For the sake of completeness, it should be mentioned that the advent of the Solow-Swan model did trigger a number of applications of Ramsey-like optimal growth models, again focusing on the question of optimal savings. At least six economists independently derived something called the 'golden rule' of economic growth: namely, that the optimal rate of investment (hence savings) should be such as to make the return on capital equal to the natural rate of population growth. None of these derivations required an assumption of intergenerational social utility in the Ramsey sense. The first to publish this interesting result was Phelps, followed by Desrousseaux, Allais, Robinson, von Weizsaecker and Swan (Phelps 1961; Desrousseaux 1961; Allais 1962; Robinson 1962; von Weizsaecker 1962; Swan 1963). Of course, the same objections raised earlier with respect to the efforts of Tinbergen and Goodwin remain applicable (for example, Bauer 1957).

But meanwhile, Koopmans and others found a way to make the intertemporal utility notion more palatable (Koopmans 1960; Koopmans et al. 1964). Along with others, including Cass, Malinvaud, Mirrlees and Shell, the Ramsey model was re-created as a formal Cass-Koopmans model of optimal growth in a single sector model (for example, Koopmans 1965; Cass 1965, 1966; Malinvaud 1965; Mirrlees 1967; Shell 1967). Once again, however, the underlying notion of an all-powerful (however altruistic)

social planner seemed increasingly anachronistic and irrelevant. Moreover, the models themselves exhibited a peculiar mathematical 'saddle point' property, with stable and unstable branches. This left a residue of doubts as to why the real economy should 'choose' an optimal trajectory.

That problem was apparently resolved in the 1980s by the advent of 'rational expectations', which seemed to provide the missing mechanism by which the economy would select a stable - rather than an unstable -trajectory from a saddle point (for example, Lucas and Stokey 1984). The fact that the economic growth trajectory seemed to be stable prior to 2008 was regarded as indirect evidence of the operation of the mechanism. As a result, optimal growth exercises are no longer considered to be normative, in the sense of explaining how things should work, but rather, as exercises in explaining how the economy really does work, as in modern business cycle theory.

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