Unit 4 Exponential and Logarithmic Functions in Economics PDF

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Unit 4: Exponential and Logarithmic Functions in Economics Effective vs. Nominal Rates of Interest  What is the nominal interest rate (𝑖 )? o Nominal interest rates can be compounded in various ways (e.g. annually, semi-annually, quarterly, monthly, daily, continuously)

 What is the effective interest rate (𝑖𝑒 )? o Effective interest rates compute the total interest rate earned over a period when there is interest compounding (1 + 𝑖𝑒 ) Example: If 𝑷𝑽 savings earn a 𝒊 nominal interest rate per annum compounded 𝒏 times, what is effective interest rate (𝒊𝒆 ) earned on 𝑷𝑽 savings over the period of one year? 𝒊

Solution: 𝑷𝑽(𝟏 + 𝒊𝒆 ) = 𝑷𝑽(𝟏 + 𝒏)𝒏(𝟏) 𝒊

𝒊𝒆 = (𝟏 + 𝒏)𝒏 − 𝟏 What happens to the value of 𝒓𝒆 when 𝒏 → ∞?

Discounting Time value of money =>A sum of money received in the future is worth less than the same sum of money received today (Why?) Discounting is the process of determining the present value (𝑷𝑽) of a future some of money (𝑭𝑽)

Example: If you want to receive amount 𝑭𝑽 in 𝒕 years at an interest rate of 𝒊 per annum compounded annually, much should you invest today? Solution: Find the present value (𝑷𝑽) of the investment 𝑭𝑽 = 𝑷𝑽(𝟏 + 𝒊)𝒕 𝑷𝑽 = 𝑭𝑽(𝟏 + 𝒊)−𝒕

Converting Exponential to Natural Exponential Functions Example: Show how the equivalent continuously 𝒊

compounded growth rate (𝒓) for 𝑷𝑽(𝟏 + )𝒏𝒕 is 𝒏

given by 𝒏𝒍𝒏(𝟏 + Solution:

𝒊 ) 𝒏...