The Lindahl Model (notes by Simon Vicary)

1. The Story so Far

Our last lecture found out that voluntary action (the market mechanism) was likely to bring about a sub-optimal provision of the public good, as a result of individuals trying to take a free ride on the contributions of others. One obvious solution to this inefficiency is to recommend some form of government intervention. Indeed many of the services provided by governments have strong public good characteristics, and these are also important in areas in which the government intervenes rather than provides (e.g. environmental regulation). Defence, as we have seen, in many respects gets close to a pure public good, and the same can be said about “law and order”. The issue is rather more blurred with areas like education and health, but even so there are certainly some public good elements to what the government provides.

However, government provision of a public good raises a whole series of new questions. In fact it is a massive area of enquiry in its own right, in some respects incorporating political science. In this module we cannot hope here to do anything other than provide some basic ideas. The traditional starting point for economists in thinking about the public sector and in particular public expenditure is the Lindahl Model. In some ways this is a rather elusive concept, as it can be looked at in a variety of different ways:

• An alternative way of representing the Samuelson analysis
• A positive theory about how government expenditure decisions are arrived at
• A normative theory about how government expenditure should be arrived at
• An analytical device for judging alternative mechanisms by which government expenditure decisions are arrived at (somewhat similar to perfect competition in a world of private goods)


2. The Benefit Approach to Taxation

Modern public economics grew out of the old sub-discipline of public finance. A key issue in this older tradition concerned the concerned the basis of what could be called a fair tax. There were two approaches:

• The Benefit Approach. Here a fair tax was based the benefit an individual received from government expenditure. People who benefited more from public expenditure would, according to this school, have to incur a higher tax bill. This line of approach was particularly popular with continental economists, especially Italians and Scandinavians, although British thinkers adopted it in the 18th century stemming from a social contract view of the state.
• The Ability to Pay Approach. Here a just tax was based on an individual’s ability to pay. Those with a higher ability to pay should pay more taxes than those with a lower ability. This school of though was more popular in Anglo-Saxon countries, especially in the later 19th century possibly because of the stronger influence of utilitarian thinking.

Modern welfare economics has rather dispensed with debates of this kind, but both lines of thought have left their mark. The key idea of the benefit approach was that taxes should represent a payment for services delivered. The relationship of the government to the citizen was therefore one not in principle different from that between individuals in the community involved in mutually beneficial exchange. The state as such is only justified to the extent to which it provides benefits to individual citizens. The mainline value judgements adopted by modern welfare economics are Paretian, and the question of ultimate “justice” is not pursued any further than this. Taxes in the modern view are desirable if they lead to Pareto efficiency (subject to any distributive objectives about which Paretianism itself remains silent).

The benefit idea was revived by the Swedish economist Knut Wicksell in 1896, and in the hands of his pupil Lindahl, together with later developments led to a re-run of the Fundamental Theorems of welfare economics when public goods are present. Wicksell started from the following presumptions:

• People should not be coerced into paying for goods or services they do not want
• Public goods are more efficiently provided by the group as a whole, so individuals need to communicate their preferences to others
• To ensure that no coercion takes place, the appropriate principle a government must make in determining its expenditure-tax decisions is unanimity.

Although Wicksell wrote before the Paretian approach got established as the mainline form of welfare economics, his thinking fits very neatly into this approach. In a way, it represents an extreme form of Paretianism. When there are collective decisions to be made, how do we guarantee that no one is going to be made worse off? Unanimity (or giving each person a veto) would seem to be the only way. Wicksell realised, of course, that unanimity would be difficult in practice, but it was a principle to aim for, and in making decisions there should be “approximate unanimity”. The difficulties can be seen if we take another two person economy, and suppose individuals are bargaining over tax bills and provision of the public good. On Figure 1 we illustrate.

