Tax Benefit Linkages in Pension Systems (a note) Monika Bütler DEEP Université de Lausanne, CentER Tilburg University & CEPR Λ July 27, 2000 Abstract This note shows that a public pension system with a fairly general individual tax benefit linkage is (computationally) equivalent toa system without linkages. The equivalent" pension system without linkages does not only facilitate simulations of policy experiments, but also offers some insight into the implied tax structure of the tax benefit linkage. Keywords: Public Pensions, Linkages JEL classification: D9, H55 Introduction The way future pension benefits are linked to contributions has consequences on individual life cycle decision making, the efficiency of the pension system, and possibly also on Λ Iwould like to thank Urs Birchler, Theo Leers and Kent Smetters for helpful comments. Address: Monika Bütler, DEEP HEC, Université de Lausanne, BFSH, CH 05 Lausanne, Switzerland; email Monika.Butler@HEC.unil.ch, tel + 4 2 692 345, fax + 4 2 692 3365.
redistribution among heterogeneous agents. In reality, a high variety oflinkage schemes can be found, both in fully funded systems and in pay as you go (PAYG) public pension systems. It is not necessarily true that PAYG systems havenolinkages while fully funded pensions do. In Germany, for example, individual benefits and individual (proportional) payroll taxes are proportional within each cohort. Other countries apply hybrid systems, in which pension payments are often related to individual contributions over a certain income range. Although the first pillar of most public pension systems is primarily financed on a PAYG base, manycountries link past contributions to future benefits by a specific benefit formula. The bulk of the literature on public pensions, however, assumes that workers perceive no linkage between their contributions while young and pension benefits in old age. Auerbach & Kotlikoff (987) show that there may be significant efficiency gains in tightening the connection between marginal taxes paid and marginal pension benefits received. If agents perceive such a linkage, their labor supply and savings decisions are less distorted even in PAYG systems provided the linkage is strong enough. Supplementing the life cycle model with tax benefit linkages, however, usually requires an additional state variable accumulated pension claims which may complicate the analysis considerably, especially in multi period settings. Fortunately, as is shown here, this latter problem can often be solved by transforming the model into a simpler model without linkages. Apart from computational advantages, the transformed model offers some helpful insights into the implied (marginal) tax structure of the tax benefit linkage under consideration. Moreover, the state variable associated with accumulated claims to the pension system can easily be uncovered. The note is organized as follows. Section 2 introduces the model with a general benefit formula and states the equivalence result and the implied tax structure. Section 3 provides examples and lists potential applications. Concluding remarks are given in section 4. 2
2 The model Households imize their expected lifetime utility over the life cycle. Preferences are time separable and the instantaneous utility function U[ ] depends on consumption c(t) and leisure l(t). We include endogenous labor supply to explore the potentially distorting impact of payroll taxes in our analysis. Income opportunities are non stochastic and known. There are no bequest or gift motives. Our economy is equipped with a single liquid asset which yields a constant real interest rate r. There is no wedge between borrowing and lending rates and agents can lend and borrow freely at the prevailing interest rate. The existence of a (public) pension system interacts with the individual's utility imization. Benefits B, which will be related to an agent's past earnings or other characteristics, are paid out after the legal retirement agej Λ, regardless of whether the agent leaves the workforce or not. Pensions are financed by a proportional payroll tax fi. Retirement is assumed to be induced by the age wage profile and is therefore voluntary. An agent born in period t = imizes discounted lifetime utility j= Φ j U [c j ;l j ] ; () where c j denotes consumption expenditures and l j leisure of an age j individual at time j. Φ j is an age specific discount factor and may include a pure discount factoraswell as mortality risk. In every period of their lives, agents are endowed with one unit of time which they can allocate to either leisure or labor, therefore 0» l j». Let e j denote (age j) labor productivity, and w the constant real wage rate per efficiency unit of labor. Then, the budget constraints of an individual can be written as a 0 = 0 Although the results presented in this note carry over to a setting with non constant interest rates, a constant r was chosen for expositional and notational simplicity. A short appendix deals with non constant interest rates. 3
