Pension Funds Saving Individuation.

ASSA BIRATI [*]

This paper considers a pension insurance problem using an intertemporal framework. We assume a deterministic framework in order to obtain tractable and yet revealing results regarding the propensity to save for retirement. The essential conclusions of this paper include

a condition for a single switch, that is, when the saver will decide the switching time, prior to retirement, to start saving. Because of the linear objective used in this paper, saving rates were found to be of the bang-bang type. In addition, we show that the tax effects are important. The richer the saver, the greater the tax advantages for pension savings. (JEL E20)

Introduction

Pension savers tend to underestimate the financial requirements to support future needs. This may lead to future economic distress as well as to a growing social condition where governments signal their inability to meet future needs. There are further darker clouds pending over the future of pension funds and the need to overhaul past practices. This has a particularly important effect on persons who do not have sufficient wealth to support their future needs and who rely mostly on withdrawals made by employers and payments made to social security [Blake, 1995, 1998]. This state of affairs has led the search for new approaches in saving rates and saving modes [Atkinson, 1987; Bodie et al., 1987]. Explicitly, the traditional pension saving approach consisting of investment in fixed income obligations, generally issued and backed by the government, is now amended to provide greater investment flexibility. This flexibility, it is believed, may be much more responsive to market forces on one hand and better tail ored to individual savers needs on the other. By the same token, pension funds based on pay-as-you-go (PAYGO) systems have been severely criticized. PAYGO consists of paying pensions to retired persons out of taxes collected on the working population. For example, China [Feldstein, 1998] has revised the PAYGO scheme to a flexible and mixed system involving an investment that is partly based on a defined contribution.

Samuelson [1958] showed that in an economy without capital stock and in equilibrium, a completely unfunded PAYGO pension system has a positive real rate of growth of aggregate real wages (that is, the sum of the growth rate of population and the growth rate of productivity). This has justified the practice to let one generation pay the retirement payments of another. However, when capital stock is considered, the cost of pensions may be reduced by the fact that the marginal productivity of capital is greater than the marginal productivity of work. This has justified the mixed pension system where a portion of the retirement is funded, leading to lower, long-run costs. This is particularly true today when we note that increased longevity of the population is imposing an added tax burden on the working population. This burden has other negative effects such as tax avoidance and lower productivity. These schemes, and the added tax benefits they may have, are sensitive to pensioners' wealth and their tax brackets, thereby affecting pensioners' welfare [Brugiavini, 1993; Feldstein, 1990; Samwick, 1998; Freidman and Warshavsky, 1990]. The purpose of this paper is to consider a simple dynamic savings problem to study pension retirement schemes. In particular, we will address two essential problems: taxation relief and mixture of the pension package on pensioners' welfare. The point of view taken in this paper emphasizes the individual saver. Finally, proofs to the basic propositions of the paper are relegated to a more extensive paper by the same authors.

A Dynamic Model of Individual Savings and Retirement

Assume a saver whose tax bracket is known and given by [tau] [greater than] 0. Let S(t) be the current account state of the saver at time t (this may stand for the number of points accumulated by the saver, the outstanding amount of money saved and accumulated over time, and so on). Funds are accumulated by individual savers in three ways. First, funds are accumulated through an obligatory payment (usually paid for by the employee and determined as a function of the saver's salary), a(w, t), where w(t) is the individual salary at time t.

Second, funds are accumulated through a voluntary contribution, v(t), measured in dollars and contributed by the individual saver. This contribution usually has some tax advantages that are usually regulated by governments that seek to provide an incentive for retirement saving. For our purpose, we state that the amount contributed at a given time is proportional to the salary with proportionality constant:

0 [less than or equal to] v(t) [less than or equal to] [v.sup.*] , (1)

where 0 [less than] [v.sup.*] [less than or equal to] 1 is a parameter expressing the maximal amount that a saver can devote to a pension fund at a given time.

