The Political Economy of Social Security Funding: Why Social VAT Reform?

Abstract

Recently, taxation reforms entailing a “social” valued-added tax (VAT), i.e., a social security reform shifting funding from traditional wage-based taxation to consumption taxation, have been obtaining political support in some developed countries, e.g., Japan, France, Denmark, and Germany. This paper analyzes the political economy of social security funding in an overlapping-generations economy. In particular, we consider how population aging influences the choice of wage or consumption tax financing by focusing on their differential impact on inter- and intragenerational redistribution. Our results show that population aging may drastically alter the political equilibrium: if the population growth rate is higher than the interest rate, wage taxation is the only equilibrium choice, but if it is lower, multiple equilibria are likely to emerge, in which the introduction of consumption taxation emerges as an alternative equilibrium choice.

The author would like to thank Toshihiro Ihori and Naomi Miyazato for their helpful comments and suggestions. Financial support from the Zengin Foundation for Studies on Economics and Finance and Waseda University Grant for Special Research Projects (#2013A-001) is gratefully acknowledged.

JEL classification numbers: D78, H55

Appendices

Appendix 1: Proof of Proposition 2

Suppose ${\tau}_c^y\left({E}_i,{\tau}_w,n\right)>0$ and τ _w  > 0. Then, differentiating (2.26) yields

$$ \begin{array}{c}\frac{\partial {\tau}_c^y}{\partial {E}_i}=\frac{1-{\tau}_w}{U_{cc}}<0,\\ {}\frac{\partial {\tau}_c^y}{\partial {\tau}_w}=\frac{1}{U_{cc}{\left(1-{\tau}_c^y\right)}^2{\left(1-{\tau}_w\right)}^2}\left[\frac{1+n}{1+r}\left(1+{\tau}_w\right)+{\tau}_c^y\left(1-{\tau}_w\right)\right]<0,\\ {}\end{array} $$

and

$$ \begin{array}{c}\frac{\partial {\tau}_c^y}{\partial n}=\frac{\tau_w}{\left(1+r\right){U}_{cc}{\left(1-{\tau}_c^y\right)}^2\left(1-{\tau}_w\right)}<0,\\ {}\end{array} $$

where

$$ \begin{array}{c}{U}_{cc}:=\frac{\partial^2{U}_i}{\partial {\tau}_c^2}=-\frac{1}{{\left(1-{\tau}_c^y\right)}^3}\left[\frac{2\left(1+n\right)}{1+r}\frac{\tau_w}{1-{\tau}_w}+1+{\tau}_c\right]<0.\\ {}\end{array} $$

To prove (i), differentiate (2.30) with respect to n, and we have

$$ \left[f\left({E}^o\right)\frac{\partial {E}^o}{\partial {E}_i}+\left(1+n\right)f\left({E}_c^{\ast}\right)\right]\frac{\partial {E}_c^{\ast }}{\partial n}=\frac{1}{2}-F\left({E}_c^{\ast}\right)-f\left({E}^o\right)\frac{\partial {E}^o}{\partial n}>0. $$

(2.32)

The sign follows because $F\left({E}_c^{\ast}\right)<F\left({E}_m\right)=1/ 2$ and

$$ \begin{array}{c}\frac{\partial {E}^o}{\partial n}=\frac{\partial {E}_j}{\partial {\tau}_c^y}\frac{\partial {\tau}_c^y}{\partial n}<0,\\ {}\end{array} $$

the latter of which comes from (2.28) given $\partial {E}_j/ \partial {\tau}_c^y>0$ and $\partial {\tau}_c^y/ \partial n\le 0$. From (2.28), on the other hand, we have

$$ \begin{array}{c}\frac{\partial {E}^o}{\partial {E}_i}=1+\frac{1+{\tau}_c^y}{\left(1-{\tau}_w\right){\left(1-{\tau}_c^y\right)}^3}\frac{\partial {\tau}_c^y}{\partial {E}_i}=-\frac{2\left(1+n\right){\tau}_w}{\left(1+r\right){U}_{cc}{\left(1-{\tau}_c^y\right)}^3\left(1-{\tau}_w\right)}>0.\\ {}\end{array} $$

Thus, the bracketed term on the left-hand side of (2.32) must be positive, and hence $\partial {E}_c^{\ast}/ \partial n>0$. From (2.31), then,

$$ \begin{array}{c}\frac{\partial {\tau}_c^{\ast }}{\partial n}=\frac{\partial {\tau}_c^y}{\partial {E}_i}\frac{\partial {E}_c^{\ast }}{\partial n}+\frac{\partial {\tau}_c^y}{\partial n}<0\\ {}\end{array} $$

whenever ${\tau}_c^{\ast }>0$.