a j = ( + r)a j +( l j ) e j w ( fi)+i [j J Λ ]B c j (2) a J 0 where a j are the end of period asset holdings of an age j individual. B are social security benefits and fi denotes payroll tax. I [:] is an indicator function, which is if the condition in square brackets is satisfied and zero otherwise. To close the model, we have to specify a benefit formula, linking contributions and future benefits. Future benefits are allowed to depend on individual characteristics, such as labor productivity and survival probabilities, and aggregate variables, such as the wage rate and fiscal parameters. From an individual perspective, the most important determinant, however, is the control variable labor supply l. We therefore consider a benefit formula of the following general nature: B = B 0 + ff j= ( l j )f j (3) where B 0 is a minimum pension level paid out regardless of previous contributions, and f j represents relevant (individual) characteristics to which the benefit level is related. If B 0 (ff) is given, the parameter ff (B 0 ) will be determined by the relevant constraints of the system, which will depend on demographics and whether the system in funded or not. Although the sum in (3) is taken over all periods, the final contribution year usually coincides with the period prior to official retirement, such that f j =0for j J Λ. In general the optimization problem cannot be solved analytically. As is obvious from the budget constraints and equation (3), computing the optimal life cycle consumption and labor supply profile for an individual agent involves two state variables, asset holdings a j and a measure of accumulated claims to the system, p j P j i= ( l i)f i. However, we can simplify the problem as is shown in the following proposition. Proposition: The optimization problem as stated in equations (), (2), and (3) is computationally equivalent to imizing () without a tax benefit linkage with the following adjustments: 4
ffl B is replaced by B 0 ffl the constant tax rate fi is replaced by ( + r) fj J Λ +g ~fi j fi fff j e j w r( + r) J j : (4) Proof: From (2), we can compute the individual's lifetime budget constraint by recursive substitution as n o ( l ( + r) j j )e j w( fi) c j + B 0: (5) j ( + r) j= Using the benefit formula (3) and the formula for arithmetic series, the second term can be expressed as ( + r) B j ( + r) fj J Λ +g = r( + r) J = j= " ff i= ρ ( + r) fj ( + r) j r( + r) J j ( l i )f i # + J Λ +g ( + r) B j 0 ff( lj )fj ff + Substituting (6) into (5) yields the transformed lifetime budget constraint,» ρ ( + r) fj J Λ +g fffj ( l ( + r) j j )e j w ( fi)+ r( + r) J j e j w j= ( + r) B 0(6) j ff + I [j J Λ ]B 0 c j 0: (7) Equation (7), however, also represents the lifetime budget constraint of an agent without a tax benefit linkage, who faces an age specific payroll tax ~fi j, where ( + r) fj J Λ +g Ξ ~fi j fi fff j e j w r( + r) J j 5 :
Note that in the presence of a tax benefit linkage the effective (distortionary) payroll tax rate is smaller or equal to the statutory level fi, and can well be negative for certain agents and/or ages. By transforming the system, we have arrived at a unique state variable ~a. While uncovering the claim variable p is straightforward, the original level of asset holdings a in the untransformed system can be uncovered as follows: In every period j, ~a j contains a certain amount of virtual savings" VS j, corresponding to the present value of pension claims; VS j = ( l j )e j w(fi ~fi j ) = ( l j )fff j ( + r) fj J Λ +g r( + r) J j : (8) In the transformed system without linkages, VS j represent the accumulation of funds for retirement. These amount to an aggregate virtual savings" AVS j at the end of period j, AVS j = = j ( + r) j i ( l i )fff i ( + r) fj i= ( + r) fj J Λ +g r( + r) J j j i= J Λ +g r( + r) J i Pure asset holdings of an age j agent are therefore a j = ~a j AVS j. ( l i )fff i (9) 3 Examples and applications Many existing benefit tax linkages can be mapped into equation (3). Figure illustrates the effective tax rate implied by a number of different linkages over the life cycle. An unfunded pension system, in which constant benefits are paid out regardless of past contributions, corresponds to (3) with B 0 B and ff 0. Consequently, payroll taxes levied to finance pensions are fully distortionary to the statutory level fi. As a second example, consider the case in which the system represents or mimics a fully funded system. The present value of all benefits is a fraction fl of the present value 6
of all contributions. For a fully funded system without (administration) costs, fl =, while the solvency condition of a less than fully funded system in general implies fl < (for a stable population) due to the absence of accrued interest. Formally, we can write Solving for B yields J J Λ B B = fl ( + r) J j= j = fl J Λ j= ( + r) j ( l j)we j fi: j r( + r) J ( l j )fiwe j ( + r) J J Λ + and the corresponding values of the tax benefit linkage (3) are therefore B 0 0, ff = fl, ; and f j fiwe j r( + r) J j =f( + r) J J Λ + g for j» J Λ. As a consequence, the implied effective tax rate amounts to ~fi j =( fl): Even in a PAYG, a tight tax benefit linkage might thus reduce labor supply distortions considerably. In a non growing economy with a stable population ~fi j illustrated in Figure. is above 0, as is In our third example benefits are linked to an unweighted sum of past earnings, B = ff P J Λ i= ( l i)e j w for j J Λ j. The benefit formula, reflects the empirical regularity that contributions towards the rest of an agent's working life often carry a greater weight in computing the benefit level than contributions at the beginning of one's career. The effective tax rate then becomes ~fi j = fi ( + r) fj J Λ +g r( + r) J j Effective tax rates are relatively high at the beginning of an agent's working life and then decline, but are independent of his/her earnings profile. The example in Figure shows that the effective tax rate may even be negative for elderly agents. ff: 7