Third, investment yield of the fund generates an income which is reinvested. For our purpose, we will assume that the investment yield is known and given by a rate of return, [rho]. As a result, we can represent the saving process by the following differential equation:

dS(t)/dt = a(w, t) + [theta](t)v(t)w(t) + [rho]S(t); S(0) = [S.sub.0] , (2)

where [theta](t) is a parameter expressing the transaction cost associated with an individual contribution. Now assume that at time T, a retirement time, the saver stops working and contributing, then collects a pension over his remaining lifetime, which is a function of the accumulated fund at time T, S(T). In particular, if 1 - F(z) is the probability that the saver is still alive at time z (after retirement), then the current value at retirement of the pension is given by:

[[[integral].sup.[infinity]].sub.T][alpha](S(T))[1 - F(z)] [e.sup.-r(z-T)]dz, (3)

which, for simplicity, is written by:

[pi][alpha](S(T)), [pi] = [[[integral].sup.[infinity]].sub.T][1 - F(z)] [e.sup.-r(z-T)]dz.

As a result, we can formulate an individual saver's objective by:

[max.sub.v(t)[epsilon]V] V(T) = [e.sup.-rT] [pi] [alpha](S(T)) + [[[integral].sup.T].sub.0] Q(t) [e.sup.-rt]dt, (4)

where Q(t) denotes the utility of consumption of the individual saver. Let c(t) be consumption. If all disposable income is consumed, then w(t) = c(t) + (1 - [tau])v(t)w(t) and, therefore:

Q(t) u(c(t)) = u(w(t)(l - (1 - [tau])v(t))), (5)

where u(*) is a utility function, while (1 - [tau])v(t)w(t) is the individual's voluntary contribution net of taxes. The wealthier the individual, the larger the taxation rate. A solution to the optimal control problem defined by (1) through (4) provides initial results for the individual's saving problem. We will summarize these results by the following proposition. Further, since there is no uncertainty, we consider a linear utility function.

Proposition 1: Assume a linear utility of consumption and, for simplicity, assume fixed wage rates and fixed tax brackets. Then:

1) if r [greater than] [rho], it is optimal not to save in the time interval, [0, [T.sup.*]), and save as much as allowed, [v.sup.*]w, subsequently, with a single switching time [T.sup.*], given by the solution at equality of:

(1-[tau])/[theta][alpha][pi] [less than] [[e.sup.(r-[rho])t] (1-[e.sup.-[tho](T-t)]) + [e.sup.-r(T-t)]];

and

2) if [less than or equal to] [rho], the policy is always optimal to save for retirement if:

(1 - [tau])/[theta][alpha][pi] [less than] 1.

Otherwise, it is optimal to never save for retirement.

Proof: See Tapiero and Birati [1999].

This proposition clearly sets out the conditions for retirement savings. Of course, if tax brackets (deductions) are sufficiently large, life after retirement is long, and the benefits are large, it will always pay to save. For this reason, it is more interesting if this is not the case and:

(1 - [tau])/[theta][alpha][pi] [greater than] 1.

In this case:

1 [less than] (1 - [tau])/[theta][alpha][pi] [less than] [[e.sup.(r-[rho])t] (1 - [e.sup.-[rho](T-t)]) + [e.sup.-r(T-t)]],

and for r [greater than] [rho], the optimal time to start saving is equal to:

1 [less than] (1 - [tau])/[theta][alpha][pi] [approximate] [[e.sup.(r-[rho])t]] for T [greater than][greater than] t,

or

[T.sup.*] = 1/(r - [rho]) 1n [(1 - [tau])/[theta][alpha][pi]] for T [greater than][greater than] [T.sup.*] and (1 - [tau])/[theta][alpha][pi] [greater than] e.

With this simple approximate formula, we see that the amount of saving decreases the larger the difference between the discount rate and the savings return rate, or (r - [rho]). Further, the larger the taxation rate, the smaller [T.sup.*] (meaning that we start saving earlier). Simple partial derivatives indicate as well:

[partial][T.sup.*]/[partial]r [less than] 0; [partial][T.sup.*]/[partial][rho] [greater than] 0; [partial][T.sup.*]/[partial][tau] [less than] 0; [partial][T.sup.*]/[partial][alpha] [less than] 0; [partial][T.sup.*]/[partial][theta] [less than] 0; [partial][T.sup.*]/[partial][pi] [less than] 0.

Consider next the optimal retirement time, T. For simplicity, concentrate on the case r [greater than] [rho], which is more interesting. In this case, the optimal objective of the saver is summarized by Proposition 2. The exact solution requires some numerical analysis.

