Pure Theory of Public Expenditure

7 months ago

Atiqur Rahman

10 minutes

Read First: Public Expenditure: Concept, Objectives, & Public vs Private Expenditure

Pure Theory of Public Expenditure

It was given by Paul A. Samuelson, though the tradition goes back to Italian and Austrian writers who are responsible for the revival of the benefit approach. Their main concern was the presentation of efficient resource allocation and optimal provision of public goods. The next topic of discussion is how to determine the optimum provision of public goods.

Private goods are provided on the basis of preferences revealed by individuals in the market. However, in the case of public goods, individual preferences are not known. So, public goods cannot be provided by private organizations and hence, are provided collectively. How can the market principle then be applied to the determination of the optimal provision of financing of public goods?

One important characteristic, as described by the definition of public goods, is that they are non-rival in consumption, meaning the consumption of the good by 1 individual does not decrease the benefits derived by all other individuals from the same good. Therefore, one resultant implication of this characteristic is that when finding out the aggregate demand for the good, an individual’s demand curves must be added vertically rather than horizontally, as in the case of private goods.

Consider the supply and demand curve for private and public goods as shown in the figure.

Figure 1: Aggregate Demand in Case of Private and Public Good

Part- (a)

Let D_A and D_B be the demands of individuals A and B, respectively. MC is the marginal cost of production. The aggregate demand curve is found by simply adding the quantity demanded by each individual at a given price. If producers price at the marginal cost, then each individual will consume the good when the marginal benefit (MB) from doing so exceeds the price. Hence, D_A+B is the aggregate demand curve at price P. At this price, the quantity demanded by individual A is q_A, the quantity demanded by individual B is q_B, and the total market demand is q_A+B.

From the diagram, it is clear that,

MC=P=MB_A=MB_B

Part- (b)

Let D’_A and D’_B be the demands of individuals A and B, respectively. MC is the marginal cost of production. Since the good is public, each individual can consume the same quantity. The aggregate demand curve, in this case, is found by adding the price that each individual is willing to pay for a given amount of the good. Therefore, D’_A+B is the aggregate demand curve for the public good. For quantity q of the public good, t_A is the price paid by A, t_B is the price paid by B, and P is the price of the public good.

The optimality condition, in this case, becomes

MC=MB_A+MB_B

As can be clearly seen from the above discussion, the aggregate demand curve in the case of private goods is found by horizontally adding the individual demand curves, whereas, in the case of public goods, the aggregate demand curve is found by vertical summation of the individual dd curves. The summation in the case of private goods is over quantity at any price, whereas in the case of private goods, the summation is over price at any quantity.

The question now is what combination of private and public goods should be produced?

Some important terms need to be discussed in order to answer the above question.

Transformation Curve: It is a graphical illustration of the maximum amount of 1 good or service that an economy can produce by reducing the production of another good or service and transferring the resources saved to the production of the former good.

Let us take the case of a commodity world. An economy is capable of building and equipping fifty hospitals if it does not build any schools or eighty schools if it builds no hospitals. Either of these combinations would be a point on the transformation curve, which can be plotted on a graph with the number of hospitals built and equipped on one axis and the number of schools on the other, as shown in the figure below. The Transformation Curve traces the no. of schools that can be built and equipped using the resources needed for any given level of hospital building.

Figure 2: A Transformation Curve

Marginal Rate of Transformation (MRT): It is the rate at which 1 good must be given up in order to produce a single extra unit (or marginal unit) of another good, assuming that both goods require the same scarce inputs. The absolute value of the slope of the Transformation Curves gives the MRT.

Pareto-optimality: It is the state of an economy in which no one can be made better off without someone being made worse off. For this to be the case, three types of efficiency must hold:

(a) productive efficiency, in which the output of the economy is being produced at the lowest cost;
(b) allocative efficiency, under which resources are being allotted to the production of the goods & services the society most values;
(c) distributional efficiency, in which output is distributed in such a way that consumers would not wish, given their disposable income and market prices, to spend these incomes in a different way.

Now, consider the given diagram.

In Part-(a) of the figure, the transformation curve TT shows the production possibilities of the economy for private and public goods. The slope of the transformation curve gives the Marginal Rate of Transformation (MRT). Pareto optimality requires that the economy operates at a point where it is impossible to make one individual better off without making someone worse off.

Samuelson’s approach is to decide just how well off one individual is going to be and then to search for that combination of private and public good, which makes the other individuals as well off as possible. In this way, he is interpreting the requirements of Pareto optimality as making any individual as well off as possible, provided that the other individual is made no worse off than some reference level of welfare.

In parts (b) and (c), the indifference curve maps are shown for individuals A and B. A is held on the indifference curve I²_A, and in this way, A’s level of welfare is fixed. Now, each individual must consume the same quantity of the public good. The indifference curve I²_A shows the alternative quantities of private & public goods that keep A on a fixed level of welfare. Therefore, when A consumes Og of public good, he will also consume g1 of the private good, and B would similarly be able to consume Og of the public good and (g2- g1) of the private good if the economy operated efficiently.

Repeating the process for every possible output of the public good produces a curve CC in part- (c). This curve is the set of consumption possibilities available to individual B, assuming that A is kept on I²_A. This curve has been derived by deducting I²A from TT. Now, the problem is to find out the output level for private and public goods, which maximizes the welfare of individual B. This is shown by the tangency point between I³_B and the curve CC.

It is clear that at point 3 in part (c), the tangency has been found by subtracting (at Og) the marginal rate of substitution (MRS) for individual A from MRT and then equating the MRS of individual B to the slope of CC which is given by (MRT-MRS_A).

Therefore, the Pareto optimality condition in an economy that produces public and private goods becomes

MRS_B=MRT-MRS_A or,

MRT=MRS_A+MRS_B

Figure 3: Production of Public and Private Goods in an Economy

The above discussion gives the conclusion that Pareto optimality in an economy producing both public and private goods depends only on MRT and MRS between the two goods for all the individuals. This method is obviously individualistic as it focuses on the MRS of the individuals.

However, would individuals’ preferences be the same as their preferences revealed in choices between goods in the private sector?

Kolm (1956) offers this criticism as a consideration in assessing the optimal provision of public goods. Musgrave and Musgrave (1989) argue that Samuelson’s approach would be improved if the decision on the correct income distribution could be separated further from the efficiency conditions in the analysis. It is also worth considering that the efficiency conditions involve different MRS for individuals, suggesting that no single common tax price is appropriate for all. This increases the problem of preference revelation.

Pure Theory of Public Expenditure

Read First: Public Expenditure: Concept, Objectives, & Public vs Private Expenditure

Pure Theory of Public Expenditure

Read More in: Theory of Public Finance

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Read First: Public Expenditure: Concept, Objectives, & Public vs Private Expenditure

Pure Theory of Public Expenditure

Read More in: Theory of Public Finance

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