Why Libertarians Don’t Win: The Pareto Optimality of a Majority Loss

Robert Gmeiner Florida State University

Abstract

Contemporary American politics is a case of a two-party system with substantial dissatisfac- tion on both the left and the right. There has long been an unsuccessful libertarian minority, despite occasional claims of widespread support. Treating these claims as valid I develop a model of political competition relying on asymmetric preferences to show why libertarian candidates and positions still tend to fail. This occurs even in the presence of majority support for the libertarian position on any given issue. A key result is that Pareto efficiency of the outcome may not be what is best for society.

1 Introduction

Political parties generally have comprehensive platforms and their successful candidates for office tend to be very similar to one another in their positions. In the United States, candidates choose their respective party and obtain its nomination through a primary election, but the winners within each party tend to hold very similar positions to one other. A common talking point of the Libertarian Party is that many people have libertarian positions. , the 2016 presidential nominee for the Libertarian Party made a statement to this effect near the beginning of his campaign.1 In this paper, I treat this claim as valid and show why libertarian policies are still unlikely to be adopted and their candidates unlikely to be elected. Whether or not Johnson’s claim is true is irrelevant because the outcome likely would not be very different. Not only does the Libertarian Party seldom win elected offices, but also libertarian-leaning politicians in other parties do not enjoy much legislative or policymaking success, even when elected. In the United States, where

1Chandra, Sho “Gary Johnson Says Most Americans Are Libertarian But Don’t Know It” Bloomberg, 4 June 2016, Bloomberg “Gary Johnson Says Most Americans Are Libertarian But Don’t Know It” http://www.bloomberg.com/politics/articles/2016-06-04/gary-johnson-says-most-americans-libertarian-but-don- t-know-it, accessed 6 March 2018

1 candidates choose the party (as opposed to parliamentary systems in which party leaders select candidates for the party), some libertarians may win office, but do not exert considerable influence. Listening to contemporary American political discourse, one could easily be led to believe that the Democratic Party supports big government and the Republican Party supports . However, comparing the overall positions of each party on different issues reveals that each supports big government in its own way. For example, Democrats tend to support more government involvement in health care, welfare programs, and tobacco regulation. Republicans tend to support more government involvement national defense, and recreational drugs (to restrict and interdict them). Many libertarians support less military spending, less health care and welfare spending, and less regulation of both tobacco and other recreational drugs. The question of why each party adheres to a given position on specific issues is not relevant to this paper. What is relevant is that each major party supports some intervention and , but in different issues. My goal is to explain why more libertarian policies are not adopted and why intervention is so prevalent across the board when each individual intervention is benefits only minority of voters. For the purposes of this paper, Democrat and Republican each refer to the established base of the respective party, while libertarian refers to persons of any affiliation, but hold libertarian views. I do not use the term libertarian to refer exclusively to the Libertarian Party. In this paper, I develop a model of political competition that explains both elections for legislators and the outcomes of the legislative process. My model is one of political interaction that explains the stability observed by Tullock (1981), nonconvergence of positions which numerous authors have noted, and it relies on ideology and preferences more than on specific policies. I explain why some ideologies succeed and why some fail, particularly libertarian ideologies. The framework I adopt is one of consumer choice with assumptions consistent with collective decision making as opposed to individual. The main point of this model is that there are multiple political factions, each of which desires intervention in a few areas and nonintervention in others. As long as the groups numerically capable of forming a majority coalition desire their favored interventions more than other desired noninterventions, the result will be more intervention in all areas. Very importantly, this holds even if a majority favor nonintervention in all areas when including those not in the majority coalition. The model I develop shows to some degree why libertarian candidates do not win, it also shows why interventionist policies across the board are successful even if any single intervention is favored only by a minority. Asymmetric preferences for government involvement are central to the ideas I develop in this paper. I contend that the two major political parties will support more government involvement in different things, but will both favor a fairly large, heavy handed government. This can be seen in the administrations of Republican Presidents Ronald Reagan and George W. Bush, both of whom were considered to be conservatives, but who presided over very large administrations with large budget deficits and significant government involvement in many spheres of life. While there was less government involvement in the some things, there was still a considerable

2 amount in many others. In contrast, the administration of Democratic President Bill Clinton was smaller in terms of personnel and spending, but still showed higher tax rates and considerable government involvement in different areas. I first provide an overview of the assumptions underlying utility functions and cost functions, which necessarily have characteristics that differ from individual market exchange. I then provide a mathematical proof of the aforementioned main point. Following the proof, I outline a static model and caveats that could present problems. I conclude with examples from recent American political history and implications of the model for societal welfare.

2 Relevant Literature

There is much literature on spatial models of voting. Downs (1957) presented a two-party system model in which the median voter’s preferred outcome will be chosen as candidates move closer to the median voter’s position in order to attract more votes. There have been arguments for and against the validity of this median voter model, but considerable evidence indicates that it is valid in many circumstances (Holcombe, 1989). Stokes (1963) offers criticism of the Hotelling-Downs model and does not reject it outright, but believes that there are missing elements that must be treated as explicit variables. Political conflict may be focused on one single issue, but often it is not. The model I formulate in this paper accounts for issues of varying importance as well as multidimensionality. Tullock (1981) raises the issue that models of multidimensional competition predict instability in government, but much stability is observed in practice. This has been modeled in the marketplace by Hotelling (1929). Taking this reality of stability to political interactions, Tullock discusses coalition building and logrolling. Tullock correctly states that much logrolling occurs, but notes that formal logrolling coalitions are rare. Tullock also cites examples from European legislatures, specifically the English legislature since 1900. Government favors do not exclusively go to constitutencies represented by members of the winning, or government, coalition. Marginal constituencies do particularly well, with fiercely loyal constituencies faring worse. He mentions R.H. Crossman’s memoir which states that safe Labour districts receive housing funds consistently, but then says that there is no noticeable geographic shift in spending when Labour is replaced by the Conservatives, but leaves open the question of why this is so. The model I develop addresses this question. In Tullock’s view, observed equilibria are inefficient because conditions needed to change the equilibria are too strong. The conclusion I reach in this paper is that equilibria are not good for society regardless of any efficiency criterion, but that this is not because of excessively strong conditions on changing the outcome. Rather, it is a natural outgrowth of the preference structure. Enelow and Hinich (1989) present a model in which uncertainty about candidate positions prevents the convergence predicted by Downs’s model. Others, such as Berger, Munger, and Potthoff (2000) explain why Downs’s median voter model’s prediction of convergence does not

