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Unformatted text preview: Chapter 13. Optimal Savings: a General Approach In chapter 10 we described the serious di¢ culties entailed by the use of utility functions in the de&nition of optimal savings policies. We then suggested to look for optimal investment paths that would maximize the sum of timediscounted consumption ¡ows. This is what we will now do. We will be led to show that the theory of economic growth should not be separated into two strands: positive, or descriptive theory on the one hand, and normative theory on the other. Traditionally, the literature has indeed asked two di/erent types of questions: what would be the evolution of the economy if the investmentsavings rate were a given, exogenous, rate s ? And another question was: what should be the savings rate if some global welfare objective were to be attained? It is essential here to understand that when asking the &rst question, we assume the economy to be competitive, and in consequence the basic marginal equalities apply: the marginal productivity of labour is equal to the wage rate; the marginal productivity of capital is equal to the real rate of interest. But those very hypotheses are in fact the answer to the apparently unrelated question: how can a society maximize the sum of the discounted consumption ¡ows it can acquire? We will show that the optimal savings rate is endogenous to a competitive economy. In this chapter, we will deal in detail with this important issue. The main result is that the optimal savings rate will now have reasonable, very reachable values. We will then be able to carry out comparative dynamics. A change in the elasticity of substitution will be shown to have more importance on the optimal savings rate than a change in any other parameter. And, even more surprising, the ultimate bene&ts that society can derive from an increase in the elasticity of substitution are considerably larger than those which would be generated by the same increase in the rate of technical progress. 1 1 Competitive equilibrium and its resulting savings rate Our starting point is the fundamental equation of equilibrium on the com petitive &nancial and capital goods markets. We can express it either in di/erential form or in integral form. In di/erential form, it is i ( t ) = F K ( t ) + _ p ( t ) p ( t ) (1) In integral form, it can be written equivalently as p ( t ) = Z 1 t F K ( & ) e & R & t i ( z ) dz d& (2) or p ( t ) = Z t + h t F K ( & ) e & R & t i ( z ) dz d& + p ( t + h ) e & R t + h t i ( z ) dz ; (3) depending upon the constant of integration we have chosen, as we have seen in chapter 9. In equation (2) the price of capital is valued as the sum of discounted cash¡ows over an in&nitely long horizon. In equation (3), p ( t ) is equal to the sum of discounted cash¡ows over a time span of length h , plus the discounted residual value of capital at time t + h ....
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 Fall '08
 OLIVIERDELAGRANDVILLE
 Interest Rates, Interest, Interest Rate, Kt Kt Kt, optimal savings rate

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