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Click to edit Master title style Click to edit Master subtitle style * * * Chapter 10: Exponential and Logarithmic Functions Alpha Chiang, Fundamental Methods of Mathematical Economics 3rd edition * * * Exponential functions * * * Exponential functions * * * Properties of exponential functions * * * The number e * * * The number e * * * Economic interpretation of e it can be interpreted as the result of a special process of interest compounding. * * * 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 * * * Interest Compounding and the function Aert A = reflects change in principal from previous level of P1 r/m = means that in each of the compounding periods in a year, only 1/m of the nominal interest will actually be applicable. mt = since interest is to be compounded m times a year, there should be a total of mt compounding in t years. * * * Interest Compounding and the function Aert Alterative form: * * * Instantaneous Rate of Growth * * * Discounting and Negative Growth Discrete: Continuous: * * * Logarithms * * * Common log * * * Natural log * * * Rules: * * * Logarithmic Functions Logarithmic Functions are functions whose variables are expressed as a function of the logarithm of another variable. Log functions are inverse functions of certain exponential functions * * * Derivatives of Exponential and Logarithmic Functions Log function rule: * * * Exponential function rule * * * The rules generalized * * * Examples: * * * Examples: * * * Case of base b * * * Higher derivatives * * * Application One of the prime virtues of the logarithm is its ability to convert a multiplication into an addition and a division into a subtraction Example: * * * Cont’d * * * Another example: * * * Optimal Timing Application to Value of wine = grows over time Problem: when to sell the wine to maximize profit. Assumption: no storage cost Need to discount each V to its present value. Interest rate has to be specified : r * * * * * * Application to Timber Cutting * * * Application of exponential and logarithmic derivatives * * * Examples Find the rate of growth of Find the rate of growth of * * * Rate of growth: Combination of functions * * * Rate of growth: Combination of functions Example: C grows at rate of α, H grows at rate of β , * * * * * * Example 4: Exports G=G(t) has a growth rate = a/t and export services S=S(t) has a growth rate = b/t * * * Finding Point Elasticity: * * * Example: Find the point elasticity of the demand function
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