Public Choice96: 381–393, 1998. 381c 1998Kluwer Academic Publishers. Printed in the Netherlands.

Predicting the institutional effects of term limits

DANIEL FRANKLIN & TOR WESTIN1Department of Political Science, Georgia State University, Atlanta, Georgia 30303–3083U.S.A.;2Alexander & Alexander Consulting Group, Inc.

Accepted 21 March 1996

Abstract. In this paper we develop a model to predict the seniority turnover, and transitionconsequences of term limit reforms for any institution with a regularized procedure for rotatingmembership. With this model we can predict the number of members who will be serving intheir last term at any given time once an institution reaches a stable state under term limitreforms. For example, our results show that for the U.S. Senate current term limit proposalswill result in a substantial increase in the number of “lame duck” members and a significantreduction in average seniority. We make no claims as to the public policy effects of termlimit proposals. However, our model can be used to design a proposal that will maximize anybenefits or minimize any public policy effects found to be associated with term limit reforms.

1. Introduction

The United States Federal Government and a number of states have movedto limit the terms of elected public officials. While it is not clear what theeffect of these laws will be or have been in the case of the Presidency or inthe states, these reforms are promoted on the general assumption that politi-cians who are not contemplating a career in office will act in a more “publicspirited” manner (See Lott and Bronars, 1993). In addition, it is assumed thatterm limits will encourage reform through the banishment of incumbents andtheir replacement by political amateurs with “fresh ideas” (Fiorina, 1994;Petracca, 1993; Frenzel, 1992).

While these assumptions are clearly subject to challenge, whatever the con-sequences of the limitation of terms, those effects will be most keenly felt ininstitutions or under term limit plans that produce members who, on average,are more junior and more likely to be “lame ducks”. In this regard, much ofthe work that has been done in the modeling of term limits has been focusedon specific institutions (Reed and Schansberg, 1994; Foster, 1994; Thomas,1993) and on the public policy effects of term limits for specific groups andindividuals (Gilmour and Rothstein, 1994; Thompson and Moncrief, 1993;Gay, 1993; Kristol, 1993; Dick and Lott, 1993; Crain and Tollison, 1993).

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The most direct modeling of term limits to date has been done for theHouse of Representatives by Reed and Schansberg (1994). In their work onthe House they conclude that should congressional turnover persist at ratesequal to the 99th through 102nd congresses, the average expected service ofa member of the House serving under term limits would be 6.32 years, downfrom the 17.8 years a freshman member can expect to serve under the currentregime (1994: 82). Furthermore, at any given time under term limits, justover 20 percent of the membership of the House would be serving in theirlast term (1994: 86).

For this article we propose to go beyond the House of Representatives indesigning a general model for forecasting the institutional effects of term lim-it recommendations. Such a model should be useful for predicting the resultsof any term limit proposal, in any institution, at any turnover rate. We makeno claims as to the public policy consequences of term limit reforms. How-ever, if there can be identified positive and negative consequences of termlimits, our model will be useful as a tool for designing term limit proposalsthat maximize the positive and minimize the negative effects of such reforms.

2. Methodology

There are a number proposals for term limits in Congress, the most promi-nent of which imposes a two term limit on the Senate and six term limiton the House.1 While it is not entirely clear what legislation would emergefrom Congress and be ratified by the states, for the purpose of this paper it isassumed that when term limits are imposed, membership seniority will not becounted retroactively and that the term limit clock will begin to count startingfrom the first election after ratification forbothnew and current members.2

Thus, there will be a transition period in which a declining number of mem-bers will have seniority of greater than two terms.