The proof of (ii) is quite similar. Differentiating (2.30) with respect to τ _w yields

$$ \left[f\left({E}^o\right)\frac{\partial {E}^o}{\partial {E}_i}+\left(1+n\right)f\left({E}_c^{\ast}\right)\right]\frac{\partial {E}_c^{\ast }}{\partial {\tau}_w}=-f\left({E}^o\right)\frac{\partial {E}^o}{\partial {\tau}_w}>0 $$

(2.33)

whenever ${\tau}_c^{\ast }>0$, because from (2.28)

$$ \begin{array}{ccc}\frac{\partial {E}^o}{\partial {\tau}_w}& =& \frac{\tau_c^y}{{(1-{\tau}_c^y)}^2{(1-{\tau}_w)}^2}+\frac{1}{1-{\tau}_w}[\frac{1}{{(1-{\tau}_c^y)}^2}+\frac{2{\tau}_c^y}{{(1-{\tau}_c^y)}^3}]\frac{\partial {\tau}_c^y}{\partial {\tau}_w}\\ {}=& \frac{(1+n)[1+{\tau}_w+{\tau}_c^y(1-{\tau}_w)]}{(1+r){(1-{\tau}_c^y)}^5{(1-{\tau}_w)}^3{U}_{cc}}<0\\ {}\end{array} $$

whenever ${\tau}_c^y>0$. Given that the bracketed term on the left-hand side of (2.33) is positive, we obtain $\partial {E}_c^{\ast}/ \partial {\tau}_w>0$. From (2.31), then,

$$ \begin{array}{c}\frac{\partial {\tau}_c^{\ast }}{\partial {\tau}_w}=\frac{\partial {\tau}_c}{\partial {E}_i}\frac{\partial {E}_c^{\ast }}{\partial {\tau}_w}+\frac{\partial {\tau}_c}{\partial {\tau}_w}<0\\ {}\end{array} $$

whenever ${\tau}_c^{\ast }>0$. | | 

Appendix 2: Proof of Proposition 3

Let ${\tau}_w^w$ and ${\tau}_w^c$ be the wage tax rates at W ₁ and C ₂ in Fig. 2.4, respectively. Formally, they are defined by ${\tau}_w^w:={\tau}_w^{\ast}\left(0,n\right)$ and ${\tau}_w^c:= \min \left\{{\tau}_w\Big|{\tau}_c^{\ast}\left({\tau}_w,n\right)=0\right\}$. If we let ${T}_w^w:=1/ (1-{\tau}_w^w)$ and ${T}_w^c:=1/ \left(1-{\tau}_w^c\right)$ to simplify the notations, then the equilibrium conditions, (2.14), (2.26), (2.28), and (2.30) yield

$$ kE-{E}_w^{\ast }+{T}_w^w\left(1-k{T}_w^w\right)=0 $$

(2.34)

and

$$ E-{E}_m+k{T}_w^c\left(1-{T}_w^c\right)=0, $$

(2.35)

where $k:=\left(1+n\right)/ \left(1+r\right)$. In (2.35), we make use of the fact that ${E}_c^{\ast }={E}_m$ when ${\tau}_c^y=0$, following from (2.28). An equilibrium with ${\tau}_w^e>0={\tau}_c^e$ exists if and only if ${T}_w^w\ge {T}_w^c$. Because (2.35) is quadratic, subtracting (2.35) from (2.34) after substituting ${T}_w^c$ for ${T}_w^w$ in (2.34), we can rewrite the necessary and sufficient condition for ${T}_w^w\ge {T}_w^c$ into

$$ {E}_w\le {E}_H:={E}_m-\left(1-k\right)\left(E-{T}_w^c\right). $$

(2.36)

