Top 3 Approaches to Public Goods Allocation

This article throws light upon the top three approaches to public goods allocation. The approaches are: 1. The Marginal Utility Approach of Public Goods Allocation 2. Voluntary Exchange Approach 3. Samuelson’s Approach to Public Goods Allocation.

1. The Marginal Utility Approach of Public Goods Allocation:

Professor A. C Pigou has made a heroic attempt to reconcile the benefit and ability to pay approaches. Pigou while developing the marginal utility theory of public goods allocation provided an analy­sis which comprehensively included both the tax and expenditure sides of public fiscal operation.


The benefits accruing to an individual are in the form of his consumption of public goods. The costs im­posed are in the form of income which taxes take away. Pigou ex­pressed these relationships in terms of the marginal utility and mar­ginal disutility created by devoting additional amounts of income to certain goods. In his analysis Pigou used the principle of marginalize.

That is the marginal sacrifice from the tax side and marginal benefit from the expenditure side, Professor A. C. Pigou relates the mar­ginal utility derived by individuals in a society from the consumption of public goods to the marginal disutility suffered by these individu­als in the payment of taxes to the public sector for the financing of the good.

If the marginal utility for the production of each good is equal to the marginal disutility caused by paying for it, either in the market place or via taxes, then optimum allocation of resources has been achieved. This optimum allocation refers to the choice between different public goods as well as between good in the public and private sectors. Figure No. 2.9 Shows the optimum allocation on the basic of considerations.

Public Goods Allocation: The Marginal Utility Approach:

Production of public good ‘X’ has been carried to point ‘M’ where the marginal utility derived from ‘X’ is just equal to the marginal disutility imposed by the taxes necessary to pay for its production.


As long as there is positive net benefit from devoting more and more re­sources to the production of good ‘X’, expenditure and taxes should be increased. Therefore optimal intersector resource allocation and optimal supply of both public and private goods occur at the point where marginal utility of public food X(MUx) is equal to marginal disutility of tax payment (MD Tax).

Moreover intersector optimality in resource allocation under this approach requires that marginal utility of the various economic goods allocated by each sector be equal within that sector. In the diagram marginal utility of public good ‘X’ and marginal disutility of tax payment are measured on the vertical axis.

Government budgetary operation including both tax and ex­penditure programme is measured on the horizontal axis.

Point M represent optimal supply of public good ‘X’, Since at output OM of public goods, the marginal utility of public good ‘X’ is equal to the marginal disutility of tax payments. Therefore an expansion of public sector allocation towards point ‘M’ is desirable.


Point ‘L’ represent under allocation of public good and over allocation of private good. Whereas point ‘N’ represent over allocation of public goods and under allocation of private goods. Therefore at point ‘M’, the economy can realize optimal intersectional allocation of public and private goods. The attainment of optimal allocation implies two things.

Firstly the marginal utilities both within the private sector and the public sectors have been equalized, Expenditure on private goods in the private sector are distributed in such a way that the marginal unit of money spends on them gives the same marginal utility to the con­sumer.

In the public sector, expenditure is distributed between pub­lic goods in such a way that the last rupee devoted to each of them yields the same return.

Secondly optimization principle requires that the equalization of the marginal utilities of expenditure between the two sectors. In this case Pigou was concerned with the equilisation of the aggregate marginal social benefit derived by the community as a whole from the expenditure programmes in the private and pub­lic sector and not with the equilisation of individual benefits in the two sectors.


Pigou’s analysis is useful because the ability to pay approach is extended to the expenditure as well as the revenue side of the bud­get. However the problems inherent in aggregating from individual pref­erences to social preferences are not solved.

Another problem re­lates to the absence of an effective means to quantify utility and disutility. Therefore it is difficult to determine precisely the optimal intersector allocation of resources. The problem of attaining equity is considered by the distribution of income, wealth, and political voting system.

2. Voluntary Exchange Approach:

The voluntary exchange theory of public goods allocation derives itself from the benefit received principle governing the distribution of tax burden. This approach suggests that resources should be allo­cated to the public sector in a manner analogous to their allocation in the market with its price system.

An individual should buy public goods through taxes just as he elects to purchase private goods through market price. An individual becomes a “tax payer-buyer” who pays taxes for public gods in accordance with the benefits received from them.

The individual would equate the marginal ratios of tax prices to public goods benefits within the public sector. Likewise he will equate the cost-benefit rates on an intersector basis, between public goods and private goods.

The Lindahl Johanson and Bowen model of voluntary exchange theory of public goods allocation is demonstrated in figure No 2.10. The figure shows the voluntary approach to the public goods alloca­tion.

The quantities of public good ‘Z’ demanded and supplied are measured on the horizontal axis and the demand price and supply costs of public good ‘Z’ are measured along the vertical axis.