3 hold and actually predicts divergence. This explains why candidate positions and ideologies do not converge, but it leaves open the question of how decisions are actually made and policies chosen. This is a relevant question because political decision making is a repeated game and, regardless of candidate divergence, one position must be chosen. Relying on this conclusion that candidate positions intuitively should diverge, even in a unidimensional setting, I explain electoral and legislative outcomes. Hinich and Munger (1992) presented a spatial theory of ideology as opposed to a spatial model of voting that incorporates the uncertainty discussed by Enelow and Hinich. They state that models in the tradition of Downs and the median voter have supplanted a discussion of politics with a discussion of decision theory that can be applied mechanically. Ideology, in their view, is an internally consistent set of propositions that makes both proscriptive and prescriptive demands on human behavior. They believe that tension between two opposing sets of ideas creates a relevant unidimensional space in which the decision will be made Hinich and Munger believe that voters want actions, but often depend on ideology for comparing candidates because future actions of politicians are hard to predict, resulting in uncertainty. It is easier for a candidate to project an overarching ideology than to articulate detailed, specific positions and then consistently follow through on them. Hinich and Munger believe that there is a need for a “theory of purposive, strategic political interaction that accounts for uncertainty.” The model I develop moves toward filling this gap by explaining results of political competition in a coherent fashion. They ask what an ideology must posess and why some succeed and some fail. My model does not impose as strict a requirement on what an ideology (or preference set) must possess as does theirs, but it does answer the question of why some succeed and some fail. It also accurately describes observed political interactions and explains why success is generally partial and not total and absolute.

3 Theoretical Model

Electoral competition mostly unidimensional, particularly in single-member districts, but political action within the legislature is not unidimensional. In a legislative election, a voter votes for one candidate. The winner subsequently casts many votes on many different issues before the next popular election. To begin, consider three equally sized ideological blocs - Republican, Democrat, and libertarian and two issues of government intervention, one supported by the Republicans and one by the Democrats. They could be defense spending and welfare spending, although the choice of issues is immaterial so long as there is ideological divergence. Suppose the libertarians oppose spending more in either area. In this scenario, two thirds of the people oppose spending more in each area. Of the two thirds, one half (one third of the total) are libertarians and the remainder are from one of the interventionist blocs. If the legislature is designed to mirror the electorate, either through proportional representation

4 or some method of districting, there will be three equally sized blocs in the legislature. In practice, this may not happen, which would summarily show why libertarian position fails. Assuming that it does happen, once the legislature is chosen, it would stand to reason that there should be less spending in both areas because two-thirds of the legislature favors less spending. However, if the preferences of the interventionist blocs are asymmetric and each favors its desired intervention rather than its desired lack thereof, they can collude and each obtain some of what they want. Either interventionist bloc can agree to some of the other’s spending in exchange for some of its own. The libertarian bloc has nothing to offer as it wants less spending across the board. Anything it desires from one interventionist bloc is also desired by the other, yet it has no desired intervention of its own to sacrifice in exchange. Collusion only happens between the interventionist blocs. This paradoxical result rests on asymmetric preferences for intervention. Within a legislature, interaction among legislators is dynamic and they can logroll, allowing for compromises on issues. In contrast, voters in an election pick all of their desired policies upfront, legislators vote on individual issues. Provided that the voters have asymmetric preferences for desired interventions over noninterventions, the result in an election will be that an interventionist candidate wins, just as interventionist policies win in the legislature. The median voter model indicates that more libertarian policies should be adopted if the claim that most American support them is true. While it may not be true, but assuming that it is, the relevance of the model I develop is to explain the empirical reality of the success of interventionist policies. In section 4, I show that this outcome is likely and is often Pareto optimal. At this point, it is worth remembering that Pareto optimality is only an economic construct indicating that nobody can be made better off without making someone else worse off. It does not imply anything about the normative question of what is best for society or what maximizes total utility, to the extent that it can be aggregated.

4 Mathematical Model

From the foregoing model sketch, I proceed to develop a mathematical framework. To do so, I utilize a consumer choice model. A consumer choice model provides a suitable framework because preferences over policy alternatives can be expressed in a similar fashion to commodities. In this section, I outline the assumptions pertaining to utility functions over political issues and the space in which they exist. This is a critical beginning point as the space and utility functions are fundamentally different from those pertaining to commodity exchange. When using a consumer choice model to explain political action, different assumptions must be used in order to reach a valid conclusion consistent with reality. Like in commodity space, costs and benefits are subjective and specific to the individual or bloc utility function. The fundamental differences is that consumption of the chosen bundle is mandatory for everyone, thus ruling out free disposal, so even if the sum of everyone’s utility is maximized,

5 there are whose utility is not maximized. It follows that each utility function has a unique maximum and is not strictly increasing.

4.1 Issue Space Holcombe (2003) defines issue space as distinct from commodity space. Issues, or policies in issue space, must be consumed in their entirety by everyone. Individuals are forced to consume government policies and may not pick and choose which policies they want, unlike commodities in markets. Any issue or policy brings a cost and a benefit, of which consumption of both is mandatory, thus implying that free disposal cannot be assumed. This presupposes that it is possible to measure government policies. This is naturally a fairly subjective measure and is likely to be ordinal as far as normative analysis is concerned. I impose a cardinal measure on government policies in order to show marginal effects of changes. The imposition of cardinal measures of government policy is not relevant to the outcome as the direction of a change can still be determined. The precise size of the marginal effects is generally unknown, but the direction is not. As in most other cases, utility remains an ordinal measure. The lack of free disposal in government policies is a very realistic assumption. It can plausibly be assumed that most, if not all people, would like their own actions always to be unconstrained completely. However, many people support government intervention in various issues. The logical reason for this is that a supporter of such intervention must get utility from having other people act in accordance with the intervention. Succinctly, people often want to be free themselves, but they also want to regulate other people. Benefits or costs from a policy may or may not be monetary. A cost could be in the form of a tax or it could be disutility from excessive regulation. Any cost must be built into the utility function because any policy in issue space has both a cost and a benefit. In order for an optimal point of the utility function to exist, there must be a unique maximum ab- sent any constraints. The optimal point refers to a desired point of both production and consumption as opposed to consumption only. Because a unique maximum must exist, a utility function in issue space must be globally strictly concave, but not globally increasing, meaning that at some level of regulation, costs exceed benefits. It follows that local nonsatiation is no longer a valid assumption.

