Fairness, social norms and the cultural demand for redistribution

Abstract

When studying attitudes towards redistribution, surveys show that individuals do care about fairness. They also show that the cultural environment in which people grow up affects their preferences about redistribution. In this article we include these two components of the demand for redistribution in order to develop a mechanism for the cultural transmission of the concern for fairness. The preferences of the young are partially shaped through the observation and imitation of others’ choices. More specifically, observing during childhood how adults have collectively failed to implement fair redistributive policies lowers the concern during adulthood for fairness or the moral cost of not supporting fair taxation. Based on this mechanism, the model exhibits a multiplicity of history-dependent stationary states that may account for the huge and persistent differences in redistribution observed between Europe and the United States. It also explains why immigrants from countries with a preference for greater redistribution continue to support higher redistribution in their destination country.

Appendices

Appendix A: Proof of Propositions

Proposition 1

From Eq. (9) it follows that the preferred tax rate of an individual of talent a is as follows:

$$\begin{aligned} \tau _{a,t}=\left\{ \begin{array}{c} \frac{\bar{a}-a+2\Phi \left( \mathcal {S}_{t-1}\right) \tau ^{f}}{2\bar{a} -a+2\Phi \left( \mathcal {S}_{t-1}\right) } \\ 0 \end{array} \right. \left. \begin{array}{l} \quad \text {if }a-\bar{a}\le 2\Phi \left( \mathcal {S}_{t-1}\right) \tau ^{f} \\ \quad \text {otherwise} \end{array} \right. \end{aligned}$$

Accordingly, as $\mathcal {S}_{t-1}=\left[ \tau ^{f}-\tau _{t-1}^{*} \right] ^{2}$ (Eq. 5), it yields:

$$\begin{aligned} \frac{\partial \tau _{a,t}}{\partial \tau _{t-1}^{*}}=\left\{ \begin{array}{c} -4\Phi ^{\prime }\left( 2\bar{a}-a\right) \frac{\tau ^{f}-\frac{\bar{a}-a}{2 \bar{a}-a}}{\left( 2\bar{a}-a+2\Phi \left( \mathcal {S}_{t-1}\right) \right) ^{2}}\left[ \tau ^{f}-\tau _{t-1}^{*}\right] \\ 0 \end{array} \right. \left. \begin{array}{l} \quad \text {if }a-\bar{a}\le 2\Phi \left( \mathcal {S}_{t-1}\right) \tau ^{f} \\ \quad \text {otherwise} \end{array} \right. \end{aligned}$$

As $\tau ^{f}\ge \tau _{a_{\inf }}^{s}$ yields $\tau ^{f}\ge \frac{\bar{a} -a}{2\bar{a}-a}$ $\forall \left( a_{\inf }\le \right) a\le 2\bar{a}$, it follows that if $\tau ^{f}\ge \tau _{a_{\inf }}^{s}$ and$\tau _{t-1}^{*}\le \tau ^{f}$, $\frac{\partial \tau _{a,t}}{\partial \tau _{t-1}^{*}} \ge 0$ $\forall \left( a_{\inf }\le \right) a\le 2\bar{a}$.

Proposition 2

In the text.

Appendix B: The case with $\Phi \left( \mathcal {S}\right) =\frac{\alpha }{\beta +\mathcal {S}}$

Let us define $\Phi \left( \mathcal {S}\right) =\frac{\alpha }{\beta + \mathcal {S}}$ and $\delta _{t}=\tau ^{f}-\tau _{t}^{*}$ the difference between the fair and the effective level of taxation. Assuming first that $\beta =0$, the dynamics of redistribution expressed by Eq. (10) can be rewritten as:

$$\begin{aligned} \delta _{t+1}=\frac{1+\Delta }{1+\Delta +\frac{\alpha }{\delta _{t}^{2}}} \left( \tau ^{f}-\tau ^{s}\right) \end{aligned}$$

(12)

and stationarity is then defined by:

$$\begin{aligned} \delta ^{3}-\left( \tau ^{f}-\tau ^{s}\right) \delta ^{2}+\frac{\alpha }{ 1+\Delta }\delta =0. \end{aligned}$$

(13)

