# ps2_sol - EC314-Fall 2010 Solutions Problem Set 2 Matt...

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EC314-Fall 2010 Solutions – Problem Set 2 Matt Turner (updated: 27 September 2010) . 1. Prove that t = 1 ( 1 1 + r ) t = 1 r . n t = 1 1 1 + r t = 1 1 + r + ... + 1 1 + r n 1 1 + r n t = 1 1 1 + r t = 1 1 + r 2 + ... + 1 1 + r n + 1 Subtracting the second equation from the first gives 1 - 1 1 + r n t = 1 1 1 + r t = 1 1 + r - 1 1 + r n + 1 Simplifying gives n t = 1 1 1 + r t = 1 + r r " 1 1 + r - 1 1 + r n + 1 # = 1 r - 1 r 1 1 + r n = 1 r - 1 r 1 1 + r n Taking a limit gives us the result. t = 1 1 1 + r t = lim n n t = 1 1 1 + r t = 1 r 2. One problem with discounting is that is counts future benefits very little after we get more than a few years ahead. To see this, conduct the following three exercises: a. Write the expression for the present value of a one hundred dollar payoff 100 years from now as a function of the interest rate. b. Plot this present value as a function of r . c. Policy makers actually use discounting when they are trying to make decisions. Commonly used discount rates range between 3% and 10%. What is the value of our 100\$ payoff 100 years from now at these two interest rates.

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