The Demand and Supply of Public Goods
Many Public Goods, Many Persons:

Figure 6.1 
If the two men should have identical ordinal utility functions, no decision problem would arise. They would agree immediately and without conflict on a single unique value for the temperature setting. If, however, they should have different utility functions, they will disagree, and some way must be found to make a decision. The problem is illustrated in Figure 6.1. The "quantity of public good," in this case, room temperature settings, is measured along the horizontal axis, and the ordinal preference rankings of the two persons along the vertical axis. Curve P_{t} shows Tizio's preference rankings, while curve P_{c} shows the same for Caio. Note that Tizio's most desired setting is at T_{t}, while Caio's most desired setting is at T_{c}.
The range of conflict or disagreement is shown between T_{t} and T_{c} as extremes. If the initial setting should be either below or above these two limits, the two persons would agree quickly to move to some point within this range. Once settled at a point along this range, however, the situation becomes one of pure conflict. Since by our assumption, there is no numeraire, no side payments or trades can be arranged, and the location of a final solution within this range will be arbitrary. It will depend largely on the skill and strength of the parties to the decision. There is no direct way, in this model, for the relative intensities of preference of the two parties to be expressed. Tizio may be relatively indifferent about room temperature over a considerable range, whereas Caio may be highly sensitive to differences in temperature. If side payments in a numeraire could, in fact, be made, this difference in relative intensities could be expressed and Caio's more intense desires manifested. Without such a numeraire, no such result is predictable.
The two men may, of course, agree on some rule for making the final decision, such as flipping a coin each day to determine who shall decide, or, more likely perhaps, splitting the difference between their two preferred settings.
Few difficulties are encountered in extending this model to include more than two persons. As more persons are added, the range of extreme values for preferred or desired positions is likely to be extended. Again there will be no means of individual expression for relative intensity of preference, and the final rule for decision must be arbitrary, in some sense.
This model, which may seem highly unreal in the twoperson context, has considerable realworld relevance in the manyperson version. It is precisely this model which many political scientists more or less implicitly assume in their discussion of voting processes. If individuals in referenda, or their representatives in legislative assemblies, are expected to reach agreement on only one issue and to do so in complete isolation from all other issues, this is the basic model for analysis.
If specific rules for making decisions are postulated, determinate results can be predicted. For example, if simplemajority voting prevails, the outcome will be that most desired or preferred by the voter whose preference curve reaches its peak at the median among all such peaks arrayed as in Figure 6.1 (see Figure 6.4 below). The stability of this outcome is guaranteed if all preference curves are singlepeaked, like those in Figure 6.1. If curves cannot be arrayed in singlepeaked fashion, the outcome of majority voting is not normally stable, and the familiar cyclical majority problem arises. This problem will be more fully analyzed later in this chapter.
Consider the TizioCaio model again, but this time with two public goods rather than one. What differences in result will the introduction of a second public good make? In addition to the thermostat setting, a decision must be made on the time to turn the lights out each evening; and, again, both persons must adjust to the same value also for this variable.
For simplicity in the initial geometrical exposition, let us assume that the two "public goods" are completely independent, one from the other, in both utility functions. There are no relationships of complementarity or substitutability. Neither Tizio nor Caio will desire to modify his optimally preferred temperature with a change as to the time for lightsout.
Figure 6.2 
If the twoperson group tries to arrive at a decision on the second common variable, lightsout time, separately from the decision on temperature, we can think of a range of conflict, such as that already shown for temperature in Figure 6.1, if the two utility functions differ with respect to this second variable. The whole situation can be illustrated in Figure 6.2. On the horizontal axis, as before, we measure units of the first public good, Q_{1}, in this example, room temperature. On the vertical axis, we measure units of the second public good, Q_{2}, time for lights to go out. Each of the two persons will have some optimally preferred values for these two commonly shared items, some most desired combination. That for Tizio is indicated by D_{t}, that for Caio by D_{c}. These are peaks on the two preference or utility surfaces, assuming smoothness and continuity, and the functions can be represented by the standard indifference mappings. The set of curves enclosing D_{t} are indifference contours for Tizio; the set of curves enclosing D_{c} are indifference contours for Caio. The analysis that follows does not depend upon the particular locations of D_{c} and D_{t}.