3. Turnover

In modeling the turnover effects of term limits, we find that the general equa-tion produced is a “geometric progression”. Table 1 shows the pattern ofturnover under a two term limit. The columns in Table 1 represent electoralcycles and the rows represent the terms in which members serve. For exam-ple, the intersection of the third column and the second row denoted N2;1 * r,is the number of members serving in their second term in the third electoralcycle after a two term limit has been imposed. It should be noted that because

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Table 1. Modeling a two term limit

T1 T2 T3 T4

Term 1 a N2;1 N2;2 N2;3

Term 2 0 a * r N2;1 * r N2;2 * r

Term 3 0 0 0 0

Table 1 models a two term limit, all cells in rows of three and higher containzeros.Let a be the number of members serving in their first term after the term limitclock has been imposed.3 At T2, the fractionr of the originala membershipremains, while N2;1 members left office after one term. Sincea * r membersare reelected, N2;1 = a(1 - r) members need to be replaced. At T3, thea *r members who were serving at T1 and T2 are forced to leave due to theterm limit restriction, while a fractionr of the new members at T2, N2;1, stillremain at T3. Thus, the number of members who have to be replaced at T3,N2;2, equalsN2;1(1 - r) + a * r . SinceN2;1 = a(1 - r), the expression for N2;2

can be simplified to:

N2;2 = a(1� r)(1� r) + a� r = a(1� r + r2) (1)

Expressions for N2;3, N2;4, ... can be derived in the same manner:

At T2 N2;1 = a(1� r)T3 N2;2 = a(1� r + r2)T4 N2;3 = a(1� r + r2� r3)T5 N2;4 = a(1� r + r2� r3 + r4)Tu+1 N2;u = a(1� r + r2� r3 + r4� r5 + ::: + ru);

where u is an even numberTv+1 N2v = a(1� r + r2� r3 + r4� r5 + :::� rv);

where v is an odd number

(2)

The expressions for N, above, are examples of a geometric progression. Thegeneral form of a geometric progression is:

S = a+ ar+ ar2 + ar3::: + arn�1

= a(1�rn)1�r

(3)

A geometric progression with a negative value for “r” produces an oscillat-ing curve with decreasing or increasing amplitude depending on whether theabsolute value ofr is less than or greater than one. Because death, retire-ment, and reelection rates, which together determine the replacement rate,

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will always produce a fractional rate of less than one, the geometric progres-sion in the case of term limits produces an oscillating curve of decreasingamplitude. So that with a two term limit:

N2 = a� ar+ ar2� ar3 + ar4� ::: (4)

where, for term limits:

NL = Number of new=leaving membersL = Maximum number of terms alloweda = Number of seats up for electionr = f1f2=Retention rate1� r = Replacement ratef1 = Fraction of members who seek reelectionf2 = Reelection rate

(5)

Using Equation (3), the expression for N2 can be written:

N2 =a[1� (�r)n]

1+ r(6)

While the oscillation of the curve never mathematically disappears, the curvereaches an infinitesimal amplitude (at current retention rates in the Senate) bythe end of 10 electoral cycles. This occurs because the term r10 approacheszero. Therefore, if we allow the number of electoral cycles to go to 10 andbeyond, Equation (6) will reduce to:

At T10:::1 :

N2:= a

1+r

(7)

For example, after 10 elections at historic turnover rates in the Senate4:

N2;10 = 33[1�(�:8879�:7620)10

1+:8879�:7620 = 19:29

N2;1

:= 33

1+:8879�:7620 = 19:68

* a(r10� r11 + r12� :::) � 0:4

(8)

Because only a third of the Senate is up for reelection in any given electoralcycle, the results in Equation (8) represent only the number of Senators whowould have to leave at the end of a Congress. In the other two Senate electioncohorts not up for reelection, however, there would still be between 19 and 20members serving in their last term. Thus, under a two term limit, lame duckmembership in the Senate will stabilize at somewhere in between 57 and 60percent (members), 60 years (10 electoral cycles) after the implementationof term limits.

