1 Introduction
In the past ten years, following Prescott (
2004)’s claim that the differences in work habits in the US and in Europe were largely due to the differences in the tax systems, a number or researchers have estimated labor supply elasticities both at the microeconomic and macroeconomic levels. It seems that an important, perhaps previously neglected, element is the extensive margin and its reaction to financial incentives at the beginning and at the end of the working life. This has led to a number of models with endogenous retirement dates in a life cycle setup, e.g. Prescott et al. (
2009), Rogerson and Wallenius (
2009) and Ljunqvist and Sargent (
2014). However there is still little work on the interaction between nonlinear taxes and pension schemes. Indeed Diamond (
2009) states
Apart from some simulation studies, theoretical studies of optimal tax design typically contain neither a mandatory pension system nor the behavioral dimensions that lie behind justifications commonly offered for mandatory pensions. Conversely, optimizing models of pension design typically do not include annual taxation of labor and capital incomes. Recognizing the presence of two sets of policy institutions raises the issue of whether normative analysis should be done separately or as a single overarching optimization.
To make progress in this direction, the present study takes as a premise that two simple policy instruments are available to the (utilitarian) government: an age-independent nonlinear tax schedule and a pension scheme that depends on a single aggregate statistic of the work life. We adopt a Ramsey approach rather than a Mirrlees one. In this respect, we sharply differ from the dynamic optimal taxation literature that allows for very general tax instruments, possibly depending on the agents’ age and on their entire labor income histories (Grochulski and Kocherlakota
2010; Weinzierl
2011; Shourideh and Troshkin
2012; Michau
2014). To study the redistributive power of pensions in this Ramsey environment, we regard pension benefits as transfers to the agents that are conditioned upon a verifiable variable. In practice, the government offers a menu of pension plans, and each agent chooses her preferred option depending on her characteristics.
Our main goal is to understand how the incentive constraints associated with pension choices affect the design of optimal taxes. To this aim, following the optimal taxation literature (Saez
2001,
2002; Jacquet et al.
2013), we proceed by evaluating welfare effects of small tax reforms around the optimum. The literature defines the mechanical or equity effect as the welfare change that would occur if there were no behavioral responses. Studies differ in the kind of behavioral responses they consider. Responses at the intensive margin give rise to substitution effects while responses at the extensive margin give rise to participation effects. The literature derives optimal tax formulae in these various contexts, sometimes allowing for income effects in labor supply. As noticed by Diamond (
1998), however, the formulae are much simpler in the absence of income effects. Here, we assume away income effects, which are difficult to handle when a retirement system is in force.
We focus on the extensive margin, appropriate to discuss the retirement decision.
1 In a first step, we examine how labor supply behavior depends on taxes and pensions. Under a pension system, agents face constraints on labor supply because they need to achieve a certain lifetime performance to receive their pension benefits. These constraints tend to reduce the (positive) labor supply responses to tax cuts compared to the case without pension. There is, however, a distinct, subtler effect which turns out to go in the same direction. The strength of the labor supply constraints placed by the pension system is itself endogenous, a function of the tax schedule: a tax cut reduces the pressure placed by the pension regime on labor supply because it becomes easier for the agent to meet a given pension requirement. This second effect, which we call “
feedback effect of taxes on pension incentives”, further reduces the labor supply response to tax cuts.
Our main contribution is to analyze the welfare effects of tax reforms when pensions are optimally designed by the government. Redistribution requires extracting rent from agents with high productivities and/or low opportunity costs of work. Rent extraction, however, is to be weighed against efficiency concerns. In principal-agent models, the rent-efficiency tradeoff is best described with the notion of virtual surplus, first introduced by Myerson (
1981) in the context of auctions.
2 Using this methodology, we distinguish the efficiency effect of taxes (effect of taxes on net production) from their redistributive effect (effect on lifetime consumption inequalities), while accounting for endogenous pension choices.
In a redistributive economy, we find that the presence of a pension scheme attenuates the usual forces that govern the welfare effects of tax reforms. While usually taxes are useful to redistribute wealth across agents and are detrimental to efficiency, the feedback effects of taxes on pension incentives introduce new forces that play in the opposite directions, making taxes both less redistributive and less detrimental to productive efficiency. In a specialized framework—same marginal utility of consumption for all agents, decreasing productivity and increasing pecuniary cost of work with age—we show that an optimal combination of tax and pension eliminates all rents coming from observable productivity differences. Furthermore it suppresses all upward labor supply distortions and reduces downward distortions. At the optimum the two instruments fully specialize: pensions provide the incentives to work, while taxes do all the redistribution.