The area ABCD represents the area of Pareto improvement over the initial position without government. To see this note that the horizontal axis represents the provision of the public good, and the vertical individual A’s share of the total cost of whatever provision is decided upon. The indifference curve going through the origin represents the utility A receives in a state of anarchy (no government) (no taxes no government expenditure). With a veto over any decision A can be sure of never getting a utility level below this. The indifference curve going through the point (0, 1) fulfils a similar function for individual B.

The “model” of public expenditure determination goes as follows. The government announces a tax-expenditure package, say, point F. If both agents are better off than their initial position then they vote to accept this package, and point F is the next starting point. If the proposed package lies outside ABCD then at least one person will veto it, and it will be rejected. The government then produces another package. Starting from F another area of Pareto improvement can be traced out by the indifference curves going through F, and the government reformulates another package in the hope of finding another Pareto improvement. Once such a package has been found to which no one objects (vetoes) then this becomes the new starting point, and the process starts again. Equilibrium is finally reached when a package has been found for which for any counter proposal made there is always an objector. Such a point will be found somewhere on the set of Pareto optima, the dashed line (locus of tangencies) on Figure 1. It is not too difficult to show that tangency of the indifference curves on Figure is equivalent to the fulfilment of the Samuelson condition for Pareto optimality.

This, in simple terms is how the Wicksell model of ideal public expenditure should go. No one would argue for this to taken literally, but in a curious way it is quite instructive:

• Taken literally it is a process that guarantees Pareto optimality. However, its very artificiality may suggest that there are serious difficulties in the way of achieving this in practice.
• If each iteration is costly, as we would expect, then this would be a very expensive way of determining public expenditure
• Although stylised, the model picks up an important aspect of politics in modern democratic states. The determination of public expenditure is a major issue, as is the related determination of the tax burden. The process of determining these variables is accompanied by a lot of bargaining, both within and between government departments, and also, significantly, by lobbying by private interests
• Although Wicksell’s model is in a way within the benefit tradition, in one respect he is quite modern. He was careful to point out that his scheme was valid only once the question of the optimum distribution of income had been settled. Hence he separates out allocation and distributional questions in very much the same way as the fundamental theorems aim to do
• There is an obvious problem with the model even on its own terms. Why should people be honest in responding to successive government packages? Why pretend not to want the public good in an attempt to lower one’s own tax bill? This is a problem to which we shall return
• The final package (t, G) is indeterminate. We could end up anywhere on the line BD. There is no obvious way within the limits of the model to say where this might be. Incidentally, this point shows too that Wicksell’s point about income distribution has not been fully settled: the actual way this process proceeds could have quite significant implications for the distribution of welfare between individuals.

The Buchanan-Tullock Optimal Constitution

Wicksell’s idea proved quite influential, and, as is the way with these things, it developed in a number of differing ways. One such was the “optimal constitution” approach developed by Buchanan and Tullock in their 1962 book The Calculus of Consent. A central part of this book was an optimal voting scheme justified by unanimity. The idea was that while it would be too costly to rely on unanimity to reach every single tax-expenditure made by governments, the principle of unanimity could be preserved if everyone gave their consent to the process by which such decisions were made.

It is worth digressing a little to see this development of Wicksell’s idea. Buchanan and Tullock’s approach can be characterised in two ways:

• Like, Wicksell, they took Paretian value judgements seriously
• They consciously adopted a view of the state that revived the social contract tradition that originated in England in the 17th century.

The practical counterpart of their theory was the formation of the constitution of the United States in 1789.

Starting from Wicksell’s thinking they took it that Paretian value judgements should be treated seriously. No one should be coerced into submitting to laws or, for our purposes here, having to pay taxes without their consent. If we start from this point then in principle each person should have a veto over any law proposed, or any proposal to raise taxes, as with Wicksell. However, the cost of doing this is likely to be prohibitive, and probably going to satisfy no one. One country which did use the veto in its parliament (sejm) was Poland in the 17th and 18th centuries. This was hardly a good example of unanimity in action. The veto (liberum veto) was abolished in 1791 just four years before the country disappeared in a (final) partition between Austria, Prussia and Russia.