4.2 Utility Functions Utility functions in issue space are fundamentally different from those in commodity space because more is not always better and costs of excessive consumption must be included. The precise amount of a policy is not chosen by the individual as it is a collective choice and maximization is not subject to any budget constraint because not all government policies involve spending, particularly on social issues. If the amount the individual is required to consume exceeds the optimum, some harm is incurred which must be represented by a cost. Costs may represent monetary or nonmonetary harm.

6 0

U 10 5 15 10 0 5 0 5 10 5 x y − − −

Figure 1: Example Utility Function

Utility functions exist for individuals as they decide for whom to vote in an election. Legislators presumably have utility functions that mirror those of the marginal voters who elected them and act accordingly to maximize their chances of reelection.

4.2.1 Marginal Utility of Intervention The marginal utility of an issue is the first partial derivative of the utility function with respect to that issue, just as in commodity space. Since the utility function already includes all costs, monetary and nonmonetary, the optimal point is found when the marginal utility is equal to zero. Any government policy is a form of government intervention. If the slope of the marginal utility is steeper for one issue than another, then a unit decrease in intervention in the issue with the steeper marginal utility will result in greater utility loss than a unit decrease in the issue with the flatter marginal utility. The slope of the marginal utility is analogous to the intensity of preferences. For example, if the slope of the marginal utility of intervention in one issue is steeper, then less intervention is desired. While precise functional forms may be unrealistic, an ordinal measure of which issues are most important to any given voter is possible. For example, consider the following simple separable utility function of two issues, x and y.

1 U(x,y) = 4x x2 + 4y y2 (1) − − 2 The marginal utilities are

MU = 4 2x (2) x − MU = 4 y (3) y − As is clearly seen, the marginal utility for x is steeper. The optimal point is (x,y) = (2,4), at which point U = 12. Holding the other issue constant, a unit decrease in x results in U = 11, but a

7 unit decrease in y results in U = 11.5. Graphically, this is seen by the elliptical shape of the level sets, which are wider along the y-axis, showing that marginal utility declines more rapidly as x is decreased than when y is decreased. Note that indifference sets are not strictly decreasing as in commodity space. Level sets will generally be elliptical because of a unique maximum. In the case of symmetric preferences, they are circular. Different functional forms may cause them to be of other shapes. Without loss of generality, the same reasoning applies to nonseparable utility functions.

4.3 Interpersonal Utility and Aggregation Since utility is generally an ordinal concept, interpersonal comparisons of utility are difficult. Despite these problems, any political process is designed in some way to aggregate utility. This empirical reality necessitates a theoretical approach, necessitating a theoretical approach. In legislatures, a simple majority is required to pass a bill in most cases. In popular elections for legislators, various rules are used, although a plurality is the most common requirement in the United States. The outcome is determined and each voter has some measure of utility of the outcome. While this does not take account of the intensity of preferences in awarding voting power, it is the only suitable method of aggregating preferences for this analysis because this is what is actually used in practice. Going forward, I assume homogeneous blocs and treat them as individuals. Abstractions are used in nearly all models and this is no exception. The only requirement for the results to hold is for the blocs to be sufficiently cohesive.

4.4 Contract Curves for Political Compromise Just as in an exchange economy, compromise must be reached for political matters in issue space. Contract curves in issue space show the Pareto optimal points of compromise on issues just like in commodity space. A contract curve consists of the locus of points for which the marginal rates of substitution for two issues are the same for two groups. The marginal rate of substitution between issues is the ratio of the marginal utility of an additional unit of government intervention in one issue to the marginal utility of government intervention in another issue. The shape of the utility functions resulting from asymmetry of preferences has powerful im- plications for contract curves in issue space. Suppose there are two blocs, one of which favors intervention in issue y and lack of intervention in issue x and the other is precisely the opposite. The extent to which each bloc prefers intervention in one issue as opposed to lack thereof in the other issue determines the shape of the contract curve. For example, if those who favor intervention in y prefer intervention in y less than nonintervention in x, the elliptical indifference curves will be wider along the y-axis, as shown in the above utility function. This is because the marginal utility is steeper in x, indicating a greater loss in utility from intervention in x than gain from an equal intervention in y from a reference point at which more intervention is desired in y and less in x.

8 0 100 U −200 − 24 24 8 16 16 0 8 0 8 8 x y − −

Figure 2: Two Utility Functions

If both groups prefer lack of intervention in one more than they want intervention in another issue, the contract curve will be bowed toward the origin, as seen below with the following utility functions.

1 1 U (x,y) = 5x x2 + 2y y2 1 − 4 − 2 1 1 U (x,y) = 2x x2 + 5y y2 2 − 2 − 4 The optimal points for bloc 1 are (10, 2) and for bloc 2 they are (2,10). Marginal utility for bloc 1 is steeper for y than x and the opposite is true for bloc 2, which is shown by the visible level curves of the plotted functions. The marginal rates of substitution are

5 1 x MRSx,y = − 2 1 2 y − , 2 x MRSx y = − , 2 5 1 y − 2 so the contract curve is

5 1 x 2 x − 2 = − . 2 y 5 1 y − − 2 This can be plotted

As is seen, the contract curve is bowed toward the origin, graphically showing a preference for nonintervention. Consider the contrasting case, in which intervention in one issue is preferred more than noninter- vention in another, for both blocs. The ability of this model to explain empirical reality rests on the