By our assumption that the two variables are wholly independent in each utility function, the lines of optima enclose a rectangular area, shown here as D_{t}MD_{c}N. A single line of optima represents the locus of points at which an individual's set of indifference curves cut a horizontal or a vertical line. These lines are dotted in Figure 6.2, and are labeled with the 0Ps, the subscripts denoting the individuals, t and c, and the superscripts denoting the goods, Q_{1} and Q_{2}. Hence, the line 0Pe_{c}^{2} depicts Caio's line of optima with respect to the good, Q_{2}. It indicates quantities of Q_{2} that he would optimally prefer for all possible quantities of Q_{1}. By our assumption of independence, this is a horizontal line.
We propose now to examine the result of one particular decision rule. Consider the arbitrary but plausible rule that each one of the two men is given the authority to make one of the two decisions. Suppose, for illustration, that Tizio is allowed to decide what the temperature in the room shall be, while Caio is allowed to decide on the time for lightsout in the evening. The outcome will be that shown at N; if the two roles should be reversed, this outcome would be changed to M.
Once this rule is chosen, and the allocation of responsibility for each decision settled, the outcome at either N or M is an equilibrium one. So long as preferences do not change, this outcome will tend to be stable. In the simplified twoperson model examined here, this rule for decision is, of course, only one from many that might be selected. It is singled out for some discussion because its analogue becomes important in the manyperson model, the model which must be introduced when political choice is seriously analyzed. Majority voting rules produce results that are similar to those suggested here when all preferences exhibit singlepeakedness and when issues are independently considered. In this situation, majority voting amounts to the delegation of decisionmaking power to one man, this man being, in each case, the member of the group whose preferences are median for the whole group.
It will be useful now to contrast the results reached under the arbitrary decision rule in the twoperson case with those that might emerge when the two variables are simultaneously considered. When the two parties recognize that there are two variables to be settled, not one, and when they attempt to agree on a twovalued outcome, the range of possible results is dramatically narrowed to the positions along the contract locus D_{t}D_{c}. This is a contract locus of the ordinary sort, and some position on this locus will dominate any position off the locus, for both persons. Along the locus itself, no single point dominates any other point, for both persons. Note that this locus can be reached with an implicit unanimity rule for making final decision. The set of points along the locus are comparable in this sense with the set enclosed by the whole rectangle formed by intersecting lines of optima. Without joint consideration of the two variables, a solution anywhere in this rectangle becomes possible if common agreement is required for any change. With joint consideration, a solution anywhere on the locus becomes possible. The final position on the contract locus will remain indeterminate unless an arbitrary selection rule is adopted. In the absence of such a rule, the final outcome will depend on the skill and bargaining strength of the two parties.
Consider the position N, which we defined as one of equilibrium under the alternative decision rule examined. Assume that Tizio and Caio, finding themselves at N, now recognize that both variables may be simultaneously selected. Clearly, it is to the interest of both to reach agreement on moves from N in a generally northeasterly direction. Any move that lies within the lozenge that is shaded in Figure 6.2 will tend to be approved by both parties.
The construction demonstrates that, even in this model where both goods are purely public, simultaneous consideration allows the introduction of "economic" evaluation that is not possible under the alternative rule. The model is not one of exchange in the ordinary sense, since there is no transfer of goods between contracting parties. What they exchange here is agreement. Moving from N, in either a vertical or a horizontal direction, will make one of the two persons worse off. He is compensated for this worsening in his position, with respect to one of the two variables, by an improvement in his position with respect to the second variable.
It should be noted that this sort of exchange is not the same as vote trading, which we shall consider fully in a more inclusive model. The exchange depicted here is more closely described by "compromise" in a politicaldecision terminology. By simultaneously considering two variables rather than each variable separately, the possibility for mutual agreement between the parties is enhanced and there is less need for reliance on arbitrary decision rules. The results are more efficient than under any such rules, in that the preferences of the parties are more fully satisfied.