See Wikipedia http://en.wikipedia.org/wiki/Sejm for some details.

Wishing for unanimity as a principle to underlie collective decisions, but recognising that it was impractical to have unanimity in every decision that has to be made, Buchanan and Tullock shifted the principle back one stage. People give their consent to each collective decision not if they approve of the decision itself, but if they have given their consent to the manner in which it was made. That is, everybody gives their consent to the “constitution”. This, as we have already seen, is to be thought of as the procedure by which this actual decisions (here most relevantly on taxation) are made. The picture they have is therefore quite parallel to the formation of the constitution of the USA, and, as a more recent possible example, to the establishment of democracy in the Eastern Europe.

This idea, though, raises the question of what sort of constitution would emerge from unanimous consent. Not much is said about the legal details in their book, but the principle by which decisions should be made is set out in the central chapter of the book, and illustrate on Figure 2 Here we have two cost functions. The decision-making function represents the cost of making a decision as determined by the percentage of individuals whose consent would be required. Quite possibly this becomes infinite as this approaches unanimity. The second function, the so-called “external cost” represents the possibility of lost of benefits from not being able to achieve Pareto optimality, or perhaps the expected cost of have a decision that makes the individual worse off. However we look at it, unanimity guarantees that no one will be made worse off or exploited by others, and it is reasonable to assume that this is downward sloping. Total cost is simply the sum of these two. So our individual would want a constitution that required the degree of consent given on the horizontal axis, the “optimal constitution”.

Various points can be made about this piece of analysis:

• If we think in terms of voting rules, the “optimal constitution” point could well vary as between the degree of consent required for different types of measure. It is notable, for example, that often a higher percentage of a vote is required for constitutional amendments.
• The schedules in Buchanan are not well defined. It is not clear what they mean in other than a very general sense.
• There is no reason why “majority rule” should have any special status as a voting scheme under this approach. This is argued by Buchanan and Tullock themselves, although they may overplay their hand because……..
• Voting rules that require more than 50% consent are not reversible. That it, they cannot be overturned by another coalition of voters. It 40% approve of a measure, then 40% could also vote to have it reversed. It is possible therefore that even though the schedules are ill-defined they have not been drawn accurately. There may be a jump in the decision-making function at 50%.
• In one sense Buchanan and Tullock do not solve the problem they pose. If our diagram represents what one individual thinks, there is no reason why the “optimal constitution” will not differ for another individual. It is not clear what happens at this point.
• One can regard the Buchanan-Tullock optimal constitution as an attempt to escape the dilemma of the Arrow Theorem. Unanimity might be easier to achieve for a constitution than for actual day-to-day decisions. However, there is no guarantee of this within the framework B-T develop.
• Buchanan and Tullock compound this last problem by asserting that at the constitutional stage people have already established a set of property rights. They negotiate about the constitution on this basis. This makes the possibility of unanimity less likely. To take the US example, an unpleasant one, it is reasonable to suppose that slaves in the 18th century USA would, had they been asked, have expressed rather different views about the constitution from the slave owners who were instrumental in writing the constitution.
• One can interpret Rawls’ 1971 A Theory of Justice as a way of clearing up this obscurity in B-T. His view was that a constitution should be based on what individuals would agree to behind a veil of ignorance. That is, people would know how society would be run, what laws and rules it would have, but would not know their identity within that society. They would have an equal chance of being any “named” individual. So they could be a slave owner, if it were a case of joining 18th century USA, but it would be much more likely that to be a slave. Who would consent to slavery in this case? Behind the veil of ignorance, it becomes more plausible that unanimity could be achieved. We could therefore interpret Rawls as trying to find a way round the Arrow problem

Another approach which leads into modern general equilibrium theory was initiated by Wicksell’s fellow Swede Erik Lindahl. To this we now turn.