Similarly, let ${\tau}_w^c$ and ${\tau}_c^c$ be the consumption tax rates at W ₂ and C ₁ in Fig. 2.4. Their formal definitions are ${\tau}_c^w:= \min \left\{{\tau}_c\Big|{\tau}_w^{\ast}\left({\tau}_c,n\right)=0\right\}$ and ${\tau}_c^c:={\tau}_c^{\ast}\left(0,n\right)$. Let us denote ${T}_c^w:=1/ \left(1-{\tau}_c^w\right)$ and ${T}_c^c:=1/ \left(1-{\tau}_c^c\right)$ for simplicity. Then the equilibrium conditions, (2.14), (2.26), (2.28), and (2.30) yield

$$ E-{E}_w+\left(k-1\right)E{T}_c^w+{T}_c^w\left(1-k{T}_c^w\right)=0 $$

(2.37)

and

$$ E-{E}_c+{T}_c^c\left(1-{T}_c^c\right)=0. $$

(2.38)

The existence of an equilibrium with ${\tau}_c^e>0={\tau}_w^e$ is guaranteed if and only if ${T}_c^c\ge {T}_c^w$. Subtracting (2.38) from (2.37) after substituting ${T}_c^c$ for ${T}_c^w$ in (2.37) reduces the condition to

$$ {E}_w\ge {E}_L:={E}_c-\left(1-k\right){T}_c^c\left(E-{T}_c^c\right). $$

(2.39)

Rewriting the equilibrium conditions, (2.14), (2.26), and (2.30), we find that if an equilibrium with ${\tau}_w^e>0$ and ${\tau}_c^e>0$ exists, then the tax rates, τ _w and τ _c , and the ability of the median worker–voter in voting on consumption tax, ${E}_c^{\ast }$, are determined through the following system of equations:

$$ E-{E}_w+E\left(k-1\right){T}_c+{T}_c{T}_w\left(1-k{T}_c{T}_w\right)=0 $$

(2.40)

$$ E-{E}_c^{\ast }+k{T}_c^2{T}_w\left(1-{T}_w\right)+{T}_c{T}_w\left(1-{T}_c\right)=0 $$

(2.41)

and

$$ F\left({E}^o\right)+\left(1+n\right)F\left({E}_c^{\ast}\right)=\frac{2+n}{2}, $$

(2.42)

where ${T}_w:=1/ \left(1-{\tau}_w\right)$ and ${T}_c:=1/ \left(1-{\tau}_c\right)$. The definition of E ^o is given by ${E}^o=E+k{T}_c^2{T}_w\left(1-{T}_w\right)$, which we obtain from (2.26) and (2.28) in the case of ${\tau}_c^y>0$. Then, subtracting (2.41) from (2.40) yields

$$ \begin{array}{c}\left(1-k\right){T}_c\left(E-{T}_c{T}_w\right)={E}_c^{\ast }-{E}_w.\\ {}\end{array} $$

Given that $E-{T}_c{T}_w>0$ owing to a positive labor supply and ${E}_c^{\ast }>{E}_w$, it follows that k < 1 is necessary for the existence of an equilibrium with ${\tau}_w^e>0$ and ${\tau}_c^e>0$.

We next show that as illustrated in Fig. 2.5, the consumption tax reaction curve is steeper than the wage tax reaction curve at their intersection.

Differentiating (2.40), we have the slope of the wage tax reaction curve,

$$ \begin{array}{c}\frac{\partial {T}_c}{\partial {T}_w}=-\frac{T_c\lambda }{T_w\lambda +E\left(k-1\right)}<0,\\ {}\end{array} $$

where $\lambda :=1-2k{T}_c{T}_w<0$ owing to the second-order condition spelled out in (2.16). Similarly, differentiating (2.41) and (2.42), and rearranging the terms, we obtain the slope of the consumption tax reaction curve,

$$ \begin{array}{c}\frac{\partial {T}_c}{\partial {T}_w}=-\frac{T_c\lambda +\alpha }{T_w\lambda +\beta }<0,\\ {}\end{array} $$

where

$$ \begin{array}{c}\alpha :={T}_c^2\left(k-1\right)+\frac{f\left({E}^o\right)}{\left(1+n\right)f\left({E}_c^{\ast}\right)}k{T}_c^2\left(1-2{T}_w\right)<0\\ {}\end{array} $$

and

$$ \begin{array}{c}\beta :=2{T}_w{T}_c\left(k-1\right)+\frac{2f\left({E}^o\right)}{\left(1+n\right)f\left({E}_c^{\ast}\right)}k{T}_w{T}_c\left(1-{T}_w\right)<0.\\ {}\end{array} $$

