Pure Theory Of Public Expenditure:
The pure theory of public expenditure was propounded in 1954/1955 by Paul Anthony Samuelson in a general equilibrium model with existence of one private good and one public good. The idea is that different individuals choose different combinations of private good and public good. Since the whole of public good is equally available to all the individuals, because of non-rivalness in consumption, it is only the sacrifice in terms of private good by different individuals that matters. This sacrifice is actually the tax paid by different individuals.
The basic idea can be captured in a little simpler manner than using production possibility curve of the society and indifference curves of the individuals constituting society, which is the framework adopted by Samuelson. In contrast to general equilibrium framework, we are adopting here partial equilibrium framework. Let there be two individuals A and B in the society, which produces private good X and public good Y. Price of purchase of public good Y is in terms of X, call it tax. In the case of a pure private good, we know there prevails one single price at different individuals normally buy different amount. By contrast, we can see that there can be only one quantity of a pure public good, which is equally available to (or enjoyed by) everybody but they may pay different prices (taxes).
In the case of a private good, we carry out horizontal summation of individual demand curves. The equilibrium quantity produced is allocated to different consumers at equilibrium price according to strength of their individual demand curves. Noting that demand curves reflect marginal revenues and supply curves reflect marginal costs, we may write the condition of equilibrium in a competitive market as MRA=MRB=MC.
In the case of a public good, we can carry out vertical summation of individual demand curves for whatever quantity is produced, because of non-rivalness in consumption, will be as a whole consumed, and therefore equally, by everybody.
Which means that individual marginal revenues are summed up to equal the marginal cost of production. The competitive equilibrium condition could then be written as
MRA+MRB=MC.
In terms of general equilibrium model, one may recall, for two private goods X1 and X2 and two individuals A and B the condition of equilibrium is MRSA=MRSB=MTS.
By contrast, for a world of a private good X and a public good Y and two individuals A and B, the condition of equilibrium would be MRSA=MRSB=MTS. MRS and MTS stand respectively for marginal rate of substitution and marginal rate of technical substitution.
Wicksell had long ago realized that the individual demand curves might be a pseudo- character as people may be tempted to conceal their preferences. Whatever public good is supplied and in whatever quantity, it is all available by definition to all, an individual may keep quiet about his need as he can enjoy its fruit without paying. It may mean that public goods may be collectively underprovided.