3. The Lindahl Model: Two Person Economy

Lindahl’s work published in 1919 refined that of Wicksell. We can get an idea of the key points by looking again at the Wicksell diagram. Consider Figure 3. The indifference map is reproduced as in Figure 1, but now individuals are asked a different question. Instead of “Do you approve this total tax-expenditure package?” they are asked, “If you had to pay x% of the tax bill what quantity of the public good would you want to see provided?” Although the fundamental features of the Wicksell approach are preserved, there are one or two significant differences in the way we can look at the model. The key difference is that after setting personal tax rates the government/planners look at the responses individual make. If people are unanimous then the process stops and the tax rates and provision of the public good are determined accordingly. If people are not unanimous, then the government adjusts tax rates and puts the question again to the electorate. A natural procedure would be to raise the tax rates for those who want a high provision of the public good, and lower the tax rate for those who want low provision. The process continues until people are unanimous.

To see how this might work out, first imagine we alter the share of tax that individual A must pay, and trace out her responses. This we do on Figure 3. Clearly, as the tax share falls in some sense the price of the public good to our individual must also fall. Hence by tracing out the locus of tangencies as we do we must derive some sort of a demand curve for the public good. We can repeat this procedure for individual B. As lowering tA means tB = 1 – tA must rise the demand curve we draw will be upward sloping. The result is shown on Figure 4. Naturally something significant must be happening at the point at which the two demand curves intersect. This point indeed represents the (unique) Lindahl equilibrium on the diagram where people are unanimous about how much of the public good should be provided. Various comments follow…

• The significance of the intersection is that again as with Wicksell individuals are unanimous about the tax expenditure package. However, in this case the unanimity takes a slightly different form. Given the tax shares no one wants to alter the quantity of the public good. If you could imagine this in a political system, this could be interpreted as meaning that no one would want to lobby to alter the level of government expenditure.
• This last point can be illustrated by drawing in the indifference curves for point L, the Lindahl equilibrium on Figure 4. Given the way the demand curves were constructed, not only will we have tangency but the each indifference curve will be horizontal at this point. Given the tax share, each person consumes exactly the quantity of the public good they would want to consume. With Wicksell it is possible to have an equilibrium in which one person would want more G given their tax share, and their partner less (Draw a diagram to confirm this point).
• Suppose, to keep life simple, we have a constant cost economy, so the public good price, pG, is fixed. If the tax share for individual is tA then the expression tApG represents the effective price of the public good for individual A. A similar point applies to B. (The assumption of constant cost is not essential to this argument. These personalised prices can simply be defined in the same way for a variable cost economy. In this case it might perhaps be better to refer to t as the cost share rather than the share of the tax bill).
• The “personalised prices” are referred to a Lindahl Prices. Usually the writer uses this term to refer to the equilibrium prices, but this semantic issue is not important. The key point is that with (equilibrium) Lindahl prices individuals get exactly the quantity of the public good that they would choose if they had to pay their Lindahl price. In practice, of course, people cannot choose the quantity of a public good in the same way as they do the quantity of a private good. However, the Lindahl model produces an outcome in which things are no different from what they would be if individuals had this choice.
• This last point means that there is a sense in which the Lindahl model converts a public good economy to one which is analytically equivalent to a private good economy.

This being the case, a natural question to ask is whether the Lindahl model has the equivalent of the two Fundamental Theorems of Welfare Economics. That is, is it the case that all Lindahl equilibria are Pareto optimal? And is it also true that any specified Pareto optimum can be realised as a Lindahl equilibrium? Not only do the last couple of points suggest the questions, they also suggest a way of answering them. For if a notional equivalence between a Lindahl economy and a private goods economy can be established then the proofs of the original fundamental theorems can be used to shown that the Lindahl model is the public goods analogue of the competitive mechanism for a private goods economy.


