Competitive equilibrium with indivisible goods and generic budgets

Competitive equilibrium from equal incomes (CEEI) is a classic solution to the problem of fair and efficient allocation of goods (Foley 1967, Varian 1974). Every agent receives an equal budget of artificial currency with which to purchase goods, and prices match demand and supply. However, a CEEI is not guaranteed to exist when the goods are indivisible even in the simple two-agent, single-item market. Yet it is easy to see that, once the two budgets are slightly perturbed (made generic), a competitive equilibrium does exist. In this paper, we aim to extend this approach beyond the single-item case and study the existence of equilibria in markets with two agents and additive preferences over multiple items. We show that, for agents with equal budgets, making the budgets generic—by adding vanishingly small random perturbations—ensures the existence of equilibrium. We further consider agents with arbitrary nonequal budgets, representing nonequal entitlements for goods. We show that competitive equilibrium guarantees a new notion of fairness among nonequal agents and that it exists in cases of interest (such as when the agents have identical preferences) if budgets are perturbed. Our results open opportunities for future research on generic equilibrium existence and fair treatment of nonequals.