global" IC slack) - exogenous types

24 Dynamic Mechanism Design Most work: - Myersonian/first-order approach ("global" IC slack) - exogenous types Robust Predictions (to binding global IC constraints)?

25 Dynamic Mechanism Design Most work: - Myersonian/first-order approach ("global" IC slack) - exogenous types Robust Predictions (to binding global IC constraints)? Novel effects due to endogeneity of types?

26 Plan 1 Robust Predictions

27 Plan 1 Robust Predictions 2 Endogenous Types

28 Plan 1 Robust Predictions 2 Endogenous Types 3 Conclusions

29 Static example Price discrimination (Mussa-Rosen, Maskin & Riley, Myerson)

30 Static example Price discrimination (Mussa-Rosen, Maskin & Riley, Myerson) Principal: seller

31 Static example Price discrimination (Mussa-Rosen, Maskin & Riley, Myerson) Principal: seller Agent: buyer

32 Static example Price discrimination (Mussa-Rosen, Maskin & Riley, Myerson) Principal: seller Agent: buyer Quasilinear payoffs U P = p c(q) and U A = θq p with θ drawn from F (density f ), privately observed by Buyer

33 Static example Price discrimination (Mussa-Rosen, Maskin & Riley, Myerson) Principal: seller Agent: buyer Quasilinear payoffs U P = p c(q) and U A = θq p with θ drawn from F (density f ), privately observed by Buyer Incentive compatibility: V A { (θ) θq(θ) p(θ) = sup θq( ˆθ) p( ˆθ) } ˆθ

34 Static example Price discrimination (Mussa-Rosen, Maskin & Riley, Myerson) Principal: seller Agent: buyer Quasilinear payoffs U P = p c(q) and U A = θq p with θ drawn from F (density f ), privately observed by Buyer Incentive compatibility: V A (θ) θq(θ) p(θ) = sup ˆθ { θq( ˆθ) p( ˆθ) } Envelope Th. θ V A (θ) = V A (θ) + q(s)ds with q( ) nondecreasing θ

35 Static example Transfer (revenue equivalence) p(θ) = θq(θ) { θ } V A (θ) + q(s)ds θ

36 Static example Transfer (revenue equivalence) p(θ) = θq(θ) { θ } V A (θ) + q(s)ds θ Optimal quantity schedule maximizes expected "virtual surplus" [( ) ] E θ 1 F(θ) q(θ) c(q(θ)) s.t. q( ) nondecreasing (M) f (θ)

37 Static example Transfer (revenue equivalence) p(θ) = θq(θ) { θ } V A (θ) + q(s)ds θ Optimal quantity schedule maximizes expected "virtual surplus" [( ) ] E θ 1 F(θ) q(θ) c(q(θ)) s.t. q( ) nondecreasing (M) f (θ) Robust predictions (e.g., Hellwig, 2010): 1. participation constraint binds only for lowest type: V A (θ) = 0 2. no distortion at the top: q( θ) = q FB ( θ) 3. downward distortions elsewhere: q(θ) < q FB (θ)

38 Static example Transfer (revenue equivalence) p(θ) = θq(θ) { θ } V A (θ) + q(s)ds θ Optimal quantity schedule maximizes expected "virtual surplus" [( ) ] E θ 1 F(θ) q(θ) c(q(θ)) s.t. q( ) nondecreasing (M) f (θ) Robust predictions (e.g., Hellwig, 2010): 1. participation constraint binds only for lowest type: V A (θ) = 0 2. no distortion at the top: q( θ) = q FB ( θ) 3. downward distortions elsewhere: q(θ) < q FB (θ) Binding (M): ironing (just more pooling)

39 Dynamic Environment t = 1,...,T (possibly infinite) Intertemporal payoffs U P = t δ t 1 (p t c(q t )) and U A = t δ t 1 (θ t q t p t ) θ t privately observed by agent at beginning of period t

40 Type process type θ t drawn from (exogenous) Markov chain on Θ = [θ,θ] R + transition probability kernels F (F t ) F t ( θ): cdf of θ t, given θ t 1 = θ F 1 : cdf of initial distribution; density f 1 stochastic monotonicity (FOSD): θ > θ F t ( θ ) FOSD F t ( θ) ergodicity:! invariant distribution π s.t., for all θ Θ sup F n (A,θ) π(a) 0 as n A B(Θ) stationarity: F 1 = π and F t = F s all t,s > 1.