4. The Fundamental Theorems of Welfare Economics Revisited

It is easy to show that at any Lindahl equilibrium the Samuelson condition must hold:

Given the replication property referred to in the last section, at any Lindahl equilibrium, and given any individual i, with a private good x and a public good G, the following condition must hold:



pG is the actual price paid for delivery of the public good. We assume that the public good is produced by competitive firms. Hence:



Now sum this equation over all n individuals:



This is the Samuelson condition.

Technically, however, this does not provide us with a proof of the first Fundamental condition. The Samuelson condition is a necessary, not a sufficient, condition for a Pareto optimum. However, this argument suggests, correctly, that in all standard cases the analogue of the first theorem holds. In fact under pretty much the same conditions as with a private good economy we have:

Theorem 1
Any Lindahl Equilibrium is Pareto Optimal.

The second theorem is a bit more difficult, but in its technical details the proof goes through in pretty much the same way as the original theorem. The key technical assumption is again convexity, which is needed to ensure the existence of a set of (Lindahl) prices. Once these have been found, then utility and profit maximisation do the rest of the work. We can show the second theorem diagrammatically using the diagram developed by Cornes and Sandler, Figure 5.

The dotted line in this diagram is a 45o line, and the line AC is the set of Pareto Optima. Our problem is this. Can we specify any point on AC, and then (at least in principle) achieve this point as a Lindahl equilibrium?

Assuming constant cost, a Lindahl set of prices for this economy would be shown as a straight line going through the origin. (Individuals pay a fixed share of whatever it is that is delivered). Suppose we are interested in reaching point A. As the diagram shows, the tangent going through point A does not go through the origin. Hence if we are to reach point A some prior redistribution must occur. To see how this happens on the diagram, recall the Warr Neutrality proposition. If we re-distribute from individual 2 to individual 1 there will be no change in final utility and all that happens will be that individual 2 cuts donations to the public good by the exact amount of the transfer, whereas individual 1 raises donations by the same amount. What does all this mean for Figure 5?

When there is a re-distribution the indifference curves must shift. Suppose say $10 is transferred from individual 2 to individual 1, and suppose 2 cuts donations to the public good by $10. In this case individual 1 would be as well off as before if she raised her contribution to the public good by $10. Now suppose we start at one particular point on 1’s indifference curve. After the transfer the equivalent point on 1’s new indifference map must be on the 450 line below and to the right of the old point (recall that individual 2 is lowering donations by $10, and 1 is raising them by the same amount). Our conclusion is:

When a transfer from 2 to 1 occurs on the diagram 1’s indifference map shifts down to the right along a 450 line.

A similar argument shows that 2’s indifference curve shifts down along the same 45o line, and by the same amount.

Armed with this conclusion, return to Figure 5. Suppose now we redistribute from individual 2 to individual 1. The indifference maps for the two individuals, and point A move down in a 450 line. As they do so the tangency going through point A follows. When point A reaches point B, the tangency as drawn goes through the origin. Hence if the re-distribution succeeds in moving point A to point B (or more precisely if the re-distribution ensures that the tangency goes through the origin), then the specified Pareto optimum can be achieved as a Lindahl equilibrium. As no matter which point we choose on the (dotted) line of Pareto optima, this will always be possible the second welfare theorem is proved for this sort of economy. To summarise

Theorem 2
Any specified Pareto optimum can be achieved as a Lindahl equilibrium.

Paradise regained it seems. However, things can’t be as easy as this, and although the Lindahl mechanism does indeed have a technical equivalence to the competitive mechanism, there are, even abstracting from the costs of operating the system a number of serious problems it faces in being implemented. We consider these in the next section.


5. The Incentive Problem

There are two points to make. One is rather technical, and just worth noting. The other is easier to understand and highlights a problem we shall take up later on in the module.

