Redistributive Capital Taxation Revisited

Ozlem Kına, EUI

(with Ctirad Slavık, CERGE-EI, Pragueand Hakkı Yazıcı, Sabancı University)

Midwest Macroeconomic Meetings, Fall 2019

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Motivation

Large literature on optimal taxation of capital income.

Most papers in the literature assume substitutability betweencapital and different labor types are identical.

But empirical evidence suggests different types of labor havedifferent elasticity of substitution with capital:

Capital-skill complementarity (CSC): Griliches (1969), Fallon, Layard(1975), KORV (2000), Flug, Hercowitz (2000), Duffy et al (2004).

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This paper

This paper analyzes optimal capital income tax in a richquantititative environment with CSC.

For comparison, also compute optimal tax in a model with astandard Cobb-Douglas production function.

Find that CSC quantitatively important for optimal taxes:

In baseline analysis, optimal capital tax rate is 13 pp. higherunder CSC relative to comparable Cobb-Douglass economy.

Intuition: Under CSC,

Capital tax ↑⇒ Capital accumulation ↓⇒ skill premium ↓⇒indirect redistribution from skilled to unskilled.

Additional motive to tax capital for redistributive government!

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Related Literature

1 Optimal Ramsey tax literature: Judd (1985), Chamley (1986).

2 Quantitative optimal Ramsey tax literature in heterogenousagent economies: Aiyagari (1995), Domeij-Heathcote (2004),Dyrda-Pedroni (2018), Acikgoz et al (2018).

3 Smaller optimal tax literature with CSC (not rich dynamicquantitative environments):

Jones et al (1997), Slavık and Yazici (2014), Angelopoulos etal (2015), Tsai et al (2018).

Taxation of robots: Guerreiro et al. (2017), Thuemmel(2018), Costinout and Werning (2018).

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Rest of the Talk

Environment.

Quantitative results.

Conclusion.

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Environment

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The Two Models

∞ horizon heterogeneous agent incomplete market models,Aiyagari (1994):

Government, measure 1 of workers and a firm.

2 types of labor: skilled and unskilled (exogenous, and fixedover time).

πs + πu = 1

1 Model 1 without complementarity: One type of capital.

2 Model 2 with complementarity: 2 types of capital; equipmentsand structures, and equipment capital-skill complementarity.

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Workers

Each period each worker of skill type i ∈ {s, u} drawsidiosyncratic productivity shock zi .

Worker of skill type i and productivity zi receives a wage ratewi = wi · zi per unit of time, with wi = MPLi .

Preferences over stochastic (ci ,t , li ,t)∞t=0 are given by

Ei

[ ∞∑t=0

βti (u(ci ,t)− v(li ,t))].

Incomplete markets: partial insurance through a risk-freebond.

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Workers

Type i’s problem:

max(ci,t ,li,t ,ai,t+1)≥0

Ei

[ ∞∑t=0

βti (u(ci ,t)− v(li ,t))]

ci ,t + ai ,t+1 ≤ wi ,tzi li ,t − T (wi ,tzi li ,t) + Rtai ,t

where Rt is the after-tax return, T (.) is the nonlinear tax function

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Production Sector

1 F (K , Ls , Lu) = A · K θ(µLs + Lu)1−θ,

A is TFP and µ controls the skill premium.

2 F (Ks ,Ke , Ls , Lu) = Kαs

(ν [ωKρ

e + (1− ω)Lρs ]ηρ + (1− ν)Lηu

) 1−αη

,

(equipment) capital-skill complementarity, KORV (2000):

MPLsMPLu

increasing in Ke (independent of Ks).

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Firm Problems

Representative firm hires labor and rents capital to maximizeprofits ∀t:

1 maxKt ,Ls,t ,Lu,t

F (Kt , Ls,t , Lu,t)− rtKt − ws,tLs,t − wu,tLu,t

2 maxF (Ks,t ,Ke,t , Ls,t , Lu,t)− rs,tKs,t − re,tKe,t − ws,tLs,t − wu,tLu,t

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Government

non-linear taxes on labor income {Tt(y)}∞t=0.

linear taxes on capital income {τt}∞t=0.

taxes are used to finance government expenditures {Gt}∞t=0

and to repay government debt {Dt}∞t=0.