The study which is closest to ours is that of Cremer et al. (
2004). Our framework, however, differs from theirs in a number of important dimensions. First, Cremer et al. (
2004) adopt a mechanism design approach and when it comes to implementation allow for a general, hybrid policy instrument that incorporates both pensions and taxes. Second, at the implementation stage, they restrict their analysis to a simple form of heterogeneity (two types). Third, their pension is based on retirement age. Here, the pension scheme lets entitlements depend on the agents’ entire work history (through an aggregate statistic), and consequently the pension system exerts pressure on labor supply all over the agents’ life cycles. As a consequence, the interaction between the fiscal and pension instruments is more complex than the implicit tax on prolonged activity that is well explained by e.g. Gruber and Wise (
1999) and Cremer and Pestieau (
2016).
Finally, it should be noted that the present study focuses on redistribution and ignores a number of important aspects of pension systems, such as uncertainty and insurance (Diamond and Mirrlees
1978; Golosov et al.
2016), political considerations (Cremer et al.
2008), and demographics (Brett
2012).
The article is organized as follows. Section
2 presents the economic environment, the government instruments, and the second-best problem. Section
3 explains how a pension system may constrain labor supply decisions and how this constraint depends on the shape of the nonlinear tax schedule, thus introducing the “feedback effects” of taxes on pension incentives. Section
4 shows that at the second-best optimum of a redistributive economy interior types have their labor supply constrained by the pension system, and that the pension feedback effects increase the redistributive role and decrease the efficiency role of income taxes. Section
4.6 provides a calibrated example with two types.
2 Model
We consider a deterministic overlapping generations model in continuous time, where all agents have the same length of life, normalized to one. At each date labor supply is extensive, either 0 or 1. Dynasties differ in their (deterministic) profiles of productivity and pecuniary cost of going to work, as well as by their instantaneous utility for consumption. The government wants to redistribute lifetime welfare across dynasties.
Formally, agents have different types, indexed by
\(\theta \), where
\(\theta \) is distributed with the cdf
F on the set
\(\Theta \). At age
\(a, 0\le a \le 1\), an agent of type
\(\theta \) produces at most
\(w(a;\theta )\) units of a single homogeneous good but suffers a pecuniary cost
\({\delta }(a;\theta )\), measured in units of good, if she works. Agent
\(\theta \) lifetime utility is
$$\begin{aligned} \int _{0}^1 u[c(a;\theta );\theta ] \,\mathrm{d}a, \end{aligned}$$
where
\(u(.;\theta )\) is an increasing concave utility index and
\(c(a;\theta )\) denotes consumption at age
a.
An agent is thus characterized by a couple of exogenous, nonnegative functions \((w(\cdot ;\theta ),{\delta }(\cdot ;\theta ))\) and by the instantaneous utility index \(u(\cdot ;\theta )\). The pair \((w(a;\theta ),{\delta }(a;\theta ))\) as age a varies describes a curve in the \((w,\delta )\)-space, that we call a trajectory. We assume that the functions w, \({\delta }\), and u are differentiable. To illustrate general qualitative properties, we use in some parts of the paper a specialized framework, which may be seen as describing the end of the agents’ life, when productivity declines and cost of work rises. In practice, this is presumably during that period that most agents retire.
At each date t, for each \(\theta \) in \(\Theta \), the economy contains a continuum of agents of type \(\theta \) of all ages a in [0, 1]; overtime the older agents die and are replaced by newborn of the same type. All cohorts are of the same size, with one agent of each type, and the economy is stationary. An allocation specifies the nonnegative consumption \(c(a;\theta )\) and the labor supply \({\ell }(a;\theta )\) in \(\{0, 1\}\) of all types \(\theta \) along their lives.
Furthermore we assume that there are perfect markets for transferring wealth across time, with a zero interest rate. The agents use these markets to smooth their consumption overtime, so that we can remove the age argument in consumption and write simply \(c(a;\theta )= c(\theta )\). Also we note \(y(\theta )\) the lifetime net output produced by agent \(\theta \), i.e., \(y(\theta )= \int _0^1 \left[ w(a;\theta )-{\delta }(a;\theta )\right] {\ell }(a;\theta )\,\mathrm{d}a\).