Figure 6.3 
For analytical completeness, the twoperson, twogood model should be modified to allow for complementarity and substitutability between the two purely public goods in one or both of the individual utility functions. One such geometrical construction, similar to Figure 6.2 but encompassing complementarity between the two goods in both utility functions, is shown in Figure 6.3. Note that the lines of optima for both parties now slope upward and to the right. If the arbitrary decision rule previously noted is chosen, with Tizio being granted authority to choose his preferred level of Q_{1} and Caio being granted authority to choose his own preferred level of Q_{2}, an initial position at T' will be selected if Tizio acts first. Note, however, that this initial choice of Q_{1} would no longer be stable. Finding himself at position N', after Caio has selected the indicated level of Q_{2}, Tizio would modify his initial choice. Caio would also shift his preferred position for Q_{2}. Equilibrium under this rule is finally attained, as before, at the intersection of the two lines of optima, shown at N_{e} in Figure 6.3. If the decision authority should be reversed for the two goods, the solution would shift to M_{e}.
As the construction of Figure 6.3 suggests, in this case when the two goods are complements in both utility functions and when the contract locus exhibits a negative slope, the delegationofdecision rule is highly inefficient in comparison to a rule that allows simultaneous consideration of both goods and, hence, leads to some solution along the contract locus. If the two goods should be substitutes in both utility functions and the optimal positions should lie in the same relation to each other, the delegationofdecision rule is, relatively, less inefficient than in the complementarity case. This conclusion is reversed if the contract locus exhibits a positive slope.
Geometrical elaboration of the argument to include the various relationships of complementarity and substitutability is omitted here for two reasons. The first is the obvious one of space; the second is the possible value that his own effort at geometrical construction can have for the student who desires to understand (and possibly to refute) the conclusions reached here, as well as those to be developed later in the chapter.
As the size of the group is expanded to include a third person, the analysis of agreement is necessarily modified. The first case, that in which all three must adjust to a single public good or "issue," can be covered briefly. To stay with our example, Tizio and Caio now have an additional roommate, Sempronio, who has a utility function that differs from either one of the other two. All three men must now adjust finally to the same quantity of the single variable, room temperature.
Figure 6.4 
A threeperson construction analogous to the twoperson construction of Figure 6.1 is presented in Figure 6.4. The most preferred levels of temperature range from T_{t} to T_{s}, with T_{c} occupying a median position. We want to examine a single decision rule, that for simplemajority voting. So long as the ordinal preferences are singlepeaked, as in Figure 6.1 or 6.4, and so long as individuals are free to suggest any quantity of the good to be chosen, the majorityrule solution will be T_{c}, the most preferred level for the median person with respect to this good. This can be easily seen by reference to Figure 6.4. Two out of three persons will approve all motions to increase quantities so long as these remain to the left of T_{c}; two out of three persons will approve all motions to decrease quantities, so long as these remain to the right of T_{c}. As a motion, T_{c} will defeat any alternative suggestions as to the quantity of the public good.
In the absence of any explicit decision rule, all three persons will agree only to limit the quantity to the range shown between T_{t} and T_{s}. Without a specific rule for choice other than general agreement, any point within this range becomes possible.
Figure 6.5 
A more interesting, and more general, model emerges when we increase the number of public goods or issues to two, and extend the size of the interacting group to three. As we did with the twoperson analysis, let us assume initially that the two goods are independent in each of the three utility functions. Figure 6.5 is a partial reproduction of Figure 6.2, with the addition of the third utilityfunction mapping. The most preferred combination for Sempronio is shown at D_{s}, and curves could be drawn enclosing this peak indicating his indifference contours (these are not drawn for economy reasons). The two lines of optima for Sempronio are shown as 0Pe_{s}^{1} and 0Pe_{s}^{2}.
We now want to examine group decisionmaking when the two public goods, or issues, are considered separately. Under a rule of simplemajority voting, each issue will be decided as if it were the only one. The model previously analyzed for the single good is sufficiently explanatory. In terms of the construction of Figure 6.5, the solution will, in each case, lie somewhere on the middle line of optima. Considered separately, majority rule will produce decision on Q_{1} somewhere along the line 0Pe_{c}^{1}, and a decision on Q_{2} somewhere along the line 0Pe_{s}^{2}. The combination selected will be that shown at the intersection of middle lines of optima, indicated by V in the figure. By our restrictive assumption that the two goods are wholly independent in each utility function, the majorityrule choice for one good is not modified by the quantity of the other selected. This suggests that an initial majorityrule solution determining the quantity of one good will tend to be stable. The analysis here can be extended without difficulty to encompass any degree of complementarity or substitutability. After a series of votes, a combined solution is indicated at the appropriate intersection of middle lines of optima.
