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A Historical Perspective on the Theories Related to the Competitive Equilibrium Allocation

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Marginal Revolution in Economics

Part of the book series: Monographs in Mathematical Economics ((MOME,volume 6))

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Abstract

Since the Marginal Revolution, mainstream economics has developed around the framework of general equilibrium theory and the fundamental theorems of welfare economics. However, we have to remember the prehistory of the marginal utility theory, in which ample pioneering works had been accumulated, which we call as a whole the theory of utility and scarcity. Tracing the development of the marginal utility theory since its prehistory, we can find out that there are three theoretical streams relating to the competitive equilibrium allocation. The theory of voluntary exchange was developed by Jevons and Edgeworth into the theory of core. The development of the theory of competitive markets has a theoretical jump from Walras’ general equilibrium theory. Modern theory does not refer to the theory of social value developed by the Austrian school, but its pioneering contributions to the history of welfare economics should be appreciated.

This paper is a revised edition of my papers in Japanese [29, 30].

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Notes

  1. 1.

    It is now known that there are many pioneering studies with similar theoretical structures to neoclassical economics since the Marginal Revolution, other than those mentioned by Menger [11, 18, 26] .

  2. 2.

    The term “Marginal Revolution” has become firmly established in the history of economics, but it is a very shallow expression. First, the Marginal Revolution is not a scientific revolution, which is a concept defined by Kuhn [32], As historians of economics agreed at the Bellaggio conference [3], the Marginal Revolution is quite different from the process of scientific revolutions and should not be called a revolution. Second, the term “Marginal” does not exactly describe the content of economic theories as they are actually constructed. “Marginal” in those theories merely express operations that mathematically characterize rationality and optimality. However, as pointed out by Hicks [17], Jaffé [22] and clearly formulated by Negishi [44], the theories of Jevons, Menger, and Walras are based on quite distinct approaches in addition to marginal analysis.

  3. 3.

    Quoted from p. 283 of an English translation of Galiani [14, pp. 54–91] by Monroe [41, pp. 281–299].

  4. 4.

    We suppose that if \( \left( x_{1}, x_{2} \right) \geqq 0 \) it is possible for the individual.

  5. 5.

    Turgot [55] calls the individual value “esteem value”.

  6. 6.

    Feldman and Serrano [12, pp. 33–49] calls it barter exchange.

  7. 7.

    See Mas-Colell et al. [38, pp. 515–686] and Feldman and Serrano [12, pp. 33–77].

  8. 8.

    If there are an allocation \( x = \left( x_{A}, x_{B} \right) \) and another allocation \( z = \left( z_{A}, z_{B} \right) \) such that \( U_{i} \left( z_{i} \right) \ge U_{i} \left( x_{i} \right) \) for all \( i = A, B \), and \( U_{i} \left( z_{i} \right) > U_{i} \left( x_{i} \right) \) for at least one \( i = A, B \), then z is called Pareto superior to x or Pareto improving of x, namely a Pareto improvement of x. An allocation x is Pareto efficient if there is no Pareto improvement of x.

  9. 9.

    It is clear that there is no inheritance between the theory of utility and scarcity and the theory of Jevons. After investigating his pioneering work on the priorities of marginal utility theory, he first mentioned Condillac [4] as a good value theory without mathematics, and highly appreciated his theory. Jevons did not know the manuscript of Turgot [55] because it had not been published.

  10. 10.

    Full-fledged Edgeworth boxes do not appear in Edgeworth’s own writings. As Jaffé [21] points out, perhaps it is Pareto [46] who drew the Edgeworth box for the first time.

  11. 11.

    Consider a group of persons S, called a coalition. S might be one person, several persons, or all. Let \( \left\{ s_{i} \right\} _{i \in S} \) represent a set of bundles of goods, one bundle for each member of S. We say that \( \left\{ s_{i} \right\} _{i \in S} \) is feasible for S if

    $$\begin{aligned} \sum _{i \in S} s_{ij} = \sum _{i \in S} \omega _{ij} , \; \text {for every good} j. \end{aligned}$$

    If x is a proposed allocation and S is a coalition, we say that S can block x if there is a feasible set of bundles \( \left\{ s_{i} \right\} _{i \in S} \) such that:

    $$\begin{aligned} & U_{i} \left( s_{i} \right) \ge U_{i} \left( x_{i} \right) \; \text {for all} \; i \; \text {in} \; S , \\ & U_{i} \left( s_{i} \right) > U_{i} \left( x_{i} \right) \; \text {for at least one} \; i \; \text {in} \; S . \end{aligned}$$

    The core is that set of allocations which cannot be blocked by any coalition. See Feldman and Serrano [12, pp. 37–40] and Mas-Colell et al. [38, Chap. 18].

  12. 12.

    Desai [8] investigates a pioneering analysis of the core by Turgot [55].

  13. 13.

    Jevons uses the concept of a trading body as a representative or average individual, because “questions which appear, and perhaps are, quite indeterminate as regards individuals, may be capable of exact investigation and solution in regard to great masses and wide averages” Jevons [24, p. 16].

  14. 14.

    It is not difficult to find the germ of this theory even in very early economic theory such as Pietro Verri [57, pp. 14–22].

  15. 15.

    If we interpret this idea in the general equilibrium theory, we must refer to the law of constant marginal utility of money. However, before Walras [59], the demand function was still a primitive notion not integrated into a systematic theory of competitive markets.

  16. 16.

    A Walras equilibrium of an exchange economy is an allocation and a price system in which each individual chooses a consumption so as to maximize her utility under the budget constraint and the total consumptions equals the total resources for each good.

  17. 17.

    Smith’s claim [53, pp. 454–456] does not mean the fundamental theorems of welfare economics [38, p. 549, Proposition 16.C.1], but a proposition that means the efficiency of decentralized production [38, p. 148, Proposition 5.E.1].

  18. 18.

    Galiani [14, pp. 89–90] states that “ this equilibrium fits in wonderfully with the just abundance of the conveniences of life and earthly happiness, although it results, not from human prudence or virtue, but from the base incentive of sordid gain: Providence, out of infinite love for men, having so ordered the relations of things that even our base passions, as if in spite of us, are often arranged for the good of the whole” [41, p. 298]. See Kawamata [28, pp. 38–39] about Turgot’s idea.

  19. 19.

    Debreu [6, pp. 50–59] shows that under a certain set of assumptions, if the preference preordering is complete, transitive, and continuous, then there exists a continuous utility function.

  20. 20.

    Pareto assumes ordinalism in economics, but he clearly reveals a cardinalist position in sociology that includes the field of welfare economics [54].

  21. 21.

    I confine my demonstration within the framework of the exchange economy. For more general cases including production, see Negishi [44, pp. 289–294].

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    Masahiro Kawamata

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Kawamata, M. (2023). A Historical Perspective on the Theories Related to the Competitive Equilibrium Allocation. In: Maruyama, T. (eds) Marginal Revolution in Economics. Monographs in Mathematical Economics, vol 6. Springer, Singapore. https://doi.org/10.1007/978-981-99-4342-5_6

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