Simple calculation then demonstrates that the consumption tax reaction curve is steeper, because

$$ \begin{array}{c}\frac{T_c\lambda +\alpha }{T_w\lambda +\beta }-\frac{T_c\lambda }{T_w\lambda +E\left(k-1\right)}=\frac{\varDelta }{\left({T}_w\lambda +\beta \right)\left({T}_w\lambda +E\left(k-1\right)\right)}>0,\\ {}\end{array} $$

where

$$ \begin{array}{cc}\varDelta :=& {T}_c(k-1)\{\lambda (E-{T}_w{T}_c)+E{T}_c(k-1)\}\\ {}+\frac{kf({E}^o){T}_c^2}{(1+n)f({E}_c^{\ast })}\{-2\lambda {T}_w+E(k-1)(1-2{T}_w))\}>0.\\ {}\end{array} $$

Finally, given that the consumption tax reaction curve is steeper, it follows that the equilibrium with ${\tau}_w^e>0$ and ${\tau}_c^e>0$ is unique and that it exists if and only if ${T}_w^w>{T}_w^c$ and ${T}_c^c>{T}_c^w$. This condition is reduced to ${E}_L<{E}_w<{E}_H$. | | 

Appendix 3: Subgame Perfection of the Equilibrium Outcomes

Following Conde-Ruiz and Galasso (2005), we show that every structure-induced equilibrium obtained in the main text under the assumption of policy commitment is established as a subgame-perfect equilibrium outcome in an infinitely repeated voting game without policy commitment.

Suppose that a voting game takes place in each period, where each worker and retiree announces a pair of wage and consumption tax rates. Let τ _wt and τ _ct be the voting outcomes in period t ≥ 1, which are defined respectively as the medians of wage and consumption tax rates announced by voters. Let h ₁ be the history at the start of the game, and let h _t be one at the start of period t. The latter is a combination of h ₁ and the outcomes having been realized until period t. The set H collects all possible histories, and H ^c contains only h ₁ and h _t such that ${\tau}_{ws}={\tau}_w^e$ and ${\tau}_{cs}={\tau}_c^e$ for all $s\le t-1$. We will denote by H _t the set of possible histories until period t.

Each voter’s strategy in period t is a mapping from H _t to the set of the pairs of the two tax rates. Let ${\sigma}_i^o\left({h}_t\right)$ be the strategy of a retiree with ability E _i when voting in period t. Following (2.7), her payoff function is defined as

$$ \begin{array}{c}{V}_{it}=\left(1+r\right)\left[{A}_i-{\tau}_{ct}\left(1-{\tau}_{wt-1}\right)\left({E}_i-E\right)\right]+\left(1+n\right){\tau}_{wt}N\left({\tau}_{wt},{\tau}_{ct+1}\right),\\ {}\end{array} $$

where A and A _i are constant, satisfying A _i  < A if and only if E _i  < E. Similarly, let ${\sigma}_i^y\left({h}_t\right)$ be the strategy of a worker with ability E _i when voting in period t and, following (2.10), define her payoff function as

$$ \begin{array}{c}{U}_{it}=\left(1-{\tau}_{ct+1}\right)\left(1-{\tau}_{wt}\right){E}_i- \ln \left(1-{\tau}_{wt}\right)\left(1-{\tau}_{ct+1}\right)+\frac{B_{t+1}}{1+r},\\ {}\end{array} $$

where

$$ \begin{array}{c}{B}_{t+1}=\left(1+r\right){\tau}_{ct+1}\left(1-{\tau}_{wt}\right)N\left({\tau}_{wt},{\tau}_{ct+1}\right)+\left(1+n\right){\tau}_{wt+1}N\left({\tau}_{wt+1},{\tau}_{ct+2}\right).\\ {}\end{array} $$