It will be useful to examine the independent majorityrule result, shown at V, somewhat more closely. Note that, as in the twoperson case, this position is one of the extreme corners of the rectangle formed by lines of optima of the two decisive members of the group, Caio and Sempronio. As suggested earlier, this shows that majority voting, when preferences are singlepeaked and when issues are considered separately, amounts to the delegation of choice on each issue to the person whose preferences are median for the group.
There are interesting differences between the twoperson and threeperson model in the comparison of results attained under the delegationofchoice rule with issues considered separately and under the simultaneous consideration of both issues. In the twoperson model, the alternate delegation of choice produced a result (at N in Figure 6.2) which was obviously inefficient in the Pareto sense once simultaneous consideration of both variables was recognized. Both members of the group found that their situations could be improved by shifting from N to a point on the contract locus. The threeperson case is quite different. Majority rule does not delegate choicemaking power arbitrarily; it delegates power to the person with the median preference. This insures that, given indifference contours of normal shape, the intersection of middle lines of optima, such as V in Figure 6.5, will lie within or upon the boundaries of the Paretooptimal set of points, enclosed by the three contract loci. This set of points in the threeperson case is equivalent to the contract locus in the twoperson model.*8 The independent consideration of each issue, with simplemajority voting on each issue, will generate an outcome that will be Paretooptimal, provided, of course, we remain within the model that denies the existence of a privategoods numeraire. Without such a numeraire, even potential side payments may not be brought into the discussion. The result will also tend to be an "equilibrium" one. So long as tastes do not change, and so long as the issues are not considered simultaneously, the rule will produce an outcome that will be stable over time.
If the two goods or issues are considered simultaneously, and simplemajority voting rules remain in effect, this equilibrium is rudely shattered. Note that, at V, both Caio and Sempronio can improve their own positions by suggesting combinations that lie within the shaded lozenge. They can, by active discussion of both issues simultaneously, move toward the contract locus, D_{c}D_{s}. Eventually, they will get to some position on this locus, say H. No further gains can be made by "exchanges" between these two persons, and, if they could be assured that Tizio will accept this result quietly, the position would be stable. However, note that, at H (as at V), gainsfromtrade can be made as between Tizio and either one of the two other persons. Recognizing this, Tizio will propose a motion shifting the outcome to some combination, say that shown at G, and this will secure majority approval. Once having arrived at G, however, Sempronio may propose the combination F, which will, in turn, secure majority approval. At F, Caio may once again propose H, and this will, in its turn, win by a majority. When the two goods or issues are simultaneously included in motions, and when simplemajority voting rules remain in force, the outcome is likely to be a cycle among separate alternatives. This familiar phenomenon, that of a rotating or cycling majority outcome, need not take place only among positions on the boundaries of the Paretooptimal area as in our example here. Cycles can occur as among combinations within the boundaries, but the latter set limits to the cyclical pattern if discrete "jumps" over these limits are ruled out. If, in each instance, the two members of the decisive majority coalition maximize the potential exploitation of the third, proposals or motions will tend to be those combinations along the contract loci.
The inconsistency represented by a cyclical majority is sometimes interpreted to be a serious limitation on the operability of a majorityvoting rule, but, in terms of the model examined here, one fact must be noted. All points within the cycle are Paretooptimal. Since it is not possible, without external criteria, to evaluate or weigh one Paretooptimal position against the other (or even against certain nonoptimal positions), there is nothing that is necessarily inefficient about the cyclical majority pattern, except, of course, the inefficiencies introduced by the multiplicity of votes.
Let us now examine what might happen if there were no explicit decision rule in existence other than one requiring general agreement, and the situation is as depicted in Figure 6.5. If the two goods or issues are simultaneously considered, general agreement would produce a result within the Paretooptimal area, bounded by the three contract loci. This is true almost by definition since we are ignoring, at this stage, costs of reaching agreement and also bargaining difficulties. No one result from within this area is any more plausible than any other in the general absence of a rule. If one of the three men assumes dictatorship, the outcome will, of course, be at one of the three optima. Many other possibilities might be examined, but space does not permit an elaboration of these.