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4. Three term limits and beyond

Gilmour and Rothstein (1994) in their study of the partisan effects of termlimits suggest that it would be possible to manipulate the effects on the insti-tutional level of term limits by increasing or decreasing the number of termsthat members can serve. For example, the more terms provided under a termlimits proposal, the more the effects of term limits approach a nullity. Inorder to determine the consequences of differing term limit proposals, wewill need to develop a generic model for term limits that will predict turnover(and seniority) rates. The following model is presented as a generic equationfor determining the institutional effects on turnover of different term limitproposals. The pattern of replacement under a two term limit after the firstelectoral cycle is:

N2;1 = a(1� r) (9)

After the second electoral (T2) cycle:

N2;2 = a(1� r + r2) (10)

In general, after the nth (Tn) cycle:

N2;n = a(1� r + r2� r3::: + =� rn) (11)

The last term in Equation (11) is positive or negative ifn is even or odd,respectively.

As we have seen, the general expression for the two term limit (once sta-bility is achieved) is:

N2 =a

1+ r(12)

The progression of turnover under a three term limit is developed in Table 2.

Table 2. Modeling a three term limit

T1 T2 T3 T4 T5 T6 T7 T8 T9

Term 1 a N3;1 N3;2 N3;3 N3;4 N3;5 N3;6 N3;7 N3;8

Term 2 0 ra rN3;1 rN3;2 rN3;3 rN3;4 rN3;5 rN3;6 rN3;7

Term 3 0 0 r2a r2N3;1 r2N3;2 r2N3;3 r2N3;4 r2N3;5 r2N3;6

Term 4 0 0 0 0 0 0 0 0 0

Deriving expressions forN, the number of new/leaving members, using thesame method as in the case of the two term limit, we find:

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386

At T2 N3;1 = a(1� r)T3 N3;2 = a(1� r)T4 N3;3 = a(1� r + r3)T5 N3;4 = a(1� r + r3� r4)T6 N3;5 = a(1� r + r3� r4)T7 N3;6 = a(1� r + r3� r4 + r6)T8 N3;7 = a(1� r + r3� r4 + r6� r7)

(13)

The general expression for the three term limit, after stability has beenachieved, can be simplified to:

N3 =a

1+ r + r2 (14)

Appendix A outlines the derivation of the three term limit expression in Equa-tion (13).

The general formula for the turnover consequences of term limits wouldbe:

NL =a

1+L�1Pn=1

rn

(15)

For example, under a three term limit in the Senate:

N3 =33

1+ (:8879� :7620) + (:8879� :7620)2 = 15:46 (16)

Under a three term limit plan, between 45 and 48 Senators would be in theirlast term, 60 years after the imposition of term limits. The more terms that areadded to the limit, the closer the equilibrium point comes to no term limitsat all. Because the expression r10 is very close to 0 at current retention rates,any term limit plan will stabilize at T10 as long as retention rates remain thesame.

5. Transition

It should be noted, however, that term limits may, themselves, affect reten-tion rates. Term limits may reduce the likelihood of voluntary retirementand/or effective opposition in the next election. Or the opposite may be true(Reed and Schansberg, 1995; Greene, 1995; Jacobson, 1995; Oppenheimer,1995; Upshaw, 1995; Reed and Schansberg, 1995b). In any case, whateverthe retention rate, and the percentage of lame duck members and, thus, their

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387

average seniority will be affected accordingly. First, the higher the retentionrate, the longer the system will take to stabilize. At retention rates of 90 per-cent or more thern term only approaches zero after about 25 electoral cycles(50 years in the House and 150 years in the Senate). At retention rates of70 percent, however, the system stabilizes at 9 terms (18 years in the Houseand 54 years in the Senate). Second, depending on the retention rate, theHouse will experience what has become known as the “superclass” phenom-enon when the first congressional cohort to be placed under term limits isforced to retire. At a 90 percent retention rate, 275 members (63 percent ofthe membership) will be forced to retire six terms after the first imposition ofterm limits.

Figure 1 is the graphic representation of the transition period for our termlimit model (six terms) for the House at 70, 80, and 90 percent retention rates.

