Further Explorations in Sequence Analysis
Hammarskjold made two important contributions to sequence analysis. The first was his “Outline of an algebraic method for dynamic price analysis” (1932), the second his doctoral dissertation on “The transmission of economic fluctuations” (1933); none of them was ever translated into English.
Hammarskjold (1932: 171-6) was very critical of Myrdal’s (1931 [1939]) definition of monetary equilibrium and based his “algebraic method” on Lindahl’s (1929a [1939]) temporary-equilibrium approach. Yet he translated Myrdal’s critique of Lindahl into a “continuation analysis”, by which consecutive periods of the dynamic process are explicitly linked through the reactions of economic agents to changes in their respective “strategic factor”. The strategic factor for the firms is their net profit, and - in the perspective of Hammarskjold in particular, and the Stockholm School in general - the entrepreneurs as leaders of the firms are the key agents in the system. Hence, (unexpected) windfall profits are the strategic factor in the system, which in Myrdalian terms could now be described as the discrepancy between the ex ante and ex post values of the same variable. The changes in plans and uses of windfall profits, which Hammarskjold captured through reaction functions, provide the links between the periods, within which the plans are coordinated through market processes (see Lindahl 1939: 152; Hansson 1982: 157-66).At the microeconomic level, and closer to Myrdal (1927, 1931 [1939]), Svennilson (1938) used sequence analysis to discuss the typical firm’s intertemporal planning under risk. His focus was set on developing a toolkit of formal concepts of probability and constrained optimization in the short term and the long, rather than on providing a theory of coordination (see Siven 1991). Svennilson’s dissertation, too, remained untranslated.
Lundberg’s dissertation was directly published in English.
His Studies in the Theory of Economic Expansion (1937) are generally regarded as the peak in Stockholm-style sequence analysis. The core part of the book is chapter 9, “The construction of model sequences”, where Lundberg (1937: 183) declared:The subject of our analysis is an economic system during a period of expansion... Production, consumption, income, savings, and investments are all increasing at certain rates, and we ask whether this growth can continue in some sort of dynamic equilibrium, or whether discrepancies must automatically come into being within the system itself, which have the tendency to interrupt the process. The conditions for continued growth may not be reconcilable to the cumulative effects of the expansion during previous periods... Cassel’s simple assumption of a [uniformly progressive economy] is hence viewed as a problem to be investigated.
Lundberg (1937: 185, 240) formalized Cassel’s conditions for steady-state growth, clearly anticipating the core of the Harrod and Domar growth models by several years (see Berg 1991). He employed these conditions as benchmarks for the construction of numerical model sequences, in which he first used multiplier analysis with exogenous investment and the famous Lundberg-lag, then sequences in which the multiplier interacts with accelerators related to variations of inventories and investments in fixed capital, and finally sequences in which investment is endogenous to a variable rate of interest. Lundberg reconstructed Wicksell’s cumulative process as a general framework of disequilibrium sequence analysis, in which he nested Keynesian “oversaving” as well as Hayekian “undersaving” sequences. Even though he was well versed in mathematics and used systems of difference equations for his dynamic analysis, he did not attempt a formal generalization of his approach. This was done in a reduced fashion in 1939, when “Harvard professor Alvin Hansen gave a talented young student with mathematical skills the assignment to formalize Lundberg’s argument.
The student, 23 years of age, was Paul Samuelson, and the result was the famous article about the interaction between the multiplier and the accelerator that led to Samuelson’s international breakthrough” (Lindbeck and Persson 1990: 276, author’s translation).Lindahl (1939: pt 1) had the final word about sequence analysis in the era of the Stockholm School, at least in terms of the publication date. It has already been mentioned that his methodological focus had turned from temporary equilibria to disequi- libria after 1930. While he had earlier assumed that markets clear within all periods, though not always at the expected prices, he now favoured an approach in which “no price movements occur during the periods themselves” (Lindahl 1939: 61), while quantities adjust to unforeseen events. Excess demands and supplies manifest themselves in unplanned orders and inventories of producers and traders, who will then change prices between the periods. While Lundberg (1937) had taken sequence analysis to a general framework for casuistic modelling, Lindahl (1939: pt 1) offered the nucleus of a general theory of out-of-equilibrium dynamics, based on a systematic comparison of the different approaches to dynamic theory developed in the Stockholm School, to which he appended an “algebraic discussion of the relations between some fundamental concepts”, both at micro- and macroeconomic levels. The ground for this formalization was laid by a “Note on the dynamic pricing problem”, which he had circulated in 1934 (see Steiger 1971: 204-11). Upon a visit at the London School of Economics in the same year, Lindahl and his note had made an impression on John Hicks, inspiring him to develop his concept of temporary equilibrium in Value and Capital (1939). Hicks and his wife Ursula also cooperated with Lindahl in the publication of the latter’s Studies, which were published simultaneously (see Hicks 1991).