The threeperson, twogood model is helpful in understanding the elements of collective decisionmaking in larger groups. The geometrical exercises included in this chapter are selective, and only a skeletal group of configurations of utility functions and of relationships between goods in utility functions can be discussed in detail. The model previously used can, however, be extended to clarify an additional distinction, that between (1) simultaneous consideration of two goods or issues, and (2) explicit vote trading on single issues. To this point, we have examined the results predictable under simplemajority voting when the two decisions are made separately, and, secondly, the change that might be anticipated in these results when the two decisions are made simultaneously, when combinations are voted on as alternatives.
Neither of these institutions of group decisionmaking involves explicit vote trading. Simultaneous consideration of two variables allows agreement to be reached under exchange of a sort, but there is no explicit delegation of voting authority, no proxy transfer as it were. Such explicit trade, however, is a third possibility, and we may examine this within the threeperson, twogood model.
The first point to be emphasized is that at least two goods or issues must be recognized to be present before vote trading can take place. This is an obvious point, especially to an economist, but it requires stress nonetheless because vote trading in an explicit sense requires a recognition of two issues but separate voting choices on each one of the two. It is not the same thing, therefore, as combining the issues and voting on a package, combination or bundle.
Under what circumstances would the three members of the group, as depicted in Figure 6.5, find it advantageous to trade votes explicitly? As we have seen, when the two decisions, on Q_{1} and Q_{2}, are made separately, position V tends to be established by simplemajority voting. In this setting, Caio is the effective swingman, the decision maker, on Q_{1}; Sempronio is the effective decision maker on Q_{2}. Tizio is left out of account; he is an "extremist" on both issues; he desires more of Q_{2} than anyone else and less of Q_{1} than anyone else. Tizio might, in this situation, be quite happy to trade away his vote on either one of these two issues (he loses both in any case when no trades are made) in exchange for support of his own position on the second. Note, however, that as the utility functions are drawn in Figure 6.5, neither Caio nor Sempronio would be likely to agree to a trade offer from Tizio. Caio, for example, is already decisive with respect to Q_{1}; he would hardly give up this power of group choice in exchange for Tizio's vote on Q_{2}.
This notrading result arises because, as we have mapped them onto Figure 6.5, the utility functions of all three potential traders exhibit relatively even strengths of preference as between the two issues. Geometrically, this means that the general shapes of the contours surrounding each optima are roughly similar. If the utility functions are different, and if at least one of the three persons should exhibit a relative intensity of preference for one of the two goods, explicit vote trading becomes a possibility even in this highly limited model.
Figure 6.6 
To show this geometrically, a modified construction similar to Figure 6.5 is presented in Figure 6.6. For purposes of comparative analysis, the lines of optima are identical with those of the earlier figure, but the utility functions are now different. If the two decisions are made separately, the majorityrule outcome is, as before, that shown at V. Note, however, that both Tizio and Sempronio are better off at V' than at V. This preferred position, at V', which is the intersection between 0Pe_{t}^{2} and 0Pe_{s}^{1}, can be attained by an explicit trade of votes. Recognizing that there are two decisions to be made, Tizio offers to support Sempronio's motion with respect to the amount of Q_{1} in exchange for Sempronio's reciprocal support for Tizio's motion with respect to Q_{2}. As drawn in Figure 6.6, there are mutual gains from such trade. In the exchange, Sempronio gives up his power of effective decision over Q_{2} because, relative to Tizio, he is more interested in Q_{1}.
Faced with this coalition between Tizio and Sempronio, there is nothing that Caio can do so long as the two issues are voted upon separately. He may, of course, denounce the exchange of votes as unethical, but he is powerless to offer terms more favorable to either member of the coalition, as the configurations drawn in Figure 6.6 indicate. He could offer his own decisive vote on Q_{1} to Tizio, but the latter is relatively uninterested in this. The trading outcome represented at U is not likely to emerge. Or, alternatively, Caio might offer to trade with Sempronio, generating a possible trading outcome at U'. This would be a plausible result under slightly different configuration of Sempronio's utility function.
