employee contributions to local government ...

0

7he purpose of this arriclr is to corurrucr a rheoretical t~toclelr a l ~ a b kof Abstract assessitt~ the e.rrcttr 10 ~vltichrrqtrired e t t t p l o ~ ecotirribtlriotts ~ r t ~a local gorcrtit~tenrreriret~tenrsysro~tt~ccessirarrincreased kvcls of et~tploj~ee rot~tpensarion. I r isfoutldrhar increased cot~rpcttsationis reqtrirerl ijrlre before-ta.r ~ i e lott i rcrirrntetir sj.stet~tassets is less rkatt rltr before-ra.rj.iek1 o n alrerttarive assets lltal \c,ou/d be /tcltl by rt~tplojees.I t is also fotoitl rho1 such it~crcasrd levels of cot~tpensariongenerally rake rhp/ort~tof incrcasctl pettsiotts ratlter rhatt incrrased \rages. lltes~lrlteorcrical rcstrlrs arc ron~parccl~ c i r l tsot~re recent entpirical restrlrs obtained by Eltrenbcrg.

EMPLOYEE CONTRIBUTIONS TO LOCAL GOVERNMENT RETIREMENT SYSTEMS: A THEORETICAL ANALYSIS OF EFFECTS G E N E E. h f Uhf Y Ohio State University

Recent theoretical and empirical work on local government retirement systems (Mumy, 1978; Ehrenberg, 1978) generates the suspicion that most of these retirement systems contain a counterproductive element. The counterproductive element is that employees are required t o make contributions to the retirement system out of wages.' The suspicion here is that required contributions increase the level of compensation that governments must pay in order to cover their employees' market opportunities. The basic intuition behind this suspicion is based on the fact that employee contributions come out of after-tax income, whereas pension accruals are not taxed until they are realized.2 As a result, an employee can receive the same amount of retirement income while the government can reduce its outlays simply by paying the employees' contribution directly to the retirement system as a pension accrual and reducing wages by a n even greater amount because of the tax deferral that is now received by the employee. I'UIILIC FISASCE QU,\KTERI.\'. O 19x0 Sngc Iauhlication\. Inc.

Vol. X So. 4. October 1980 443-456

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PUBLIC F I N A N C E QUARI'ER1.Y

The purpose of this article is to construct a model that allows us to judge the validity of the basic intuition discussed above. That is, the model will be used t o address the question of the extent to which required employee contributions necessitate increased compensation from employers and, further, t o examine whether changes in compensation take the form of changes in wages or in pensions. This will be accomplished by constructing a model of wage and pension equilibrium in Section I and then examining the comparative statics of this equilibrium with respect to changes in the level of employee contributions in Section 11. Section I11 contains a comparison of the theoretical results obtained in Section I1 with some recent empirical results obtained by Ehrenberg (1978).

I. THE MODEL'

In our model, let us assume that employees are hired for one period, after which they retire and collect retirement benefits for one period. In a particular geographical area, there is a perfectly elastic supply of identical employees to the local government at utility level U, where

with CIand Cz being private consumption expenditures during the period of employment (period I ) and the period of retirement (period 2), respectively. T o consider the wage-pension combinations that will keep a n employee at utility level we begin with the employees' maximization of equation 1 subject to the following budget constraints for periods 1 and 2, respectively:

c,

where W = the current wage income received by an employee during period 1; R = the fixcd amount an employee is required to contribute to the

-

rctirement system; S = the amount of discretionary saving by an employee; TI = the amount of taxes paid by an employee during period I; P = the pension benefits reccived during period 2;

r, = the rate of return earned by the rctirement system on its assets: r, = the rate of return earned by employees on their discretionary savings; and T2 = the amount of taxes paid in period 2. It is important t o note that benefits from the retirement system are broken into two constituent parts: the accumulated value of the employee's own contribution and the employer's pension properly defined. A n important part of the model is the tax structure which determines T1and TI.A progressive rate structure in this model can be described by

and

where li is the amount of taxable income in the iIhperiod. According t o the tax provisions covering pension accruals. we have 11 = W and

12

=P

+

r s + r,S.

