The individual and the market: Paul Samuelson on (homothetic) Santa Claus economics

Abstract

Paul Samuelson often used the term “Santa Claus economics” for mathematical models with empirically unrealistic assumptions. I focus on one particular member of the Santa Claus family that Samuelson was very sceptical about: homothetic general equilibrium models (where all agents have identical homothetic preferences). I argue that Samuelson's concerns about these models provide insights into how he viewed the relationship between the individual and the market, a relationship that has implications for not only his economic theorising, but also his broader political–economic vision. His criticisms are also relevant to some ongoing debates within contemporary economic theory.

Keywords:

• Paul Samuelson
• demand theory
• representative agent
• neoclassical synthesis

Acknowledgements

I would like to thank John Davis, Pedro Duarte, Ivan Moscati, and two anonymous referees for helpful comments on earlier drafts of this paper.

Notes

¹ All page references to Samuelson's works reprinted in the seven volumes of The Collected Scientific Papers of Paul A. Samuelson will be to the reprinted versions.

² The literature is far too extensive to attempt to cite. This said, Arrow and Hahn (1971) is the canonical text.

³ Again the literature is extensive but Arrow and Hahn (1971) is probably the single best source: see Chapters 11 and 12 on stability, Chapter 9 on uniqueness, Chapter 10 on comparative statics, and Chapters 6, 8, 13 and 14 on other popular areas of research.

⁴ Debreu (1974), Mantel (1974, 1977) and Sonnenschein (1973). See Rizvi (1998, 2006) for historical discussion.

⁵ The paper that Samuelson himself referred to as the first example of the homothetic Santa Claus case was his 1956 paper on “Social Indifference Curves.” This paper does present the model, but he called it the “Robinson Crusoe” case in this early paper and began calling it Santa Claus later. It is important to note that Samuelson's discussion of this class of models appeared in different contexts off and on throughout his career; he never changed his position, but neither did he ever dedicate an entire paper to the topic. As a result, the Samuelson citations I use will come from papers on a number of different topics and spread out over decades. As discussed in Appendix 1, the main results on homothetic preferences were Eisenberg (1961) and Gorman (1953), but some of the ideas can be found as far back as Antonelli (1886).

⁶ And more critically he adds: “Much of what we have regarded as interesting and important would be lost” (Hahn 1983a).

⁷ See Hurwicz (1971) and Hurwicz and Uzawa (1971) for the technical results on integrability and Hands (2006, 2011a) for historical discussion.

⁸ Since Wald (1951) was the first person to formally impose revealed preference assumptions on market demand or excess demand functions, such conditions were often called Wald's restrictions in the literature of the period.

⁹ It is useful to note that the later literature (Varian 1983) has developed a homothetic version of revealed preference – the homothetic axiom of revealed preference (HARP) – that guarantees that demand function can be rationalised by homothetic preferences.

¹⁰ The empirical inadequacy of the assumption of homothenticity is a common feature of general equilibrium theory during this period. For example, Arrow and Hahn:

“It is apparent that the restrictions involved are pretty stringent. It is hard to believe not only that all households are alike in this sense, but also that all Engel curves pass through the origin so that as an individual gets poorer and poorer he nonetheless continues to mix holidays, say, in the same ratio to bread, as he did when he was richer” (Samuelson 1971, p. 220).

¹¹ As noted in Appendix 1, such conversion requires that certain symmetry conditions hold on the cross-partial derivatives of the relevant functions and that is often not the case (or impossible to determine if it is the case).

¹² As Hahn summarised this aspect of Foundations:

We owe it to Samuelson more than to anyone else that at the level of the individual agent, the maximization hypothesis has been fruitfully put to work … But our main interest here is the economy as a whole, or at least in market predictions. Here matters are less satisfactory … But when households are brought into the picture, then generally things fall apart unless we can abstract from income effects; that is, unless agents have identical homothetic utility functions (Gorman, 1953). In that rather special case, the economy can be viewed as if a single maximizing household were involved, and so the maximization hypothesis can deliver at full power. In general however, the equilibrium cannot be converted into the solution of an as if maximization. (Hahn 1983b, p. 34)

¹³ “Not working with an explicit dynamical model, Professor Hicks probably argued by analogy from well-known maximum conditions, whereby a maximum must hold for arbitrary displacements and through any transformation of variables.” (Samuelson, 1947, p. 273)

¹⁴ The cross-partial derivatives of factor demand functions are always symmetric (one of the important additional restrictions that homothenticity imposes on market demand functions for consumer goods).