41 Principal s problem Principal designs χ = q,p to maximize [ ] E δ t 1 (p t (θ t ) c(q t (θ t ))) t subject to IR-1 and IC-t, all t 1 Stronger (periodic) IR Complexity: - different types have different beliefs about future - multi-period deviations IC-IR-extended

42 State representation and impulse responses Eso-Szentes (2007), Pavan, Segal, Toikka (2014) Auxiliary shocks, orthogonal to initial private information θ t = Z t (θ 1,ε) where ε (ε t ) are iid r.v.s Integral-transform-probability theorem (F 1 t inductively) Impulse responses: I t (θ 1,ε) = θ 1 θ t = Z t(θ 1,ε) θ 1

43 Examples AR(1): θ t = γθ t 1 + ε t = Z t (θ 1,ε) = γ t 1 θ 1 + γ t 2 ε γε t 1 + ε t I t (θ 1,ε) = γ t 1

44 Examples AR(1): θ t = γθ t 1 + ε t ARIMA: = Z t (θ 1,ε) = γ t 1 θ 1 + γ t 2 ε γε t 1 + ε t I t (θ 1,ε) = γ t 1 θ t = Z t (θ 1,ε) = a t,1 θ 1 + a t,2 ε a t,t 1 ε t 1 + ε t I t (θ 1,ε) = a t,1

45 Examples AR(1): θ t = γθ t 1 + ε t ARIMA: = Z t (θ 1,ε) = γ t 1 θ 1 + γ t 2 ε γε t 1 + ε t I t (θ 1,ε) = γ t 1 θ t = Z t (θ 1,ε) = a t,1 θ 1 + a t,2 ε a t,t 1 ε t 1 + ε t I t (θ 1,ε) = a t,1 Multiplicative shocks θ t = Z t (θ 1,ε) = θ 1 ε 2 ε t I t (θ 1,ε) = ε 2 ε t

46 Examples AR(1): θ t = γθ t 1 + ε t ARIMA: = Z t (θ 1,ε) = γ t 1 θ 1 + γ t 2 ε γε t 1 + ε t I t (θ 1,ε) = γ t 1 θ t = Z t (θ 1,ε) = a t,1 θ 1 + a t,2 ε a t,t 1 ε t 1 + ε t I t (θ 1,ε) = a t,1 Multiplicative shocks θ t = Z t (θ 1,ε) = θ 1 ε 2 ε t I t (θ 1,ε) = ε 2 ε t Continuous-time (Bergemann and Strack, 2015)

47 Examples AR(1): θ t = γθ t 1 + ε t ARIMA: = Z t (θ 1,ε) = γ t 1 θ 1 + γ t 2 ε γε t 1 + ε t I t (θ 1,ε) = γ t 1 θ t = Z t (θ 1,ε) = a t,1 θ 1 + a t,2 ε a t,t 1 ε t 1 + ε t I t (θ 1,ε) = a t,1 Multiplicative shocks θ t = Z t (θ 1,ε) = θ 1 ε 2 ε t I t (θ 1,ε) = ε 2 ε t Continuous-time (Bergemann and Strack, 2015) More generally, I t = s t θ F 1 s (ε s θ s 1 )

48 Local IC heuristics Assume T = 2 Fix period-1 report, ˆθ 1, and priod-2 reporting strategy, σ(ε) Agent s payoff U A (θ 1, ˆθ 1 ;σ) = θ 1 q 1 ( ˆθ 1 ) p 1 ( ˆθ 1 )+δe [ Z 2 (θ 1,ε)q 2 ( ˆθ 1,σ(ε)) p 2 ( ˆθ 1,σ(ε)) ] If χ = q,p is IC, then V1 A (θ 1) = sup U A (θ 1, ˆθ 1 ;σ) ˆθ 1 ;σ Envelope theorem V1 A [ ] = U A (θ 1,θ 1 ;σ truth Z2 (θ 1,ε) ) = q 1 (θ 1 ) + δe q 2 (θ 1,ε)) θ 1 θ 1 θ 1 [ ] = E δ s 1 I s q s θ 1 s 1