Feasibility An allocation is
feasible if and only if total consumption does not exceed total output net of production cost:
$$\begin{aligned} \int _{\Theta } c(\theta ) \,\mathrm{d}F(\theta )\le \int _{\Theta } y(\theta ) \,\mathrm{d}F(\theta ). \end{aligned}$$
(1)
Efficiency An allocation is
efficient whenever output net of production costs is maximized, i.e.,
\(\ell (a,\theta )=0\) if
\({\delta }(a;\theta ) > w(a;\theta )\) and
\(\ell (a,\theta )=1\) if
\({\delta }(a;\theta ) < w(a;\theta )\).
Utilitarian optimum (first-best) The utilitarian optimum is the allocation that maximizes
\(\int _{\Theta } u([c(\theta ); \theta ]\,\mathrm{d}F(\theta )\) subject to the feasibility constraint (
1). It is the feasible efficient allocation such that marginal utilities are equal: for all
\(\theta \) in
\(\Theta \)
$$\begin{aligned} u_{c}[c(\theta ), \theta ]=\lambda . \end{aligned}$$
Laissez-faire Laissez-faire induces the agents to maximize their lifetime consumption
$$\begin{aligned} c(\theta )= \int _{0}^{1} \max (0, w(a;\theta )-{\delta }(a,\theta )) \,\mathrm{d}a. \end{aligned}$$
They work whenever their productivity is larger than their opportunity cost of work, so the laissez-faire equilibrium is efficient. In general, laissez-faire yields an allocation that differs from the utilitarian optimum.
In all the paper we suppose that the utilitarian government observes the employment status of the agents and, when they work, their productivity w. It never observes the pecuniary cost \(\delta \), which is private information.
The government instruments The government has access to two policy instruments, an income tax and a retirement scheme. The first policy instrument is a time invariant income tax schedule. The tax schedule is described by a nondecreasing function R(w), the age-independent after-tax income of a worker with before tax wage w.
The second instrument is a pension scheme that relates a lifetime statistic
Z, to a (possibly negative) government lifetime transfer
P(
Z), which represents the present value of all contributions and benefits associated with the retirement plan. An agent is entitled to receive
P(
Z) provided that her lifetime performance is at least equal to
Z:
$$\begin{aligned} \int _0^1 z({w}(a;\theta )) \,{\ell }(a;\theta )\,\mathrm{d}a \ge Z. \end{aligned}$$
(2)
As the above equation shows, we assume in this study that the pension system relies on a single performance indicator that is linear in labor supply. The contribution of working at age
a to the pension requirement,
\(z({w}(a;\theta ))\), depends positively—and possibly nonlinearly—on the observed productivity at that age. To illustrate, we consider three stripped down legislations:
-
Regime L, \(z(w)=1\): all working years bring identical contributions to the pension requirement, which here coincides with total time worked over life;
-
Regime W, \(z(w)=w\): working years are weighted by the corresponding productivity, and the pension statistic is lifetime gross earnings;
-
Regime N, \(z(w)=R(w)\): the pension statistic is net lifetime earnings.
In practice, the pension transfers depend on individual labor histories through a number of channels. The regimes that we analyze are far from exhausting the existing legislations.
3 Note in particular that we do not allow the tax schedule to be age dependent, contrary to Weinzierl (
2011), nor do we deal with the financial market imperfections that underlie some of the pension regimes in practice. Our pension regimes can be seen as a restricted way of introducing age-dependent transfers.
Second best program Facing the tax schedule
\(R(\cdot )\) and a pension regime associated with transfers
\(P(\cdot )\), the consumer chooses her labor supply
\({\ell }(a)\in \{0, 1\}\) and pension plan
Z so as to maximize her lifetime utility
$$\begin{aligned} c(\theta )=\max _{{\ell }, Z} \ \ \int _0^1[R({w}(a;\theta ))) -{\delta }(a;\theta )]{\ell }(a)\,\mathrm{d}a +P(Z) \end{aligned}$$
(4)
under the pension requirement (
2). The second best program consists in designing the tax and pension schedules to maximize the sum of utilities
$$\begin{aligned} \int _{\Theta } u[c(\theta );\theta ]\,\mathrm{d}F(\theta ) \end{aligned}$$
under the feasibility constraint (
1) when the agents choose their optimal consumption production and work levels according to program (
4).