Trading outcomes will be located at the intersection of lines of optima so long as the exchanges take what might be called a proxy form. This means that the trade involves an agreement between two parties to exchange reciprocal support on undefined motions as to the quantities of specific goods. Under this restriction, outcomes, once attained, will tend to be reasonably stable. Trade may also be of a different sort and without this stability element. Faced with the outcome V', Caio may offer to Sempronio, not an exchange of proxies, but an exchange of specifically defined motions. He may agree to support Sempronio's motion for a quantity of Q_{1} represented at T_{s} in exchange, not for his own optimally preferred quantity of Q_{2}, which would be L_{c}, but for a quantity measured by the distance T_{s}Z. Sempronio will find this trade advantageous since, at Z, his own position is clearly improved over that at V'. In turn, this may lead Tizio to make a further concession to Sempronio, and, by a series of exchanges on specific motions, Sempronio may actually approach his own optima for both variables. He is placed in this strategically favorable position here because Tizio is relatively interested in Q_{2}, not in Q_{1}, and Sempronio is an extremist with respect to Q_{1}, not Q_{2}.
Explicit vote trading of the proxy form tends to shift the majorityrule result outside the boundaries of the Paretooptimal area enclosed by the three contract loci. To the extent that the trade departs from the proxy form and takes on that of exchanges of support on specific motions, the outcome shifts in the direction of the Paretooptimal area, and, in one sense, the votetrading equilibrium is attained at D_{s}, which is Paretooptimal.
At either this outcome, D_{s}, or at V', or at any other outcome along the vertical from T_{s}, Caio is in a considerably worse position than at V, where no vote trading takes place. Rather than engage in a competition with Tizio for the favors of Sempronio, Caio might try to secure an institutional change that will allow both issues to be treated simultaneously rather than separately. If, faced with an outcome V', he can secure such a change, any offer of a combined package falling between V' and the contract locus within the shaded lozenge in Figure 6.6 will be approved by all three persons. However, once a position on the contract locus has been reached, Caio can proceed to form a new majority coalition with either of the other two persons, offering motions represented perhaps by either G' or H'.
Figure 6.7 
One interesting configuration of utility functions is shown in Figure 6.7, which contains only the lines of optima. The same person exhibits median preferences for each good. If the two decisions are made separately, and if no vote trading takes place, he will reach his own optimal position. If, however, his two fellow citizens should differ from each other in relative intensity of preference as between the two goods, explicit vote trades may generate an outcome at either U or U' and the average man may be left out in the cold with neither of his median preferences honored. This model has considerable realworld suggestiveness, especially in the budgetary process. Congressmen from California are intensely interested in waterresource projects in the West; congressmen from West Virginia are interested in waterresource projects in Appalachia. Vote trades between these two may secure substantial appropriations for both, leaving the Iowa congressmen, who are mildly interested in both projects and with moderate preferences on each, without an effective voice in decisions.
Throughout this discussion of threeperson models we have remained within the confines of the independence assumption. If the two goods are complements or substitutes in any of the individual's utility functions, the appropriate changes in results can be traced out with similar, but more complex, geometrical constructions. Basically, the conclusions reached under the model examined here are not modified. The exercises should have made clear that the outcome will depend not only on the relations between the two goods in individual utility functions, but, also, on the relationships among the separate utility functions of the separate persons, and on the institutions and rules for group decisionmaking. Until and unless these elements are specified, indeterminacies remain. Even when these are fully specified, outcomes may be unstable in the cyclicalmajority sense.
Figure 6.8 
So long as we remain in the twogood or twoissue model, the analysis can be extended without undue complexity to include any number of persons. Additional utility functions can be mapped onto the constructions developed in Figures 6.2, 6.5, 6.6 and 6.7. Consider a fiveman group, as illustrated in Figure 6.8. The positions D_{1} through D_{5} are the optimally preferred combinations. For purposes of economy, indifference contours surrounding each peak are not drawn.
