Roger L. Faith, Department of Economics , Arizona State University


Does Abstention Matter? A significant portion of James Buchanan’s academic career has been devoted to the study of the constraints, legislative and market that limit the power of government. Some of these studies have dealt with legislative constraints such as constitutional limits to taxation and a balanced budget amendment. Other studies have focused on market-based constraints where citizens’ ability to "vote with their feet", or even to secede, affect the distribution of government output and place constraints on governmental behavior. In Buchanan and Faith [American Economic Review 77 (5), 1023], for example, a monopolistic government is constrained in its ability to extract resources from the citizenry by the ability of subgroups of the population to secede. In particular, the opportunity cost to a coalition of secessionists of setting up a new jurisdiction determines the extent to which that same group of persons could be subject to discriminatory taxation. Their model assumed the existence of a monopolistic government that attempted to maximize its wealth by redistribution.

The question asked in Buchanan and Faith was: what limits the ability of the government to use its power to tax for exploitative purposes? This short, exploratory paper asks a similar question but from a different perspective. Assuming a representative democracy, what limits the ability of an elected representative to discriminate against those who abstain from voting? For example, suppose that a certain subgroup population is observed not to vote in the general election for representatives? Would such groups be expected to receive a smaller quantity of government-provided benefits than those groups that do vote? After all, why would a representative work to produce benefits for a group that does not respond at polls? To put it another way, does abstention matter? This paper outlines an analytical framework for addressing these questions. The basic model, though quite simple, seems capable of generating implications with respect to the level and distribution of government output under majority rule when voting is voluntary and when voting is mandatory.

I. The Model.

The model consists of N + 1 agents, an elected representative and N voter/consumers. The elected representative seeks to maximize his or her utility, which depends on two arguments, leisure and votes in the election. Votes, both for and against the representative, depend on the quantity of governmentally supplied output consumed by the voters. Greater government output generally means greater voter support but also requires greater legislative effort on the part of the representative. The more time the representative spends securing output for his constituents the less leisure he consumes. Thus, the representative faces a tradeoff in the form of a time constraint

Given the level of output procured by the representative, each voter decides whether or not to vote and, if the voter decides to participate, how to vote. The decision to vote depends on the degree of voter satisfaction with respect to her consumption of government outcome as compared to some alternative and the cost of voting. How one votes, given one does vote, depends on the whether the voter is better off under the representative output level or some exogenous alternative output level. In our model there is no presumption that those who choose not to vote are necessarily disgruntled citizens.

We will consider two environments. In the first, all voters vote; that is, there is no abstention. In the second environment, voters are free to participate or not to participate in the election of their representative. Abstention is permitted.

a. No abstention.

Let N represent the number of eligible voters in the representative’s district. The representative utility function is U (L, V), where L is leisure measured in units of time V is the representative’s vote margin in the next election. U(· ) has the usual curvature, U is increasing concave function of both L and V. We will assume that the vote margin is completely predictable so that there is no uncertainty about how a person will vote given the level of output.

The number of favorable votes that the representative receives in the next election is a function of the quantity of public output consumed by each of the N voters in the representative’s district. Voter i’s consumption level is denoted xi. For this first exercise, we shall assume that X is a pure public good. Thus, xi = X. Output X maps into votes in the following way. Each voter i has some most preferred, or ideal, quantity of X, which we will denote Zi, given by i’s preferences, which we assume are single-peaked. There also exists an exogenous alternative output level Xa that will automatically be supplied if the current representative is defeated in the next election. Voter i will vote for (against) the representative

if ˝ X – Zi˝ <(>) ˝ Xa - Zi˝ . The number of voters who support the representative is denoted n, and non-supporters are denoted N-n. Next we normalize the population and let f(X) be the distribution of ideal points with its median denoted Zmed. Let F(Xa) be the cumulative density function of f(X). If X < Xa, then the fraction of voters who prefer X to Xa, n/N, equals F((Xa+X)/2), and 1- F(· ) equals the fraction of voters who prefer Xa to X. If X > Xa, then the fraction of voters who prefer Xa to X equals 1 - F((Xa+X)/2), and the fraction of non-supporters equals F(· ) .

The representative produces government output X by allocating some fraction of his time to the legislative process. The more time spent proposing, lobbying other representatives, etc., the greater the output. Let the total time cost of producing X be C(X)X where C(X) is the average time cost of producing the output level X. We assume that C˘ > 0.

The representative’s problem is to maximize U(L, V(X)), by choice of X, subject to the time constraint, T = L + C(X)X, F(Xa), and V = n/N – (N-n)/N. The last two constraints can be combined into V = 2F([Xa + X]/2) – 1. Substituting V into the utility function and maximizing with respect to X, the first-order condition to is: U/ L (XC˘ +C) = 2 U/ V F/ X, or UL/UV = 2F˘ /(XC˘ +C).

If the representative were interested only in votes, the vote-maximizing level of X would occur at, or arbitrarily close to, Xa. )). (If Xa = Zmed, the vote- maximizing point would be Xa. If Xa < (>) Zmed, the vote-maximizing point would be Xa + (-) e , where e is an arbitrarily small number. This is because the fraction of voters voting for the representative increases (decreases) discontinuously at Xa if Xa < (>) Zmed, and declines thereafter. At X= Xa, voters are indifferent between the representative’s choice and the default choice of X, so we assume that n/N equals ˝ at this point.) But, since utility depends on leisure as well as votes, the utility-maximizing level of X need not equal Xa. The actual outcome will depend on f(X), C(X), and U(L, V(X)).

b. Abstention permitted.

The question now becomes: what happens when the act of voting is voluntary? Using the same model as before we add two new considerations. First, we give the representative two choices – how much output to supply and who to supply it to. Second, we impose a cost of voting on all voters. This second consideration now requires more assumptions on the voters decision process.

