Public Goods Demand - Median-Voter Model

Public Goods Demand and the Median Voter Model

The theory of public-goods demand is an integral aspect of contemporary public-choice theory. Further, it is a good example of how economic analysis developed to handle one problem can often be applied to new problems. In this case the theory of public-goods demand is analogous in most respects to the Mill-Marshall joint-supply theory developed to analyze simultaneous produc­tion of such items as beef and hides, mutton and wool, and so on . Originally articulated by Howard Bowen in 1943, the necessary conditions for allocative efficiency in the provision of a public good were developed by Paul Samuelson in 1954 in a classic paper "The Pure The­ory of Public Expenditures." A public good in this context may be distin­guished from a private good in that, in the public-good case, an individual's consumption of the public good does not reduce all other individuals' simulta­neous consumption. In the private-good case, if XT is the total consumption of shoes, then XT — x, + x2 + ... + xn, where x, + x2, etc., is the sum of all in­dividuals' consumption of shoes. In the public-goods case, Xp may be total consumption of, say, national defense, and Xp = x1 = x2 = ... = xn, where all individuals consume the same amount of defense. In the latter case, one individual's consumption of defense does not detract from another's, and all consume the same quantity of defense.

Here, units of measurement are important. A "unit" of a good is defined as the minimum quantity of that good required to provide more than one con­sumer simultaneously with that particular bundle of services that serves to dis­tinguish the good in question from all other goods. Accordingly, a dozen pen­cils would not be considered a unit of a public good even though twelve individuals could consume this good simultaneously. The reason is that one pencil is capable of providing the unique bundle of services (writing, erasing, etc.) usually associated with the term "pencil." A unit of pencils would be a private good because its services are provided to only a single individual.

A Polaris submarine, on the other hand, can be viewed as a unit of a public good because it provides "safety from nuclear attack" simultaneously to more than one individual. While the provision of "safety from nuclear attack" as a private good might be possible (individual concrete underground silos, for ex­ample), the cost per individual presumably is less when the service is provided as a public good.

Some other characteristics of public goods are important though they are not unique to public goods. For instance, in the public-good case described by Samuelson, the marginal cost of supplying additional users would be negligi­ble—sometimes zero—and the exclusion of nonpaying consumers would be impossible. Some goods in the private sector approximate the above cost con­ditions (a bus trip for a particular journey, perhaps). Moreover, it may always be possible to exclude consumers. Even in the case of national defense it would theoretically be possible to remove nonpayers to (nonprotected) islands in the Pacific Ocean although such exclusion would be costly. The conceptual difficulties of defining a pure public good are many, therefore, but these mat­ters need not detain us here. Let us assume that joint-consumption, zero mar­ginal cost, and nonexcludability conditions apply and turn to the Bowen-Samuelson equilibrium of Figure 1.


The two upper quadrants of Figure 1 depict the demands for a public good (education, Polaris submarines, etc.) on the part of a closed community of two individuals. These demands are summed vertically in order to get the total demand for the public good shown (with a constant-cost supply curve) in the lowest quadrant of Figure 1. Vertical summation of individual demand curves is called for in the public-goods case since consumption between indi­viduals is noncompeting. Individual A's consumption of nuclear submarines does not compete with individual B's. Consumption is simultaneous and "complementary." Most importantly, note that the equilibrium described in the public-goods case with simultaneity of consumption requires (in exact con­trast to the private-goods example) that the same quantity of the good be con­sumed by each consumer (quantity Q* in Figure 1). Different prices are re­quired in equilibrium to get different individuals with different demands to hold Q* of the commodity. The equilibrium prices would not be equal except in the unlikely event that the two individuals' demands are identical.

Samuelson's description of the demand for public goods is perfectly ab­stract and general, but in fitting the principle to real-world applications several difficulties emerge. When the good' in question is not purely public in Samuelson's sense, the optimal size of the consuming group will not be known, and the question that begs answering is: What quantity should be pro­duced (i.e., what Q*)? In his 1943 paper, Howard Bowen reviewed this last issue and answered: "It is, of course, no more difficult to obtain information on the cost of producing social goods than to get data on individual goods; but to estimate marginal rates of substitution [public-goods demands] presents se­rious problems, since it requires the measurement of the preferences for goods which, by their very nature, cannot be subjected to individual consumer choice" ("The Interpretation of Voting in the Allocation of Resources " dd 32-33).

Some sort of proxy for public-goods demands is needed, in other words, and Bowen suggested that, under certain conditions, voting (in a democratic setting) is the closest substitute for consumer choice. This so-called median-voter model (actually a whole set of models) became the major tool of public-choice theorists in the 1960s and 1970s owing in large part to the pioneering efforts of Duncan Black and Kenneth Arrow. While this literature is central to modern public-choice theory, it is fairly technical and would take us too far afield. Nevertheless, the Bowen model and its variants (along with possible complications and problems) may be presented in simple terms.

Any individual's demand for public goods will be determined by two things:
(1) the satisfaction he or she expects to receive from various amounts of it, and (2) the cost to the individual of alternative amounts of the public good. In order to look at even a basic model of voting behavior, we must invoke simplifying assumptions. First, assume that all members of a community actually vote and thereby correctly reveal their individual preferences for the social good. Second, suppose that the total and average cost of the good to the community is known and that it is divided equally among all citizens. Finally, assume with Bowen "that the several curves of individual marginal substitution [i.e., the individual demand curves] are distributed according to the normal law of error" ("The Interpretation of Voting," p. 34). This simply means that there are a large number of demand curves and that, for any quantity of the public good provided, there will be demands clustered symmetrically about a mode. Such a community may be illustrated easily in terms of Figure 2, which shows the clustering of demands about the demand of the median voter. The pro rata tax share (AC/N) is the same for each voter-consumer. Now consider a provision of some quantity of the public good 2, in Figure 2. Clearly for the same quantity of the good, different demanders would be willing to pay different tax shares. Thus, for Q1,, those who value the good highly would be willing to pay D7, those placing little value on the public good would only be willing to pay D1, and so on. The median voter, however, values Q1, at some rate D4, which is higher than the pro rata tax share to all taxpayers who re­ceive the public good AC/N (MC/N). Thus in, say, a town-meeting process em­ploying majority rule, any Q proposed above Q1, will win approval; any Q pro­posed above Q*, such as Q2, will fail to carry. In this process, the quantity preferred by the median voter, Q*, will always defeat any other motion. The median-voter process, under certain circumstances, can yield similar results in other variants of the model, such as voting for marginal increases of a public good in a referendum process or through elected representatives.


In the latter case, if the people are consulted on particular policies and if repre­sentatives identify with specific issues, the results of the process can approx­imate those of Figure 2. Many factors affect voting. Public officials working through certain institutions may upset the results of Bowen equilibrium by ma­nipulating the agenda or simply by representing and voting on a large variety of issues. Thus, majority-rule election processes do not ensure that voter prefer­ences for public goods will be optimized. It does seem to be a practical system for approximating preferences, however.