The fundamental theorems of welfare economics

The social optimum is well described through the fundamental theorems of welfare eco­nomics, which formalize some ideas already present in Pareto’s works (especially for the first theorem) and in Walras’s works (especially for the latter).

Oskar Lange (1942), Abba Lerner (1934), and Harold Hotelling (1938) have pro­vided the first order conditions for economic efficiency, and the primary proofs of the first theorem. The problem of maximizing overall welfare amounts to maximizing the utility of each individual under the constraints of others’ utility, possible allocation and transformation functions, which has up to three conditions. First, individual utilities are maximized if the marginal rates of substitution for two given commodities between two different individuals are equal. Second, the aggregate output and the optimal allocation of goods among individuals are obtained by equalizing the marginal rates of substitution with the marginal rates of transformation between the two given commodities. Finally, the marginal rates of transformation of the different firms among any two commodities must be equal to guarantee the efficiency of production for the various technologies. Notice these results, now rigorously established, resume some economic laws previously discovered by the precursors of the marginalists, such as Hermann Heinrich Gossen in 1854, and by the marginalists themselves, such as William Stanley Jevons in 1871.

Kenneth Arrow (1951), Gerard Debreu (1951), and then the two together (Arrow and Debreu 1954) have generalized the proofs and these results. In formal terms, they have overcome the use of calculus, though intuitive to use for demonstrations related to opti­mization problems, by now using set theory. They have shown, with very few conditions, that the optimum more fundamentally derives from the price system. That is how they have formulated what is now called the two fundamental theorems of welfare economics.

The first theorem of welfare economics states that competitive equilibria are Pareto- optimal, if individual preferences are monotonic and if there are complete markets.

The second fundamental theorem of welfare economics states that one can achieve any Pareto-optimal allocation in a competitive equilibrium when the social planner under­takes an appropriate redistribution of endowments. Among several Pareto optima, some are probably more satisfactory than others. The theorem points out that the preferred social optimum can be achieved by a competitive equilibrium if accompanied by proper redistribution policy which shall establish the new “initial” allocations. An important consequence of this theorem is that it is not necessary to alter the competitive system to obtain Pareto optimality. A trade-off between efficiency vs. equity is not any more required; however, the issue of the redistribution is pregnant.