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P_CHAPTER9 - Chapter 9 First Applications to Optimal Growth...

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Chapter 9. First Applications to Optimal Growth Our aim is now to look for optimal growth paths, given initial conditions of the economy. The reader will see that for most problems the calculus of variations is quite su¢ cient; but we feel that we should present applications of the maximum principle as well because it is so widely used. Most important however, is the following caveat: in this chapter we will present the traditional results of optimal growth theory as they have been expounded in the literature for more than four decades. This bulk of litera- ture is an outgrowth of the seminal paper by Frank Ramsey (1928) in which the author was looking for optimal investment trajectories maximizing a sum of utility °ows entailed by consumption. We should stress that the great part of this literature remained very much theoretical in the sense that it just posited the existence of a concave utility function that would be accepted by society as a whole. Results were discussed from a qualitative point of view, on the basis of the phase diagram that gave the directions of the fundamental variables of the economy ±in general, the capital-labour ratio, on the one hand, and consumption per person on the other. Whenever the di/erential equations were solved (through numerical methods) at the beginning of the sixties, the strangest of results appeared whatever the utility functions used: for instance, exceedingly high savings rates (in the order of 60-70%). The consequence of this dire situation is that optimal economic growth always remained in the realm of theory, and no serious attempt to compare optimal investment policies to actual time paths was ever carried out. In the next chapter we will show that the culprit is the very utility func- tion itself. We will analyze in a systematic way the consequences of using any member of the whole spectrum of utility functions, and we will show the damage done by them. We will then suggest another, much more direct, way of approaching optimal growth, leading to applicable results. For the time being, however, we want to stick to the traditional approach because the reader should be familiar with the hypotheses, methods and results of this mainstream approach. 1
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1 The mainstream problem of optimal growth: a simpli°ed presentation We will now present a problem of optimal growth with a single aim in mind: to illustrate the methods of economic dynamics that were presented in the last chapter. For that purpose, we shall deal with the simplest model we can imagine; it is stripped of any complication which might interfere with the exposition of the way methods work 1 . Thus we will suppose that capital is the only factor of production; net income (net of depreciation) is given by Y t = F ( K t ) , where F ( : ) is strictly concave ( F 0 ( : ) > 0 ; F 00 ( : ) < 0) . The problem is to determine K t such that Max K t Z 1 0 U ( C t ) e ° it dt (1) subject to the constraint _ K t ° I t = Y t ± C t (2) where i is a discount rate which applies to all utility °ows U ( C t ) . The utility function, U ( : ) , is supposed to be concave (thus U 0 ( : ) > 0
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