Budget constraint:

1 Gt + RtDt = Dt+1 + τk,t(rt − δ)Kt +∑

i=u,s

πiEi

[Tt(li,twi,tzt)

],

2

Gt + RtDt = Dt+1 + τk,t [(re,t − δe)Ke,t + (rs,t − δs)Ks,t ] +∑i=u,s

πiEi

[Tt(li,twi,tzt)

]where Rt = [1 + (1− τk,t)(rs,t − δs)] = [1 + (1− τk,t)(re,t − δe)]

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Definition of competitive equilibrium

Define zt ≡ (z0, ....., zt ), zt ∈ Z ti , and Pi,t (zt ) :

Definition: A competitive equilibrium consists of a policy (Tt (.), τk,t ,Dt , Gt )∞t=0, an allocation

((ci,t (zt ), li,t (zt ), ai,t+1(zt ))zt∈Zti,i ,Ks,t ,Ke,t , Ls,t , Lu,t ), and a price system (rs,t , re,t ,ws,t ,wu,t , Rt )∞t=0

such that:

1 Given the policy and the price system, the allocation ((ci,t (zt ), li,t (zt ), ai,t+1(zt )),zt )∞t=0 solves type i’s

problem ∀t, zt :

max(ci,t (zt ),li,t (zt ),ai,t+1(zt ))≥0

∞∑t=0

∑zt∈Zt

i

Pi,t (zt )βti (u(ci,t (zt ))− v(li,t (zt )))

s.t. ci,t (zt ) + ai,t+1(zt ) ≤ wi,t zt li,t (zt )− T (wi,t zt li,t (zt )) + Rtai,t (zt−1),

where Rt = [1 + (1− τk,t )(rs,t − δs )] = [1 + (1− τk,t )(re,t − δe )]

2 Given prices, the allocation {Ke,t ,Ks,t , Ls,t , Lu,t} solves the firm’s problem for each period.

3 Markets clear ∀t.4 The government’s budget constraint is satisfied ∀t.

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Optimal Taxation Problem

Optimal taxation problem: to choose capital income tax tomaximize total welfare taking transition into account

maxτk

∑i∈{s,u}

πi Ei

[(1−βi )(u(ci ,0)−v(li ,0))+

∞∑t=1

βti (u(ci ,t)−v(li ,t))]

s.t. the corresponding allocation is a competitive equilibrium

- λ clears the government budget for a fixed G and D

- Utilitarian social welfare function with equal weights

Numerical implementation: grid search

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Quantitative Analysis

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Quantitative Analysis

Overview:

Calibrate parameters of the two model economies in SRCE tothe U.S. economy so that they are comparable.

Calculate optimal capital income tax taking transitiondynamics into account.

Compare optimal taxes, macroeconomic quantities, welfaregains and distributional consequences.

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Production Functions

Parameterizations (calibrations) to make models comparable:

(2) Kαs

(ν [ωK ρ

e + (1− ω)Lρs ]ηρ + (1− ν)Lηu

) 1−αη

Use η, ρ, δs , δe from KORV(2000). Details

Calibrate ω and ν s.t. skill premium = 1.9, labor share = 2/3.Calibrate α so that the share of equipments in total capital is1/3 as in the data.

(1) A · K θ(µLs + Lu)1−θ

µ = 1.9, labor share 1− θ = 2/3,A calibrated so that Y1 = Y2, δ = 1

3δe + 23δs = 0.0787.

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Workers

Comparable in the two models:

Utility function:

u(c)− v(l) =c1−σ

1− σ− φ l1+γ

1 + γ.

In benchmark, use σ = 2, γ = 1, and calibrate φ s.t. averagelabor supply = 1/3.

Calibrate βs and βu to match capital to output ratio andrelative skilled wealth

πs = 35.44% (CPS 2018, males aged 25-60, with earnings).

Type specific skill processes as in Krueger, Ludwig (2015).Details

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Status Quo Government Policy

τk = 0.36 as in Trabandt, Uhlig (2011).

Government expenditure G/Y = 0.16 (NIPA).

Government debt D/Y = 0.60 (FRED).

Following Heathcote, Storesletten, and Violante (2017):

T (y) = y − λy1−τl

τl = 0.18λ clears the government budget.

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Benchmark Calibration

Parameter Symbol CSC Cobb-Douglas Target Source

prod. func. parameter ω 0.3332 - labor share (2/3) NIPA

prod. func. parameter ν 0.6205 - skill premium ( wswu

=1.9) CPS

prod. func. parameter α 0.1920 - share of equipments ( KeK

=1/3)

TFP A 1 0.4037 Y2

skilled discount factor βs 0.9415 0.9415 KY

= 2 NIPA, FAT

unskilled discount factor βu 0.9365 0.9365 rel. skilled wealth=2.68 US Census

tax function parameter λ 0.8142 0.8142 govn. budget balancedisutility of labor φ 65.9705 65.9705 labor supply (1/3)

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Initial Steady States

Variable CD CSC

Y 0.1478 0.1478K 0.2955 0.2955C 0.1009 0.1009Ls 0.2765 0.2765Lu 0.3646 0.3646ws 0.4595 0.4595wu 0.2419 0.2419R 1.0563 1.0563λ 0.8142 0.8142τk 0.36 0.36G 0.0236 0.0236D 0.0887 0.0887