Given this description of the tax structure, it is important t o note in connection with equations I a n d 2 that a n employee*^ required contribution t o the retirement system and discretionary savings are taken from after-tax wage income, while accruing pension benefits are not taxed during period I. Now, by solving

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PUIlI.IC I:INANCE Q U A R T E R L Y

equation 2 for S and substituting the result into equation 3 and the expression for 11, we obtain the intertemporal budget constraint

and

T o maximize utility subject to the intertemporal budgel constraint, we form the Lagrangian

and set its partial derivatives with respect to C1 and C2 equal to zero. Combining these two equations gives

where r: is the marginal net yield on a n employee's discretionary savings and is given by

Equations 1, 4, and 5 define a n indirect utility function

u = v (P,W)

V, = A ( l - Ti)

From the supply side of the market, we know that

U - U (Cl, C2) = 0.

PI

so for a wage-pension function, W = f(P), that generates it follows that

6,

which results in

Equation 7 is the wage-pension trade-off, and d W / d P is negative as long as the marginal tax rates, T i and T5, are less than 100%. which is the ease. It can also be shown that the curvature of the wage-pension function is such that d 2 w / d p 2> 0 because a s the pension size increases and the current wage decreases, the employee is moved into a higher period 2 marginal tax bracket and a lower period I marginal tax bracket. As a result, the value of the tax. deferral on pension accruals decreases. This means that incremental pension increases result in current wagc decreases of progressively smaller amounts.' A representative wagepension function for employees is shown in Figure 1, where W, is the period I wagc income necessary - t o compensate a n employee without a pension when U = U and R T o determine the equilibrium values of W and P we must now briefly consider the behavior of the governmcntal employer. The government has to pay each employee a current wage of W and provide a pension of P during retirement. I~owcver,the government can currently fund a n employee's pension by contributing P/I + r, dollars to the retirement system because r, is the tax-free rate at which the system's asscts accumulate. As a result, the present value of unit labor costs is given by

=x.

Let us assume that one of the government*^ goals is to minimize unit labor costs. By taking the derivative of L with respect to P and setting the result equal to zero, we obtain

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IDUBl.IC FINANCE QUAHTER1.Y

Figure 1

By substituting for dW/dP from equation 7, wc obtain the condition that

Equation 9 means that the optimum pension size is reached when an additional dollar increase in pension size leads to an additional reduction in current wages, which, if contributed to

hfumy / Ehflll.OYEE CONTRIBUTIONS

419

the retirement systcm, would accumulate a t a rate just sufficient to meet the additional pension liabilities. Until this point is reached, additional wage rcductions accumulate at a rate greater than that ncccssary to meet thc additional liabilities, thus allowing the government to reduce its labor costs. We now have four equations, 4,5,6, and 9, which are functions of CI, C2, W, and P only and can be solved simultaneously for the optimizing values C t Cz? \V*, and P*. With these four equations, then, we can obtain comparative static results for variations in the levcl of employee contributions, R.

11. THE EFFECTS OF EMPLOYEE CONTRIBUTIONS

T o analyze the effects of employee contributions on Clf ~ 2 f W*, and P*, we use comparative statics t o compute d ~ ? / d K , d ~ ; / d R , dW*/dR, and d ~ * / d K where , d R is the variation in the required levcl of employee contributions. We have seen that W*, and P*. equations 4, 5, 6, and 9 can be solvcd for Clt We can now rewrite these equations in general functional forms as equations I' to 4'. respectively, where:

CZ

If we take the total differentials of equations 1'to 4'with respect to C I , Cz, W, P, and x , w e know that the equilibrium conditions 1' t o 4' are maintained if

We can use Cramer's Rule t o solve equations 5' for d y / d k , where y = CI, C2, W, o r P. Now define D a s the determinant of

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PUBLIC FIXAKCE Q U A R T E R L Y

the partial derivatives o n the lefthand side of equations 5'and A: a s the determinant ivhich is formed by replacing the appropriate F; column in D with the elements -Fk. By Cramer's Rule we have

T h e solutions t o 6' turn out t o be extraordinarily simple if \ve actually compute the ~b a n d -Fk elements and compare them. By making these computations we find:

W e can now observe that -FLk = Fb(rt - rs). It is known that if each element in a row o r column of a determinant A is multiplied by a constant, k, t o form the determinant A*, then k A = A*. This means we can rewrite equation 6' a s

where A, is now formed by replacing the elements of the appropriate F: column with the elements F;. Now consider the solution t o equation 7' for the cases where Y # P. If we expand A, by the co-factors of the elements in the column that we have substituted for in D, then it is clear that this is a n expansion of D by alien co-factors because we have replaced F; with Fa. That is, we a r e multiplying the co-factors

of the elenients Ff by the elements As a result. it follo\\*sthat A, = 0 a n d we obtain the result that ~ 1 1 .

O n the other hand, \\@heny = I' it is clear that A,=, = D because A,=, is obtained by replacing the elements F; in D with the elements Fa. The resulting solution for equation 7' is

We have now obtained the important result that W* is invariant with respect t o variations in R, whereas the variation in P* depends on the magnitudes of r,, the rate of return earned on a n employee's discretionary savings, and r,, the rate of return earned o n a n employee's contribution t o the retirement system. It turns out, then, that we cannot say a priori that required contributions necessitate an increased level of employee compensation. T h e reason for this is that required contributions simply cause employees t o substitute retirement system assets for discretionary assets in their portfolios, and increased compensation will only be necessary if this substitution causes a decline in investment yield. I n the case where r, = r,, there is no yield decline and d ~ * / d R= 0, and, of course, if r, \\.ere actually greater than r,, then P* would even decline. However, there may still be reason to believe that required contributions necessitate higher levels of employee compensation. It might well be the case that r, is less than r, because of conservative norms of retirement system asset management and statutory restrictions that limit ~ this case, the investment authority of system r n a n a g e r ~ . In d ~ * / d R= r, - r, > 0,s o the optimum pension size increases as the required contribution increases. The above results can be given additional intuitive appeal in conjunction with Figure 2. In the initial situation, the employee receives W*, I>* and makes a required contribution of E.This essentially gives the employee a n endowment a t point A in the (CI, CZ)space of Figure 2, with CI = W* - TI - R and Cz = P* +

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PUBl.IC FINANCE QUARTERLY

Figure 2

r,R - T2.The employee can move from point A along the intertemporal opportunity locus dd'. T h e locus dd' is concave from below because a s savings increase the increase in period 2 income increases the marginal tax rate, thus decreasing the net yield on savings, r:. The employee moves from point A t o point B, the point of intertemporal consumer equilibrium, and consumes (CI: which generates the necessary utility level Now suppose that increases by one dollar. If W* remains unchanged, the employee can still attain Cf by reducing discretionary savings, S, by one dollar. When r, = r, this alteration in the employee's portfolio has no effect on period 2 income, s o the employee can also attain ~ F e v e nwithout a n increase in P*. O n the other hand, when r, < r, the shift of the one dollar from discretionary savings t o the retirement system results in a re-

a,

u.

hlumy / EhlPLOY EE CONTRIBUTIONS

453

duction of period 2 income by the amount (r, - r,), and as a result P* has to be increased by exactly that amount to enable the employee to attain Cf as well C: and hence utility level 0. In either case, period 2 taxable income remains the same, s o point B remains the point of consumcr equilibrium because the slope of dd' is unaltered. O u r comparative static results have shown that the effect of employee contributions on pension size is ambiguous and essentially a n empirical question. On the other hand, it is analytically unambiguous that employee contributions should have no effect on wages. This is a result that needs to be reconciled with some recent empirical results obtained by Ehrenberg that seem to suggest that required contributions t o the retirement system cause wages to increase.

111. A COMPARISON OF THEORETICAL A N D EMPIRICAL RESULTS

In recent empirical work, Ehrenberg has attempted t o ascertain the determinants of uniformed public employees' wages. In what follows, data and statistical problems are not at issue; rather, we will be interested in the analytical implication of Ehrenberg's estimates for the model developed in Section 11. Since we are interested only in the impact of retirement benefits and employee contributions. we can represent Ehrenberg's estimating equation in terms of our model as being

where X Iis the proportion of retirement benefits to wages and X1 is the proportion of employee contributions to wages. The constant term cro summarizes the impact of all other independent variables on wages. Equation 10 should be thought of as an analytical structure where we import Ehrenberg's results to give us the signs of a1 and al. T h e important result for us is that Ehrenberg estimates that a*is positive. This seems to imply that as employeecontributions increase, employees must be compensated with a higher wage.