49 Local IC general case More generally, Theorem (Pavan, Segal, Toikka, 2014) If χ = q,p is IC, then, for every truthful history θ t 1, t 0, Vt A is equi-lipschitz-continuous in θ t and [ ] Vt A = E θ t δ s 1 I t s q s θ t a.e., s t (ICFOC) where I t s = d dθ t θ s (with I t I 1 t ) ICFOC-proof

50 Suffi ciency and Integral Monotonicity Envelope conditions necessary but not suffi cient

51 Suffi ciency and Integral Monotonicity Envelope conditions necessary but not suffi cient Appropriate monotonicity conditions

52 Suffi ciency and Integral Monotonicity Envelope conditions necessary but not suffi cient Appropriate monotonicity conditions Theorem (PST, 2014) Mechanism χ = q,p is IC iff, for all t 0, [ ] Vt A = E θ t δ s 1 I t s q s θ t a.e., s t (ICFOC) and, for all θ t and ˆθ t, θ t ˆθ t [D t ((θ t 1,x);x) D((θ t 1,x); ˆθ t )]dx 0 (INT-M) where [ ] D t (θ t ; ˆθ t ) E δ s 1 I t s q s (θ s t, ˆθ t ) θ t s t

53 Suffi ciency and Integral Monotonicity Envelope conditions necessary but not suffi cient Appropriate monotonicity conditions Theorem (PST, 2014) Mechanism χ = q,p is IC iff, for all t 0, [ ] Vt A = E θ t δ s 1 I t s q s θ t a.e., s t (ICFOC) and, for all θ t and ˆθ t, θ t ˆθ t [D t ((θ t 1,x);x) D((θ t 1,x); ˆθ t )]dx 0 (INT-M) where [ ] D t (θ t ; ˆθ t ) E δ s 1 I t s q s (θ s t, ˆθ t ) θ t s t Int-M one-stage deviations suboptimal

54 Suffi ciency and Integral Monotonicity Envelope conditions necessary but not suffi cient Appropriate monotonicity conditions Theorem (PST, 2014) Mechanism χ = q,p is IC iff, for all t 0, [ ] Vt A = E θ t δ s 1 I t s q s θ t a.e., s t (ICFOC) and, for all θ t and ˆθ t, θ t ˆθ t [D t ((θ t 1,x);x) D((θ t 1,x); ˆθ t )]dx 0 (INT-M) where [ ] D t (θ t ; ˆθ t ) E δ s 1 I t s q s (θ s t, ˆθ t ) θ t s t Int-M one-stage deviations suboptimal Int-M + Markov + OSDP all deviations suboptimal Int-M-Proof

55 Stronger suffi cient conditions Int-M holds if expected future output, discounted by impulse responses [ ] D t (θ t ; ˆθ t ) = E δ s 1 I t s q s (θ s t, ˆθ t ) θ t s t is nondecreasing in current report ˆθ t. Output need not be monotone history by history, enough to have monotonicity on average over time and states. Literature typically checks strong monotonicity (i.e., q t (θ t ) nondecreasing in θ t ), but that s stronger than necessary.

56 Full program Principal s full program [ ] max E t δ t 1 (p t c(q t )) χ= q,p subject to IR: V A 1 (θ 1) 0 all θ 1 ICFOC-(t): V A t (θ t ) θ t = D t (θ t ;θ t ) Int-M: θ t ˆθ t [D t ((θ t 1,x);x) D t ((θ t 1,x); ˆθ t )]dx 0.

57 Relax program Myersonian/First-Order Approach Principal s relaxed program [ ] max E t δ t 1 (p t c(q t )) χ= q,p subject to IR: V A 1 (θ 1) 0 all θ 1 V A 1 (θ) 0 ICFOC-(t): V A t (θ t ) θ t = D t (θ t ;θ t ) Int-M: θ t ˆθ t [D t ((θ t 1,x);x) D t ((θ t 1,x); ˆθ t )]dx 0.

