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Unformatted text preview: Chapter 7. Optimal Growth Theory: an Introduction to the Calculus of Variations We know how an economy might behave under a number of hypotheses regarding the way factors of production are linked to output, and the way each of those factors are modi&ed over time: labour through population growth; capital through investment. Also, technical progress can enter the picture in a number of di/erent ways. Finally, the structure of the production function itself may be modi&ed through time due to changes in the elasticity of substitution between capital and labour. The question we now want to ask is the following: among all possible time paths that we might want to choose by setting a given investment pol- icy, is one of those optimal in a sense to be de&ned? This question is at the heart of optimal growth theory, and has given rise to a huge volume of literature, sometimes controversial. We &rst have to determine the objective function, and for historical reasons which go back to Ramsey (1928), the objective supposedly de&ned by society is to maximize the sum, over a time span which may be &nite or in&nite, of discounted utility ¡ows pertaining to consumption, given a production function constraint. We will &rst give an example of such an objective; the example is vol- untarily simpli&ed in order for the reader to grasp easily the nature of the problem at stake. Suppose that the production function is given by Y t = F ( K t ;L t ;t ) , where, as before, Y is the net domestic product (net of capital depreciation); so investment is I t = dK t =dt & _ K t , and consumption is C t = Y t ¡ I t = Y t ¡ _ K t . Let U ( C ) be a strictly concave utility function; we thus have U ( c ) > and U 00 ( c ) < . The objective of society may be to choose the investment time path _ K t or, equivalently, taking account an initial condition K (0) = K ; the capital time path K t that maximizes the integral Z 1 U [ C t ] e & it dt (1) where i is a discount rate, subject to the constraint C t = Y t ¡ _ K t = F ( K t ;L t ;t ) ¡ _ K t ; (2) 1 assuming that the trajectory L t is known. Replacing the constraint into the objective function, our task is to determine the entire time path K t such that max K ( t ) Z 1 U [ F ( K t ;L t ;t ) & _ K t ] e & it dt: (3) This integral de&nes a relationship between an entire time path ( K t ) and a number, the de&nite integral. Such a relationship is a functional . Tradi- tional di/erential calculus deals with optimization of functions, relationships between a number (or several numbers) and a number. In this case we have a relationship between a function and a number, which will require a new methodology. In traditional calculus, the basic idea of &nding a point that is a candidate to maximize a function f ( x ) is to give an increase to the independent variable, determine the limit of the resulting rate of increase of the function, and examine whether there are any values of the variable where that limit is equal to zero. Geometrically, this implies looking for points of the functionequal to zero....
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- Fall '08
- Derivative, dx, Leonhard Euler, Euler equation, Akira Takayama