Optimal Taxation of Entrepreneurial Income

Transcription

1 Optimal Taxation of Entrepreneurial Income Ali Shourideh University of Minnesota February 14, 2010

2 Introduction Literature on optimal taxation: focus on labor income risk. Main lesson: Capital income should be taxed to get the incentives right. Question: What about idiosyncratic investment risk? Motivation: A significant fraction of wealth is held in risky capital. Most models of labor income risk fail to match the wealth distribution.

3 Introduction Main idea: Each individual has a project with a privately known future return. Planner has to balance equality vs efficiency. Project returns are subject to idiosyncratic risk over time.

4 New Results Main contribution: Using a first order approach to characterize allocations. Modified Inverse Euler Equation In general, capital taxes tend to be lower at the top i.i.d. case: Negative capital taxes at the top and the bottom.

5 A Two Period Example

6 Environment A continuum of agents, live for two periods, t = 0,1. At t = 0 agents types are realized: θ [θ, θ], θ F(θ). Each agent has access to a private investment technology: Invest k 1 at t = 0. Output y 1 = θ α k α 1 at t = 1. θ: Entrepreneurial Ability - Lucas span of control.

7 Environment Agents: consume at t = 0, 1, invest at t = 0. All agents are endowed with e 0 at t = 0. Preferences: u(c 0 ) + βu(c 1 )

8 Environment Allocation: (c 0 (θ),c 1 (θ),k 1 (θ)) Feasibility: θ θ [c 0 (θ) + k 1 (θ)] df(θ) e 0 θ θ c 1 (θ)df(θ) θ θ θ α k 1 (θ) α df(θ)

9 Full Info Efficiency Productive Efficiency: Equalize return qαθ α k 1 (θ) α = 1 q: relative intertemporal price of consumption Ex-ante efficiency: c t (θ ) = c t (θ) t = 0,1. t = 0: Higher transfer to higher types t = 1: Higher transfer to lower types Euler Equation: u (c 0 ) = β q u (c 1 ) = βαθ α k 1 (θ) α 1 u (c 2 (θ))

10 Introducing Private Information θ, consumption and investment are private. Income at t = 1 is observable. Full Info Efficient allocation is not incentive compatible.

11 Incentive Compatibility Revelation Principle: Focus on direct mechanisms Incentive Compatibility: u(c 0 (ˆθ) + k 1 (ˆθ) ˆθ θ k 1(ˆθ)) + βu(c 1 (ˆθ)) }{{} u(θ,ˆθ) u(c 0 (θ)) + βu(c 1 (θ)) (IC)

12 Local Incentive Compatibility Local IC: or ˆθ u(θ, ˆθ) ˆθ=θ = 0 [ u (c 0 (θ)) c 0(θ) 1 ] θ k 1(θ) + βu (c 1 (θ))c 1(θ) = 0 (LIC)

13 Local vs Global Incentive Constraints Theorem If c 1 (θ) and θk 1 (θ) are increasing in θ, (LIC) implies (IC). Second Order Condition: Some linear combination of c 1 (θ) and θk 1 (θ) has to be increasing. (SOC) + (LIC) (IC) Focus only on (LIC).

14 Planning Problem subject to θ max [u(c 0 (θ)) + βu(c 1 (θ))] df(θ) c 0 (θ),c 1 (θ),k 1 (θ),c θ θ θ θ θ [c 0 (θ) + k 1 (θ)]df(θ) e 0 c 1 (θ)df(θ) θ θ θ α k 1 (θ) α df(θ) u(c 0 (θ)) + βu(c 1 (θ)) = u(c 0 (θ)) + βu(c 1 (θ)) + θ θ 1 ˆθ k 1(ˆθ)u (c 1 (ˆθ))dˆθ

15 Modified Inverse Euler Equation Proposition Any constrained efficient allocation satisfies: q βu (c 1 (θ)) = 1 u (c 0 (θ)) + u (c 0 (θ)) u (c 0 (θ)) 2k 1(θ) [ qαθ α k 1 (θ) α 1 1 ] }{{} v 0 (θ) where q is the relative intertemporal price of consumption. v 0 (θ): measure of distortions in investment.

