Lectur20 - Lecture XX Applications of Calculus of...

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Lecture XX Applications of Calculus of Variations I I. Example 7: Maximize Utility over Time A. An individual seeks the consumption rate over some time T. The utility function is assumed to obey the typical conditions U’(x)>0, U’’(x)<0: max ( ( )) eUC td t rt T - 0 subject to a cash flow constraint. B. The cash flow constraint. 1. Assume that the individual receives current income v(t) and a return on capital ik(t) where k(t) is the level of capital invested. 2. The uses of income are consumption c(t) and the change in investment k’(t). 3. Thus, the cash flow equation becomes vt ikt ct k t () += + with boundary conditions k(0)=k 0 and k(T)=k T . C. Solution: e Uvt k t d t rt T max (() ) =+ - + - - 0 The Euler equation is then formed as: ftkt kt e Uvt ikt kt fi e U C t fe U C t rt k rt k rt (, () , ’ ) ) ’( ( )) - = =- - - - Using the integral approach [] ie U C s ds de U C t ds ds i eUCsd s e UC t t t ieUCsd t rs t t rt t t rs t t rt r t rt rs t t - + - + - + -+ - -- + ∫∫ = - + + + ’( ( )) ∆∆ 1. Interpretation t rt - is the marginal utility of consuming an additional dollar at time t. While i eUCsd t rs t t - + + is the marginal value of the postponed dollar. Implicitly, ied s rs t t - +
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is the small change in dollars resulting from investing the dollar for consumption later rather than consuming it in the current period. Thus, ieUC sd s rs t t - + ’( ( )) is the change in utility from investing the dollar of consumption and receiving the consumption value in the next period. Similarly, UC t e rt ’( ( )) () + -+ is the marginal value of consumption at the next increment in time.
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This note was uploaded on 11/08/2011 for the course AEB 6533 taught by Professor Moss during the Fall '08 term at University of Florida.

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Lectur20 - Lecture XX Applications of Calculus of...

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