Resource Utilization

We study resource utilization in an exchange economy. Is an allocation of consumption to consumers weakly efficient? Could a positive amount of each good be discarded, and the remaining supply reallocated to attain the same levels of preference? If not, then the utilization of resources is weakly efficient. We model this issue by three alternative optimizations. The resource-utilization primal solves for the quantity such that this quantity of every good can be discarded, and the remaining supply is sufficient to attain the same levels of preference. From this primal, we calculate an expenditure dual (Luenberger, D. G. (1995). Microeconomic theory. New York: McGraw-Hill). The expenditure dual chooses prices to minimize the value of supply less the sum of the expenditure functions for all consumers. The resource allocation is weakly efficient if the optimum value is zero. Using this duality, we prove the first and second theorems of resource utilization, akin to the first and second theorems of welfare economics. In a compensated equilibrium, the resource utilization is weakly efficient. Conversely, an allocation in which the resource utilization is weakly efficient can be realized as a compensated equilibrium. From the expenditure dual we derive the equivalent formulation by (Debreu, G. (1983). The coefficient of resource utilization. In W. Hildenbrand (Ed.), Mathematical economics: Twenty papers of Gérard Debreu (pp. 30–49). Cambridge: Cambridge University Press. (Originally 1951)). We restate the expenditure dual as a standard positively homogeneous fractional program. We find the equivalent concave maximization and its Fenchel dual. The dual interprets the fractional program as Debreu's coefficient of resource utilization: its optimum value is the fraction of resources that is just sufficient to attain the same levels of preference. The resource allocation is weakly efficient if the value is one. Whether the resource allocation is benefit-efficient is a third formulation. The dual of the benefit-efficiency problem reduces to the expenditure dual. That the utilization of resources is weakly efficient is thus equivalent to benefit efficiency. We restate the first and second theorems of resource utilization in terms of benefit efficiency.