The (Non-shrinking) Core

In examining competitive equilibrium, mathematical economists developed the idea of the core. The concept itself is not too difficult to grasp. Suppose any set of agents was able to make any set of binding contracts they liked amongst themselves. What sort of allocation would result? The idea of the core results from the following observation. Suppose some allocation, call it A, is about to be realised as a result of a set of binding contracts individuals are about to make with one another. Suppose, however, that there exists a sub-set of individuals who can make a set of contracts amongst themselves that ensures they will all be better off than in A regardless of what the rest of the community does. In this case we would expect this subset to sign the relevant set of contracts. Allocation A would then not be realised. In thinking about what set of contracts people might agree upon, we should therefore require for that any set of contracts or allocation B to be a candidate for the final outcome for the economy, no such coalition should exist. That is, there should be no group of individuals who, by making a suitable set of contracts amongst themselves, can make themselves better off than they would be at B.

The set of allocations/contracts with this property is called the core of an economy. The interest of the core lies in its absence of any institutional detail, apart from the enforcement of contracts. If competitive allocations all lie inside the core we would expect them to have some sort of stability property. No group of individuals will be able to make alternative arrangements of mutual benefit to themselves. On the other hand, if allocations other than competitive ones lie in the core it is possible that outcomes other than the competitive ones are conceivable. So, given that people can make any set of contracts they like how do allocations in the core compare to the competitive allocations?

Using a box diagram it is easy to show that not all core allocations are competitive (starting from an initial endowment of goods). However, in 1963 Scarf and Debreu proved a result that had been suspected for some time. Take a private goods economy. As the number of individuals in the economy increases to infinity the set of core allocations shrinks to the set of competitive allocations.

Given that the Lindahl mechanism has an analogous role to the competitive mechanism for a public goods economy, it is natural to ask whether the same results hold. The answer is simple:

• Lindahl Equilibria allocations are in the core
• In general the core contains allocations other than Lindahl allocations
• The core does not shrink to the set of Lindahl allocations as community size expands

The analogy does therefore not go through completely, because of the last proposition. Intuitively, this can be understood in the following way. To eliminate an allocation from the core, there must exist a group of individuals who can make themselves better off regardless of what others in the community might do. In a private goods economy, the rest of the community can attempt to force some allocation of the “rogue” subgroup by refusing to trade with them, but this is the limit of what they can do. With a public good economy, the rest of the community has an extra weapon: they can refuse to supply any public goods (remember the rogue group will still benefit from any provision by the rest of the community). The extra sanctions available mean that other allocations are possible (Think for example of a Wicksell equilibrium that is not a Lindahl equilibrium).

This subsection is more in the nature of a footnote. We now turn to a more important point (for our purposes).

The Incentive Problem

Recall the way the Lindahl process works. People are asked to say how much of a public good they might want given the tax price (or Lindahl price) they face. To achieve the Pareto optimal outcome we must suppose that individuals are honest about how much G they want at each stage of the process. In this way the individual demands for the public good can be traced out accurately. But how plausible is that individuals will reply honestly? The answer is not very likely. Figure 6 illustrates.

Here we reproduce Figure 4 with the demand functions for individuals A and B. The Lindahl equilibrium is marked L, and individual A’s indifference curve at L is also drawn in. Remember that the direction of preference for A on the diagram is downwards. Other things being equal he wants to pay lower taxes. Individual B’s demand curve is drawn in. We can suppose she is being honest, but this is not needed. All we need is that B’s demand apparent curve is upward sloping, as drawn. Will A be honest? Suppose A responds to the government’s questions dishonestly, revealing the dotted line marked DA’ as his demand curve. If so the Lindahl process will end up at point Q. Given B’s declared demand curve, A has maximised utility reaching ICA’. For A, utility rises as a result of his dishonesty. It is also worth bearing in mind that A is being dishonest about something only he knows, so there is no danger of being caught out!

The conclusion is that there is no reason, apart from self-imposed morality, for individuals to behave honestly. This being so we would expect people to understate their preference for the public good (compare the false and true demand curves for A on Figure 6), and that if anything like a Lindahl process was ever used it would lead to under-provision of the public good.