9 y 12 10 8 6 4 2 0 x 0 2 4 6 8 10 12

Figure 3: Convex Contract Curve

existence of such preference and the resulting contract curve. The example utility functions are

1 1 U (x,y) = 10x x2 + y y2 1 − 2 − 4 1 1 U (x,y) = x x2 + 10y y2 2 − 4 − 2 The optimal points are the same, (10,2) for bloc 1 and (2,10) for bloc 2. The marginal rates of substitution are

, 10 x MRSx y = − 1 1 1 y − 2 1 1 x MRSx,y = − 2 , 2 10 y − and the contract curve is

10 x 1 1 x − = − 2 , 1 1 y 10 y − 2 − which is plotted below

As is clear from the above equations and illustration, if intervention in one issue is more important than nonintervention in the other for both parties, the contract curve is concave to the origin. Using second derivatives to show concavity, this can be shown more generally for any utility function U(x,y). This is irrespective of whether utility is separable or nonseparable, provided that it meets the required assumptions, namely strict concavity and a unique maximum. Utility functions

10 y 12 10 8 6 4 2 0 x 0 2 4 6 8 10 12

Figure 4: Concave Contract Curve

must always be strictly concave, but the contract curve will only be concave to the origin if certain characteristics of marginal utilities are satisfied. For this analysis, it is assumed one must prefer x to y and the other must prefer y to x and whichever bloc prefers x must desire more x than the other bloc and the quantity of x demanded by that bloc must exceed the other bloc’s demand of y at the optimal point. More succinctly, using to ∗ denote optimal points, I assume x1∗ > y1∗; x1∗ > x2∗; and y1∗ < y2∗, meaning that the slope of the line between the two optimal points must be negative. Intuitively, this is not a problem because there is little point to an analysis of a situation in which one group desires more intervention than the other group in both issues. In a space with two blocs and two issues, with each bloc preferring intervention in one and liberty in another, if the marginal utility of the desired intervention for both groups is greater than the marginal utility of liberty in the other issue, the contract curve between the two blocs will be concave to the origin as in Figure 4. This is shown rigorously below. Consider two utility functions, U1(x,y) and U2(x,y), which may take different forms while still having these assumptions satisfied. The marginal rate of substitution is the ratio of marginal utilities, which are partial derivatives with respect to intervention in the two issues, x and y. The contract curve is

∂U1 ∂U2 ∂y = ∂y ∂U1 ∂U2 ∂x ∂x which equivalently can be written as

11 ∂U ∂U ∂U ∂U 1 2 = 1 2 . (4) ∂y · ∂x ∂x · ∂y

Define the contract curve, denoted by C below as a function of the utility functions. C is only defined when its value is equal to zero as the contract curve in eq. (4) holds with equality by definition. C is written as an explicit function. Strictly speaking, the contract curve in eq. (4) is generally not explicit because the indifference curves from which it is formed are implicit functions. The indifference curves surround the optimal point and may be circular, elliptical, or of other shapes. The locus of their tangency points which define the contract curve are often not explicitly defined, so C as defined below (often an implicit function) is, strictly speaking, only the relevant portion of the function that lies between the two optimal points, which can be defined explicitly.

∂U ∂U ∂U ∂U C = 1 2 1 2 (5) ∂x · ∂y − ∂y · ∂x

Because the utility function has a unique maximum, its first partial derivatives of both issues must be locally decreasing in the neighborhood of the maximum. A well-behaved utility function that has no local maxima other than the unique global maximum will have strictly decreasing first partial derivatives with respect both variables. At this point, I assume that the utility function is well-behaved in this regard. Implicitly, the first derivative of the contract curve with respect to x is

∂C ∂ 2U ∂U ∂U ∂ 2U ∂ 2U ∂U ∂U ∂ 2U = 1 2 + 1 2 1 2 1 2 (6) ∂x ∂x2 · ∂y ∂x · ∂y∂x − ∂y∂x · ∂x − ∂y · ∂x2

In the case of separable utility, all cross partial terms are equal to zero and the equation is greatly simplified. For the general case, when cross partial terms may be nonzero, the implicit second partial derivative of the contract curve with respect to x is

∂ 2C ∂ 3U ∂U ∂ 2U ∂ 2U ∂ 2U ∂ 2U ∂ 3U ∂U = 1 2 + 1 2 + 1 2 + 2 1 ∂x2 ∂x3 · ∂y ∂x2 · ∂y∂x ∂x2 · ∂y∂x ∂y∂x2 · ∂x ∂ 3U ∂U ∂ 2U ∂ 2U ∂ 2U ∂ 2U ∂U ∂ 3U 1 2 1 2 1 2 1 2 (7) − ∂y∂x2 · ∂x − ∂y∂x · ∂x2 − ∂y∂x · ∂x2 − ∂y · ∂x3

If this second partial derivative is decreasing, then the contract curve is concave to the origin, just as in figure 4. As this function is an implicit curve, the only relevant portion is between the two optimal points. If utility functions are strictly concave, all second partial terms must be negative. At the optimal point, the first partial derivative of U1 or U2 is zero and it may be either positive or negative away from the optimal point. Because of strict concavity, all second partial derivatives must be negative. Third partial derivatives may or may not exist, depending on the chosen functional

12 form. For the case in which third partial derivatives do not exist, the contract curve is concave if

∂ 2U ∂ 2U ∂ 2U ∂ 2U 1 2 < 2 1 , (8) ∂x2 · ∂y∂x ∂x2 · ∂y∂x

provided that the equivalent result holds from differentiating the contract curve twice with respect to y. Under the same circumstances, it is convex if

∂ 2U ∂ 2U ∂ 2U ∂ 2U 1 2 > 2 1 . (9) ∂x2 · ∂y∂x ∂x2 · ∂y∂x

In the presence of defined third partial derivatives, the contract curve is concave if

∂ 3U ∂U ∂ 2U ∂ 2U ∂ 2U ∂ 2U ∂ 3U ∂U 1 2 + 1 2 + 1 2 + 2 1 < ∂x3 · ∂y ∂x2 · ∂y∂x ∂x2 · ∂y∂x ∂y∂x2 · ∂x ∂ 3U ∂U ∂ 2U ∂ 2U ∂ 2U ∂ 2U ∂U ∂ 3U 1 2 + 1 2 + 1 2 + 1 2 (10) ∂y∂x2 · ∂x ∂y∂x · ∂x2 ∂y∂x · ∂x2 ∂y · ∂x3 and convex if