Now we will show that every structure-induced equilibrium $\left({\tau}_w^e,{\tau}_c^e\right)$ presented in Proposition 3 is established as a stationary subgame-perfect equilibrium outcome of the infinitely repeated voting game by the combination of strategies, ${\sigma}_i^o\left({h}_t\right)=\left({\tau}_{wi}^o\left({h}_t\right),{\tau}_{ci}^o\left({h}_t\right)\right)$ and ${\sigma}_i^y\left({h}_t\right)=\left({\tau}_{wi}^y\left({h}_t\right),{\tau}_{ci}^y\left({h}_t\right)\right)$ for t ≥ 1, such that

$$ {\tau}_{wi}^o\left({h}_t\right)={\tau}_w^o\left({\tau}_c^e\right),\kern2em {\tau}_{ci}^o\left({h}_t\right)={\tau}_c^e $$

(2.43)

for ${h}_t\in {H}_t$ and ${E}_i\in \left[\underset{\bar{\mkern6mu}}{E},\overline{E}\right]$;

$$ {\tau}_{wi}^y\left({h}_t\right)=\left\{\begin{array}{cc}{\tau}_w^e& \mathrm{if}\ {h}_t\in {H}^c\\ {}0& \mathrm{otherwise},\end{array}\right.\kern1em {\tau}_{ci}^y\left({h}_t\right)=\left\{\begin{array}{cc}{\tau}_c^e& \mathrm{if}\ {h}_t\in {H}^c\\ {}0& \mathrm{otherwise}\end{array}\right. $$

(2.44)

for ${E}_i\in \left[\underset{\bar{\mkern6mu}}{E},{E}_c^e\right]$; and

$$ {\tau}_{wi}^y\left({h}_t\right)=\left\{\begin{array}{cc}{\tau}_w^y\left({E}_i,{\tau}_c^e,n\right)& \mathrm{if}\ {h}_t\in {H}^c\\ {}0& \mathrm{otherwise},\end{array}\right.\kern1em {\tau}_{ci}^y\left({h}_t\right)=\left\{\begin{array}{cc}{\tau}_c^y\left({E}_i,{\tau}_w^e,n\right)& \mathrm{if}\ {h}_t\in {H}^c\\ {}0& \mathrm{otherwise}.\end{array}\right. $$

(2.45)

for ${E}_i\in \left({E}_c^e,\overline{E}\right]$, where ${E}_c^e$ is the ability level that the equilibrium median worker–voter has in voting on consumption tax rates, implicitly defined by ${\tau}_c^y\left({E}_c^e,{\tau}_w^e,n\right)={\tau}_c^e$.

These strategies have the following properties. First, as (2.43) shows, concerning voting on wage tax rates, the equilibrium strategy of a retiree stipulates the same behavior as she chooses in the structure-induced equilibrium with policy commitment. As regards consumption tax rates, every retiree votes for ${\tau}_c^e$, whatever happens in the past. Second, as (2.44) shows, the votes of workers with ${E}_i\le {E}_c^e$ cluster at the pair of tax rates realized in the structure-induced equilibrium with policy commitment, as long as it has been repeatedly realized in the past, and otherwise they all vote for abolishing the social security system. Third, as (2.45) shows, workers with ${E}_i>{E}_c^e$ vote as described in the text whenever the outcome $\left({\tau}_w^e,{\tau}_c^e\right)$ has been repeatedly realized, but otherwise they will vote for abolishing the social security system. Under these strategies, $\left({\tau}_{wt},{\tau}_{ct}\right)=\left({\tau}_w^e,{\tau}_c^e\right)$ if ${h}_t\in {H}^c$, and otherwise $\left({\tau}_{wt},{\tau}_{ct}\right)=\left(0,0\right)$.

Let us check whether these strategies constitute a subgame-perfect equilibrium, assuming that even a single vote can affect the voting outcome.