58 Relax program Myersonian/First-Order Approach Principal s objective as "Dynamic Virtual Surplus" [ ( ] max E t δ t 1 θ t 1 F 1(θ 1 ) I q f 1 (θ 1 ) t )q t c(q t ))

59 Relax program Myersonian/First-Order Approach Principal s objective as "Dynamic Virtual Surplus" [ ( ] max E t δ t 1 θ t 1 F 1(θ 1 ) I q f 1 (θ 1 ) t )q t c(q t )) Pointwise maximization: period-t virtual value = θ t 1 F 1(θ 1 ) I f 1 (θ 1 ) t = c (q t ) = marginal cost distortions driven by impulse responses I t

60 Validity of First-Order-Approach Remaining IR constraints slack under FOSD and q 0

61 Validity of First-Order-Approach Remaining IR constraints slack under FOSD and q 0 Remaining IC constraints (equivalently, Int-M) slack if [ ] E δ t I t s q s (θ t s, ˆθ t ) θ t nondecreasing in ˆθ t all t s t

62 Validity of First-Order-Approach Remaining IR constraints slack under FOSD and q 0 Remaining IC constraints (equivalently, Int-M) slack if [ ] E δ t I t s q s (θ t s, ˆθ t ) θ t nondecreasing in ˆθ t all t s t Suppose c(q) = 1 2 q2. Solution to relaxed program Monotone enough? q t = θ t 1 F 1(θ 1 ) I f 1 (θ 1 ) t

63 Validity of First-Order-Approach Remaining IR constraints slack under FOSD and q 0 Remaining IC constraints (equivalently, Int-M) slack if [ ] E δ t I t s q s (θ t s, ˆθ t ) θ t nondecreasing in ˆθ t all t s t Suppose c(q) = 1 2 q2. Solution to relaxed program Monotone enough? q t = θ t 1 F 1(θ 1 ) I f 1 (θ 1 ) t Example (AR-1) q t = θ t 1 F 1(θ 1 ) φ f 1 (θ 1 ) t 1 suffi ces that F 1 log-concave

64 Robust predictions in Dynamic Screening Garrett-Pavan-Toikka (2015) Predictions that do not hinge on FOA

65 Robust predictions in Dynamic Screening Garrett-Pavan-Toikka (2015) Predictions that do not hinge on FOA Full program: hard to solve

66 Robust predictions in Dynamic Screening Garrett-Pavan-Toikka (2015) Predictions that do not hinge on FOA Full program: hard to solve Idea: Let q be optimal allocation process. Any perturbation preserving (Int-M) and IR constraints must be suboptimal

67 Robust predictions in Dynamic Screening Garrett-Pavan-Toikka (2015) Predictions that do not hinge on FOA Full program: hard to solve Idea: Let q be optimal allocation process. Any perturbation preserving (Int-M) and IR constraints must be suboptimal Variational approach robust predictions for average distortions Existence

68 Robust predictions Assume IR binds only at θ 1 = θ (always under FOSD and q 0) and interior solutions.

69 Robust predictions Assume IR binds only at θ 1 = θ (always under FOSD and q 0) and interior solutions. Simple perturbation: add constant a R to period-t allocation (equivalently, Gateux derivative in direction (0,...,0,1,0,...))

70 Robust predictions Assume IR binds only at θ 1 = θ (always under FOSD and q 0) and interior solutions. Simple perturbation: add constant a R to period-t allocation (equivalently, Gateux derivative in direction (0,...,0,1,0,...)) FOC for optimum at a = 0: [ ] E θ t 1 F 1(θ 1 ) I f 1 (θ 1 ) t = E [ c (q t ) ] average virtual value equals average marginal cost

71 Robust predictions Assume IR binds only at θ 1 = θ (always under FOSD and q 0) and interior solutions. Simple perturbation: add constant a R to period-t allocation (equivalently, Gateux derivative in direction (0,...,0,1,0,...)) FOC for optimum at a = 0: [ ] E θ t 1 F 1(θ 1 ) I f 1 (θ 1 ) t = E [ c (q t ) ] average virtual value equals average marginal cost Same prediction as under FOA, but only in expectation! E[period-t distortion] E[θ t c (q t )] [ ] 1 F1 (θ = E 1 ) I f 1 (θ 1 ) t

72 Handicap Dynamics Theorem (Garrett, Pavan, Toikka, 2015) Assume F is ergodic. Then [ ] 1 F1 (θ 1 ) E I t 0. f 1 (θ 1 ) Moreover, if F satisfies FOSD, then convergence is from above. If, in addition, F is stationary, then convergence is monotone in t. Handicap-proof

73 More general bounds When IR binds only at bottom and q interior E[distortion] = E[handicap] = E [ ] 1 F1 (θ 1 ) I f 1 (θ 1 ) t