Rather than engage in the debate of instrumental versus expressive voting, we shall simply assume that a person votes when the expressive benefit from voting exceeds the cost of voting. However, we assume that the benefit from voting is a monotone function of the difference in utility between consuming the representative-supplied output X and some alternative, default output level, Xa. In particular, voter i will vote with positive probability if and only if Bi(˝ Ui(Xi) - Ui(Xa)˝ ) > ki , where B is i’s expressive value of voting function, Ui is i’s utility function and k is i’s cost of voting. More precisely, we assume that the probability that a person votes is a monotone function of (Bi-ki). Obviously, if Ui(Xi) - Ui(Xa) is positive (negative) then i prefers the representative’s (alternative) output level. Further, the larger the absolute differences in utilities the greater the probability that the voter will vote (yea or nay). We shall denote the probability that a person votes for the representative given the representative’s output level, X, by f i(X,Xa,ki) and the probability that a person votes against the representative, g i(X,Xa,ki).

For simplicity, assume there are two classes of voters, favored and non-favored. The n favored voters each receive a common output level x, and the N-n non-favored voters receive no government output. Because all voters only vote probabilistically, the representative’s utility function depends on leisure and the expected vote margin, EV = å i=1 to n [f i (x)] – å j=n+1 to N [g j(0)]. The representative can chooses both size of the favored group n and the level of output X, where X = nx. The representative’s problem is to maximize U(L, EV(X)), by choice of x and n, subject to EV, f , g and the budget constraint, T = L + nxa(x), where a(x) is the average time cost of supplying x. We assume that a˘ > 0.

First-order conditions for maximizing representative utility are:


- U/ L[n(a˘ x+a)] + U/ E [ å i=1 to n [ f i/ x]] = 0, and

- U/ L[ax] + U/ E[ (å f i)/ n - (å g i)/ n] = 0

The second first-order condition says that the representative will expand the coverage of benefits (n) to the point where the marginal disutility of lost leisure equals the marginal expected vote gain as the marginal voter moves from being a "no" voter with probability of voting g to a "yes" voter with probability of voting f . The choice of n, then, depends on the distributions of f i and g j . We can still say some things, however, regardless of the densities of f and g . If (f (x)+ g (0)) is close to zero, then the additional expected votes from extending the range of the public good are near zero as well, while if (f (x) + g (0)) is relatively large, for all k, then the favored group will increase in size. Output x may also be small for these voters. This is because the cost of producing X rises with n. Thus, for a given group size n, x will be larger. This that abstention matters. Those with low probabilities of voting tend to get kept out of the favored group. But, since the representative utility depends on both leisure and expected votes, increasing the size of the favored group increases the expected vote margin but also implies greater time costs, nxa(x). If inframarginal members of n are not very responsive to changes in x, then the representative may reduce x as he increases n. In this case, reduced abstention (at least probabilistically) may reduce the benefits of those with high participation rates. It is also possible that increasing the output of the non-favored group may significantly reduce the probability non-favored voters vote, which benefits them (and for the same reasons as given above reduce the amount of x going to the favored group.) So, it appears that, indeed, the possibility of abstention affects the distribution and size of government-supplied output, but the magnitude and direction of the effect is ambiguous.

Would mandatory voting increase the plight of the non-favored group relative to the voluntary voting equilibrium? Coverage should go up. Since voting is mandatory, each unit increase in n will increase the vote margin by necessarily more than the voluntary case since voting is no longer probabilistic and belonging to the favored group is preferable to belonging to the non-favored group. Since everyone votes there is no return to choosing an output level larger than what would keep the voter indifferent between x and xa , and greater x reduces the representative’s leisure, we would expect x to be smaller and n to be greater under mandatory voting . At the same time, mandatory voting increases total voting costs. Thus, it is it is possible that every voter could be worse off in the mandatory voting equilibrium.

II. Remarks.

1. In our setup, we have assumed that the representative possesses a great deal of information. Individual voter participation probabilities depend on Bi(Xi,Xa), and ki. And we have assumed that the representative knows the probability functions f i and g i. In reality, the representative likely only knows demographic distributions like income, race, education, occupation, and industry. The representative would need to know the probability of voting given X conditional on some observable characteristic and the frequency distribution of the characteristic over the constituent population, and how the observable characteristic affects the valuation of a particular type of government output. For example, income (a proxy for the value of time), or education (a proxy for cost of getting informed) are used to predict voter participation rates. On the output side, observable differences in output may be geographic (certain locations get more government output than others), industrial , occupational, age-based, religion-based and so on. If voter turnout rates for these demographic categories are known, and the distribution of these characteristics in the population are known, the representative is better placed to discriminate among voter groups.

2. The ability of the politician to treat subgroups of voters differentially is limited by the existence of voters just on the margin between voting and not voting. If the distribution of k’s is discrete, then it’s possible that non-voters can be treated less well fiscally than higher-probability voters can. These results are essentially similar to those in the contestable markets literature where the cost of entry of current out-of-market firms affects the pricing of in-market firms. The same is true with respect to voting.

3. In our model, the alternative output level Xa was treated as exogenous. A more a more complex analysis would derive Xa . For example, Xa might represent the rationally chosen output of the representative’s opponent in the next election.

In this case, the choices of Xa and X would represent a Nash-Cournot equilibrium to a two-candidate election game. The standard Downsian result is that both candidates would settle on the median ideal point. It would be interesting to see if that result held up in the current model.

4. Given the ambiguity of the analytical results, it may prove fruitful to assume some specific distributional forms (uniform, normal, etc.) for the probability distributions f and g as well as specific functional forms for the representatives utility and cost functions.