6. Conclusion

The Lindahl mechanism can be looked at in a number of ways. In these notes we have focused on its role as public good analogue to the competitive mechanism for private goods. As such, it provides a notional ideal against which actual allocation mechanism can be measured. Although rather abstract, there are number of issues in taxation and political economy that it highlights.

Public Attitudes to Taxation
One key problem in determining the optimal level of government expenditure is the views the public or, from the politicians’ point of view, the electorate has on public goods. If say the question is about an expansion of health or education expenditure, what does the public want? In fact a common feature of opinion surveys is a form of schizophrenia on taxation. People value health, education etc., but are somewhat cagey about the question of taxation to pay for extra expenditure. A common response is: “Yes, I think extra expenditure on health is a good idea”. However, when asked about how this should be paid for, it is not uncommon to get the reply that “the rich” (i.e. someone else) should pay for it. This is unhelpful in working out what the optimum provision should be. Even with benign politicians, if we think the political process does in some remote way resemble the Lindahl model then there are going to be inefficiencies. The cause in this case is not dishonest politicians, but a dishonest electorate.


Not one key feature of the Lindahl model: government expenditure is linked directly to taxation. The only way you would get extra health-care expenditure in a Lindahl world is through being willing to pay for it, just as in an “ordinary” market.

Hypothecated Taxation
One proposal to overcome this problem is to hypothecate taxes. That is, to assign certain tax revenues to certain public expenditures. This is not common in the UK. Possible examples are the revenue from the road fund (car licence) which is supposed to go on road maintenance etc., and national insurance payments which finance the state pension and other social insurance benefits (job seekers’ benefit). In neither case is the link taken very seriously. More significantly, in the USA local elections are held to determine whether say a rise in sales tax should be enacted so as to finance extra expenditure on schools.

Whilst hypothecation is some way from a true Lindahl system, it is a step in that direction, and shares the key feature that tax and expenditure decisions are linked.

Implicit Lindahl Prices
Here we have not explored the political economy aspects of the Lindahl model in any detail. They are in any case a little diffuse. However, one insight from our analysis is worth mentioning. A key feature of most political systems is lobbying by various interest groups. This can take various forms, but often the lobbying is for some form of public provision of a good or service (which often does have some public characteristic). What determines who lobbies, and what sort of cause would they lobby for?

The Lindahl model gives us a clue as to where to start. If the government increases expenditure on say health, it will raise taxes to do so, or in Gordon Brown’s case national insurance contributions (usually taken to be a tax in practice). The point is that implicitly there is a price each person pays for the extra health expenditure. How does the net marginal benefit compare across individuals? Are there consistent differences between individuals we might expect, based perhaps on income levels?

There are many issues here. The Lindahl model does provide a basis for understanding the process of lobbying in democratic systems. Even when the system is not “democratic”, there are likely to be some albeit implicit interest groups. It would be foolish to suppose, though, that a Lindahl mechanism would end lobbying. It is true that given Lindahl prices no one would want to lobby for an extra amount of a public good. However, you do not need to pay more than casual attention to political debates to realise that much lobbying concerns taxation. A clear case of this was the fuel tax protests a few years ago, not to mention the debate over Ken Livingstone’s imposition of a road charge in London. Note, however, how the Lindahl model highlights the dishonest nature of much of this lobbying. “I want to pay lower taxes, so you (non-motorist or whoever) must pay more.” The second clause of the sentence is usually left out.

The point here is that much lobbying is for income distribution purposes. Under a Lindahl mechanism this problem emerges, as we have seen, in the incentive of individuals have to understate their preference for a public good. Hillman rightly places the Lindahl model at the centre of the economist’s attempt to understand the public sector, and refers to it as providing a “consensus” solution. It is unclear, though, how far we can take this idea if people are willing to use the “democratic” process to alter income distribution in their favour.
Pectus Excavatum