Prices and Pareto optima

Abstract

We provide necessary conditions for Pareto optimum in economies where tastes or technologies may be nonconvex, nonsmooth, and affected by externalities. Firms can pursue own objectives, much like the consumers. Infinite-dimensional commodity spaces are accommodated. Public goods and material balances are accounted for as special instances of linear restrictions.

Keywords:

• First and second welfare theorem
• Weak and strong Pareto optimum
• Nonconvex tastes or technologies
• Public goods
• Externalities
• Local separation
• Subdifferentials
• Normal cones

Keywords:

• MSC Classifications: 90C26
• 91B50

Keywords:

• JEL Classifications: C60
• D50
• D60

Acknowledgments

The authors thank the referee for careful reading of the article. His comments led to substantial improvements in the presentation. The first author thanks the Université de Bourgogne for support and great hospitality.

Notes

¹ The assumptions run against various observable phenomena such as indivisibilities, scale economies, and risk-loving behavior. So there are ample reasons to relax them 4, 40, 47.

² Broadly speaking, this corresponds to the situation in mathematical programming. There optimal solutions to sufficiently smooth problems must locally be supported by “prices” (dual vectors or Lagrange multipliers). Conversely, granted convexity, the existence of a dual optimal solution suffices for global support – and for decomposition (decentralization, relaxation) of the program.

³ Otherwise, at least two divisions would benefit by bilateral exchange. This observation points to cooperative production games and core solutions thereof; see 14–16.

⁴ While Samuelson (1947), Debreu (1954), Negishi (1960) already accommodated nonsmooth data, studies of nonconvex items appeared later; see 4, 5, 8, 28, 29.

⁵ Useful cones were introduced by Bouligand, Clarke, Dubovickii-Miljutin, Ioffe, Mordukhovich, Rockafellar, and others 9, 22, 36, 42. See 2 for a good overview.

⁶ Those could occur in preference or production. We thus accommodate consumers displaying say, altruism or envy and firms affected by such phenomena as congestion or pollution.

⁷ We shall not replace original data, represented as sets, locally by cone approximations.

⁸ In fact, those objects will here be taken as primitives, exemplified by well-studied prototypes.

⁹ Even more general spaces could be – and have been – considered 12, 29, but we find the Banach setting largely sufficient and most tractable. Anyway, the reader not appreciating so much generality can construe E as a finite-dimensional Euclidean.

¹⁰ Some agent, say a firm, might come without objectives. We shall, however, without loss, equip each k ∈ K with a preference . For example, the specification could be one extreme and quite innocuous choice. This remedy provides additional generality, and it renders the role of consumers and firms more symmetric.

¹¹ Thus in microeconomic jargon, a consumer (producer) would have a consumption (production) correspondence.

¹² For more about see 27.