Optimal taxation in the presence of income-dependent relative income effects

Abstract

Unlike neoclassical economics in which humans are typically described as self-interested, more and more social studies strongly support the notion that individuals derive utility not only from their own status, but also from comparisons with others. While prior studies have shown that relative income effects matter for optimal income taxation, most assumed either homogeneous relative income effects for the entire population, or relative income effects that differ only on the basis of whether one is above or below their comparison group. We investigate the importance of relative income effects within the context of an optimal income tax model with a broader form of heterogeneity. Specifically, we assume income-dependent relative income effects, which follow in the spirit of empirical evidence. Simulation results show that the optimal tax system becomes more progressive to the extent that the relatively wealthy have stronger concerns regarding others’ income than the relatively poor. This is an important result because it may provide theoretical evidence that increasing progressivity can be efficiency-enhancing.

Appendices

Appendix 1

Given that

$$\begin{aligned} x_i =w_i y_i -T\left( {w_i y_i } \right) +transfer \end{aligned}$$

(2)

Take derivative w.r.t. $w_i$, we’ll have:

$$\begin{aligned} \frac{\partial x_i }{\partial w_i }=\left( {1-t\left( {w_i y_i } \right) } \right) \left( {y_i +w_i \frac{\partial y_i }{\partial w_i }} \right) \end{aligned}$$

(A1)

Given that

$$\begin{aligned} u_i =\log \left( {x_i } \right) +\varphi _i \log \left( {\frac{x_i }{\mu }} \right) +\log \left( {1-y_i } \right) \end{aligned}$$

(3)

and

$$\begin{aligned} \frac{\partial u_i }{\partial x_i }=U_{x_i } =\frac{1+\varphi _i }{x_i } \end{aligned}$$

(A2)

Individual i chooses his/her labor supply $y_i$ to maximize own utility:

$$\begin{aligned} \frac{1+\varphi _i }{x_i }\cdot w_i \cdot \left( {1-t\left( {w_i y_i } \right) } \right) =\frac{1}{1-y_i } \end{aligned}$$

(A3)

Still start with (3), take derivative w.r.t. $w_i$ gives us:

$$\begin{aligned} \frac{du_i }{dw_i }=\left( {\frac{1+\varphi _i }{x_i }} \right) \frac{\partial x_i }{\partial w_i }-\frac{1}{1-y_i }\frac{\partial y_i }{\partial w_i } \end{aligned}$$

(A4)

Substitute (A1) and (A3) into (A4) gives us the incentive compatibility condition:

$$\begin{aligned} \frac{du_i }{dw_i }=\frac{y_i }{\left( {1-y_i } \right) w_i }=g \end{aligned}$$

(A5)

Following Kanbur and Tuomala (2013), it is easier to think of government as choosing $x_i$, $y_i$ and $\mu$ to maximize the utilitarian social welfare function (5). First, we invert the utility function and it gives us:

$$\begin{aligned} x_i =h\left( {u_i ,y_i ,\mu } \right) \end{aligned}$$

(A6)

Calculating derivatives based on (3):

$$\begin{aligned} h_{y_i }= & {} \frac{x_i }{\left( {1-y_i } \right) \left( {1+\varphi _i } \right) } \end{aligned}$$

(A7)

$$\begin{aligned} h_{u_i }= & {} \frac{x_i }{1+\varphi _i }\end{aligned}$$

(A8)

$$\begin{aligned} h_\mu= & {} \frac{x_i \varphi _i }{\left( {1+\varphi _i } \right) \mu } \end{aligned}$$

(A9)

The government is maximizing (5) subject to conditions (1), (6), and (7). Their Lagrangian multipliers are $\gamma$, $\lambda$, and $\alpha \left( w \right)$, respectively. After integrating by parts, the Lagrangian becomes (dropping i thereafter):

$$\begin{aligned} L&={\int _{\underline{w}}^{\bar{w}}} ufdw+\lambda {\int _{\underline{w}}^{\bar{w}}} \left( {wy-x} \right) fdw+\gamma \mu -\gamma {\int _{\underline{w}}^{\bar{w}}} xfdw\nonumber \\&\quad +\,\alpha \left( w \right) {\int _{\underline{w}}^{\bar{w}}} \left( {\frac{du}{dw}-g} \right) dw\nonumber \\&={\int _{\underline{w}}^{\bar{w}}} \left[ {uf+\lambda \left( {wy-x} \right) f-\gamma xf-{\alpha }^{\prime }u-\alpha g}\right] dw\nonumber \\&\quad +\,\gamma \mu +\alpha ({\bar{w}})u({\bar{w}})-\alpha \left( {\underline{w}} \right) u\left( {\underline{w}} \right) \end{aligned}$$