74 More general bounds When IR binds only at bottom and q interior More generally, E[distortion] = E[handicap] = E [ ] 1 F1 (θ 1 ) I f 1 (θ 1 ) t

75 More general bounds When IR binds only at bottom and q interior More generally, E[distortion] = E[handicap] = E Theorem (Garrett-Pavan-Toikka, 2015) If F is ergodic, then limsupe[θ t c (q t )] 0 t If, in addition, q eventually strictly interior, then lim E[θ t c (q t )] = 0 t Finally, if distortions are eventually downward, then [ ] 1 F1 (θ 1 ) I f 1 (θ 1 ) t (limit upward distortions) q t p q FB t

76 More general bounds When IR binds only at bottom and q interior More generally, E[distortion] = E[handicap] = E Theorem (Garrett-Pavan-Toikka, 2015) If F is ergodic, then limsupe[θ t c (q t )] 0 t If, in addition, q eventually strictly interior, then lim E[θ t c (q t )] = 0 t Finally, if distortions are eventually downward, then [ ] 1 F1 (θ 1 ) I f 1 (θ 1 ) t (limit upward distortions) q t p q FB t Corollary Failure to converge over-consumption and exclusion eventually infinitely often.

77 Dynamic Managerial Compensation: A Variational Approach Garrett and Pavan, JET 2015 Suppose now with U P T δ t 1 [y t c t ] t=1 U A T [ ] δ t 1 v A (c t ) ψ(y t,θ t ) t=1 ψ(y t,θ t ) = 1 2 (y t θ t ) 2 and θ t = ρθ t 1 + ε t Non quasi-linear environment Theorem (Garrett and Pavan, JET 2015) When the agent s risk aversion and productivity persistence are low, distortions decrease, on average, over time. Opposite is true when they are suffi ciently high. distortions increasing over time reduce compensation risk

78 Managerial Turnover in a Changing World Garrett and Pavan (JPE, 2013) Under risk neutrality and declining impulse responses Effort gradually converges to FB Suppose firm can replace its managers and that each new employment relationship is affected by same information frictions as with incumbent Theorem (Garrett and Pavan, JPE 2013) Firm s optimal contract (i) either induces excessive retention (ineffi ciently low turnover) at all tenure levels (ii) or excessive firing early on followed by excessive retention in the long run.

79 Plan 1 Robust Predictions 2 Endogenous Types 3 Conclusions

80 Motivation/Perspective Most of DMD literature: exogenous types/information

81 Motivation/Perspective Most of DMD literature: exogenous types/information Many problems of interest: information/types endogenous - bandit auctions for sale of experience goods Pavan, Segal, Toikka, dynamic matching w. unknown values Fershtman and Pavan, habit formation/addiction - taxation under learning-by-doing (Makris and Pavan, 2015)

82 Incentives for Endogenous Types: Makris and Pavan (2015) Design of incentive schemes in Markovian setting with - endogenous types - non-quasilinear payoffs - (imperfect) type persistence

83 Incentives for Endogenous Types: Makris and Pavan (2015) Design of incentive schemes in Markovian setting with - endogenous types - non-quasilinear payoffs - (imperfect) type persistence Flexible framework embedding - taxation under LBD - managerial compensation -...

84 Incentives for Endogenous Types: Makris and Pavan (2015) Design of incentive schemes in Markovian setting with - endogenous types - non-quasilinear payoffs - (imperfect) type persistence Flexible framework embedding - taxation under LBD - managerial compensation -... Theory + quantitative analysis

85 Questions Effect of endogeneity of types on distortions and their dynamics?

86 Questions Effect of endogeneity of types on distortions and their dynamics? (Mirrleesian) taxation: - Effect of LBD on progressivity and dynamics of marginal taxes - Welfare gains of tax reforms - Loss from neglecting LBD

87 Related literature (incomplete) Dynamic Mechanism Design... New Dynamic Public Finance Farhi and Werning (2012) Kapicka (Restud 2013) Golosov, Troshkin, Tsyvinski (2014)... Taxation w. Human Capital Accumulation / LBD Kapicka (2014) Kapicka and Neira (2014) Stantcheva (2014) Krause (2009) Best and Kleven (2013) Heathcote, Storesletten and Violante (2014)...

