Statics—Production and Consumption

Samuelson was entirely correct to say that he expanded his thesis as well as revised it, for about half the material was new and yet it is striking how little the text of the thesis was changed.^41,f However, one change was very significant.

f. He added four new chapters and an appendix, which amounted to 181 of the book’s 439 pages. A further 47 pages of new material were added elsewhere to the book. terms of an assumed dynamical system. This implies no ideological or nor­mative significance.”⁴² In the book, that became:

Instead, the dynamical properties of the system are specified, and the hypothesis is made that the system is in “stable” equilibrium or motion. By means of what I have called the Correspondence Principle between comparative statics and dynamics, definite operationally meaningful theorems can be derived from so simple a hypothesis. One interested only in fruitful statics must study dynamics.

The empirical validity or fruitfulness of the theorems, of course, cannot surpass that of the original hypothesis. Moreover, the stability hypothesis has no teleological or normative significance.⁴³

Similarly, in the chapter “Stability and Comparative Statics,” after explaining that dynamics and comparative statics were connected, he added the sentence, “This duality constitutes what I have called the correspondence principle.”⁴ The content of his argument had not changed, but Samuelson introduced a name for the idea and elevated it to the status of a “principle.”⁴⁵ Naming the idea gave it a prominence that it would otherwise not have had; it drew attention to Samuelson’s claim that it was important to specify a dynamic system, for without this the derivation of comparative statics results might be impos­sible.

He gave an example to illustrate this point in the first substantial addi­tion to the text of his thesis—a section titled “The Calculus of Qualitative Relations.” Samuelson argued that even if we know the signs of all the parameters in a system (and typically economic theory says nothing about the size of coefficients, merely whether they are positive or negative), it can be impossible to work out comparative statics results. For example, in the Keynesian system, with only three equations determining three variables, even if we knew the signs of all the parameters, it would still be complicated, if not impossible, to work out comparative static results. What he was doing, thus, was explaining to the reader why it was necessary to use mathematical methods such as he was about to present. “Intuition and a general feeling for the direction of things does not carry one far in the analysis of a complex many-variable system.”⁴⁶

g. Even Hicks, in Value and Capital, had analyzed dynamics, but without specifying equations governing the behavior of a system out of equilibrium.

Having sought to shape his work in this way, most of the thesis's dis­cussion of maximization, cost and production, and the consumer remained unchanged. Samuelson added two pages on “indeterminacy in purest com­petition?” but this was essentially an aside.^h He clarified a few passages and expanded his discussion of the analysis of the consumer in Hicks's Value and Capital, a book he had by now had time to read more carefully. He sig­nificantly shortened his discussion of the constancy of the marginal utility of money on the grounds that, though it had once been central to debates about the consumer, it was of no more than historical interest. He had to cover it because Hicks and others were trying to reestablish the concept of consumer's surplus as providing a way of doing welfare economics.^{[53] [54] However, Samuelson argued that the concept of consumer's surplus, though it raised some inter­esting mathematical problems, was unnecessary for welfare economics and was a vestige of an age when economists used literary methods and failed to see the underlying mathematical structures.}

The direction in which Samuelson thought consumer theory should move was indicated by a completely new chapter, “Transformations, Composite Commodities and Rationing.” He was even more critical than he had been in his thesis of Marshall's concept of elasticity; it served to conceal problems and was of little use “except possibly as mental exercises for beginning stu­dents.”⁴⁷ Index numbers were more important because they were central to national accounting, where they were needed for measuring concepts such as industrial production or the price level that involved aggregation over many commodities.

A major problem with index numbers was whether they could be used to ascertain whether an individual consumer would be better or worse off after a change had taken place. Using arguments about revealed preference that went back to his first articles, he argued that there was an inevitable area of igno­rance, in the sense that no simple index number could be a perfect measure of well-being. With any index number, there would either be some circumstances where, for example, a measure of real income would rise even though someone were worse off, or fall even though someone were better off. This indeterminacy, which many economists tackling the problem in the 1920s and 1930s had tried to eliminate, was inescapable. He stressed this point:

I should like to state as strongly as possible that this final indeterminacy is intrinsic and inherent. No amount of ingenuity can remove it, grounded as it is in the fundamental convexity properties of the indifference field, or more accurately in the consistency behavior of the individual.⁴⁹

He also noted that the problem arises only when finite changes are involved, echoing the point Wilson had stressed, following Gibbs, about the importance of considering finite changes. Though the theory might be abstract, it led to the practical conclusion that, in the absence of a perfect index number, it was neces­sary either to work with more than one index number or to choose the one that was the best approximation for the purpose at hand.

The same mathematics could be used to analyze the problem of rationing, which was topical during wartime. Rationing might involve either a limit on how much of a good could be bought, or it might involve “points rationing,” where goods required payment of cash plus points. In both cases, the equations describing equilibrium needed to be modified, and when this was done, the Le Chatelier Principle, discussed in his thesis, enabled Samuelson to deduce that when rationing is introduced, the consumer would be less responsive to price changes than in the absence of rationing. He also investigated whether allow­ing people to trade their ration points for cash would increase or decrease their well-being.