(A10)

The first order conditions are:

$$\begin{aligned} \frac{\partial L}{\partial u}= & {} f-h_u \left( {\lambda +\gamma } \right) f-{\alpha }^{\prime }u=0 \end{aligned}$$

(A11)

$$\begin{aligned} \frac{\partial L}{\partial y}= & {} \lambda \left( {w-h_y } \right) f-\gamma h_y f-\alpha \left( w \right) \cdot \frac{1}{w\left( {1-y} \right) ^{2}}=0\end{aligned}$$

(A12)

$$\begin{aligned} \frac{\partial L}{\partial \mu }= & {} -\lambda {\int _{\underline{w}}^{\bar{w}}} h_\mu fdw+\gamma -\gamma {\int _{\underline{w}}^{\bar{w}}} h_\mu fdw=0 \end{aligned}$$

(A13)

Traditional transversality conditions are satisfied:

$$\begin{aligned} \alpha \left( {\bar{w}} \right) =\alpha \left( {\underline{w}} \right) =0 \end{aligned}$$

(A14)

Integrating (A11) gives us:

$$\begin{aligned} \alpha \left( w \right) =\int _{w}^{\bar{w}} \left( {\frac{\lambda +\gamma }{U_x }-1} \right) f\left( \theta \right) d\theta \end{aligned}$$

(A15)

Re-arrange (A3) we can get:

$$\begin{aligned} \frac{t}{1-t}=\frac{w}{h_y }-1 \end{aligned}$$

(A16)

Substitute (A15) into (A12), we have:

$$\begin{aligned} \lambda wf-\left( {\lambda +\gamma } \right) h_y f=\frac{1}{w\left( {1-y} \right) ^{2}}\int _w^{\bar{w}} \left( {\frac{\lambda +\gamma }{U_x }-1} \right) f\left( \theta \right) d\theta \end{aligned}$$

(A17)

Re-arrange (A17) we can get:

$$\begin{aligned} \frac{1}{h_y }=\frac{\lambda +\gamma }{\lambda w}+\frac{U_x }{\lambda w^{2}f\left( w \right) \left( {1-y} \right) } \int _w^{\bar{w}} \left( {\frac{\lambda +\gamma }{U_x }-1} \right) f\left( \theta \right) d\theta \end{aligned}$$

(A18)

Substitute (A16) and (8) into (A18) and rearrange:

$$\begin{aligned} \frac{t}{1-t}=\frac{\gamma }{\lambda }+\frac{U_x }{wf\left( w \right) \left( {1-y} \right) }\left[ \int _w^{\bar{w}} \frac{1+\frac{\gamma }{\lambda }}{U_x }f\left( \theta \right) d\theta -\left( {1-F\left( w \right) } \right) {\int _{\underline{w}}^{\bar{w}}} \frac{1}{U_x }f\left( \theta \right) d\theta \right] \end{aligned}$$

(9)

where (A13) indicates that $\frac{\gamma }{\lambda }=\frac{ {\int _{\underline{w}}^{\bar{w}}} h_\mu fdw}{1- {\int _{\underline{w}}^{\bar{w}}} h_\mu fdw}$.

Appendix 2

For agent i, recall that the utility function is as follows:

$$\begin{aligned} u_i =\log \left( {x_i } \right) +\varphi _i \log \left( {\frac{x_i }{\mu }} \right) +\log \left( {1-y_i } \right) \end{aligned}$$

(3)

The Frisch Elasticity of labor supply holds the marginal utility of consumption constant. Following the traditional setting, regarding our utility specification, the FOCs (omitting all i thereafter) are:

$$\begin{aligned} \frac{\partial U}{\partial C}= & {} \frac{1+\varphi }{x}={\upzeta } \end{aligned}$$

(A19)

$$\begin{aligned} \frac{\partial U}{\partial y}= & {} \frac{1}{1-y}=\upzeta \hbox {w} \end{aligned}$$

(A20)

where ${\zeta }$ is the multiplier of the budget constraint for the individual.

Combing (A19) and (A20), we have the Frisch Elasticity of labor supply $\eta ^{\upzeta }$:

$$\begin{aligned} \eta ^{\upzeta }=\frac{\partial y}{\partial w}\cdot \frac{w}{y}=\frac{x}{w\left( {1+\varphi } \right) -x} \end{aligned}$$

(A21)

Hoding everthing else constant, clearly low-skill jobs have larger elasticities. As either w or $\varphi$ increases, $\eta ^{\upzeta }$ decreases. Agents with large relative concerns or high wage levels are less elastic: taxing them more heavily does not reduce labor supply much and causes relatively small efficiency loss.