Notice that the above program does not involve a subsistence income
s, a benefit often paid to the unemployed in extensive models. This is because the consumption equation would then take the form
$$\begin{aligned} c(\theta )=\max _{{\ell }, Z} \ \ s+ \int _0^1[\tilde{R}({w}(a;\theta )) -{\delta }(a;\theta )-s]{\ell }(a)\,\mathrm{d}a +\tilde{P}(Z), \end{aligned}$$
and the subsistence income appears to be superfluous by letting
\(R= \tilde{R} -s\) and
\(P=\tilde{P} +s\).
3 Labor supply
We first explain how labor supply depends on the tax schedule
R and the pension requirement
Z. It is useful to introduce the optimal lifetime earnings, the function
\(\gamma (Z;\theta )\) which is the maximum of
$$\begin{aligned} \int _0^1 [R({w}(a;\theta )) -{\delta }(a;\theta )]{\ell }(a;\theta ) \,\mathrm{d}a \end{aligned}$$
(5)
over
\({\ell }(.)\), subject to the pension requirement (
2). Thus, lifetime consumption is
\(c(Z;\theta )=\gamma (Z; \theta ) + P(Z)\). We also denote lifetime net output by
\(y(Z;\theta )\). Letting
\(\pi (Z;\theta )\) be the Lagrange multiplier associated with the pension constraint (
2) for agent
\(\theta \), we rewrite optimal lifetime earnings as
$$\begin{aligned} \gamma (Z;\theta )= & {} \max _{{\ell }(a;\theta )} \left\{ \int _0^1 \left[ R({w}(a;\theta ))-{\delta }(a;\theta )\right] {\ell }(a;\theta )\,\mathrm{d}a \right. \nonumber \\&\left. +\; \pi (Z;\theta ) \left( \int _0^1 z({w}(a;\theta )){\ell }(a;\theta )\,\mathrm{d}a - Z\right) \right\} . \end{aligned}$$
(6)
Agent
\(\theta \)’s labor supply at age
a if she picks plan
Z is therefore given by
$$\begin{aligned} {\ell }(a;Z;\theta )={\mathbb {1}}_{\displaystyle R({w}(a;\theta )) +\pi (Z;\theta ) z({w}(a;\theta ))-{\delta }(a;\theta )\ge 0}. \end{aligned}$$
(7)
For our purpose, it is conceptually important to distinguish objects that are function of the pension requirement
Z from objects that are evaluated at the agents’ preferred plans. In this section, we take for granted that agent
\(\theta \) has chosen her preferred pension plan
\(Z(\theta )\) by solving the full program (
4). Accordingly, we use the notations
\({\ell }(a;\theta )={\ell }(a;Z(\theta ),\theta )\) for labor supply,
\(c(\theta )=c(Z(\theta );\theta )\) for lifetime consumption,
\(y(\theta )=y(Z(\theta );\theta )\) for lifetime net output, and
\(\pi (\theta )=\pi (Z(\theta );\theta )\) for the pension multiplier. Later in the analysis, we shall need to make the dependence in the pension requirement
Z apparent.
Equation (
7) clearly shows the absence of income effects in labor supply over the life cycle. In particular, labor supply is independent of the level of pension benefits, which affects the agents’ lifetime consumption.
4 To choose her labor supply agent
\(\theta \) must consider her adjusted tax schedule or financial incentive to work
\(R(w)+\pi (\theta ) z(w)\). She works in regions where her trajectory is located below her incentive schedule
\((w, R(w)+\pi (\theta ) z(w))\), i.e., her opportunity cost of work
\(\delta \) is smaller than the financial incentive to work. The first component
R(
w) represents the instantaneous after-tax income while the second term
\(\pi (\theta ) z(w)\) represents the (deferred) pension benefit associated with before-tax earning
w. The multiplier
\(\pi (\theta )\) can be thought of as an implicit conversion rate between after-tax earnings and pension benefits for agent
\(\theta \).
A change in after-tax schedule R has two effects. First, at a given level of the pension multiplier \(\pi (\theta )\), agent \(\theta \) is subject to the incentive schedule \((w,R(w)+\pi (\theta ) z(w))\) as productivity varies. Her work status changes at “switch points” where her trajectory crosses her incentive schedule. The static effect of a change in R outside switch points is zero. We show in the Appendix that a marginal tax rise around a switch point w, \(\,\mathrm{d}R<0\) on \([w,w+\,\mathrm{d}w]\), directly decreases agent \(\theta \)’s labor supply around w by \(\eta |\mathrm{d} R|\), where \(\eta >0\) is the “static” elasticity.