If $\frac{\alpha }{\left( \tau ^{f}-\tau ^{s}\right) ^{2}}\le \frac{ 1+\Delta }{4}$, Eq. (13) exhibits three real roots $\delta _{1}^{*}=0$,$\delta _{2}^{*}=\frac{\tau ^{f}-\tau ^{s}+\sqrt{\left( \tau ^{f}-\tau ^{s}\right) ^{2}-\frac{4\alpha }{1+\Delta }}}{2}$ and $\delta _{3}^{*}=\frac{\tau ^{f}-\tau ^{s}-\sqrt{\left( \tau ^{f}-\tau ^{s}\right) ^{2}-\frac{4\alpha }{1+\Delta }}}{2}$.

In addition, as $\frac{1+\Delta }{1+\Delta +\frac{\alpha }{\delta _{t}^{2}}} \left( \tau ^{f}-\tau ^{s}\right)$ is continuous and monotonous in $\delta _{t}^{2}$, as $\frac{\partial \left[ \frac{1+\Delta }{1+\Delta +\frac{\alpha }{\delta _{t}^{2}}}\left( \tau ^{f}-\tau ^{s}\right) \right] }{\partial \delta _{t}^{2}}\ge 0$, and as $\lim \nolimits _{\delta _{t}^{2}\rightarrow 0}\frac{\partial \left[ \frac{1+\Delta }{1+\Delta +\frac{\alpha }{\delta _{t}^{2}}}\left( \tau ^{f}-\tau ^{s}\right) \right] }{\partial \delta _{t}^{2}}=0$, if $\frac{\alpha }{\left( \tau ^{f}-\tau ^{s}\right) ^{2}}\le \frac{1+\Delta }{4}$, there exist two stable stationary states characterized by $\left| \frac{d\delta _{t+1}}{d\delta _{t}}\right| \le 1$ which are $\delta =\delta _{1}^{*}$ and $\delta =\delta _{2}^{*}$, where $\tau ^{s}=\frac{\Delta }{2\left( 1+\Delta \right) }$ and $\Delta =\bar{a} -a_{m}$. In addition, if $\delta _{0}\ne \delta _{3}^{*}$, as long as $\left| \delta _{0}\right| <\frac{\tau ^{f}-\tau ^{s}-\sqrt{\left( \tau ^{f}-\tau ^{s}\right) ^{2}-\frac{4\alpha }{1+\Delta }}}{2}$, $\lim \nolimits _{t\rightarrow \infty }\delta _{t}=\delta _{1}^{*}$, otherwise $\lim \nolimits _{t\rightarrow \infty }\delta _{t}=\delta _{2}^{*}$.

Equivalently, as $\delta _{t}=\tau ^{f}-\tau _{t}^{*}$, we can assert that assuming $\frac{\alpha }{\left( \tau ^{f}-\tau ^{s}\right) ^{2}}\le \frac{1+\Delta }{4}$, $\tau _{0}\ne \hat{\tau }=\tau ^{f}-\delta _{3}^{*}$, if $\tau _{0}\in \left] \tau ^{f}-\delta _{3}^{*},\tau ^{f}+\delta _{3}^{*}\right[$ then $\tau _{H}=\lim \nolimits _{t\rightarrow \infty } \tau _{t}^{*}=\tau ^{f}$, otherwise $\tau _{L}=\lim \nolimits _{t\rightarrow \infty } \tau _{t}^{*}=\frac{1}{2}\left( \tau ^{f}+\tau ^{s}-\sqrt{ \left( \tau ^{f}-\tau ^{s}\right) ^{2}-\frac{4\alpha }{1+\Delta }}\right)$, $\tau _{H}>\tau _{L}$.

Besides, the dynamic process exhibits only one stationary state $\tau _{L}$ if $\beta$ is large enough such that $\lim \nolimits _{\beta \rightarrow +\infty }\xi =1\Leftrightarrow \lim \nolimits _{\beta \rightarrow +\infty }\tau _{L}=\tau ^{s}$.

