Foundations of Economic Analysis

Foundations of Economic Analysis is a book by Paul A. Samuelson published in 1947 (Enlarged ed., 1983) by Harvard University Press. It is based on Samuelson's 1941 doctoral dissertation at Harvard University. The book sought to demonstrate a common mathematical structure underlying multiple branches of economics from two basic principles: maximizing behavior of agents (such as of utility by consumers and profits by firms) and stability of equilibrium as to economic systems (such as markets or economies). Among other contributions, it advanced the theory of index numbers and generalized welfare economics. It is especially known for definitively stating and formalizing qualitative and quantitative versions of the "comparative statics" method for calculating how a change in any parameter (say, a change in tax rates) affects an economic system. One of its key insights about comparative statics, called the correspondence principle, states that stability of equilibrium implies testable predictions about how the equilibrium changes when parameters are changed.
Introduction.
The front page quotes the motto of J. Willard Gibbs: "Mathematics is a language." The book begins with this statement:
Its other stated purpose (p. 3) is to show how "operationally meaningful theorems" can be described with a small number of "analogous methods". Thus, "a general theory of economic theories" (1983, p. xxvi).
Topical outline.
The body of the book is 353 pages. Topics and applications covered (all in terms of theory) include the following.
Methods and analysis.
Samuelson's "Foundations" demonstrates that economic analysis benefits from the parsimonious and fruitful language of mathematics. In its original version as a dissertation submitted to the David A. Wells Prize Committee of Harvard University in 1941, it was subtitled "The Observational Significance of Economic Theory" (p. ix).
One unifying theme, on the striking formal similarities of analysis in seemingly diverse fields, occurred only in the course of writing on them—from consumer's behavior and production economics of the firm to international trade, business cycles, and income analysis. It dawned on the author that he was prodigal "in proving essentially the same theorems" over and over. His failure of initial intuition, so he suggests, might be less surprising in light of the few economic writings then extant concerned with formulating "meaningful theorems" – hypotheses "about" empirical data—that could conceivably be refuted "by" empirical data (pp. 3–5).
Samuelson (pp. 5, 21–24) finds three sources of meaningful theorems sufficient to illuminate his purposes:
Part I conjectures that meaningful theorems for economic units and for their respective aggregates are almost all derivable from general conditions of equilibrium. The equilibrium conditions can in turn be stated as "maximization conditions". So, meaningful theorems reduce to maximization conditions. The calculus of the relations is at a high level of abstraction but with the advantage of numerous applications. Finally, Part I illustrates that there "are" meaningful theorems in economics, which apply to diverse fields.
Part II concentrates on aggregation of economic units into equilibrium of the system. But the symmetry conditions required for direct maximization of the "system", whether a market or even the simplest model of the business cycle, are lacking, in contrast to an economic "unit" or its corresponding aggregate. What can be hypothetically derived (or rejected in some cases) is a "stable" equilibrium of the system. (This is an equilibrium of the system such that, if a variable disturbs equilibrium, the system converges to equilibrium.) "Stability of equilibrium" is proposed as the principal source of operationally meaningful theorems for economic systems (p. 5).
Analogies from physics (and biology) are conspicuous, such as the Le Chatelier principle and correspondence principle, but they are given a nontrivially generalized formulation and application. They and mathematical constructions, such as Lagrangian multipliers, are given an operational economic interpretation. The generalized "Le Chatelier principle" is for a maximum condition of equilibrium: where all unknowns of the function are independently variable, auxiliary constraints ("just-binding" in leaving initial equilibrium unchanged) reduce the response to a parameter change. Thus, factor-demand and commodity-supply elasticities are hypothesized to be lower in the short run than in the long run because of the fixed-cost constraint in the short run. In the course of analysis, "comparative statics", changes in equilibrium of the system that result from a parameter change of the system, is formalized and most clearly stated (Kehoe, 1987, p. 517). The "correspondence principle" is that the "stability" of equilibrium for a system (such as a market or economy) implies meaningful theorems in comparative statics. Alternatively, the hypothesis of stability imposes directional restrictions on the movement of the system (Samuelson, pp. 258, 5). The correspondence is between comparative statics and the dynamics implied by stability of equilibrium.
Chapter VIII on welfare economics is described as an attempt "to give a brief but fairly complete survey of the whole field of welfare economics" (p. 252). This Samuelson does in 51 pages, including his exposition of what became known as the Bergson–Samuelson social welfare function. Theorems derived in welfare economics, he notes, are deductive implications of assumptions that are not refutable, thus not meaningful in a certain sense. Still, the social welfare function can represent any index (cardinal or not) of the economic measures of any logically possible ethical belief system that is required to order any (hypothetically) feasible social configurations as "better than", "worse than", or "indifferent to" each other (p. 221). It also definitively elucidates the notion of Pareto optimality and the "germ of truth in Adam Smith's doctrine of the invisible hand" (Samuelson, 1983, p. xxiv; Fischer, 1987, p. 236).
The final pages of the book (pp. 354–55) outline possible directions analytical methods might take, including for example models that show how:
Samuelson closes by expressing hope in the future use of "comparative dynamics" to:
Appendices.
There are two mathematical appendices totalling 83 pages. The first gathers and develops "very briefly" and "without striving for rigor" results on maximization conditions and quadratic forms used in the book and not conveniently collected elsewhere (p. 389). The other is on difference equations ("for the dynamic economist") and other functional equations.
Enlarged edition.
The 1983 enlarged edition includes an additional 12-page "Introduction" and a new 145-page appendix with some post-1947 developments in analytical economics, including how conclusions of the book are affected by them.