The present study raises the following questions: To what extent is axiomatic general equilibrium analysis a rational reconstruction of Scottish Political Economy as defined by the writings of David Hume and Adam Smith? How much is gained and how much lost by the axiomatic transformation of the invisible-hand proposition? What are the implications of negative results like the Sonnenschein-Mantel-Debreu demonstrations for the Scottish point of view? Did it reach deadlock, or is there still hope for the dominant trajectory in the history of economics? In contrast to the rich historical literature on the invisible-hand proposition, the present study does not level any paradigmatic criticism at neo-Walrasian analysis. Rather, by focalizing the most important results against the backdrop of Scottish Political Economy, it provides some flesh to the bones of axiomatic economics and, insofar, may inform theory choice within the neo-Walrasian paradigm. Naturally, the answers to the questions raised are complex and do not fit into an abstract. Instead, the reader is referred to the final section, which lists, interrelates, and discusses the major results of the study.I'm not sure that Smith would have believed in any invisible-hand proposition so I'm not sure that its even meaningful to ask what is "lost by the axiomatic transformation of the invisible-hand proposition?" In a working paper of my own I write,
For Smith competitive markets were the most prominent mechanism for coordinating and motivating people to maximise the grains that result from increased specialisation and an expanded division of labour. Well functioning market institutions leave individuals free to pursue self-interested behaviour, but guide their choices by the prices they pay and receive. For economists, the 200 years following Smith involved a search for conditions under which the price system would function well, conditions under which it would not descend into chaos.But I add the footnote,
The formal (neoclassical) model that arose from this search is one which abstracts completely from any form of centralised control in the economy.
For Adam Smith this would be an abstraction too far. Smith knew of the importance of institutions to the proper functioning of the market economy. Mark Blaug points out that “[ . . . ] Smith’s faith in the benefits of ‘the invisible hand’ has absolutely nothing whatever to do with allocative efficiency in circumstances where competition is perfect `a la Walras and Pareto; the effort in modern textbooks to enlist Adam Smith in support of what is now known as the ‘fundamental theorems of welfare economics’ is a historical travesty of major proportions. For one thing, Smith’s conception of competition was, as we have seen, a process conception, not an end-state conception. For another society, a decentralised competitive price system was held to be desirable because of its dynamic effects in widening the scope of the market and extending the advantages of the division of labour - in short, because it was a powerful engine for promoting the accumulation of capital and the growth of income”. (Blaug 1996: 60-1).Thus the question "[t]o what extent is axiomatic general equilibrium analysis a rational reconstruction of Scottish Political Economy as defined by the writings of David Hume and Adam Smith?" would have the answer, very little.