Notes-Chapter 10

Notes-Chapter 10 - Chapter 10: Exponential and Logarithmic...

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Unformatted text preview: Chapter 10: Exponential and Logarithmic Functions Alpha Chiang, Fundamental Methods of Mathematical Economics 3 rd edition Exponential functions ( ) 1 Generalized: a and c are compressing or extending agents t ct y f t b b y ab = = = Exponential functions 3 Preferred base: e = 2.7182818284591.... or exp( ) or exp(3 ) or exp( ) t t rt y e y t y e y t y Ae y A rt = = = = = = Properties of exponential functions , where Using the chain rule, rt t t rt w r w r t t y Ae y Ae w rt dy d d e e dt d Ae rAe y dw Ae r rAe dt dt dw dt = = = = × = × = = = The number e 1 1 1 1 2 1 2 3 1 3 4 1 4 1 ( ) (1 ) (1) (1 ) 2 (2) (1 ) 2.25 (3) (1 ) 2.37037 (4) (1 ) 2.44141 lim ( ) lim(1 ) 2.7182818284591.... m m m m m m f m f f f f e f m →∞ →∞ = + = + = = + = = + = = + = = = + = M The number e 2 3 4 5 ( ) 1 1 1 1 1 2! 3! 4! 5! for a special case x=1, 1 1 1 1 1 1 2! 3! 4! 5! 2 .5 0.166667 0416667 0.0083333 ..... 2.7182818.... x x x e e x x x x x e φ = = + + + + + + = + + + + + + = + + + + + = L L Economic interpretation of e it can be interpreted as the result of a special process of interest compounding. 1 1 1 2 1 2 3 1 3 4 1 4 1 (1) 1(1 100%) 1(1 ) 2 (2) (1 50%)(1 50%) (1 ) 2.25 (3) (1 ) (4) (1 ) ( ) (1 ) m m v v v v v m = + = + = = + + = + = = + = + = + M Economic interpretation of e For the limiting case, when interest is compounded continuously during the year, the value of the asset will grow in a snowballing fashion becoming 1 lim ( ) lim(1 ) 100% nominal interest 172% = effective interest m m m m v m e →∞ →∞ = + = = Interest Compounding and the function Ae rt ( ) (1 ) mt r m v m A = × + A = reflects change in principal from previous level of P1 r/m = means that in each of the compounding periods in a...
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This note was uploaded on 11/09/2011 for the course ECON 101 taught by Professor Richards during the Spring '11 term at Cambrian College.

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Notes-Chapter 10 - Chapter 10: Exponential and Logarithmic...

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