∂ 3U ∂U ∂ 2U ∂ 2U ∂ 2U ∂ 2U ∂ 3U ∂U 1 2 + 1 2 + 1 2 + 2 1 < ∂x3 · ∂y ∂x2 · ∂y∂x ∂x2 · ∂y∂x ∂y∂x2 · ∂x ∂ 3U ∂U ∂ 2U ∂ 2U ∂ 2U ∂ 2U ∂U ∂ 3U 1 2 + 1 2 + 1 2 + 1 2 (11) ∂y∂x2 · ∂x ∂y∂x · ∂x2 ∂y∂x · ∂x2 ∂y · ∂x3

Equations (8) and (9) clearly are not defined if there is no cross partial term due to separable utility. An analogous result can be found by differentiating eq. (6) with respect to y instead of x because cross partial terms will still exist in that case. All assumptions regarding concavity and a global maximum must remain satisfied. Equivalent results hold when differentiating with respect to y instead of x when all assumptions are satisfied. The clear result from eq. (8) and eq. (9) is that the contract curve is concave if one bloc prefers intervention in x more than liberty in y and the other bloc prefers intervention in y more than intervention in x, as stated above because, under these circumstances, the second partial derivatives of the contract curve are all negative for the relevant portion of the curve. This relies on the assumptions given above regarding optimal amounts of intervention, stated before the theorem. Each bloc prefers intervention in one issue and liberty in another, but concavity is the result when that desire for intervention in one issue outweighs the desire for liberty in the other for both groups. If one bloc prefers intervention in one issue more than liberty in another, but the other bloc prefers liberty to intervention, the shape of the contract curve is ambiguous and is possibly neither strictly convex nor strictly concave. Assuming there are three individuals or blocs, an enclosed area is formed by the contract curves between the three optimal points. This area has been termed a Pareto set (Holcombe, 2003) which is

13 an acceptable term because movement from a point outside this area to a point inside it is always a Pareto improvement. As I show below, Pareto improvements are not possible inside this Pareto set.

4.5 Social Utility Maximization and Pareto Optimality

I define a social utility function, Us(x,y) as the sum of all three utility functions and the Pareto optimal point is its unique maximum. This social utility function is what is maximized by the legislature or voters and it does not represent the utility of any bloc or individual. In this regard, it bears slight resemblance to Rousseau’s general will in that it exists outside any particular individual. The Pareto optimal point at which this unique maximum occurs may be defined (x∗,y∗) without subscripts. This is shown below.

Us(x,y) = ∑Ui(x,y), i = 1,2,3 (12) i

∂U ∂U s = 0 and s = 0 ∂x∗ ∂y∗ The social utility function is found by adding the individual utility functions and is therefore additively separable in its component functions, so the marginal social utility with respect to any issue is also additively separable.

s 1 2 3 MUx = MUx + MUx + MUx s 1 2 3 MUy = MUy + MUy + MUy (13)

At the optimum, marginal social utility is equal to zero. The social optimum can possibly occur at the optimum for one bloc, but that the optimal points for the other two blocs must be distinct from the social optimum. The social optimum is either equal only to one bloc’s optimum or it is distinct from all three because the three blocs are not identical. As seen above, the social utility maximizing point is found when the sum of all individual marginal utilities is equal to zero. At that point, marginal benefit equals marginal cost because the utility function includes costs. It must be stated that the social utility maximizing point is only the point at which utility is maximized when cardinally summed. When each legislator or voter makes a utility-maximizing choice and has one vote, the result of summing the votes will be this outcome. At this outcome, presumably the utility-maximizing point, some individuals may have strongly negative utility and it is possible for all three blocks to have strongly negative utility. Social utility is a result of collective decision making and the pre-existing political process of aggregating preferences. It is pertinent to few if any individuals. While other systems for aggregating preferences could produce a different outcome, the one vote per person system implicitly assumes the method I have described for aggregating preferences.

14 y 3

2

1

0 x 0 1 2 3

Figure 5: Social optimum in interior of contract curves (Pareto Set)

Note that it is possible for the social utility maximizing point to be outside the triangle formed by three straight lines connecting the individual optimal points, but it is still inside the area enclosed by the bowed contract curves. In a two-person exchange economy, any point along a contract curve is Pareto optimal by the definition of Pareto optimality. However, not all such points yield the same total utility level. At this point, the difficulty of aggregating utility across persons or blocs presents a major problem. The point at which utility is maximized is not certain without a perfect cardinal measure. Moreover, any such point is probably most optimal for very few if any individuals. A Pareto optimal point or set of points must exist, but its utility level are not certain (Buchanan, 1962). When there are more than two distinct individuals or blocs, the set of Pareto optimal points can still be defined. The unique point that maximizes total utility is Pareto optimal as are additional points at which the indifference sets of all blocs are tangent. These points may not exist. Inside the area enclosed by the contract curves, a movement toward one is a movement away from at least one. As such, this Pareto set contains all Pareto optimal points and a movement within this set is not a Pareto improvement, but the new point is still Pareto optimal. A movement into this set from the outside is always a Pareto improvement because it is a movement closer to all three optimal points. The social utility maximizing point is a unique point that is Pareto optimal, but from which social utility cannot be increased. Outside the area enclosed by the contract curves, points exist from which Pareto improvements are possible that improve or hold constant utility for all blocs or individuals as well as increasing the value of the social utility function. Inside this area or Pareto set, a movement from a given point to the social utility maximizing point is not be a Pareto improvement because it helps one or two blocs at the expense of the other(s).

