Welfare and Equilibrium

Working from the concave utility of consumers, this chapter derives first and second theorems of welfare economics for an exchange economy. General equilibrium refers to simultaneous demand and supply equilibrium in all markets. The Hicks and Marshall decompositions for utility contrast expenditure-minimizing compensated demand and utility-maximizing consumer demand. Always consumer demand is expenditure-minimizing, but compensated demand is perhaps not utility-maximizing. Compensated equilibrium is general equilibrium for compensated demand, whereas competitive equilibrium is general equilibrium for consumer demand. A competitive equilibrium is necessarily a compensated equilibrium, but not conversely. Perturbation duality leads to first and second theorems of welfare economics. Starting from a weak-efficiency primal, we derive a compensated-equilibrium dual. The primal/dual analysis yields first and second theorems of welfare economics for compensated equilibrium. Starting from an efficiency primal, we derive a competitive-equilibrium dual, resembling the compensated-equilibrium dual, but slightly different. The primal/dual analysis leads to first and second theorems of welfare economics for competitive equilibrium.