Q_CHAPTER10 - Chapter 10 Optimal Growth and the Optimal...

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Chapter 10. Optimal Growth and the Optimal Savings Rate The fall in the savings rate observed in most OECD countries in recent years propounds the perennial question of optimal savings, and foremost the problem of de°ning an optimality criterion. For nearly eighty years ±since Frank Ramsey²s seminal contribution (1928) ±the fundamental problem of optimal savings policy has been to °nd the time path of capital accumulation maximizing, over a °nite or an in°nite horizon, the sum of discounted utility ³ows pertaining to consumption. The Ramsey problem was a problem in the theory of optimal growth; unfortunately however, it remained in the realm of theory. No serious at- tempt to compare an optimal strategy of investment to the amounts actually invested in a given nation was ever carried out. In our opinion, this is due to the fact that the optimal paths resulting from the di/erential equations governing the motion of the economy were simply inapplicable because they implied unrealistic policies, in particular exceedingly high savings rates. We hold that a fundamental reason for such a lack of applicability of the traditional model is the use of an arbitrary utility function. However, the idea of introducing a concave utility function cannot be outright dismissed because it has considerable intuitive appeal. Indeed, one could well envision (as many writers did) a benevolent planner for whom adding one more unit to consumption when total consumption is low has more value for society than when total consumption is high, and who therefore would have a bias toward giving more consumption to poorer generations; hence, one could feel well justi°ed to use a concave utility function. But the outcome of those models is, as we will see, exactly the contrary of what intuition would dictate. A few warning lights had been ³ashed in the sixties: in a two-sector model, Goodwin (1961) had obtained ´optimal´savings rate in excess of 60%; and StolØru (1970), in the only numerical solution of the Ramsey problem ever given, was led to savings rates in the order of 90%. We feel that time has come to make a systematic analysis of the role played by utility functions upon optimal trajectories of the economy. We have known since the sixties that the model exhibited only saddle-point sta- bility, implying that, given an initial capital-labour ratio, one and only one 1
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initial value of consumption (or saving) would eventually lead to a steady state, the slightest departure from this initial value entailing the collapse of the economy. But we still possess only scant indications about the exact properties and the implications of the rare transition path eventually leading to the steady state. In this chapter our preliminary aim is to take a close look at optimal growth paths implied by the families of utility functions commonly referred to in the literature, the power function and the negative exponential function.
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