15 4.6 Marginal Utility Effects Despite majority support on both issues, the libertarian position will fail on both if the two interven- tionist blocs are better off colluding with each other to each get some desired intervention. Forgoing their desired intervention in favor of their desired liberty must result in less utility. As shown above, this is most likely when the contract curve between the two interventionist blocs is concave to the origin. Concavity to the origin of the contract curve occurs when both interventionist blocs’ marginal decrease in utility from a lack of intervention in the issue for which intervention outweighs the corresponding marginal decrease in utility from an undesired intervention in the other issue. This requirement for concavity only pertains to the concavity to the origin of the contract curve; I already assume strictly concave utility functions. This can be thought of as a situation in which an externally imposed uniform cost of intervention in both issues causes the desired level of intervention to decline more slowly in the issue for which intervention is desired, which is an intuitive result. Graphically, this requires that the marginal utility curve for the issue in which intervention is desired must be steeper than the curve for the issue in which liberty is desired (see figure 1). This is not an unreasonable assumption. Costs are one component of disutility. As disutility of intervention is removed, more should be demanded because the level of utility from intervention would rise. This is purely a theoretical exercise as removal of nonmonetary costs to intervention is not realistic for many issues. There must always be a unique optimal point at which a bloc’s utility is maximized, so it is reasonable that the optimal points for all issues should converge to a similar amount of intervention as disutility is removed. See the figure below, for which x is the level of intervention in each issue and the lines labeled C1 and C2 represent different externally imposed costs. The vertical axis, MB represents the positive component of marginal utility. The point of desired intervention on x will be chosen when the positive and negative components marginal utility are equal, setting marginal utility itself equal to zero and maximizing the strictly concave utility function. If the marginal utility curve for the issue in which intervention is desired were flatter than the curve for the issue in which nonintervention is desired, then an increase in the cost of the intervention would cause the level of desired intervention to decrease more rapidly for the issue for which intervention is desired, which is not intuitively pleasing. Equivalently, the levels of desired intervention would converge as costs rise across the board for all issues as opposed to when they decline, which is also not intuitively pleasing. It is a reasonable assumption that, if nonintervention is already preferred in one of two issues, increasing the cost of intervention in each should result in a more rapid decline of utility in the issue for which nonintervention already is desired. This is represented in figure 6, which shows the intuitively pleasing representation, as opposed to figure 7 below.

16 MB 6

5

4

3

2 C2 1 C1 0 x 0 1 2 3 4 5 6

Figure 6: Steeper MU for intervention in desired issue

MB 6

5

4

3

2 C2 1 C1 0 x 0 1 2 3 4 5 6

Figure 7: Flatter MU for intervention in desired issue

17 y 12 bloc 1 10 A 8 6 4 C 2 bloc 3 B bloc 2 0 x 0 2 4 6 8 10 12

Figure 8: Not all points on the contract curve can be chosen

4.7 Collusion The entire potential for majority-held libertarian ideas to fail lies in the marginal utility structure outlined above. If the three blocs are equally weighted, no single bloc will get its way on every issue. There must be compromise and each bloc chooses to compromise with one bloc or the other based on its marginal utility.. The chosen point must be within the Pareto set because all three blocs could agree on an improvement if it were not. Since two blocs can form a majority, it is most probable that the chosen point will be on the contract curve between two of the three blocs. If the contract curve between the interventionist blocs is concave to the origin, then utility for each interventionist bloc declines more rapidly along its contract curve with the libertarian bloc than with the interventionist bloc. The assumptions for a contract curve that are concave to the origin are realistic, the point chosen by majority rule should be on that curve. This is probably not the point at which social utility is maximized, but it is optimal for the two blocs that collude for a majority. This underscores the point that social utility is not pertinent to a bloc or an individual. It is simply a by-product of this model. Two colluding blocs each maximize their own utility and compromise on the contract curve between their optimal points. The question of where along the contract curve this point will be located is determined by the fact that neither interventionist bloc should be better off by colluding with the libertarian bloc instead. This can be determined from indifference curves and contract curves. As seen above, bloc 1 cannot propose point A and expect bloc 2 to cooperate because bloc 2 is better off colluding with the libertarian bloc and choosing point B. The feasible choice is for two blocs to choose a point on their contract curve such that neither can do better by colluding with the third bloc. It follows naturally that both sides have to make substantial concessions, but that the

18 libertarian bloc is left out of any meaningful compromise. A point R on the contract curve between blocs one and two is an equilibrium if there does not exist a point S on the curve between blocs two and three or point T between blocs one and three such that

U2(S) > U2(R)

U1(T) > U1(R)

U3(S) > U3(R)

U3(T) > U3(R).

In the above graphical example, it would be simple for bloc two, the libertarian bloc, to collude with bloc one and both would improve from point A. This would harm bloc 2, which would in turn make a mutually beneficial offer. This process continues until these collusions with bloc 3 reach a point at which blocs one and two are better off colluding with each other. This highlights the reason that debate over divisive issues never ceases even as change in any direction is gradual to nonexistent. Both interventionist sides fight over a few issues, but neither can demand too many concessions or the third bloc intervenes This point at which collusion between blocs one and two becomes an improvement for each over collusion with bloc 3 is found by analyzing the level sets. Blocs one and two each have level sets that are tangent to bloc three’s. The area on the contract curve between blocs one and two for which a collusive equilibrium is stable is bounded by the lowest level sets for each that are tangent to bloc three, but intersect on the contract curve between blocs one and two. If either bloc one or two proposes a different point on the contract curve, there exists a profitable deviation for the other by colluding with bloc three. In effect, the utility of bloc three limits the unilateral success that can be had by bloc one or two. This is so because bloc three, the libertarian bloc, always presents a viable alternative to the losing interventionist bloc when the other interventionist bloc demands too much. Even though the libertarian bloc cannot win, it is able to prevent the equilibrium from being too skewed towards either interventionist bloc. This is graphically shown below. In this example, the “stable equilibrium” is at the midpoint of the contract curve between blocs one and two. This is not a requirement. It arises due to the shape of level sets. If the libertarian bloc were not perfectly symmetric, the outcome would be skewed in the dominant direction of the libertarian bloc’s asymmetric preferences. Similarly, the asymmetry of preferences for each interventionist bloc may be such that the equilibrium is at a point on the curve away from the midpoint. Moreover, stable equilibria could occur in a range along the curve as opposed to only at one point if the level sets of each interventionist blocs that is tangent to the libertarian bloc is not tangent to the level set of the other interventionist bloc. It should be noted that the “stable equilibrium” may not be stable over the long term. Political