16 Modified Inverse Euler Equation Intuition: Perturbation: for a type θ, ǫ 0,ǫ 1,ǫ k such that: No change in utility: u(c 0 (θ) + ǫ 0 ) + βu(c 1 (θ) + ǫ 1 ) = u(c 0 (θ)) + βu(c 1 (θ)) Incentive Compatibility: (k 1 (θ) + ǫ k )u (c 0 (θ) + ǫ 0 ) = k 1 (θ)u (c 0 (θ)) Optimality cost of perturbation=0

17 Modified Inverse Euler Equation Cost of pert.=0 MIEE: qαθ α k 1 (θ) α 1 ǫ k ǫ k qǫ 1 ǫ 0 = 0 [ qαθ α k 1 (θ) α 1 1 ] k 1 (θ) u (c 0 (θ)) u (c 0 (θ)) u (c 0 (θ)) = q βu (c 1 (θ))

18 Distortions to Productive Efficiency No distortions at the top and bottom: qα θ α k 1 ( θ) α 1 = qαθ α k 1 (θ) α 1 = 1 For each θ, qαθ α k 1 (θ) α 1 1, with st. ineq. for some types.

19 Distortions to Productive Efficiency Intuition of Proof: Same idea as Mirrlees positive marginal tax rate. Suppose for some type qαθ α k 1 (θ) α 1 < 1 By MIEE: u (c 0 (θ)) > βq 1 u (c 1 (θ)). DRAW PICTURE ON THE BOARD

20 Investment Wedge Investment wedge: u (c 0 (θ)) = (1 τ k (θ)) βαθ α k(θ) α 1 u (c }{{} 1 (θ)) inv. wedge Can we determine its sign?

21 Investment Wedge Recall MIEE: [ qαθ α k 1 (θ) α 1 1 ] k 1 (θ) u (c 0 (θ)) u (c 0 (θ)) u (c 0 (θ)) = q βu (c 1 (θ))

22 Investment Wedge Recall MIEE: [ qαθ α k 1 (θ) α 1 1 ] k 1 (θ) u (c 0 (θ)) u (c 0 (θ)) }{{ 2 } + 1 u (c 0 (θ)) = q βu (c 1 (θ))

23 Investment Wedge Recall MIEE: [ qαθ α k 1 (θ) α 1 1 ] k 1 (θ) u (c 0 (θ)) u (c 0 (θ)) }{{ 2 } + 1 u (c 0 (θ)) = q βu (c 1 (θ)) 1 u (c 0 (θ)) Investment wedge is positive. q βu (c 1 (θ)) 1 βαθ α k 1 (θ) α 1 u (c 1 (θ))

24 Summary MIEE Individual return on capital>social return on savings Investment wedge > 0.

25 Multi-Period Model

26 Environment Time t = 0,1,,T + 1; T N { }. Continuum of agents. Agent: draw a type θ t Θ = [θ, θ] at t = 0,,T.

27 Environment θ t is a 1st order Markov Process: θ t F t (θ t θ t 1 ) F t (θ t θ t 1 ) is increasing in θ t 1 according to SOSD. Atom-less distribution: f t (θ θ t 1 ) = θ t F t (θ t θ t 1 ) > 0, θ t Θ µ t (θ t ): distribution of types at date t.

28 Environment Agent of type θ t s income at t + 1: θ α t kα t+1. Allocation (c t (θ t ),k t+1 (θ t )) T+1 t=0.(θ T+1 = ) Feasibility: [ ct (θ t ) + k t+1 (θ t ) ] dµ t (θ t ) θt 1k α t (θ t 1 ) α dµ t 1 (θ t 1 ) Θ t+1 Θ t

29 Incentive Compatibility For any reporting strategy σ t : Θ t+1 Θ t+1 : T+1 β Θ t u(c t (θ t ))dµ t (θ t ) (IC) t+1 t=0 T+1 t=0 ( β Θ t u c t (σ t (θ t )) + k t+1 (σ t (θ t ))(1 σ t(θ t ) ) ) dµ t (θ t ) t+1 θ t

30 Incentive Compatibility For any allocation let U t (θ t ) = u(c t (θ t )) + β T+1 s=t+1 β s t Θ s u(c s (θ s ))dµ s (θ s θ t ) One-Shot IC: θ u(c t (θ t )) + β U t+1 (θ t,θ t+1 )df t+1 (θ t+1 θ t ) θ ) u (c t (θ t 1,θ ) + k t+1 (θ t 1,θ )(1 θ ) θ t θ +β U t+1 (θ t 1,θ,θ t+1 )df t+1 (θ t+1 θ t ) θ (OSIC)

31 Incentive Compatibility Proposition An allocation satisfies (IC) iff it satisfies (OSIC). T < : Proof by backward induction. T = : Continuity at infinity.