88 Design problem Agent s productivity with z t increasing θ t = z t (θ t 1,y t 1,ε t )

89 Design problem Agent s productivity with z t increasing θ t = z t (θ t 1,y t 1,ε t ) Principal s (dual) problem: choose χ = y, c to maximize T ] E[ δ t (y t c t ) t=1 s.t. χ be IC and θ 1 θ 1 ( ) q V1 A (θ 1) df 1 (θ 1 ) κ where q ( V A 1 (θ 1) ) are non-linear Pareto weights

90 Design problem Agent s productivity with z t increasing θ t = z t (θ t 1,y t 1,ε t ) Principal s (dual) problem: choose χ = y, c to maximize T ] E[ δ t (y t c t ) t=1 s.t. χ be IC and θ 1 θ 1 ( ) q V1 A (θ 1) df 1 (θ 1 ) κ where q ( V A 1 (θ 1) ) are non-linear Pareto weights Rawlsian: q ( V A 1 (θ 1) ) = 0 all θ 1 > θ

91 Incentive Compatibility Analysis similar to exogenous case BUT I t s (θ s,y s 1 ) now depend on past income

92 Wedges Period-t labor wedge (equivalently, marginal tax rate) [ 1 + LDt ][1 FB;χ W t ] = ψ y (y t,θ t ) v A (c t ) where LDt FB;χ (θ t ) are FB effects of learning-by-doing

93 Wedges Period-t labor wedge (equivalently, marginal tax rate) [ 1 + LDt ][1 FB;χ W t ] = ψ y (y t,θ t ) v A (c t ) where LDt FB;χ (θ t ) are FB effects of learning-by-doing Interested in how LBD affects progressivity and dynamics of W t

94 Wedges Period-t labor wedge (equivalently, marginal tax rate) [ 1 + LDt ][1 FB;χ W t ] = ψ y (y t,θ t ) v A (c t ) where LDt FB;χ (θ t ) are FB effects of learning-by-doing Interested in how LBD affects progressivity and dynamics of W t We study Ŵ t (θ t ) W t(θ t ) 1 W t (θ t )

95 Wedges: Two-period Rawlsian Period-2 wedge where Ŵ 2 (θ) = RA 2 (θ) [ Ŵ RN 2 (θ) ] - Ŵ RN 2 (θ) is wedge under risk neutrality and no LBD - RA 2 (θ) > 0 is correction due to risk aversion

96 Wedges: Two-period Rawlsian Period-1 wedge under LBD Ŵ 1 (θ 1 ) = RA 1 (θ 1 ) [ Ŵ RN 1 (θ 1 ) + Ω 1 (θ 1 ) ] where Ω 1 (θ 1 ) is effect of LBD on period-1 wedge

97 Wedges: Two-period Rawlsian Period-1 wedge under LBD Ŵ 1 (θ 1 ) = RA 1 (θ 1 ) [ Ŵ RN 1 (θ 1 ) + Ω 1 (θ 1 ) ] where Ω 1 (θ 1 ) is effect of LBD on period-1 wedge Ω 1 (θ 1 ) takes into account how variation in y 1 affects cost of future incentives through - distribution of future productivity F 2 (θ 2 θ 1,y 1 ) - impulse responses (handicaps): I 2 (θ,y 1 )

98 Wedges: General case Theorem (Makris and Pavan, 2015) At any history θ t, optimal wedge given by Ŵ t (θ t ) = RA t (θ t ) [ Ŵ RN t (θ t ) + Ω t (θ t ) ]

99 Literature: special cases Static RN (Mirrlees-Diamond-Saez) Ŵ 1 (θ 1 ) = Ŵ1 RN (θ 1 ) = 1 F 1(θ 1 ) 1 ε ψ y f 1 (θ 1 ) θ θ (y 1(θ 1 ),θ 1 ) 1 where ε ψ y θ (y 1,θ 1 ) is elasticity of marginal disutility of effort.

