We prove existence and generic regularity of equilibria in a general equilibrium model of a completely decentralized pure public good economy. Competitive firms using private goods as inputs produce the public good, which is privately provided by households. Previous studies on private provision of public goods typically use one private good, one public good models in which the public good is produced through a constant returns to scale technology. As two distinguishing features of our model, we allow for the presence of several private goods and nonlinear production technology. In that framework, we use an homotopy argument to prove existence of equilibria and we show that economies are generically regular.
In this paper, we prove the existence and regularity of equilibria in a general equilibrium model of a completely decentralized pure public good economy. The model studied extends the standard pure exchange model with private goods by allowing households to make voluntary purchases of (or “privately provide”) a public good that is produced by competitive firms using private goods as inputs.
The interest in a general equilibrium model with private provision of public goods lies in the fact that it serves as a benchmark extension of an analysis of completely decentralized private good economies to public good economies. Moreover, there are some relevant situations in which public goods are in fact privately provided, e.g., private donations to charity at a national and international level, campaign funds for political parties or special interests groups, and certain economic activities inside a family.
Previous studies on private provision of public goods typically use one private good, one public good models in which the public good is produced through a constant returns to scale technology. With only one private good, assuming constant returns to scale amounts to assuming linear production function, and as a result there is no loss of generality in normalizing prices of both the private and the public good to one. These assumptions also allow taking profits of firms equal to zero, with the implication that the presence of firms basically plays no role in the model. Therefore, as clearly explained by Bergstrom et al. (1986), the model reduces to a simple game where households are the players and the unique strategy variable is the choice of voluntary contribution. Then, proving existence is straight forward (see Bergstrom et al., 1986, Theorem 2, p. 33). 1
As two distinguishing features of our model we have the presence of several private goods and we allow for nonlinear production technology for the public good. Therefore, the above-described price normalization will typically not be possible. Moreover, competitive firms will typically have non-zero profits, which are to be distributed among households. We are, therefore, analyzing a full-fledged general equilibrium model.
For given prices firms maximize profits. For given prices, initial endowments of private goods, ownership distribution of firms, as well as other households’ choices of private provision of public goods, each household chooses a vector of private good consumption levels and a voluntary contribution level for the public good so as to maximize utility. Finally, markets for private and public goods clear. We show both existence and generic regularity of equilibria in this model.
We provide a proof of existence using a very simple homotopy argument, whose main characteristics consist in linking the true model with a fictitious one where the public good is treated as a private good. This strategy of proof turns out to be useful to show existence of equilibria in some related models as well.2
Regularity is an indispensable tool for carrying out comparative statics analyses. We show that for any vector of utility and transformation functions and ownership shares, in an open and full measure subset of the endowments, there is a finite number of equilibria and a local smooth dependence of the equilibrium variables on the exogenous variables. To show regularity we need to strengthen the assumptions on utility functions and production technology with which we prove existence.
Previous work on existence of competitive equilibrium with externalities in consumption and production relate to the model we study here. Public goods can be seen as a special case of consumption externalities. McKenzie (1955) studies a model of externalities with production, but he considers a linear production technology. Arrow and Hahn (1971) allow for decreasing returns to scale in production, but some of their assumptions do not apply to our case. Shafer and Sonnenschein (1975) show the existence of competitive equilibria in a pure exchange framework, hence their results do not apply to our model which incorporates production.3
Section 2 presents the setup of the model. In Section 3 the existence of equilibrium is proved. Section 4 contains the proof of generic regularity of the equilibria.4