Often pension benefits are more responsive tothenumber ofcontribution years than to life time earnings. 2 As a fourth example, let us therefore consider a case in which future pensions are directly linked to total labor supply during working life. In this case f j and B 0 =0. The resulting implicit tax rate is then ( + r) fj J Λ +g ~fi j = fi ff e j w r( + r) J j The effective tax rate ~fi j is high when productivity is high. Moreover, as contributions in the form of labor supply do not bear interest, the distortion is higher at the beginning of the life cycle. As before, effective tax rates may be negative asfigure shows. As a last rather extreme example, consider a pension system in which pensions are proportional to the final wage in period J Λ. Formally we have B 0 = 0, f j = 0 for j 6= J Λ, and f j > 0 for j = J Λ. Effective taxes therefore amount up to the full statutory level fi for most of an individual's working live andare well below zero in the final period before retirement. Such a system will have an unfavorable performance in terms of labor supply distortions even if it is fully funded. : Figure Two applications of our setup are worth mention. The first concerns the assessment of the pension system's efficiency. The more individual allocations are distorted, the less efficient a system ceteris paribus. A PAYG scheme with a close linkage between taxes and benefits might perform not worse than a fully funded system withaloselinkage. The second potential application is intra generational redistribution between heterogeneous agents. A non constant effective tax rate over the life cycle implies different impacts on people with different life cycle earnings patterns. In our last three examples above, the larger weight on contributions towards the end of the working life implies a redistribution from agents with a flat to agents with a steep labor earnings profile. Our 2 In Switzerland, for example, every one year contribution gap results in a reduction in future benefits by =40 even if aggregate contributions exceed the level that normally qualifies for imum benefits. 8
analysis can, moreover, be extended to analyze the effect of different family structures and corresponding linkages on marginal tax rates. In reality, the important question is how much agents perceive a tax benefit linkage. If they do not, the linkage is virtually ineffective. As we have not assumed anything about the nature of the linkage, our results are equally valid for perceived linkages. Alternative ways to finance the pension system for example by levying consumption taxes or using general revenues necessarily weaken the linkage between contributions and benefits and thus might lessen the efficiency of the pension system. 4 Conclusions We have shown that to a pension system with an individual tax benefit linkage there exists, under certain conditions, an equivalent system without such a linkage. The equivalent system, moreover, provides a simple way to circumvent the problem of an additional state variable for policy simulations in the presence of tax benefit linkages. At the same time we can gain some insight into the pension system's implied marginal tax structure. Although our model obviously cannot capture all possible linkages in the real world, the model is general enough to be applied to investigating the efficiency and intra generational redistribution of (public) pension systems. References Auerbach, Alan J. & Kotlikoff, Laurence J.: Dynamic Fiscal Policy, Cambridge University Press, Cambridge, UK, (987). Bütler, Monika: The Political Feasibility of Pension Reform Options: The Case of Switzerland," Journal of Public Economics, 75(3): 389 46, (975). 9
0.5 0. 0.05 implicit tax rate 0 0.05 0. no linkage past earnings + interests (mimics fully funded) past earnings, unweighted (no interests) total labor supply 20 25 30 35 40 45 50 55 60 65 age Figure : Effective tax rates over the life cycle implied by a number of tax benefit linkages. Population is assumed to be stable and the statutory contribution rate is fi =0:. The chosen parameter values are Φ j = fi j Ψ j, where fi = :0 (yearly) and Ψ j is the probability of surviving to period j (implying a hump shaped discount rate profile). Labor productivity is taken from Swiss earnings data, the interest rate is r=3% per year, and the parameters of the utility function are ff = 4 and = 0:33. The calibration of the model without linkages corresponds to the one in Bütler (2000). 0
Appendix: Non constant interest rates If interest rates are not constant, (2) is replaced by a 0 = 0 a j = ( + r j )a j +( l j ) e j w ( fi)+i [j J Λ ]B c j (0) a J 0 We can then state the following result: Proposition: The optimization problem as stated in equations (), (0), and (3) is computationally equivalent to imizing () without a tax benefit linkage with the following adjustments: Proof: ffl B is replaced by B 0 ffl the constant tax rate fi is replaced by 3 ~fi j fi fff j e j w Y "ψj k= ( + r k )!ψj Q i i=j Λ l= ( + r l)!# : () From (2), we can compute the individual's lifetime budget constraint by recursive substitution as j= Q j ( + l= r l) o n( l j )e j w( fi) c j + Q j ( + B 0: (2) l= r l) 3 While the expression in square brackets could be somewhat simplified, it is chosen not to so in order to avoid notational problems for j J Λ (and f j 6= 0), i.e. for cases in which labor income achieved after the official retirement age J Λ is still taxed.
The second term can be expressed as Q j = = l= ( + r l) B J Q j l= ( + r l) J Q j " ff i= k= ( + r k) ff( l j)f j J + Q j ( l i )f i # + j Y k= ( + r k ) Q j l= ( + r l) B 0 ψj Q i i=j Λ l= ( + r l) l= ( + r l) B 0 (3) Substituting (3) into (2) yields the transformed lifetime budget constraint, j= Q j ( + l= r l) " ( l j )e j w ( ( fi)+ fff j e j w Y ψj k= ( + r k )!ψj Q i i=j Λ! l= ( + r l)!) + I [j J Λ ]B 0 c j 0: The last equation, however, also represents the lifetime budget constraint of an agent without a tax benefit linkage, who faces an age specific payroll tax ~fi j as defined in (). Ξ 2