Forced saving, redistribution, and nonlinear social security schemes.

• Author: Cremer, Helmuth

1. Introduction

   Social security systems typically fulfill several functions. They force myopic individuals (who are inclined to save less than what is reasonable, given their life expectancy) to save an appropriate amount. They also contribute to redistributing resources. Finally, they provide insurance, in particular, for longevity risks, by providing an annuity.

   In this paper, we focus on the first two functions. The "forced saving" argument is rarely disputed. What is disputed is whether one needs social security to ensure that everyone saves enough; after all, the government needs only to require that individuals save the desired amount. This would be a valid objection if first-best redistribution were available. However, in a world of asymmetric information, where productivity and degree of myopia are not publicly observable, there may well be a case for a social security scheme that pursues both functions.

   We adopt a two-period model: Individuals work in the first period and retire in the second. They save part of their earnings for their consumption in retirement. Individuals differ in two respects, productivity on the one hand and degree of myopia on the other hand. Myopic individuals may not save "enough" for their retirement because their "myopic self" emerges when labor supply and savings decisions are made. In other words, they use a discount factor that does not reflect their "true" preferences. (1) When they retire, they regret their earlier decisions. Consequently, if they could be forced to save a certain amount, they would be in favor of such an imposed commitment. We assume that the government has a paternalistic view and wants to help these individuals to overcome their myopia; when measuring social welfare, we use the rate of time preference of the individuals whose myopic self never emerges. Ex post, myopic individuals will be grateful to the government for such forced saving. (2)

   In our model, both productivity and time preference are not observable. The government will design a tax transfer policy based on what is observable: gross earnings, disposable income, and saving.

   Anticipating the results, we show that the paternalistic solution does not necessarily imply forced savings for the myopics. This is because paternalistic considerations are mitigated or even outweighed by incentive effects. In other words, the interaction between paternalism and redistribution is rather complex and may bring about results that contradict conventional wisdom. Our numerical results suggest that as the number of myopic individuals increases, there is less redistribution and more forced saving. Furthermore, as the number of myopic individuals increases, the desirability of social security (measured by the difference between social welfare with and without social security) increases.

   This paper is part of ongoing research on social security and myopia. It focuses on nonlinear schemes. In companion papers (Cremer et al. 2007, 2008b), we study the same problem using a linear schedule and took both a normative and a positive viewpoint. The closest predecessor in the literature is probably Diamond (2003, ch. 4). He studies income taxation with time-inconsistent preferences in a two-period model, which provides arguments in favor of a progressive social security system. In his setting, myopia affects only labor supply. We assume that myopia also affects savings decisions, and we provide an explicit model for an optimal social security scheme where individuals differ in both productivity and farsightedness. In another closely related paper, Tenhunen and Tuomala (2009) also analyze the design of nonlinear pension schemes with myopic individuals. There are, however, some important differences between our paper and theirs. First and foremost, our analytical results are both more precise and more general. Second, the questions dealt with in the simulations are quite different. For instance, Tenhunen and Tuomala concentrate on a comparison between the paternalistic and nonpaternalistic case, while we study the impact of the degree of myopia and the proportion of myopics. Furthermore, they concentrate on inequality in consumption measured with Gini and Lorenz criteria (which is not consistent with the utilitarian paternalistic welfare function they use), while we look at inequality in utility levels.

   The rest of the paper is organized as follows: The basic model is introduced in the next section, section 3 discusses the second-best optimum in general and then in a three-type setting, section 4 provides numerical simulations, and section 5 concludes the presentation.

2. The Model

   Myopic and Farsighted Individuals Individuals' utility is given by

   U([c.sub.i],[d.sub.i], [l.sub.i]) = u([c.sub.i]) + [[beta].sub.u]([d.sub.i]) - v([l.sub.i]), (1)

   where [c.sub.i] and [d.sub.i], are first- and second-period consumption, while [l.sub.i], is labor supplied in the first period, u and v are both increasing functions; u is concave, and v is convex. Observe that we can think of [l.sub.i] as the retirement age. Individuals differ in their wage rate, [w.sub.i] [member of] {[w.sub.L], [w.sub.H]], where [w.sub.L]

   [FIGURE 1 OMITTED]

   For all individuals, the "true" time-discount factor is given by [beta]. However, not all individuals will make their labor supply and consumption decisions according to this parameter. For some individuals, their "myopic self emerges when labor supply and saving are chosen. They take all decisions according to a time discount parameter [[beta].sub.0]

   [U.sub.i]{[c.sub.i],[d.sub.i],[I.sub.i]) = u{[c.sub.i]) + [[beta].sub.i]u([d.sub.i]) - v([l.sub.i]). (2)

   For myopic individuals we have P, = p0, while P, = P holds for the farsighted individual. (3)

   To sum, there are four types of individuals, as represented in Figure 1. Type-1 and type-3 individuals are the farsighted with low and high ability, respectively. Type-2 (low ability) and type-4 (high ability) individuals, on the other hand, are myopic. Total population size is normalized at one, and the proportion of type i = 1, ..., 4 individuals is denoted by [[pi].sub.i]. In the analytical second-best part, we provide general expressions, but for their interpretation, we concentrate on a three-type setting. The fully fledged four-type case is then solved in numerical examples (see section 4).

   First-Best Solution

   We take a paternalistic approach and consider the utilitarian optimum based on individuals' true preferences. The corresponding Lagrangian expression is given by

   [L.sub.FB] = [summation over (i)] [[pi].sub.i][u([c.sub.i]) + [[beta].sub.u]{[d.sub.i]) - v([y.sub.i]/[w.sub.i])] - [mu] [summation over (i)] [[pi].sub.i] ([c.sub.i] + [d.sub.i] - [y.sub.i]),

   where u is the Lagrangian multiplier associated with the budget constraint. This yields

   [c.sub.1] = [c.sub.2] = [c.sub.3] = [c.sub.4], [d.sub.1] = [d.sub.2] = [d.sub.3] = [d.sub.4] [l.sub.1] = [l.sub.2] = [l.sub.3] = [l.sub.4]

   With separable preferences, the utilitarian solution implies that consumption levels are equalized across types and periods and that the able individuals work more than the unable. This first-best allocation can be decentralized by using two instruments. First, we need lump-sum transfers to redistribute from high- to low-productivity individuals. In addition, a "Pigouvian" (corrective) subsidy at the rate 1 - [[beta].sub.0]/[beta] on the savings of the myopics is required to induce them to save the appropriate amount. As an alternative to the savings subsidy, one can also use a pension scheme to force myopic individuals to save. Either way, in a full-information setting, there is no conflict between paternalism and redistribution. The two objectives are addressed by separate...