¹⁵ There have, of course, been many recent attempts to introduce heterogeneous agents into DSGE models, but the representative agent is the traditional, and still standard, framework. See Athreya (2013) for a recent discussion of this literature and the key references.

¹⁶ See Hands (2011b, 2013) and Varian (2006) for discussions of these developments.

¹⁷ It is also useful to point out that there is very little revealed preference theory, even for individual agents, in Foundations. The entire discussion of consumer choice theory is conducted in terms of standard ordinal utility theory with one passing reference to Samuelson's original 1938 paper and the term “revealed preference” does not appear in the consumer choice chapter at all, but rather in his discussion of index numbers. See Hands (2014) for a discussion of Samuelson on revealed preference theory.

¹⁸ In fairness to Ross, his historiography allows him to be perfectly comfortable with this fact. His approach is explicitly rational reconstruction. As he explains:

“It searches for tendencies in the development of scientific thought that were influenced by conceptual logic. It is frank about the fact that we are in a much better position to discern these tendencies and this logic than were the people who couldn't know where the story was going to go. Such history starts from where we now find ourselves and looks for anticipations in earlier problem settings and conceptual evolution – looks, that is, for the growing seeds of the present in the relatively chaotic past where they were difficult to distinguish from seeds that didn't germinate. (Ross 2009, p. 101)

In a sense, Ross’ historiography is a version of the revealed preference theory he endorses – it is revealed preference historicism – as the “economic agent” choices can be rationalised, so can the choices of earlier economists. I believe this approach to the history of economic thought is problematic (Hands 2009), but it does make Samuelson's remarks irrelevant to Ross’ reconstruction of the history.

¹⁹ For example: that all individuals have homothetic preferences (but not necessarily identical) and proportional income (a fixed distribution of income). See Chipman (1974, 2006) or Polemarchakis (1983). There are similar results involving quasi-linear preferences (Mas-Colell et al. 1995, pp. 80–1).

²⁰ Since demand functions are invariant with respect to any monotonic transformation, the results for homothetic preferences/utility could just as well be obtained from homogeneous functions (as some authors do).

²¹ Note this is symmetry on the regular demand functions, not just symmetry of the Slutsky terms which holds for any demand function. Such symmetry was termed “Hotelling Symmetry” in Chipman and Moore (1976) because of its use in Hotelling (1932).

²² See Hoover (2012) for a recent discussion.

²³ Even when the concern is demand function aggregation, there are empirical as well as theoretical issues. The empirical issues concern the circumstances under which it is empirically appropriate to use total income (often available) to estimate market demand functions rather than the income levels of all of the individuals in the market (seldom available). The empirical and theoretical issues are of course related, but I will focus exclusively on the latter.

²⁴ This list is a modified version of the list in Mas-Colell et al. (1995, p. 105).

²⁵ Or other forms of revealed preference such as the weak axiom (WARP) or generalised axiom (GARP).

²⁶ Also see Rader (1976), Samuelson (1990) and Silberberg (1972).

²⁷ In fact, the most intensive discussion of uniqueness in Walrasian general equilibrium theory – Chapter 9 of Arrow and Hahn (1971) – is all framed in terms of this revealed preference condition. The authors discuss a number of conditions that are sufficient for uniqueness, but in almost every case there are conditions that are sufficient for Wald's revealed preference axiom that technically does all of the heavy lifting.

²⁸ Arrow and Hahn call this case – the case of the one household economy – the “Hicksian” case (p. 220) – and they demonstrate that the Hicksian case is globally, and independently locally, stable.

²⁹ Notice that while this is true for the homothetic Santa Claus (Hicksian) case it is certainly not true in general: “If the auctioneer's rule may be treated as an intelligent method of maximizing or minimizing some relevant function, then if such a function is well behaved … we should expect the rule to exhibit global stability. Unfortunately, however, except in exceptional circumstances of which the Hicksian is one instance, the price mechanism cannot be taken to act as if someone were trying to maximize or minimize some well-behaved function of prices” (Arrow and Hahn 1971, p. 278).

³⁰ The general equilibrium model discussed in Mirowski and Hands (1998) has similar properties, but starts from a different (non-budget-constrained) characterisation of the individual choice problem.