For the purpose of simplicity three assumptions are made:

(a) The produc­tion or supply cost of public good is constant,

(b) The society con­sists of only three consumers or taxpayers A, B, and C. and

(c) There is only one type of pure public good Z’ which is demanded by all these consumers.

Voluntary Exchange Approach:

In fig. 2.10 curves AA, BB and CC represent the demand curves of consumers A, B and C respectively for the public good ‘Z’.

The curve DD is the total demand curve of all the three tax-payer-buyers or consumer, tax-payers. The summation individual demand curve is vertical since each consumer consumes the same amount of the public good.

From the figure we can realize that ‘R’ is the optimal point where the aggregate supply curve FS and aggregate demand curve DD intersect with each other and each consumer consumes QQ2 quan­tity of public good. Any other point except R is suboptimal. For example point N shows under allocation and point T shows over allocation, both being sub-optimal position.

The figure further shows that at quantity Q1 at point N on the aggregate demand curve, the demand price per unit exceeds the supply cost per unit by amount EF. This represents sub-optimal allocation i.e. an indication of under supply of public good. Whereas at quantity Q3, at point T on the aggregate demand curve, the supply cost per unit exceeds the demand price by an amount FG.

This represents suboptimal allocation, i.e. an indication of oversupply of the public good. Therefore output should be reduced towards the optimal quantity Q2. At the optimal output Q2, consumer A will pay OPA, B will pay OPB and consumer C will pay OPC per unit price. It should be noted that though each consumers consumes the same quantity, they do not pay the same price.

The combined payment OF of the three tax payer consumes is equal to the cost per unit, that is also OF, as measured along the vertical axis. This is the sum and substance of the voluntary exchange theory as illustrated by Lindahl, Johansen and Bowen.

The voluntary exchange model is a refinement over the earlier approaches to optimal allocation of public goods. This approach pro­vides an exposure to the nature of public goods and the difficulty experienced in allocating them in a market process owing to the collective and joint consumption characteristics of the public good.

Moreover if the sharing group of public good is larger, some people may conveniently hide their preferences and derive the benefit as free riders. In such cases a political process rather than market driven process will be the proper vehicle for covering people preferences (Aronson).

In a sense the tax-payer buyer is a taxpayer voter who must revel his preferences” for private and quasi-public goods through the political process. The same consumers will reveal his preference for private and quasi private goods’ through the market process.

3. Samuelson’s Approach to Public Goods Allocation:

Prof. Paul. A, Samuelson’s model of public goods allocation is con­sidered as the most efficient theory of public goods allocation. The Samuelson model clearly shows the fundamental differences that ex­ist between the allocation of public goods and private goods, based upon the application of micro economic principles.

Therefore this theory is viewed as the extreme opposite of the private goods situa­tion represented by Walrasian general equilibrium case of perfect competition.

Samuelson presumes that an individual may not voluntarily re­veal his preferences for public goods. Therefore the market principle cannot be applied to the provision of public goods. The justification for government provision of public goods in a democratic society is the desire of the members of the society for such goods and activi­ties.

He therefor assumes that there exist omniscient planners to whom all the necessary data (regarding factor supplies, production functions, preference pattern of both private and public goods) are known.

In other words the omniscient planner knows each individu­als preferences for private goods as well as for public goods. There­fore the concept of Pareto optimality was applied by Samuelsson in the case of provision of social goods by the use of what he terms pseudo-demand curves and ‘pseudo prices’. Samuelson model also accepts interpersonal comparison of utility based on distributional value judgment.

These ethical judgments are then applied to an economic effi­ciency norm (i.e. the Pareto optimum norm) in the form of a social welfare function. His model can again be interpreted in terms of refection of interpersonal comparisons.

In the absence of interper­sonal comparisons, the social welfare of the community is simply a heterogeneous collection of individual welfares. This analysis takes into account the condition of Pareto optimality whereby social wel­fare can be increased only if one person’s welfare can be increased through an allocative readjustment without making another person worse or loosing welfare.

However this is a conditional approach in the sense that no judgment can be made concerning social welfare in a situation when one person loses as another gains.

Pareto opti­mum condition, therefore reflect the fundamental problem of scarcity in economic since one individual would not lose welfare while an­other gain, if all economic goods were in infinite supply. The Samuelson model is explained with the following diagrams 2.11.

X's Indifference Schedule between Private Good

Fig. 2.11 (a), (b), (c), (d) depicts the Samuelsson model of public goods allocation. In each segment of the graph, the quantity of pure public good ‘B’ collectively consumed by the community is mea­sured along the X axis and quantity of privately consumed pure pri­vate good ‘A’ on the Y axis.