As $\frac{\partial \xi }{\partial \beta }>0$ and $\lim \nolimits _{\tau ^{*}\rightarrow \tau ^{f}}\xi =0$, it follows that if $\frac{\alpha }{ \left( \tau ^{f}-\tau ^{s}\right) ^{2}}\le \frac{1+\Delta }{4}$ there exists $\hat{\beta }>0$ such that if $0<\beta \le \hat{\beta }$ the model exhibits two stable stationary states $\tau ^{US}$ and $\tau ^{EU}$, where $\tau ^{s}<\tau ^{US}<\tau ^{EU}<\tau ^{f}$. Defining the dynamics as $\tau _{t+1}^{*}=\Psi \left( \tau _{t}^{*}\right)$, $\hat{\beta }$ can be characterized by $\lim \nolimits _{\beta \rightarrow \hat{\beta }}\Psi ^{\prime }\left( \tau ^{EU}\right) =1$ whereas $\lim \nolimits _{\beta \rightarrow 0,\beta >0}\Psi ^{\prime }\left( \tau ^{EU}\right) =\Psi ^{\prime }\left( \tau ^{f}\right) =0$.

Appendix C: Endogenous perceptions

1.1 The linear case

Define $\mathcal {T}^{f}\left( \tau \right) =\underline{\tau }^{f}+\left( 1- \frac{\underline{\tau }^{f}}{\hat{\tau }^{f}}\right) \tau$, where $0< \underline{\tau }^{f}<\hat{\tau }^{f}\le 1$, $\hat{\tau }^{f}$ being the only root of the equation $\tau =\mathcal {T}^{f}\left( \tau \right)$.

In that case, utility Eq. (2) becomes $U_{it}=u_{it}-\varphi _{t}\left( \mathcal {T}^{f}\left( \tau _{t}\right) -\tau _{t}\right) ^{2}=u_{it}-\varphi _{t}\left( \underline{\tau }^{f}+\left( 1-\frac{ \underline{\tau }^{f}}{\hat{\tau }^{f}}\right) \tau _{t}-\tau _{t}\right) ^{2}$. Defining $\breve{\varphi }_{t}=\left( \frac{\underline{\tau }^{f}}{ \hat{\tau }^{f}}\right) ^{2}\varphi _{t}$ allows us to rewrite utility as $U_{it}=u_{it}-\breve{\varphi }_{t}\left( \hat{\tau }^{f}-\tau _{t}\right) ^{2}$, i.e. as if $\hat{\tau }^{f}$ was an exogenous fair level of redistribution unanimously shared in the population. Therefore, it goes straightforwardly that the dynamics of redistribution can be expressed according to Eq. (10) as $\tau _{t+1}^{*}=\breve{\xi }_{t}\tau ^{s}+\left( 1-\breve{ \xi }_{t}\right) \hat{\tau }^{f}$, where $\breve{\xi }_{t}=\frac{1+\Delta }{ 1+\Delta +\breve{\Phi }\left( \mathcal {S}_{t}\right) }\in \left[ 0,1\right)$ , $\tau _{0}^{*}=\tau _{0}\ge 0$ given, $\breve{\Phi }\left( \mathcal {S} _{t}\right) =\left( \frac{\underline{\tau }^{f}}{\hat{\tau }^{f}}\right) ^{2}\Phi \left( \mathcal {S}_{t}\right)$.

Knowing that $\mathcal {S}_{t}=\left( \mathcal {T}^{f}\left( \tau _{t}\right) -\tau _{t}\right) ^{2}=\left( \frac{\underline{\tau }^{f}}{\hat{\tau }^{f}} \right) ^{2}\left( \hat{\tau }^{f}-\tau _{t}\right) ^{2}$, if $\Phi \left( \mathcal {S}\right) =\frac{\alpha }{\beta +\mathcal {S}}$, it follows that $\breve{\Phi }\left( \mathcal {S}_{t}\right) =\frac{\alpha }{\breve{\beta }+ \mathcal {S}}$, where $\breve{\beta }=\left( \frac{\hat{\tau }^{f}}{\underline{ \tau }^{f}}\right) ^{2}\beta$. Accordingly, we deduce from Appendix B that if $0<\alpha \le \hat{\alpha }$ then there exists $\hat{\beta }>0$ such that if $0<\breve{\beta }\le \hat{\beta }$ the model exhibits two stable stationary states $\tau ^{US}$ and $\tau ^{EU}$, where $\tau ^{s}<\tau ^{US}<\tau ^{EU}<\hat{\tau }^{f}$, as illustrated in Fig. 2a.