32 Incentive Compatibility Local IC: [ u (c t (θ)) c t (θ t 1,θ t ) 1 ] u (c t (θ))k t+1 (θ t ) θ t θ t +β θ Equivalent to (OSIC) if: θ (LIC) θ t U t+1 (θ t,θ t+1 )df t+1 (θ t+1 θ t ) = 0 θ t k t+1 (θ t ), U t+1 (θ t+1 ) 0 θ t θ t

33 Planning Problem θ max U 0 (θ 0 )df 0 (θ 0 ) θ subject to [ ct (θ t ) + k t+1 (θ t ) ] dµ t (θ t ) θt 1 α k t(θ t ) α dµ t 1 (θ t 1 ) Θ t+1 Θ t U t (θ t ) = u(c t (θ t )) + β (LIC) θ θ U t+1 (θ t,θ t+1 )df t+1 (θ t+1 θ t )

34 Modified Inverse Euler Equation Proposition Any solution to the above planning problem at any date for any history must satisfy, Q t u (c t ) +Q u (c t ) tv t k t+1 u (c t ) 2 = β 1 E t where [ Qt+1 u (c t+1 ) + Q u ] (c t+1 ) t+1v t+1 k t+2 u (c t+1 ) 2 v t (θ t ) = Q t+1 Q t αθ α t k t+1(θ t ) α 1 1 Q t : Price of consumption at t Proof: same perturbation as before.

35 Discussion Why positive capital taxes with labor income risk: [ ] Q t u (c t ) = Qt+1 β 1 E t u (c t+1 ) Jensen s Inequality: Regroup MIEE: Q t u (c t ) = E t Q t u (c t ) > β 1 Q t+1 E t [u (c t+1 )] [ Qt+1 βu (c t+1 ) + Q t+1v t+1 k t+2 u (c t+1 ) βu (c t+1 ) 2 Q tv t k t+1 u (c t ) βu (c t ) 2 ]

36 MIEE - Some Heuristics No distortion at the top and bottom: v t (θ t 1, θ) = v t (θ t 1,θ) = 0. In many examples: v t (θ t ) > 0 for θ (θ, θ), v t (θ t 1, ) has an inverse U-shape. What are the implications?

37 Investment Wedge - Heuristics MIEE at the top and bottom: MIEE in the middle; Q t u (c t (θ t 1, θ)) < E t Q t u (c t (θ t 1,θ)) > E t [ Qt+1 ] βu (c t+1 ) [ Qt+1 ] βu (c t+1 ) Lowest tax perhaps at extremes Highest tax on capital perhaps in the middle. Established Results: i.i.d. case

38 Infinite Horizon Model + i.i.d. Shocks

39 Recursive Problem v(w) = subject to θ max [qθ α k(θ) α k(θ) c(θ) + qv(w(θ))] df(θ) c(θ),w(θ),k(θ) θ θ θ [u(c(θ)) + βw(θ)] df(θ) = w u(c(θ)) + βw(θ) = u(c(θ)) + βw(θ) + θ θ 1 ˆθ k(ˆθ)u (c(ˆθ))dˆθ q = Q t+1 Q t

40 Negative Marginal Tax Rate No-distortion-at-the-top: u (c( θ)) β 1 = qv (w( θ)), qαθα k(θ) α 1 = 1 When u(c) = exp( γc), it can be shown that 1 v (w) > u (c(θ))df(θ). In general, if w(θ) is increasing: 1 v (w) > u (c(θ))df(θ).

41 Negative Marginal Tax Rate u (c(θ)) > βq 1 u (c(θ,w( θ)))df(θ ) Investment wedge τ k ( θ) = 1 u (c(θ)) βαθ α k(θ) α 1 u (c(θ,w( θ)))df(θ ) < 0