Dimitrios Diamantaras and Simon Wilkie
Nash Implementation of Valuation Equilibrium
January 2000 (preliminary)

Abstract

Mas-Colell (1980) introduced the concept of valuation equilibrium for  
economies with one private good and a set of possible public projects. 
The set of public projects need not have any mathematical structure, 
so it can model a wide range of public projects. Indeed, we prefer 
"collective goods" in place of "public projects", to emphasize this 
generality. 

Diamantaras and Gilles (1996) initiated a literature that examines the  
possibilities for the extension of the valuation equilibrium concept, 
and the closely related cost share equilibrium concept, to economies 
with a finite number of private goods and an unrestricted set of 
collective goods. They showed the welfare theorems for valuation 
equilibrium and the non- equivalence of the core and cost share equilibrium. 

Valuation equilibrium is a generalization of Lindahl equilibrium. Agents 
pay personalized prices for the collective goods in valuation equilibrium, 
but since there are no units in which to measure the provision levels of 
collective goods, their prices are lump-sum prices, which can differ across 
individuals and collective good chosen. In valuation equilibrium individuals 
can receive net subsidies as well as pay net contributions for the cost of 
the collective good chosen; cost share equilibrium is a valuation 
equilibrium in which everybody pays a share: there are no subsidies allowed. 
Price vectors for the private goods depend on the collective good chosen, 
and individuals have, in equilibrium, compatible guesses on how the private 
good prices would vary if other collective goods would be chosen instead. 

In the present paper we examine the Nash implementation of the valuation  
equilibrium correspondence as generalized by Diamantaras and Gilles (1996).  
We find that the constrained version of the valuation equilibrium  
correspondence (along the lines of constrained Walrasian equilibrium) is Nash  
implementable with a market-like game. We use standard assumptions, ? and in 
particular, the assumption that private goods are essential in the sense  
that a nonnegative vector of private goods, along with any collective good, 
is preferred by all individuals to the zero vector of private goods along 
with any collective good vector. This is a decentralization result for a 
remarkably general model of economies with collective goods.