Abstract
I develop models in which a minimum winning coalition decides on the level of government spending, where the Coase theorem holds amongst members of the winning coalition. An increase in the supermajority requirement has potentially conflicting effects on spending. A higher requirement increases the tax price internalized by the minimum winning coalition, but also increases the number of districts included in this coalition. I develop separate models in which the spending in question consists of (i) a nonexcludable good, (ii) a distributive consumption good, (iii) infrastructure spending and (iv) a transfer payment. A supermajority rule has no effect on spending for nonexcludable goods and ambiguous effects on spending for distributive projects and infrastructure spending. An increased supermajority requirement does unambiguously reduce transfer spending. I also relate the supermajority rule to the law of 1/n. If the Coase Theorem holds and a minimum winning coalition forms, an increase in the number of districts n has precisely the same effect on overall expenditure as a decrease in the supermajority requirement. Thus, the ambiguous spending effects stemming from supermajority rule carry over into this version of the law of 1/n.
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Notes
Supermajority requirements can reduce spending in other ways by, for example, raising transaction costs and thereby reducing the chances that any agreement on new spending projects can be reached. While such a mechanism is potentially important, it lies beyond the scope of this paper. See Buchanan and Tullock (1962, pp. 85–116).
In some cases, non-excludability may arise from constitutional considerations requiring equal treatment of citizens. Note that throughout the analysis in this paper, districts are homogenous. If districts are heterogeneous, then a tuition subsidy might provide partially nonexcludable benefits if some districts send more students to college than others.
Lee (2016b) uses a similar model to motivate his empirical study of the interaction between a supermajority rule and a bicameral legislature.
Lee (2016a) considers a standard form of the law of 1/n in which the Coase theorem does not hold among members of the winning coalition. He then analyzes the interaction between the law of 1/n and supermajority rule. My point is that when the Coase theorem holds within the winning coalition, the law and 1/n and supermajority rule are equivalent, such that an increase in the number of districts is the same as a reduction in the supermajority requirement.
Existing theoretical and empirical work analyzes how supermajority rule affects taxation. Knight (2000) finds that supermajority rules lead to lower taxes. Heckelman and Dougherty (2010) find that supermajority rules reduce the prevalence of taxes levied on narrow bases, which tend to be redistributive, but have no effect on more broad-based taxes. The result, presented in the current paper, that redistribution unambiguously declines with the supermajority requirement appears to be consistent with that empirical finding. Gradstein (1999) develops an intertemporal model in which a supermajority rule may be a mechanism credibly to commit to a low future tax rate.
Replacing nβ in (1) with a more general functional form will have no effect on the results.
As noted previously, the result from this section is similar to the result derived in Lee et al. (2014), who do not derive their result from a utility function. Rather, the authors appeal to a reduced-form demand equation.
Infrastructure also is analyzed in McGuire and Olson (1996) in the context of their analysis of democracy and autocracy. A single “district” is assumed in their model, but what varies is the stake the ruling coalition has in aggregate output. By contrast, here infrastructure spending is directed to the district level, where it determines district level output.
Every single term in (13) is multiplied by n, so that term drops out of the expression.
Equation (14) implies that (1 − t) and F(g−z) move in opposite directions. Thus, it seems unlikely that a sufficiently large increase in their product could offset the direct effect of an increase in z on the right-hand side of (15). Nevertheless, I am unable to verify that that intuition always holds in a comparative static exercise.
The theory of the law of 1/n has been amended in the subsequent literature. See, among others, Baron (1991), Chen and Malhotra (2007) and Primo and Snyder (2008). Chen and Malhotra amend the basic theory to take into account the fact that most legislatures are bicameral. The resulting empirical literature is quite large. See, among others, Bradbury and Crain (2001), Gilligan and Matsusaka (2001), Pettersson-Lidbom (2012), Crowley (2015) and Hankins (2015). While the empirical findings on the law of 1/n have been quite mixed, recent work has found either insignificant results (Bel et al. 2017) or a negative relationship between legislature size and spending (Höhmann 2017).
If it is believed that universalism prevails in spending decisions, supermajority rules are irrelevant. If universalism prevails, but the Coase theorem holds within the (universal) winning coalition, changing the number of districts will have no effect on spending.
Of course, the supermajority requirement can take on any value of z = 0.5 to z = 1.0, but under simple majority rule only specific values of z can be realized. Thus, three districts represent the equivalent of z = 0.67 and five districts are equivalent to z = 0.60, but z cannot take on values either above 0.67 or between 0.60 and 0.67 under simple majority rule.
Thus, we can think of the theory here as the law of 0.5(1 + 1/n).
This is discussed by Primo and Snyder (2008, pp. 479–480). Unlike the current paper, Primo and Snyder assume universalism. They show circumstances under which total spending can fall in the number of districts in a model with congestion and cost sharing.
When the legislative body is large we may have a violation of individual rationality if spending requests reflect a tax price of 1/n. Assume that taxes are lump sum and that district level benefits are given by the power function B(g) = Agα, where A > 0 and 0 < α < 1. Further suppose that spending requests reflect a tax price of 1/n and that universalism prevails. It is easy to show that all legislators are made worse off by voting in favor of a spending bill when the size of the legislature n > 1/α. If we impose individual rationality on legislators, the law of 1/n will be inoperative unless α is tiny or n is very small. This result suggests that the law of 1/n is possibly relevant only for legislative bodies the size of city councils. However, with a legislature that small, the Coase theorem should hold, rendering the traditional law of 1/n inoperative.
In personal communication, Höhmann has indicated that the results in Höhmann (2017), reporting a negative relationship between legislative size and spending, are not statistically different when at-large and district-based city councils are considered separately.
