Notes-Chapter 10 (1)

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Chapter 10:
Exponential and Logarithmic Functions
Alpha Chiang, Fundamental Methods of Mathematical Economics
3rd edition
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Exponential functions
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Exponential functions
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Properties of exponential functions
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The number e 
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The number e
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Economic interpretation of e 
it can be interpreted as the result of a special process of interest compounding.
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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 
		
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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.
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Interest Compounding and the function Aert
Alterative form:
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Instantaneous Rate of Growth
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Discounting and Negative Growth
Discrete:
	
Continuous:
	
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Logarithms
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Common log
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Natural log
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Rules:
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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 
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Derivatives of Exponential and Logarithmic Functions 
Log function rule:
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Exponential function rule
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The rules generalized 
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Examples:
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Examples:
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Case of base b
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Higher derivatives
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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:
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Cont’d
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Another example:
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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
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Application to Timber Cutting
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Application of exponential and logarithmic derivatives 
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Examples
Find the rate of growth of
Find the rate of growth of 
 
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Rate of growth: Combination of functions
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Rate of growth: Combination of functions
Example: C grows at rate of α, H grows at rate of β
, 
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Example 4: Exports G=G(t) has a growth rate = a/t and 
	export services S=S(t) has a growth rate = b/t
	
 
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Finding Point Elasticity:
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Example: Find the point elasticity of the demand function