Distribution of the tax share, as noted above, is a crucial feature in the provision of public goods, since any individual will demand a good both on the basis of its (marginal) value and on the basis of its cost. The "marginal cost" is simply the share of taxes that the citizen-consumer pays for his or her portion of the output. A major problem in public choice, then, is to devise a means for providing an optimal quantity of any public good such that, for the single quantity produced, some distribution of the tax burden may be found that equates the marginal valuation of the good to the marginal tax share for each citizen-consumer. Two early writers on public choice, Erik Lindahl and Knut Wicksell, were interested in different aspects of this question and originated different paths of analysis in modern public-choice theory.
Lindahl Equilibrium
In his 1919 contribution entitled "Just Taxation—A Positive Solution" (a part of his book Die Gerechtigkeit der Besteuerung), Lindahl treats the problem of tax-share determination as one of bilateral exchange in an "isolated" community with two categories of taxpayers, one "well-to-do" and the other "relatively poor." The problem of the distribution of the tax shares is then considered to be one settled by free argument, or "a kind of economic exchange." (Lindahl of course recognized that this process was filtered through protagonists in a political process and that resultant tax-share distributions assigned would be influenced by their relative power, but he assumed initially that such political "blocs" did not influence the model under free exchange.)
Lindahl's solution is straightforward. In a "solution in which both parties have equally safeguarded the economic rights to which they are entitled under the existing property order," the price of the collective good "tends to correspond to marginal utility for each interested party" ("Just Taxation," pp. 172-173). This means that tax price will equal the affected voter's (or group of voters') marginal valuation of the public good.
Figure-3
Consider the modern adaptation of Lindahl equilibrium in Figure 3 In Figure 3, DT is the vertically summed demand curve for the public goods, with D1 and D2 being the separate demand curves of the two groups. Lindahl equilibrium would occur, through voluntary exchange, when for quantity Q*, well-to-do demanders are charged a marginal tax rate T1 and the relatively poor consumers are charged a lower tax rate T2. Under this tax system, each group is paying a marginal cost (T1, and T2 respectively) equal to its marginal valuation of the public good. Efficiency is achieved in the Bowen-Samuelson sense because a single quantity of the good is produced, Q*, which corresponds to the equation of total demand DT and marginal cost of production.
The establishment of Lindahl prices is not necessary in order to obtain efficiency in the production of public goods in the Bowen-Samuelson sense. All that is required for efficiency is that total output of the good be established at point F (producing Q*) in Figure 3. In order to understand this fact, consider the imposition of some "average" tax rate T—one that would be imposed on both groups of demanders and that would cover the costs of producing Q*.
It is easy to see that the well-to-do demanders would prefer this system and would, if possible, foist it on the poor through a political process (Lindahl considered this case). Note, however, that at tax rate T3 the poor would prefer Q2, a less-than-optimal quantity of the public good. If the poor were politically powerful, they might force society to take a less-than-optimal quantity of the good. In general, however,* a system of Lindahl-tax prices would produce Bowen-Samuelson efficiency—everyone would agree on how much of the public good should be produced. While a Lindahl system is not the only one capable of producing this result, it is also the case that a Lindahl model features unanimous agreement of the taxed parties in voluntary exchange given differential tax rates. Probably this feature of Lindahl's work is very strong because his conception of public finance was deeply influenced by his mentor, Knut Wicksell.