Higher transaction costs might be thought to reduce spending by making any deal less likely. However, transaction costs could raise spending if they make it more difficult to pin legislators to their reservation prices as the winning coalition is built. That could happen if informational asymmetries regarding legislator preferences are worse in a larger legislative body. Absent a formal model, however, it is difficult to know if such a mechanism increasing expenditures could indeed be operative.
Political parties are not present in my analysis, but are of obvious potential importance. In particular, a political imperative likely exists for including all members of the party in the winning coalition. Thus, in terms of the spending effects identified in this paper, a de facto supermajority rule may be in effect even if such a rule does not exist de jure. For example, if the majority party holds 55% of the seats in the legislative body, the winning coalition likely includes all of the members, implying that the coalition represents 55% of the districts and pays a tax price of $0.55 on the dollar. Thus, the predicted outcomes would be equivalent to those observed when z = 0.55.
Clearly 0.5 drops from (A.3), but I have kept it there to make the make the linkage with a supermajority rule of z = 0.5(1 + 1/n) clearer.
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Acknowledgements
I would like to thank an anonymous referee, William B. Hankins, Arye Hillman, Natalia Kolesnikova, Matt Van Essen, and participants at the Public Choice Society Meeting and the University of Alabama Brown Bag workshop for providing helpful comments on this paper.
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Appendix
Appendix
1.1 Proof that Result 1 holds for a more general utility function
In this section of the appendix, I will show that Result 1 holds for a continuous and strictly concave utility function U(C, g). Note that this function has the properties U C > 0, U g > 0, U CC < 0 and U gg < 0, where the subscripts indicate the partial derivatives with respect to the indicated argument. Making use of Eqs. (1) and (3) from the main body of the paper allows us to rewrite the utility function as follows: U(Y − G/n, G/nβ). For the winning coalition, we have welfare WZ = znU(Y − G/n, G/nβ). Choosing G to maximize WZ yields
Once again, z does not enter this equation, so the level of the public good chosen is independent of the supermajority requirement. Also, (A.1) is the same condition that we generate from maximizing the welfare of the entire society. When β = 0, we have the Samuelsonian condition for optimal provision of the pure public good, where the left-hand side is the sum of the marginal rates of substitution for all of the identical members of this society.
1.2 The law of 1/n and supermajority rule
In this section, I will show that, in terms of its spending effects, simple majority rule is equivalent to a supermajority requirement z = 0.5(1 + 1/n). In the analysis of supermajority rule, the population of each district was normalized to 1. Changing the supermajority requirement had no effect on the population of each district. However, if we now consider an increase in the number of districts holding total population constant, population within each district must fall. As we will see, however, the equivalency between supermajority rule and majority rule is not disrupted.
Let N be the total population of the political entity under consideration so that N/n individuals live in each district. First, it is clear that the results for nonexcludable goods are unaffected by the changing population of each district. In our earlier analyses, total population simply equaled the number of districts, but now it equals N. If you replace n in Eq. (4) with N, set z = 0.5(1 + 1/n) and note that G = NtY, the first-order condition with respect to G will show that spending is independent of the number of districts n.
Next, consider a distributive consumption good. The benefit that each individual gets from the distributive consumption good depends on the per person level of spending within the district. In the main body of the paper, district-level spending and per capita spending in the district are the same, because district level population is normalized to 1. Here we will interpret g as per person expenditure in districts that are in the winning coalition. Since 0.5 N(1 + 1/n) people live in the districts that are in the winning coalition, total expenditure G = 0.5 N(1 + 1/n)g. Since each person has income Y, the tax rate t = G/NY. If we make the appropriate substitutions in the utility function in (2), we find that the welfare of an individual in a winning district may be expressed as follows:
Choosing a level of g that is optimal for individuals in the winning districts thus is equivalent to maximizing (8) with z = 0.5(1 + 1/n).
Next consider infrastructure spending. Assume that individual income within the district depends positively on the per person spending level, which is gz in districts in the winning coalition and g−z in districts outside the winning coalition. The production function F(g) has the properties specified in the main text. The objective function for the winning coalition is to choose gz so as to maximize
where 0.5(1 + 1/n)N is the total number of individuals in the winning coalition. This is the same as the objective function in (12), except now z = 0.5(1 + 1/n) and the total population is N rather than n. Since n is constant in the analysis of supermajority rule and total population N is constant here, the presence of a different constant in front of (A.3) versus (12) will not affect the optimal choice of gz as those constants are multiplying the entire objective function.
While noting that N cancels from both sides of the equation, the government’s budget constraint may be expressed as follows:
This is equivalent to the constraint in (13) with z = 0.5(1 + 1/n).Footnote 23 Both the objective function and constraint are consistent with the analysis of supermajority rule where z = 0.5(1 + 1/n) and the outcome of the winning coalition’s optimization problem will reflect this equivalency.
Finally, consider transfer payments. The total transfer is equal to [1 − δ(t)]tNY and each individual in a winning district receives [1 − δ(t)]tNY/[0.5(1 + 1/n)N] = [1 − δ(t)]tY/[0.5(1 + 1/n)]. The welfare of a representative individual in a winning district is then
This is the equivalent of the objective function in (21) with z = 0.5(1 + 1/n). Thus, increasing the number of districts has an effect equivalent to lowering the supermajority requirement z, thus leading to a higher level of gross transfers when the number of districts increases.
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Pecorino, P. Supermajority rule, the law of 1/n, and government spending: a synthesis. Public Choice 175, 19–36 (2018). https://doi.org/10.1007/s11127-018-0512-x
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DOI: https://doi.org/10.1007/s11127-018-0512-x