## The Ramsey Theory Of Optimal Growth

In 1920 Arthur Pigou, suggested that - thanks to congenital myopia -people discount future utility; that is, they don't save enough to provide for their later wants or, in a different context, people in every generation consume too much, leaving too little for their successors (Pigou 1920). This left an unanswered question: namely what is the optimal rate of savings? Frank Ramsey tackled this problem by means of the calculus of variations (Ramsey 1928).2 He did not believe in discounting - in fact, he thought it unethical - so he devised a clever way to avoid the problem of comparing infinities. He assumed that there is a utility due to consumption but that there is a disutility arising from the need to work (labor) and a maximum utility, called 'bliss'. He also assumed that the maximum social utility for every generation would be achieved when that generation achieved bliss. The problem, then, is to minimize the distance between present utility and bliss, by choosing the best possible tradeoff between savings (investment) and loss of consumption in the early generations.

The mathematics of the Ramsey model have been extensively discussed in textbooks and need not be recapitulated here. Since there are two controls in the model (labor and capital), there are just two Euler-Lagrange equations. The first equation yields the result that the marginal disutility of labor must always be equal to the product of the marginal utility of consumption times the marginal product of labor. The second equation - as interpreted by Keynes - says (in words) that the optimum investment times the utility of consumption is equal to the distance from bliss or, more intuitively, the marginal benefit to later generations of faster approach to bliss must be balanced by the marginal loss of consumption benefits by the earliest generations. This became known as the Keynes-Ramsey rule. Ramsey's analysis confirmed Pigou's conjecture that the optimal savings rate is higher than the rate chosen by myopic agents in a market economy.

For various reasons, largely due to discomfort with Ramsey's social utility function and his unfamiliar mathematics, the notion of optimality was neglected for nearly 30 years. Jan Tinbergen and Richard Goodwin were the first to revive the idea, as applied to the Harrod-Domar growth models (Tinbergen 1956, 1960; Goodwin 1961). These attempts were criticized early on for obvious difficulties, notably that they imply an authoritarian 'social planner' which was an idea already past its time (Bauer 1957). In any case, the Harrod-Domar model was soon replaced by the Solow-Swan model.

There was one other early application of the calculus of variations by Harold Hotelling, not to growth but to the optimal extraction of exhaustible resources (Hotelling 1931). The problem, posed by Hotelling, was to maximize the total cumulative benefits from an exhaustible resource. The control variable, in this case, is the stock R of the resource. The consumption benefit can be defined as the product of the price P(t) multiplied by dRJdt, discounted by the factor exp(-St). Hotelling assumed that extraction would cease after a finite period t = z, when some 'backstop' technology would become available at a lower price. The simple integral can be integrated by parts, yielding the well-known result that (in equilibrium) prices will increase at the rate of discount, that is, P(t) = P(0)exp(-5t). Extraction costs can be introduced explicitly as a function of the remaining stock R, and the resulting integral can be solved by use of the Euler-Lagrange equations. The results in this case are similar. Hotelling's result has been the foundation of the field of resource economics. Hotelling's simple model has been elaborated in recent decades to deal with a variety of technological and geological complexities and uncertainties. However, these complications have made it difficult to verify the fundamental theory.

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