It is apparent that there is an inverse relationship between the overall levelof seniority produced by term limits and the “superclass” effect. The high-er the retention rate, the larger the superclass. The opposite would also betrue. Of course , we have no way of knowing with certainty what would beretention rate under term limits. However, once these effects become appar-ent either the term limits, themselves, or the way that they are imposed can beadjusted for a more desirable result. It should also be noted that because of itsshorter term and despite its higher retention rate, the House would stabilizemore quickly than the Senate.

6. Seniority

Under term limits, average seniority will be lowered as well. If relativelyjunior members are less likely to be under the thumb of special interests andare more likely to be fresh and creative in their thinking, term limit propos-als that generate a more junior membership, will be more desirable. WhileReed and Schansberg (1994; 1995) do not address this question directly, theyestablish that in the House of Representatives, under a six term limit, afterthe body reaches a term limit equilibrium, members will serve for an averageof 6.1 years, markedly lower than the 13–17 year expected career of a newlyelected member today.

In order to provide for a general model of the seniority consequences ofterm limits, we focus on average seniority rather than career length. After all,it is the relative inexperience of members serving under term limits that issupposed to be such a benefit. Because average seniority is dynamic, increas-ing for all members across the course of the congressional session, we havedecided to model seniority at the mid-point of the two year congressional

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Fig

ure

1.

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389

cycle. Our model can just as easily be applied at the beginning or end of acongressional term.

Table 1 (on turnover) above can be used to derive a formula for seniority ina two term limit case. For example, at time T1 there area members servingin their first term. At midterm, seniority is t/2 (wheret is the term length). AtT2, there area * r members serving in their second term. Their total seniorityat midterm isa * r * 1.5t. There are alsoa(1–r) serving in their first term.Their total seniority at midterm isa(1–r) * 0.5t. Since the total number ofmembers isa, the average seniority for the whole body is:

a� r � 1:5t+ a(1� r) � 0:5ta

= t(r + 0:5) (17)

It is possible then to show the following progression:

At T1 Sm1 = t(r + 0:5)T2 Sm2 = t[r� r2 + 0:5]T3 Sm3 = t[r� r2 + r3 + 0:5]Tp Smp = t[r� r2 + r3� r4 + :::� rp + 0:5]Tq Smq = t[r� r2 + r3� r4 + ::: + rq + 0:5]

(18)

where

Tn = Electoral cycleSmn = Seniority at midtermt = Term lengthL = Term limitf1 = Percentage of members who seek reelectionf2 = Reelection rate(for those seeking reelection)r = (f1=f2)=Retention ratep = An even numberq = An odd number

(19)

Once stability is achieved, we derive the following formula for average midtermseniority under a two term limit (For the derivation of this equation, seeAppendix B):

S2m = t�

1:5�1

1+ r

�(20)

Thus, for the Senate, pursuant to a two term limit, the following formuladetermines the average seniority of the members at the midpoint of theelec-toral term5:

S2m(Senate) = 6�

1:5�1

1+ :8879� :7620

�= 5:42 (21)

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390

This represents a substantial reduction in seniority in the Senate from themean rate in the 102nd Congress of 11.1 years (Ornstein et al., 1992: 21).Thus, if term limits are intended to produce a more junior membership, theyachieve the desired effect. In fact, a two term limit might overshoot desiredlevels of reduction in seniority. Those who are interested in term limits mightwant to design a reform that will not have the effect of producing so amateura legislature. Consequently, we have produced a general equation for pre-dicting the institutional effects of term limits on seniority, where “L” equalsthe maximumnumberof terms allowed under any term limit plan (see alsoAppendix B for the derivation of a formula for seniority under any multipleterm limit plan):

SLm = t

266641:5�

1�L�1Pn=1

(n� 1)(r)n

1+L�1Pn=1

(r)n

37775 (22)

Average Seniority during the transition period to term limits will mirrorthe pattern of transformation in turnover.

