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Public Good Provision Financed by Nonlinear Income Tax Under Reduction of Envy

Takuya Obara ORCID logo and Shuichi Tsugawa ORCID logo

Abstract

We examine optimal taxation and public good provision by a government that considers reduction of envy as a constraint. We adopt the extended envy-freeness proposed by Diamantaras and Thomson (1990. “A Refinement and Extension of the No-Envy Concept.” Economics Letters 33: 217–22), called λ-equitability. We derive the modified Samuelson rule under an optimal nonlinear income tax and show, using a constant elasticity of substitution utility function, that the direction of distorting the original Samuelson rule to relax the λ envy-free constraint is crucially determined by the elasticity of substitution. Furthermore, we numerically show that the optimal level of provision increases (decreases) in the degree of envy-freeness when the original Samuelson rule is upwardly (downwardly) distorted.

JEL Classification: D63; H21; H41

Acknowledgements

We would like to thank our supervisors, Helmuth Cremer, Jean-Marie Lozachmeur, and Motohiro Sato, for their invaluable comments and discussions. We are also grateful to Francesca Barigozzi, Takashi Kunimoto, Franco̧is Maniquet, Kaname Miyagishima, Pierre Pestieau, Takuro Yamashita, and the anonymous referees for their helpful comments. In addition, we thank the seminar participants at 18th meeting of the Association for Public Economic Theory in July 2017, Economic Theory and Policy Workshop at the Tokyo Metropolitan University in September 2017, the 23rd Decentralization Conference and Japanese Economic Association Autumn Meeting at Aoyama Gakuin University in September 2017, and the 14th Meeting of the Society for Social Choice and Welfare in June 2018.

Appendix

A  

Assume that δsH > 0 and δsL = 0. Differentiating Lagrangian (4) with respect to cL,cH,lL,lH and G,

(21)LcH=(γ+δsH)UcHδrnHδeλUˉc=0,
(22)LcL=(1+δe)UcLδrnLδsHUˆc=0,
(23)LlH=(γ+δsH)UlH+δrnHwHδeλUˉl=0,
(24)LlL=(1+δe)UlL+δrnLwLδsHwLwHUˆl=0,
(25)LG=(γ+δsH)UGH+(1+δe)UGLδrϕ(G)δeUˉGδsHUˆG=0.

Rearranging (21) and (22) yields the optimal marginal income tax rate at the top. Conversely, we can derive the marginal income tax rate at the bottom by combining eqs. (23) and (24). The provision rule for the public good is obtained by substituting eqs. (21) and (22) into (25).

B  

Differentiating Lagrangian (20) with respect to p,RL,RH, and G,

(26)Lp=(1+η)VpL+(μ+δ)VpHδVˆpηVˉp+γi=H,L[xi+(p1)xip]=0,
(27)LRH=(μ+δ)VRHηVˉRγ+γ(p1)xHRH=0,
(28)LRL=(1+η)VRLδVˆRγ+γ(p1)xLRL=0,
(29)LG=(1+η)VGL+(μ+δ)VGHδVˆGηVˉG+γi=H,L(p1)xiGγϕ(G)=0.

Equations (26), (27), and (28) give:

(30)Lp+iLRixi=0.

By using eqs. (7), (8), (10), and xˆ=VˆpVˆR, eq. (30) can be transformed into eq. (17). In addition, by substituting eqs. (27) and (28) into (29) and using eqs. (9) and (11), we can derive eq. (19).

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Published Online: 2019-08-13

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