Proposition 2: For the case r [greater than] [rho], (1 - [tau]) [greater than] e[alpha][pi][theta] and the special case, (1 - [rho]/r)[e.sup.[rho](T-[T.sup.*])] [greater than][greater than] 1, retirement time is found by a solution of:

[e.sup.-(r-[rho])(T-[T.sup.*])] = ([rho]/r) [((1-[tau]) - [alpha][pi][theta]) [v.sup.*]w]/(1 - [rho]/r)[w + [alpha][pi]a(w) - (r - [rho])[alpha][pi]S(T)],

where [T.sup.*] is an optimal switching time, a function of T, and given in Proposition 1. Further, for a solution to exist, we require:

S(T) [less than] w + [alpha][pi]a(w)/(r-[rho])[alpha][pi].

Proof: See Tapiero and Birati [1999].

The results obtained in these two propositions can be studied further. Consider first the effects of the tax brackets. Note that from the optimality condition, saving for retirement will increase when tax brackets are increased. In other words, the richer the person, the more he will save for retirement. In this sense, individuating a retirement fund will increase inequalities. By the same token, we can study the effects of an extended life of savers by the comparative static of parameter [alpha]. Other effects can be studied as well, and this will be the case in subsequent research. For example, so far, we have assumed a linear utility function of consumption. When we have nonlinear utility of consumption, we can expect greater savings when the saver is risk averse. However, this will depend on the discount rate used. Further, uncertainty regarding premiums at retirement will also affect the savings rate.

Conclusion

This paper has considered a pension saving insurance problem using an intertemporal framework. We have assumed a deterministic framework in order to obtain tractable and yet revealing results regarding the propensity to save for retirement. Introducing uncertainties in wage rates, funds rates of return, retirement time, and life provide avenues for further study, although these uncertainties are likely to have few substantial effects on the problem's conclusion. The essential conclusions obtained include the condition for a single switch. This generally means that a saver will decide to start saving at some time prior to retirement. When this decision is made, it will be maintained consistently until the time of retirement. Because of the linear objective used, saving rates were found to be of the bang-bang type. Of course, for nonlinear utility of consumption, this result is likely to be amended. This would be the case if the utility of consumption were to be some convex function in consumption, yielding in terior solutions.

The effects of taxes were also considered important, reflecting the saver's wage income. The richer the saver, the higher the tax rate. Therefore, there are advantages to high-wage earners for saving. In other words, saving for retirement will tend to maintain income inequality over time since high-wage earners will save and later on collect greater benefits. Of course, in some countries such as France, taxes are deferred to the time of retirement. At this time however, chances are that the individual's tax rate will be smaller and thereby will provide another indirect benefit to savers. If the tax rate at retirement equals [tau]', our results are still valid if we replace the parameter, [alpha], with [alpha](1 - [tau]'), which measures the income received at retirement net of taxes.

(*.) Ecole Superieure des Sciences Economiques et Commerciales--France and Bar-Ilan University--Israel.

References

Atkinson, A. B. "Income Maintenance and Social Insurance," in A. J. Auerbach; M. S. Feldstein, eds., Handbook of Public Economics: Volume 2, New York, NY: North Holland, 1987.

Blake, David. Pension Schemes and Pension Funds in the United Kingdom, Oxford, United Kingdom: Oxford University Press, 1995.

___. "Pension Schemes as Option Fund Assets: Implications for Pension Fund Management," Insurance: Economics and Mathematics, 23, 1998, PP. 263-86.

Bodie, Z.; Shoven, J. B.; Wise, D. A., eds. Issues in Pension Economics, Chicago, IL: University of Chicago Press, 1987.

Brugiavini, A. "Uncertainty Resolution and the Timing of Annuity Purchases," Journal of Public Economics, 50, 1993, pp. 31-62.

Feldstein, M. Privatizing Social Security, Chicago, IL: University of Chicago Press, 1990.

___. "Social Security Pension Reform in China," working paper, 6794, National Bureau of Economic Research, November 1998.

Freidman, B. M.; Warshavsky, M. "The Cost of Annuities: Implications for Saving Behavior and Bequests," Quarterly Journal of Economics, 420, 1990, pp. 135-54.

Samuelson, Paul A. "An Exact Consumption Loan Model of Interest with or Without the Social Contrivance of Money," Journal of Political Economy, 66, 1958, pp. 467-82.

Samwick, Andrew A. "New Evidence on Pensions, Social Security, and the Timing of Retirement," Journal of Political Economy, 70, 1998, pp. 207-36.

Tapiero, C. S.; Birati, A. "Pension Insurance Individuation," working paper, Bar Ilan University, 1999.