100 Literature: special cases Static RA (Mirrlees-Saez): Ŵ 1 (θ 1 ) = RA 1 (θ 1 )Ŵ RN 1 (θ 1 ) = [ θ 1 1 θ 1 v (c 1 (s)) ] df 1 (s) 1 F 1 (θ 1 ) v (c 1 (θ 1 )) 1 F 1 (θ 1 ) f 1 (θ 1 ) 1 ε ψ y θ θ (y 1(θ 1 ),θ 1 ) 1

101 Literature: special cases Dynamic RN (exogenous types): Ŵ t (θ t ) = Ŵt RN (θ t ) = 1 F 1(θ 1 ) f 1 (θ 1 ) I t (θ t ) ε ψ y θ θ (y t(θ t ),θ t ) t

102 Literature: special cases Dynamic RA (exogenous types; Farhi and Werning, 2013): ] Ŵ t (θ t ) = RA t (θ t RN ) [Ŵ t (θ t )

103 Literature: special cases Dynamic RA (exogenous types with training; Stantcheva, 2014): where ] Ŵ t (θ t ) = RA t (θ t RN ) [Ŵ t (θ t,b t (θ t 1 )) Ŵt RN (θ t,b t (θ t 1 )) = 1 F 1(θ 1 ) f 1 (θ 1 ) I t (θ t ) ε ψ y θ θ (y t(θ t ),θ t,b t (θ t 1 )) t

104 General case More generally: Ŵ t (θ t ) = RA t (θ t ) [ Ŵ RN t (θ t ) + Ω t (θ t ) ] Key: endogeneity of type process

105 Example of effects of endogenous types on progressivity/dynamics ψ(y,θ) = 1 1+φ ( y θ )1+φ where 1/φ > 0 is Frisch elasticity

106 Example of effects of endogenous types on progressivity/dynamics ψ(y,θ) = 1 1+φ ( y θ )1+φ where 1/φ > 0 is Frisch elasticity Pareto F 1 : f 1 (θ 1 ) 1 F 1 (θ 1 ) θ 1 = λ

107 Example of effects of endogenous types on progressivity/dynamics ψ(y,θ) = 1 1+φ ( y θ )1+φ where 1/φ > 0 is Frisch elasticity Pareto F 1 : f 1 (θ 1 ) 1 F 1 (θ 1 ) θ 1 = λ Ben-Porath specification: θ 2 = z 2 (θ 1,y 1,ε 1 ) = θ ρ 1 yζ 1 ε 2 where - ρ controls for exogenous type persistence - ζ controls for intensity of LBD

108 Example (cont d) Consider economy described above, and assume agent is risk-neutral and principal Rawlsian

109 Example (cont d) Consider economy described above, and assume agent is risk-neutral and principal Rawlsian Without LBD Ŵ t = (1 + φ)/λ all t

110 Example (cont d) Consider economy described above, and assume agent is risk-neutral and principal Rawlsian Without LBD Ŵ t = (1 + φ)/λ all t With LBD Ŵ 1 (θ 1 ) > Ŵ 2 (θ) all θ with Ŵ 1 (θ 1 ) strictly increasing in θ 1 (progressivity)

111 Kapicka Restud 2013 (Pareto)

112 Diamond, 1998, AER (Pareto-Lognormal)

113 Intuition for progressivity and wedge dynamics Positive effect of LBD on W t - reduction in y t reduces future cost of incentives through (a) impulse responses (I t s increasing in y t under Ben-Porath specification) (b) distribution of future types (handicaps increasing in future types) - importance of future cost of incentives declines with t W t decline with t Progressivity - handicaps increasing in types reduction in cost of incentives most pronounced for higher types

114 Conclusions Dynamic Mechanism Design useful tool for positive and normative analysis of environments in which - information arrives over time - sequence of decisions

115 Conclusions Dynamic Mechanism Design useful tool for positive and normative analysis of environments in which - information arrives over time - sequence of decisions Myersonian and Variational Approach - robust predictions

116 Conclusions Dynamic Mechanism Design useful tool for positive and normative analysis of environments in which - information arrives over time - sequence of decisions Myersonian and Variational Approach - robust predictions Endogeneity of private information important qualitative and quantitative implications - taxation - dynamic matching (Fershtman and Pavan, 2015)

117 Conclusions Dynamic Mechanism Design useful tool for positive and normative analysis of environments in which - information arrives over time - sequence of decisions Myersonian and Variational Approach - robust predictions Endogeneity of private information important qualitative and quantitative implications - taxation - dynamic matching (Fershtman and Pavan, 2015) (Much) more remains to be done - partial commitment - interaction of time-varying information w. population dynamics (e.g., Garrett 2013) - endogenous disclosure (Calzolari and Pavan (2006a,b), Kamenica and Gentzkow, 2011,...) - empirics (Handel, Hendel, Whinston, 2014, Einav et al., )