1.2 The nonlinear case

Define $\Phi \left( \mathcal {S}\right) =\frac{\alpha }{\beta +\mathcal {S}}$ and assume that $\tau ^{f}=\hat{\tau }_{j}^{f}$. It follows (see Appendix B) that if $0<\alpha \le \hat{\alpha }_{j}$, where $\hat{\alpha }_{j}=\frac{ 1+\Delta }{4}\left( \hat{\tau }_{j}^{f}-\tau ^{s}\right) ^{2}$, then there exists $\hat{\beta }>0$ such that if $0<\beta \le \hat{\beta }$ the model exhibits two stable stationary states $\tau _{j}^{US}$ and $\tau _{j}^{EU}$ , where $\tau ^{s}<\tau _{j}^{US}<\tau _{j}^{EU}<\hat{\tau }_{j}^{f}$, $\lim \nolimits _{\beta \rightarrow 0,\beta >0}\tau _{j}^{US}=\frac{1}{2} \left( \hat{\tau }_{j}^{f}+\tau ^{s}-\sqrt{\left( \hat{\tau }_{j}^{f}-\tau ^{s}\right) ^{2}-\frac{4\alpha }{1+\Delta }}\right)$ and $\lim \nolimits _{\beta \rightarrow 0,\beta >0}\tau _{j}^{EU}=\hat{\tau }_{j}^{f}$.

Consider now that perceptions are defined as $\mathcal {T}^{f}\left( \tau \right) =\left\{ \begin{array}{l} \hat{\tau }_{\inf }^{f} \\ \hat{\tau }_{\sup }^{f} \end{array} \begin{array}{l} \text {if }\tau \le \check{\tau } \\ \text {otherwise} \end{array} \right.$, where $\tau ^{s}<\hat{\tau }_{\inf }^{f}<\check{\tau }<\hat{\tau } _{\sup }^{f}\le 1$. Therefore, if $\alpha$ is low enough and $\beta$ low enough (such that $\frac{\alpha }{\beta }$ is high enough), there are four potential stable stationary states that are $\tau _{\inf }^{US}$, $\tau _{\sup }^{US}$, $\tau _{\inf }^{EU}$ and $\tau _{\sup }^{EU}$. However, we deduce from $\frac{\partial \lim \nolimits _{\beta \rightarrow 0,\beta >0} \tau _{j}^{US}}{\partial \hat{\tau }_{j}^{f}}=\frac{1}{2}\left[ 1-\frac{\hat{ \tau }_{j}^{f}-\tau ^{s}}{\sqrt{\left( \hat{\tau }_{j}^{f}-\tau ^{s}\right) ^{2}-\frac{4\alpha }{1+\Delta }}}\right] <0$ $\left( \hat{\tau }_{j}^{f}>\tau ^{s}\right)$ that, if $\beta$ is low enough, then $\tau _{\sup }^{US}<\tau _{\inf }^{US}<\hat{\tau }_{\inf }^{f}<\check{\tau }$: $\tau _{\sup }^{US}$ cannot be a stationary states associated with the fair level of redistribution $\hat{\tau }_{\sup }^{f}$. Therefore, as illustrated in Fig. 2b, if $\alpha$ is low enough and $\beta$ low enough (such that $\frac{ \alpha }{\beta }$ is high enough), the model exhibits three stable stationary states $\tau ^{US}=\tau _{\inf }^{US}$, $\tau ^{G}=\tau _{\inf }^{EU}$ and $\tau ^{F}=\tau _{\sup }^{EU}$ where $\tau ^{s}<\tau ^{US}<\tau ^{G}<\hat{\tau }_{\inf }^{f}<\tau ^{F}<\hat{\tau }_{\sup }^{f}$. Note that the observed levels of unfairness in the two high-redistribution countries are as follows: $\lim \nolimits _{\beta \rightarrow 0,\beta>0} \left( \hat{\tau } _{\inf }^{f}-\tau ^{G}\right) ^{2}=\lim \nolimits _{\beta \rightarrow 0,\beta >0}\left( \hat{\tau }_{\sup }^{f}-\tau ^{F}\right) ^{2}=0$.