To begin with, consider the strategy of retirees. If the consumption tax rate is ${\tau}_c^e$, they all want to increase the wage tax rate above ${\tau}_w^e$ because ${\tau}_w^o\left({\tau}_c^e\right)>{\tau}_w^e$. To do this, they have to increase the votes for the tax rates higher than ${\tau}_w^e$. However, their votes are already higher than the level, and thus there is no room for them to change the voting outcome. With respect to the consumption tax rate, retirees with ${E}_i<{E}^o\left({E}_c^e,{\tau}_w^e\right)$ want to increase it above ${\tau}_c^e$. However, they cannot manipulate the voting outcome in their desired direction because they already vote for ${\tau}_c^e$. A similar reasoning applies to the voting behavior of retirees with ${E}_i>{E}^o\left({E}_c^e,{\tau}_w^e\right)$.

Now turn to the strategy for workers. First, in the case of ${h}_t\in {H}^c$, a similar reasoning applies. There is no room for each worker to manipulate the voting outcome in period t in her desired direction because she already votes in that way. Next, suppose that workers with ${E}_i\le {E}_w^e$ strategically voted for a wage tax rate below ${\tau}_w^e$ and successfully reduced τ _wt in period t. Then, the voting behavior stipulated in (2.44) will yield ${\tau}_{wt+1}={\tau}_{ct+1}=0$ in period t + 1. This means that these workers receive no social security benefits. If so, their best outcome in period t is τ _wt  = 0. However, even when this happens, ${U}_i\left({\tau}_w^e,{\tau}_c^e\right)\ge {U}_i\left(0,0\right)$ holds for the following reason. First, ${U}_i\left({\tau}_w^e,{\tau}_c^e\right)\ge {U}_i\left(0,{\tau}_c^e\right)$ for ${E}_i\le {E}_w^e$ because of the single-crossing property of the utility function. Second, ${U}_i\left(0,{\tau}_c^e\right)\ge {U}_i\left(0,0\right)$ for ${E}_i<{E}_c^e$ because ${\tau}_c^e<{\tau}_c^y\left({E}_i,0,n\right)$ and ${\partial}^2{U}_i/ \partial {\tau}_c^2<0$. Accordingly, they have no incentive to deviate from (2.44). Regarding workers with ${E}_i>{E}_w^e$, because the majority of votes cluster at ${\tau}_w^e$, they cannot manipulate the voting outcome even if they change their votes on the wage tax rates. A similar reasoning applies to the voting on the consumption tax rates, and the above strategies result to form a subgame-perfect Nash equilibrium. | | 

Comment Paper to Chapter 2

Naomi Miyazato

Nihon University, 1-3-2 Misaki-cho, Chiyoda-ku, Tokyo, Japan

e-mail: miyazato.naomi@nihon-u.ac.jp

This chapter presents an analysis of the political economy of financing social security by wage tax and consumption tax by using a model of majority voting within an overlapping-generations framework. The chapter focuses on the effects of wage tax and consumption tax on intra-generational redistribution as well as on the cost of intergenerational transfer and examines how population ageing affects the political equilibrium. Analytical results show that a society with low population growth is likely to have multiple equilibria with social security financed by wage tax only, consumption tax only, or both, whereas a society with rapid population growth has a unique equilibrium with social security financed by wage tax only. The results are very interesting and seem to elucidate the reasons behind the recent popularity of financing social security with consumption tax in some countries and the diversified composition of social protection receipts across countries.

Here, I pose two questions that may appear trivial. First, people could transfer the burdens of financing social security onto the future. For example, Miyazato (2012) shows that policies implemented in the 1990s and the first half of the 2000s in Japan led to the transfer of burden onto future generations. Thus, people seem to have the policy option of transferring the social security burden onto future generations in addition to financing social security by consumption tax and wage tax. My question here is whether this transfer of burden onto future generations affects equilibrium. The next point regards the population growth rate in simulation analysis. According to the prediction of the model and simulations results, social security could be financed by taxing consumption only if the population growth rate is below 1.5 %. However, no country seems to be financing social security by levying only consumption taxes, even though many developed countries already have a population growth rate below 1.5 %.²⁰ Therefore, my second question is what factors lead to this difference between the model prediction and the real world?

Finally, I would like to consider the possibility of an empirical study of the theory presented in this chapter. The model predicts that the ratio of financing social security with consumption tax increases with population aging. Therefore, we could regress the ratio of financing social security with consumption tax on an aging indicator, as proposed by Razin et al. (2002) and Disney (2007), by using panel data from the Eurostat and OECD revenue statistics to check the main implications of the theory in this chapter.