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Optimal Policy

Calibrated Optimal Calibrated OptimalCD CD CSC CSC

τk 0.36 0.47 0.36 0.60λ 0.8142 0.8401 0.8142 0.8760

K/Y 2 1.8118 2 1.5770ws/wu 1.90 1.90 1.90 1.6716as/au 2.78 2.6424 2.78 2.3260cs/cu 1.5499 1.5302 1.5499 1.4058

Main finding: Optimal capital taxes significantly larger under capital-skillcomplementarity, and optimal labor taxes much lower on average. Details

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Welfare Effects of Optimal Policy

Optimal CD Optimal CSC

Welfare gains 0.18% 0.78%Unskilled gains 0.40% 2.40%Skilled gains -0.43% -3.42%

measure of supporters:unskilled 43.43% 50.36%skilled 16.86% 3.80%

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Naıve Government

Calibrated Optimal Complementarity withComplementarity Complementarity Cobb-Douglas Optimal Policies

τk 0.36 0.60 0.47λ 0.8102 0.8760 0.8404

ws/wu 1.90 1.6716 1.8106as/au 2.78 2.3260 2.5916

total welfare gains 0.78% 0.54%unskilled gains 2.40% 1.18%skilled gains -3.42% -1.17%

Measure of supporters:Unskilled 50.36% 49.85%Skilled 3.80% 7.55%

Naive government: welfare losses are not negligible

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Rawlsian Social Welfare

Optimal CD: Optimal CSC:Utilitarian Rawlsian Utilitarian Rawlsian

τk 0.47 0.64 0.60 0.85λ 0.8401 0.8831 0.8760 0.9481

ws/wu 1.90 1.90 1.6716 1.1274as/au 2.6424 2.3849 2.3260 1.5618cs/cu 1.5302 1.4985 1.4058 1.0935

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Deterministic Models: Optimal Policies

Calibrated Optimal Calibrated OptimalCD CD CSC CSC

τk 0.36 0.27 0.36 0.38λ 0.5328 0.5200 0.5328 0.5358

K/Y 2 2.1393 2 1.9686ws/wu 1.90 1.90 1.90 1.8891as/au 2.78 2.7586 2.78 2.8023cs/cu 1.5525 1.5585 1.5525 1.5475

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Frame Title ??????

1 F (K , Ls , Lu) = A · K θ(µLs + Lu)1−θ

2 F (Ks ,Ke , Ls , Lu) = Kαs

(ν [ωKρ

e + (1− ω)Lρs ]ηρ + (1− ν)Lηu

) 1−αη

3 F (K , Ls , Lu) = A · K θ[κLεs + (1− κ)Lεu]1−θε

where MPLsMPLu

= κ1−κ( Ls

Lu)ε−1,

θ = 1/3, A and κ calibrated so that Y3 = Y2 and MPLsMPLu

= 1.9,

11−ε = 1.41 (Katz and Murphy (1992))

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Optimal Policies

Cobb-Douglas KM ?? Complementarityτk 0.47 0.48 0.60λ 0.8401 0.8423 0.8760

ws/wu 1.90 1.8735 1.6716

total welfare gains 0.18% 0.23% 0.78%unskilled gains 0.40% 0.58% 2.40%skilled gains -0.43% -0.70% -3.42%

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Conclusion

Taxation of capital income is an important topic in macro

redistributioninsuranceefficiency

We build a standard quantitative model with capital-skillcomplementarity as supported in the data

Capital-skill complementarity calls for substantially largercapital tax rate, in benchmark 13 percentage points.

Welfare gains are larger.

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Current and Future Work

1 Sensitivity analyses

2 Optimize also over progressivity of labor income tax

3 Endogenous skill types

4 Better match the tails of the wealth distribution

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Thank you!

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Additional Slides

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Technology Parameters

Table: Production Function Parameters

Parameter Value Source

η 0.401 KORV (2000)ρ -0.495 KORV (2000)δs 0.056 Greenwood, Hercowitz, Krusell (1997)δe 0.124 Greenwood, Hercowitz, Krusell (1997)

Return

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Skills

Table: Skill Parameters

Parameter Symbol Value Source

Skill persistence skilled workers ρs 0.9408 KLSkill volatility skilled workers var(εs) 0.1000 KLSkill persistence unskilled workers ρu 0.8713 KLSkill volatility unskilled workers var(εu) 0.1920 KL

KL stands for Krueger, Ludwig (2015).

Return

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Changes in Allocations and Prices

Variable CD CSC

ws -4.82% -18.74%wu -4.82% -7.63%

K -12.24% -26.82%Y -3.12% -7.19%C -1.75% -4.93%

cs -2.43% -9.92%cu -1.18% -0.68%Ke -16.22%Ks -28.22%Ls 2.77% 8.52%Lu 1.14% 1.16%

Return

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