19 bloc 1

A

y bloc 2 12

10 bloc 3 8 6 4 2 0 x 0 2 4 6 8 10 12

Figure 9: Location of equilibrium point

decision making is an ongoing, dynamic process in which there are always opportunities to collude and slightly alter the outcome. Moreover, regular elections in democratic systems have the effect of redistributing the weight of each bloc in the legislative body, thereby making for a new arrangement of policies. In most developed countries, policies do not change dramatically very fast. Adjustments tend to be slow, although the cumulative effect can be vast as various opinions lose widespread appeal. In this model, persistent or formal coalitions are not required. As the composition of the legisla- ture varies from time to time, the location of the point at which blocs collude varies, but stability results first and foremost from the fact that large majorities are seldom if ever permanent. Moreover, any substantial libertarian bloc can prevent excessive unilateral success of either interventionist bloc. A small slice of the contract curve betwween the interventionist blocs is where any compromise will be reached. On account of this, there is no long-run stable equilibrium, but rather a general area in which fluctuations constantly occur, but overall there is stability. This is consistent with Tullock’s observation.

4.7.1 Popular Elections and Legislatures The model makes no distinction between the behavior of voters in elections and legislators in legislative bodies. In popular elections, candidates who strongly favor some and some strong interventions have had the greatest success in winning mass support. Libertarian-leaning candidates, such as Republican Senator in the 2016 U.S. presidential election Republican primaries and Libertarian party candidate Gary Johnson in the general election, have attempted to appeal across the spectrum by advertising their support for liberties desired by both sides, yet both miserably failed to win substantial support.

20 In legislative bodies, the collusion is not done in the form of campaign pitches, advertisements, and rallies, but logrolling. The idea is that each party has some desired interventions that are opposed by the other party. Each party can tone down its interventionist stance just enough to get a few votes from the other and simultaneously agree to support some undesired interventions. This is possible because the intervention, or desire to regulate other people, is stronger than the desire for liberty. The theory of logrolling is eloquently explained in Buchanan and Tullock’s The Calculus of Consent (1962). However, in this model, undesired interventions that a party agrees to support may not necessarily be trivial things or unimportant things; they must only be less important to that party than its desired intervention. The theory of logrolling is refined by this model because it highlights the fact that only parties who desire some kind of intervention can participate in logrolling. Democrats and Republicans can each forego some desired liberty for some desired intervention because a compromise can be reached. Libertarians, in contrast, have nothing to offer either bloc. They have some desired liberties in common with each bloc, but libertarians can offer nothing meaningful to one bloc that cannot be offered by the other interventionist bloc, so they have no bargaining ground. Each interventionist bloc can yield its undesired liberty to the other interventionist bloc in return for some desired intervention. Yielding to the libertarians offers much less return to the interventionist blocs compared to yielding to the other interventionist bloc.

5 Examples

The 2016 U.S. presidential election provides one of the most striking examples of this model. The Democratic candidate, , and the Republican candidate, , both had remarkably high unfavorability ratings. Each of these candidates had been involved in a long list of scandals which became highly publicized news stories. The climate of the election was such that, if a libertarian were to enjoy substantial success at the national level, 2016 would have been a good year for it. The Libertarian Party nominee, Gary Johnson, achieved less than 4% of the vote nationwide. Johnson was also the Libertarian Party nominee in 2012 when he achieved 0.99% of the vote. His share of the vote more than tripled in the presence of two highly unpopular candidates the second time he ran, but it was still incredibly small. In the 2016 election, there were many voters with a never-Clinton or never-Trump attitude and many supporters had misgivings about their own candidates. Whatever their reasons for thinking and voting as they did, it is clear that despite their misgivings about either candidate, most people still voted for either Clinton or Trump. No third party candidate commanded more support than Johnson. was the one state in which both Cinton and Trump were rejected by a substantial portion of the electorate. Trump won Utah, but only took about 45% of the vote. Evan McMullin won more than 20% of the vote in Utah. While McMullin could hardly be considered a libertarian, he was viewed as a reaonable alternative to two highly unpopular candidates. Nevertheless, McMullin still

21 finished third and Trump finished more than 20 percentage points ahead of him. In a legislative setting, this has been seen repeatedly in the United States as Democrats and Republicans fight over funding the federal government with the threat of a shutdown. The Bipartisan Budget Act of 2018, which ended a brief shutdown increased defense and nondefense spending, favoring Republicans and Democrats, respectively. A similar outcome was reached in the Continuing Appropriations Act of 2014 to resolve a government shutdown, with concessions going to each major party that mostly pertained to health care policy. It is most likely true that, even if non-intervention is supported in some areas by a majority, this is not the case for most issues. Nonintervention is not supported for all or nearly all issues by a majority of people. The relevance of the 2016 election is in the polarizing nature of two intensely disliked candidates. If libertarian ideas could command majority support, it should happen when the two interventionist candidates are most disliked. This shows the strong desire of many voters to regulate others or a feeling that some undesired intervention was better than other undesired intervention or liberty. In either case, a desire for widespread nonintervention was not seen in the 2016 election.