Figure No 2.11 (a) shows the relative preference pattern of consumer ‘X’ for private good A’ and public good ‘B’ which is drawn along indifference curves lcx1, lcx2 and lcx3. Likewise in figure 2.11 (b), the relative preference pattern of con­sumer Y for public good ‘B’ and private good A is shown along indif­ference curves lcy1 lcy2 and lcy3.

The societies production possi­bility curve is given as PP, in figure 2.11 (c). If reflects the different combinations of public good ‘B’ and private good A, that can be produced with the limited productive resources possessed by the society.

A pure public good is not divisible among consumers. It is con­sumed equally by all. Therefore it possess the same quantity scale value on each graph, because an increase in the total quantity of public goods would increase the quantities available to consumer ‘X’ and ‘Y’ by amounts equal to the total increase (A move to the right of K in 2.11 (c).

Therefore the quantity of public goods on each graph is OK. Since the exclusion principle does not apply in the case of consumption of public goods, both consumes ‘X’ and ‘Y’ individually consume the total quantity of the pure public good ‘B’.

In the Samuelon model the individual preferences by the two consumers in the society for the public and private good is given in figure 2.11 (a) and 2.11 (b). The production possibility curve in figure 2.11 (c) shows the resource constraint imposed on societies pro­duction capacities.

The basic question is, what constitute the opti­mal allocation point between public and private good for the society in other words what is the allocation division which will maximize welfare according to the preference of the two individuals of the com­munity?

The Allocative efficiency point can be determined in figures 2.11 (a), (b) and (c), based upon Pareto optimum social welfare norm. As per this norm aggregate social welfare is increased if one individual moves to a higher indifference curve, without another individual’s satisfaction level being moved to a lower indifference curve.

In order to apply this norm let us assume that one consumer is at a specified level of indifference so that his satisfaction level will not be changed. The problem of optimum allocation then becomes a matter of moving the second consumer in our analysis to his highest possible indiffer­ence curve or satisfaction level.

In figure No. 2.11 (b) consumer ‘Y’ is placed on the specified indifference curve icy, which will now be designated CD. Considering the resource constraint shown by the production possibility curve PR, in figure 2.11 (c) what is the highest level of satisfaction (the highest indifference curve) that consumer ‘X’ can attain.

The answer is shown by the tangency point Ex in figure 2.11 (a) The correspond­ing equilibrium point are at EY in figure 2.11 (b) and E in figure 2.11 (c). The equilibrium is derived by super imposing indifference curve NL from figure 2.11 (b) on figure 2.11 (c) and designating it NiLi.

Then subtract NiLi vertically from production possibility line PP,. The re­sidual gives the quantities of public goods and private goods avail­able to consume ‘X’. This amount may be placed on Figure 2.11 (a) and designated NL.

Consumer X reaches and obtains his highest satisfaction level at tangency point Ex, NL, touches the highest at­tainable indifference curve lcx2. Here NL provides the constraints of resources scarcity and lzx2 reflect the relative preferences of the consumer between public and private goods. This Pareto optimum point means that there is no movement away from point Ex, in terms of reallocation of resources.

Since an optimal state of distribution depends upon societal value judgment, an infinite number of such Pareto optimum points exist. For every infinite specified indifference curve where consumer ‘Y’ might rest, a different optimal tangency point of satisfaction would occur for consumer ‘X’.

That is Ex in figure No. 2.11a will not be the optimal allocation point, if consumer ‘Y’ consumes along an indifference curve other than Icy1 (NL) in figure 2.11b. Figure No. 2.11 d reflects the utility possibilities for consumer ‘X’ and ‘Y’ who constitute the total consumption of society for the public and private goods.

The line MN in this figure indicate utility frontier of Pareto optimum welfare points. The points within the line (shaded area) represent a less than Pareto optimum position. The slop of the Pareto-optimum line to the south east shows the conflicting consumption interest between consum­ers ‘X’ and ‘Y’.

In figure 2.11 (d), point ‘H’ indicate optimal allocation for the society… At this point utility frontier MN is tangent to the highest attainable social indifference curve S2. Line S1, S2, S3 indicates the various combinations of preferences for the members of the society, say ‘X’ and Y. This in turn can be designated as the social welfare function.

Prof, Samuelson’s analysis is an exhaustive presentation of the optimal provision of pure public good. However this analysis suffers from many limitations.

It is point out that social welfare function can­not reveal a true ordering of preferences through economic analysis. The problems associated with the existing state of income distribu­tion in any society, put a constraint on achieving allocative efficiency in the case of public goods provision.

In fact very often provisions of public good and the sharing of costs by the society is basically a political decision, where value judgment criterions cannot scientifi­cally operate.

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