When the two issues are separately considered, and simplemajority voting is the decision rule, the solution remains that shown by the intersection between middle lines of optima, position V. The contract loci connecting the separate positions of optima are drawn in Figure 6.8. For obvious reasons these take on a somewhat reduced significance in all models that include more than three persons. No longer will agreement between two persons alone constitute a majority. All points along or within the boundaries of the area enclosed by the contract loci are not, therefore, possible majorityrule outcomes when the two goods are simultaneously considered, as was the case with the threeperson group. This outside boundary encloses, as before, the Paretooptimal set of positions. Under a rule of unanimity, some position within or on the boundaries of this area would be attained. With majority rule, however, simultaneous consideration of both issues will produce solutions contained within a subset of this larger Paretooptimal set. Under specific assumptions about the shapes of the indifference contours, assumptions that are within the normal constraints of convexity and continuity, the "majorityrule area" can be isolated. Within this area, the voting process will produce cyclical results with constantly changing coalitions. Under plausible assumptions about the range of variation in alternatives presented for the pairwise voting comparisons, this area may approximate that of the shaded fivesided inner figure. The details of analysis need not concern us here. The main point is that the majorityrule set is now a subset of the Paretian set, an important difference between all models with more than three persons and the threeperson model.
As the number of persons in the interacting group expands, the Paretooptimal set of positions or points also expands, as the construction indicates, but the majorityrule set as a proportion of the Paretooptimal set is reduced in size. This suggests that even in large groups, although the problem of cyclical majorities will remain, the set of alternative combinations over which results will cycle tends to become smaller and smaller, in some sense relative to the total possible range of decisions. With very large groups, the discrete area over which majority outcomes may cycle may substantially disappear. Wide and discrete shifts in outcomes are not likely to emerge under the operation of simplemajority voting unless institutional barriers prevent the offering of compromise motions. For extreme shifts to occur, the alternatives must be largely restricted to those that are themselves extreme, relative to the particular configuration of preferences among members of the group.
Vote trading may, of course, take place in such manyperson models, especially if the number of parties is not overwhelmingly large. Given specific preference configurations among subgroups, results akin to those developed in connection with the threeperson model can emerge. The group of individuals whose preferences dictate an extreme position with respect to one of the two variables along with a relative indifference as to the other variable and a decisive median position with respect to the latter is advantageously placed for strategic trading. Interesting examples can be developed by considering npersons subdivided broadly into a relatively small number of pressure groups.
The most general of all possible models is one in which the interacting group contains a large number of persons and where there are many goods, each one of which is purely public. The twodimensional confines of plane geometry are no longer helpful, even in the threeperson, threegood case. Pictures of threedimensional space may be attempted, and threedimensional constructions are helpful in classroom presentation. But even the standard economists' calculus provides little assistance here, since the required conditions for equilibrium, under any rule for decision, cannot readily be stated.
We do not propose to examine this model in detail; references in the appendix are provided for those whose intellectual curiosity prompts them to follow up the suggestions made. The summary comments will be limited to specific relationships that are more or less intuitive.
As the size of the group expands, the Paretooptimal set of positions also expands, but the majorityrule set contracts, relative to the Paretian set. This has already been shown with reference to comparisons between the threeperson and fiveperson models when only two goods are considered. The relationship holds generally as the number of variables, goods or issues, is increased.
On the other hand, as the number of goods or issues increases, the Paretooptimal set of positions tends to contract in a relative sense. This second relationship has not been discussed, but it can be shown in the threeperson model as the number of goods increases from one to two. Refer to Figure 6.4. The possible range of solutions on the single variable Q_{1} extends from 0 to T. The Paretooptimal set of positions, that set from which one position would be attained under a rule of unanimity, includes all positions along the spectrum ranging from T_{t} through T_{s}. If we can assume that positions are discrete and that possible outcomes are uniformly measured, the Paretooptimal set clearly makes up more than onehalf of the total set of possible outcomes. Now, by comparison, refer to Figure 6.5, when a second dimension has been added. The possible solutions for the variable, Q_{2}, are shown by the range 0L. Conceptually, therefore, all possible combinations of Q_{1} and Q_{2} are contained in the rectangle 0LMT. If we again assume discreteness and uniformity over the whole space, it is clear that the set of positions or outcomes enclosed by the contract loci, the Paretooptimal set, makes up considerably less than onehalf the total set of prospects. Again this relationship is a general one, and as the number of goods is increased, the solution set, given any decision rule, tends to contract, relative to the total set of attainable alternatives.