6 Implications of the Model

Wittman (1989) develops a model explaining why democracies produce efficient results, an assertion that is generally not accepted in disciplines such as . In the model I develop, the outcome is Pareto optimal, indicating some measure of efficiency. The model in this paper can be seen as consistent to a degree both with Wittman’s assertion of democratic efficiency and also with the common public choice assertion of political inefficiency. Political outcomes are a product of intensity of preferences and negotiation or compromise. In the and in the absence of externalities, this same interaction of negotiation (exchange) with preferences does produce efficient outcomes. However, a central element of the model I develop is that any outcome affects everyone, not just those who make the decision. Because the outcome affects many others beyond those who actually make the decision, any “efficiency” is not optimal for some people. Moreover, under most circumstances, very few people ever get precisely what they want. The efficiency Wittman describes is simply the expected outcome based on this model. Regulations on various policies in the United States tend to be very complicated. At teh federal level, the tax code is very long and detailed and is difficult for anyone other than tax accoun- tants and tax lawyers to understand. Numerous agencies, such as the Consumer Products Safety Commission; the Consumer Financial Protection Bureau; Securities and Exchange Commission; Federal Trade Commission; Bureau of Alcohol, Tobacco, Firearms, and Explosives; and many others have put forth numerous regulations authorized by acts of Congress. Regarding spending and government programs, programs favored by different parties remain highly funding. For example, the Departments of Defense and Health and Human Services have the two largest departmental

22 budgets. Ranking third is the Department of the Treasury, which has a large budget due to interest on the federal debt, which covers spending favored by both parties. At the state level, numerous occupations require licensing and many are heavily regulated. The general situation of regulation in the United States is one of widespread regulation, yet neither side seems to be satisfied. There are regulations on guns and abortion, yet Democratic candidates persisently want more regulations on guns and less on abortion and Republicans tend to want the opposite. Regarding spending, Democrats tend to want to expand welfare programs such as , , and food stamps while Republicans push for cuts. Likewise, many Democratic politicians lean toward cutting defense spending while increases are often called for by Republicans. The general resolution has been to do more of both as far as spending is concerned. Regarding issues like and abortion, which mostly involve rules rather than spending, the status quo seems to get maintained fairly well as the model indicates. These results of the model are consistent with empirical regularities and demonstrate the pitfalls of relying solely on efficiency. The efficient outcome helps some people in some ways, but in some way harms nearly everyone. The assumption of intensity of preferences in that people desire their desired intervention more than liberty in another issue has the converse of the fact that failure to achieve the desired intervention brings greater harm than failure to obtain the desired liberty. A similar proposition is that the undesired intervention harms that group more than it would be harmed by its undesired liberty. These two propositions may not always hold and it likely depends on the issue in question. When viewed from this angle, the results produced through this type of democratic negotiation may satisfy economic efficiency, but they are manifestly worse for society than liberty. The notion that policies are implemented because of concentrated benefits and widely dispersed costs and subsequent incentive to lobby is consistent with this analysis. This is just a situation in which multiple regulations must be done to get a package agreeable to a majority of people. The result is that few policies favored by everyone are actually adopted and there is a minority that disfavors nearly all implemented policies that is severely hurt. This may be the efficient Pareto optimal outcome, but it’s usefulness is questionable. Without cardinal utility rankings, it is difficult to know just how much libertarians are harmed. Pareto optimality and efficiency may not correspond to utility maximization in these cases. In a democratic system, each person generally has one vote regardless of utility or the level of harm experienced. As long as the interventionist blocs are better off colluding with one another, the collusive outcome can be implemented regardless of the harm imposed on libertarians. The Pareto efficient outcome can be very far from the utility maximizing point.

7 Conclusions

Pareto optimality is not likely to satisfy everyone because a policy always affects everyone and there are winners and losers with every policy. There is no requirement that Pareto optimality be

23 the best for each individual, but rather that it maximize total utility for the group. The number of persons in the group and their different utility structures can produce outcomes that harm others, yet are Pareto optimal. This model explains why majority opinions can fail and why this result is likely and close to Pareto optimal. While the Pareto optimal result may not actually be reached, it is still possible in this model to have a Pareto optimal result that is detrimental to everyone. Of course, two of the three individuals or blocs can collude and the result will not be on the Pareto optimal line, but the model still shows why the majority opinion can be very far from Pareto optimal. It relies on the utility structure and on relative preferences for intervention and liberty in different issues, showing that a minority of critical size and strong enough opinions has an incentive to collude with other minorities and collectively overthrow the majority opinion and thereby increase its utility and also increase total utility. Relative marginal utilities and preferences are a crucial missing link in discussing models of voting and preference aggregation. Unlike commodity space, there is no possibility in issue space of compensating the losers because costs are nonmonetary and are built into the policy. They can only be compensated by changing the policy, which in turn affects total utility, which already had been maximized. The main conclusion is that, given strong preferences for intervention among minority groups, the result is much intervention across the board that is not desired by substantial portions of the electorate. This is possible because of the democratic political process and these preference structures. The mathematical proof and empirical observations support this conclusion.

References

Berger, Mark M., Michael C. Munger, and Richard F. Potthoff. “Expository Note: The Downsian model Predicts Divergence”. Journal of Theoretical Politics Vol. 12, No. 2 (2000), pp. 228–240.

Buchanan, James M. “The Relevance of Pareto Optimality”. Journal of Conflict Resolution Vol. 6, No. 4 (1962), pp. 341–354.

Buchanan, James M. and Gordon Tullock. The Calculus of Consent. Ann Arbor: University of Michigan Press, 1962.

Downs, Anthony. An Economic Theory of Democracy. New York: Harper and Row, 1957.

Enelow, James M. and Melvin J. Hinich. “A General Probabilistic Spatial Theory of Elections”. Public Choice Vol. 61, No. 2 (1989), pp. 101–113.

Hinich, Melvin J. and Michael C. Munger. “A Spatial Theory of Ideology”. Journal of Theoretical Politics Vol. 4, No. 1 (Jan. 1992), pp. 5–30.

Holcombe, Randall G. “The Median Voter in Public Choice Theory”. Public Choice Vol. 61, No. 2 (1989), pp. 115–125.

24 Holcombe, Randall G. “The Size and Significance of the Pareto Set in Spatial Voting Models”. Public Choice Vol. 116, No. 1 (July 2003), pp. 19–29.

Hotelling, Harold. “Stability in Competition”. The Economic Journal Vol. 39, No. 153 (Mar. 1929), pp. 41–57.

Stokes, Donald E. “Spatial Models of Party Competition”. American Political Science Review Vol. 57, No. 2 (June 1963), pp. 368–377.

Tullock, Gordon. “Why So Much Stability?” Public Choice Vol. 37, No. 2 (1981), pp. 189–204.

Wittman, Donald. “Why Democracies Produce Efficient Results”. Journal of Political Economy Vol. 97, No. 6 (Dec. 1989), pp. 1395–1424.

25