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PUULlC 1:INANCE QUARTERLY

This implication is contrary to thc thcorctical rcsult, which indicates that d w / ~ E = 0. O n closcr cxamination. Iiowever. these seemingly disparate thcorctical and cmpirical results can be rcconciled. In terms of o u r modcl, Ehrenbcrg's variablc X2 is given by

In addition, recalling that rctirement benefits consist of pensions and the accumulated valuc of employee contributions, wc have

By substituting for X Iand XI in equation 10, we obtain

If a 2 is positive, we have shown only that the partial derivative of W with respect to X2 is positive. The result which is relevant for our theoretical model, howcvcr, is to take the total derivative of W with respect to in cquation 10' and to ascertain its sign. Taking the indicated total derivative yields

= 0 and If we now substitute our theoretical results that d ~ / d R d ~ ' / d E = r, - r,, we arrive a t

We can now see that Ehrcnbcrg's estimate of a2 > 0 does not subvert our theoretical results if the parameter restriction of cquation I I holds. This, of course, requires a1 < 0. T h c predicted sign of a' is negative, but Ehrenberg's estimates of a ~under ,

different definitions of the variiibles, are rarely significantly different from zero. However, whcn they are significantly diffcrcnt from zero they are negative. I t turns out then, that the theoretical results in Section I1 are plausible even in light of Ehrenberg's empirical work. I V . CONCLUSION

We have now seen that employee contributions to the retirement system require increased compensation only if the contributions are forcing employees to substitute lower-yield assets for higher-yield assets in their portfolios. If this is the case, we have also seen that the increased compensation will take the form of a pension increase sufficient to cover the yield loss while wages are unaffected. The policy implications of this are relatively clear. First, retirement systems should attempt to hold portfolios of assets that have a yield at least equivalent to the yield on employees' discretionary portfolios. If this is not possible, however, the second implication is that government costs can be reduced by eliminating required employee contributions to retirement systems altogether.

NOTES I. According to Blcakney (1972: 26). over 90% of the members of public systcms arc rcquircd t o make contributions. It is interesting t o note that only about 25% of the members of private system a r c rcquircd t o make contributions. 2. hfost of thc relevant tax provisions relating t o retirement systcms can be found in sections 401-404 of the Internal Revenue Code. For a brief discussion of the tax provisions, see Mumy (1978). 3. As in the u s e of Oakland (1978). the results in this section arc obtained with a more detailed specification of the model in Mumy (1978). 4. It is generally the case that investment earnings arc credited t o a n employee's contribution account at a ratc equivalent t o the ratc of return earned by the system a s a whole. See Blcakney (1972: 32). 5. For rigorous proof of the curvature of the wage-pension function, scc Oakland (1978: Appendix). 6. See Blcakncy (1972: ch. 7).

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REFERENCES BLEAKNEY. T. P. (1972) Retircmcnt Sjrtcrns for Public Ernployccs. Homewood, 1 ~ : Iruin. EHRENBERG, R. G. (1978) "Rctiremcnt systcrn chzracteristicsand compensating %age diffcrcntials in thc public scctor." Prcscntcd a t theannual mcctings of the Econometric Society, August. hfUhlY. G. E. (1978) "The economics of local govcrnmcnt pensions and pension fund. ing." J. of Political Economy 86. OAKLAND. W. H. (1978)'Thc tradc-off bctwccn public cmploycc pensionsand wages." (rnirneo)

Gene E. hlunly is currently Assistant Professor of Economics at the Ohio State University and tvas previously on the economics faculty of the Virginia Poktechnic Institute and State University. He bvas educated at the University of Colorado (B.A.. 1970)and the Johns Hopkins University (Ph. D..1974). His main research areas arepublicfinance andurban economics. HHir current research deals with the impact of employee pensions on capital accunrulation and economic growth.