In the specialized framework of Assumption
2.1, the static elasticity is weakly smaller when the pension requirement is binding
\((\pi >0\)) than in the absence of retirement scheme (
\(\pi =0\)). The agents respond less vigorously to tax rises because they understand that they need to meet a pension requirement to receive the corresponding benefits. Specifically, the static elasticity under regime
L is in fact independent from the pension multiplier because the derivative
\(z_{w}\) is identically zero in that case. In the other regimes, the static elasticity decreases with the pension multiplier in the particular case where productivity declines and labor cost rises as time passes, i.e., where the agent’s lifetime trajectory is decreasing in the
\((w,{\delta })\)-space.
Second, when the pension constraint (
2) is binding, a change in the after-tax schedule affects agent
\(\theta \)’s pension multiplier
\(\pi (\theta )\), and thus modifies indirectly the incentive schedule
\(R(w)+\pi (\theta ) z(w)\). We refer to
\(\partial \pi /\partial R\) as the
feedback effect of taxes on pension incentives. Increasing
R around a switch point causes
\(\pi (\theta )\) to decrease, i.e., translates into less pressure placed by the pension scheme on the agent’s labor supply. In this sense, the effects of pensions and taxes on labor supply are substitutes.
When the pension requirement is binding, a tax cut leads to lower pension multipliers. In other words, it weakens the labor supply incentives created by the pension system. The feedback effect affects labor supply in a nonlocal way, see (
7). By “nonlocal”, we mean that a change of
R around a particular switch point alters labor supply around
all switch points of the agent. We show in the Appendix that the
static effect of taxes on labor supply locally (i.e., around the productivity level where taxes are changed) dominates the
feedback effect. More precisely, labor supply around a switch point weakly increases following a local increase in after-tax income (a tax cut) around that switch point—while it decreases around the other switch points.
In the specialized framework of Assumption
2.1, trajectories are decreasing and therefore cannot intersect the incentive schedule
\(R(w)+\pi z(w)\) more than once. Hence they have at most one switch point. Suppose that an agent’s labor supply is constrained by the pension system (
\(\pi (\theta )>0\)). Then the static effect and the feedback effect of income taxes cancel out exactly for that agent. This is because when an agent’s trajectory has a single switch point, labor supply is entirely determined by the pension requirement (
2), so changing
R around the switch point has no effect on labor supply: the total labor supply elasticity in this particular case is zero.
5 Conclusion
In this article, we have uncovered a fundamental channel through which pension schemes may help redistribute welfare across agents with different intertemporal profiles of productivity and opportunity cost of work. The main intuition is that by letting the agents choose their preferred option among a menu of pension plans, the government can indirectly control their labor supply decisions. In practice, agents take into account the deferred pension benefit associated with their current earnings when deciding to work. The implicit conversion rate between after-tax earnings and pension benefits reflects the pressure placed by the pension system on an agent’s labor supply. This pressure itself depends on the shape of the income tax schedule, a phenomenon we refer to as the “feedback effect” of taxes on pension incentives.
The subtle interplay between taxes and pensions is better understood by studying the virtual surplus associated with the government problem. In particular, an adequate decomposition of that surplus allows to separate the impact of tax reforms on redistribution and on efficiency. The main insight from our analysis is that the presence of the feedback effect counteracts the usual forces in the equity-redistribution tradeoff. Through the interplay with the pension system, decreasing taxes has positive redistributive effects and increasing taxes favors efficiency. The overall impact of tax reforms, however, depends on fine details of the economy. In a particular, quite extreme example, we find that labor supply can be fully controlled by the pension system and a strong degree of taxation is therefore optimal.
An important policy question is the choice of the lifetime statistic(s) that should underlie the pension scheme. Each statistic is associated with a particular shape of the virtual surplus, but also with a distinct monotonicity condition (recall Lemma
2). As explained at the end of Sect.
4.5, basing the pension scheme on total working time allows the government to reduce the inequalities in lifetime consumption that come from productivity differences across agents.
8 However, we have been unable to obtain a formal optimality result, and leave the design of optimal pension schemes for future research.