7. Conclusion

Without entering the debate on the public policy effects of the impositionof term limits, we have developed a generic model for predicting the insti-tutional consequences of proposed term limit reforms. If the desire of termlimit proponents is to produce an institution with lower average seniority anda greater number of members serving in their first and last terms, term limitproposals achieve the desired results. Using our model, term limit proponentscan “fine tune” their proposals to produce more or less of the desired result –assuming that reelection and retirement rates remain the same. Indeed, termlimit proponents can even use our equation to model the consequences ofchanges in turnover rates that may be the indirect consequence of term limitreforms.

The fact remains, however, that the most important consequence of termlimit reforms will be the public policy effects. While we remain skeptical ofthe benefits of term limit reforms, our attempt to model term limit proposalspresents us with a number of options that could serve as alternatives to currentterm limit proposals.

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Appendix A

Term limit model for a three term limit (stable state)

N3 = a(1� r + r3� r4 + r6

� r7 + :::)

= a� a(r + r4 + r7 + :::) + a(r3 + r6 + r9 + :::)

= a� [(ar)(1+ r3 + r6 + :::)] + [(ar3)(1+ r3 + r6:::)]

= ah1� (r� r3)

�1

1�r3

�i= a

h1� r(1�r2)

(1�r)(1+r+r2)

i= a

h1� r(1+r)

1+r+r2

i= a

h1+r+r2

�r(1+r)1+r+r2

i= a

h1

1+r+r2

i

(23)

Appendix B

General model for stable state seniority under term limits

Midterm seniority under a two term limit:

S2m =�

0:5t 11+r

�+�

1:5t r1+r

�= 0:5t

1+r(1+ 3r)

= t1+r [0:5(1+ r) + r]

= t�

0:5+ r1+r

�= t

h1:5� 1+r

1+r +r

1+r

i= t

h1:5� 1

1+r

i

(24)

Midterm seniority under a three term limit:

S3m =�

0:5t 11+r+r2

�+�

1:5t r1+r+r2

�+�

2:5t r2

1+r+r2

�= t

�0:5+ r+2r2

1+r+r2

�= t

�1:5� 1�r2

1+r+r2

� (25)

Using the same methodology we can derive:

Midterm seniority under a four term limit:

S4m = t

�1:5�

1� r2� 2r3

1+ r + r2 + r3

�(26)

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Notes

1. See H.J. Res. 38, 103rd Congress, 2nd Session for the six term limit in the House and H.J.Res. 160, 103rd Congress, 2nd Session for the three term limit in the House.

2. The Grandfathering of current members under any term limit plan is the current positionof the House Republican Conference. Brenda Benjamin, House Republican Conference,Telephone interview, November 16, 1994.

3. A number of members will have been serving under the previous no-term limit regime atthe beginning of this period. For the purpose of Table 1, they are serving in their first termat T1.

4. On average (1946–1990), 29 (of 33/34) Senators seek reelection in any given electoralcycle. Of the 29 members who run for reelection, 76.2 percent are successful. On average,this means that 22 of 33/34 Senators don’t retire and survive the electoral process. Thus, onaverage, there are 11 to 12 new members of the Senate in each new Congress without termlimits. See Norman J. Ornstein, Thomas E. Mann, and Michael J. Malbin,Vital Statisticson Congress: 1991–1992(Washington: CQ Press, 1992) pp. 59.

5. The Senate presents an unusual case in determining average seniority in the sense thatonly one third of Senators stand for re/election in any given electoral cycle. Given genericturnover rates, average seniority within a cohort remains the same, but presents with muchgreater amplitude, the length of terms within a cohort being six years. Thus, the average,midterm seniority within a cohort is still 5.42 at current replacement rates, but startingat 2.42 years at the beginning of their terms and ending at 8.42 years. For the Senate asa whole, the average seniority will vary between 4.42 and 6.42 years, with the lowestvalue at the beginning of the term and with the highest value at the end. Every two years,between elections, the average seniority is 5.42 years.

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