Allocation of Resources in the Presence of Indivisibilities: Scarf’s Problem Revisited

In his article “The Allocation of Resources in the Presence of Indivisibilities,” Scarf points out that the major problem presented to economic theory by the presence of indivisibilities is the impossibility of detecting optimality at the level of the firm, or the economy as a whole, using the creation of profitability based on competitive linear prices. In the absence of such competitive prices, Scarf instead introduces a quantity test. Further development of the quantity test idea has lead to algorithms that are used to solve parametric integer programming problems. However, the quantity test is not a fully acceptable replacement of prices to analyse markets with indivisibilities. Recently, O’Neill et al. have suggested a new scheme that generates discriminatory equilibrium prices in markets with non-convexities. In this paper we elaborate this idea even further and use it to generate non-linear price functions that can be interpreted as a non-linear pricing scheme for markets with non-convexities.