118 Thank You!

119 Mechanisms and Principal s problem direct mechanism χ = q,p, with q t : Θ t Q and p t : Θ t R principal designs χ to maximize [ ] E δ t 1 (p t c(q t )) t subject to ] E [ δ t 1 (θ t q t p t ) θ 1 0 for all θ 1 Θ (IR-1) t [ ] E δ s 1 (θ s q s p s ) θ t s t [ ] E δ s 1 (θ s q σ s p σ s ) θ t (IC-t) s t for all σ, all θ t = (θ 1,...,θ t ) Θ t IC-IR-simple

120 ICFOC: Proof Sketch Agent s payoff in terms of state representation: E ] ] [ δ t 1 (θ t q t p t ) θ 1 = Ẽ [ δ t 1 ( q t (θ 1,ε t )Z t (θ 1,ε t ) p t (θ 1,ε t )) θ 1 t t Thus, where V 1 (θ) = maxu( ˆθ;θ) ˆθ [ U( ˆθ;θ) Ẽ δ t 1 ( q t ( ˆθ,ε t )Z t (θ 1,ε t ) p t ( ˆθ,ε t ) ) ] θ t For fixed ˆθ, Envelope theorem then gives result [ ] d dθ U( ˆθ;θ) = Ẽ δ t 1 q t ( ˆθ,ε t )I t θ t Corollary: q pins down V 1 up to constant even if ε publicly observable Eso-Szentes irrelevance result ICFOC

121 Integral Monotonicity: Proof sketch Int-M Fix t and θ t 1. Let U( ˆθ;θ) = continuation utility of period-t type θ from one-stage deviation to ˆθ. Markov and full support IC equivalent to Equivalently, V (θ) U(θ;θ) = maxu( ˆθ;θ) all θ Θ. ˆθ { ˆθ argmax U( ˆθ;θ) V (θ) } for all ˆθ Θ. θ ICFOC implies that, for ˆθ fixed, g(θ) = U( ˆθ,θ) V (θ) is Lipschitz with g (θ) = U 2 ( ˆθ,θ) V (θ) = U 2 ( ˆθ,θ) U 2 (θ,θ) a.e., so ˆθ g( ˆθ) g(θ) = [U 2 ( ˆθ,x) U 2 (x,x)]dx, θ Because U 2 ( ˆθ,x) = D t ((θ t 1,x); ˆθ), ˆθ maximizes g(θ) iff (Int-M).

122 Existence [ ( ( ) )] Let g(q) = E t δ t 1 q t θ t 1 F 1(θ 1 ) I f 1 (θ 1 ) t c(q t ) and consider sup g(q) q L 2 s.t. (Int-M) where L 2 = L 2 (R T ) is space [ of square ] integrable processes with discounted measure, q L 2 iff q = E t δ t 1 qt 2 <. Assume c(q) q 2 for q > q, for some q Then g(q) as q. Moreover, g is concave and Gateux differentiable, and feasible set is closed, convex, and nonempty since defined by bounded linear operators. So supremum is achieved, because in a Hilbert space, every concave Gateux-differentiable functional that is minus infinite at infinity achieves its maximum on a closed convex set. robust

123 Handicap Dynamics Proof sketch Recall that E[I t θ 1 ] = Thus, d dθ 1 E[θ t θ 1 ]. [ ] [ ] θ 1 F1 (θ E 1 ) 1 F1 (θ I f 1 (θ 1 ) t = E 1 ) E[I f 1 (θ 1 ) t θ 1 ] = (1 F 1 (θ 1 ))E[I t θ 1 ]dθ 1 θ by ergodicity. If F monotone (FOSD), = (1 F 1 (θ 1 ))E[θ t θ 1 ] θ 1= θ θ 1 =θ + θ = E[θ t ] E[θ t θ] 0 E[θ t ] E[θ t θ] 0 θ f 1 (θ 1 )E[θ t θ 1 ]dθ 1 If, in addition, F 1 = π, then [ ] [ ] 1 F1 (θ E 1 ) 1 F1 (θ I f 1 (θ 1 ) t E 1 ) I f 1 (θ 1 ) s = E[θ s θ] E[θ t θ] 0 for t > s. handicap-dynamics