There is a commonsense basis for this second relationship. As the number of goods expands, even if all of these are purely public in the sense that all persons must adjust to the same quantity, individual expressions of preference can be more fully reflected, at least for that subset of persons who are in the decisive coalition as determined by the decision rule in being. Each member of the coalition has in hand, so to speak, a more varied set of counters and this allows him to reach accord with other members more readily and at less cost. The economic analogue, which must be used with caution, is a genuine barter system of exchange. In the absence of a money commodity, an agreedon numeraire, each potential seller must seek out a buyer for the particular good he has to offer and vice versa. It seems clear that the larger the number of goods in his possession the more fully can an individual secure that final set of goods dictated by his utility function as the most desirable. The advice for caution in the use of this analogy lies in the fact that, with barter in private goods, all exchanges are bilateral. In the publicgoods model, even when there are many separate goods, individuals cannot exchange or transfer goods directly. Exchange cannot be bilateral in the standard sense. Agreements can be exchanged, or votes traded, but, in either case, the trading behavior will affect others who are not direct parties to the exchange. There remains an inherent externality in any groupchoice situation that may be absent from privategoods trading.
As stated at the outset, the analytical exercises presented in this chapter are aimed at partially bridging the gap between the economic theory of public goods, explored in Chapters 1 through 5, and the theory of political or collective choice, to be examined more thoroughly in subsequent chapters. Little if any of the material discussed in this book has yet attained the status of orthodoxy or received doctrine, and this applies with special force to the nonnumeraire models considered in this chapter. Only a handful of scholars has worked with such models. The whole analysis remains in its infancy.
The theory of committees and elections was pioneered by Duncan Black, and was exhaustively analyzed in his book [The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1958)]. This provides the background for all models containing only the single public good or issue. As developed by Black and others, the theory does not explicitly refer to "public goods," but to issues, motions or candidates in an election when only one alternative is to be chosen. It is also in this context that the discussion of the "paradox of voting," or cyclical majorities, has taken place. The nowclassic work on this, in addition to Black's, is Kenneth J. Arrow's book [Social Choice and Individual Values (New York: John Wiley and Sons, 1951; Revised edition, 1963)].
The extension of the models to two goods, again discussed in terms of issues, was initially contained in a muchneglected small book by Duncan Black and R. A. Newing [Committee Decisions with Complementary Valuation (London: William Hodge, 1951)]. This book contains extremely interesting geometrical exercises and illustrations, some of which are closely akin to those presented in parts of Chapter 6.
Otto A. Davis and Melvin Hinich have provided a formal mathematical treatment of the behavioral strategy of candidates and parties seeking election as the relevant variables are extended from one to many ["A Mathematical Model of Policy Formation in a Democratic Society," in Mathematical Applications in Political Science, II, edited by Joseph L. Bernd (Dallas: Southern Methodist University Press, 1966); "Some Results Related to a Mathematical Model of Policy Formation in a Democratic Society" (Mimeographed, Carnegie Institute of Technology, May 1966)].
In the context of formal welfare economics, Ragnar Frisch's paper ["On Welfare Theory and Pareto Regions," International Economic Papers, No. 9 (London: Macmillan, 1959), pp. 3992] develops the twogood case exhaustively and extends the analysis to many dimensions.
Building on the work of both Black and Frisch, Charles Plott completed the most rigorous statement of the necessary conditions for equilibrium under alternative decision rules in the manypersons, manygoods model [Generalized Equilibrium Conditions Under Alternative Exchange Institutions, Research Monograph No. 9 (Charlottesville: Thomas Jefferson Center for Political Economy, University of Virginia, December 1964)].
The analysis of the section which discusses the model containing many persons and two public goods has been exhaustively treated by Gordon Tullock ["The General Irrelevance of the General Impossibility Theorem," Quarterly Journal of Economics, LXXXI (May 1967), 25670]. He examines this model in particular relation to Arrow's discussion.
My own analysis of the material in Chapter 6 owes much to many discussions, extending over several years, with Duncan Black, Charles Plott and Gordon Tullock. Should they be willing to accept my interpretations, I should be happy to list them all, informally, as joint authors.
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