prelims_Macro Prelim ANSWERS July 2007

prelims_Macro Prelim ANSWERS July 2007 - Answers to the...

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Answers to the July 2007 Macro Prelim 1. What is dynamic e¢ ciency? Can the steady-state equilibrium in the Ramsey model be dynamically ine¢ cient? Why or why not? ANSWER: An allocation is dynamically ine¢ cient if the capital stock exceeds the Golden Rule level. With non-negative discounting, a Ramsey steady state cannot be dynamically ine¢ cient. If it were, the representative agent could increase utility by reducing saving and consuming some of the capital. But that contradicts the assumption that Ramsey consumers maximize utility. 2. Assume that consumption growth is random and has the following representation: c t +1 c t = (1 + g c ) " t +1 where it is assumed that ln " t is distributed normally with E (ln " t ) = 2 c = 2 and V ar (ln " t ) = 2 c : In an economy populated by identical agents with constant relative risk aversion utility, use the Euler equation associated with real bonds to derive an expression for the equilibrium real interest rate. Use the approximation that ln (1 + x ) ± x to simplify the expression. Discuss the implications of this expression and, in particular, discuss the impact that uncertainty has on the real interest rate. Also discuss the implications that your result has for the risk-free rate puzzle. (Recall that if ln z ² N ( 2 ) ; then E ( z ) = exp [ ± + 2 = 2] .) ANSWER: The Euler equation associated with a one period bond is: 1 = (1 + r t ) ²E c t +1 c t ± = (1 + r t ) ² (1 + g c ) E ² " t +1 ³ where the last expression is derived by using the assumed process for consumption growth. Under the assumption that ln " t is normally distributed, this implies that - ³ ln " t is distributed normally with mean of E ( ³ ln " t ) = 2 c = 2 and V ar ( ³ ln " t ) = ³ 2 2 c . Then E ² " t +1 ³ = exp [ 2 c = 2 + ³ 2 2 c = 2] = exp [ 2 c = 2 (1 + ³ )] : Therefore we have: 1 = (1 + r t ) ² (1 + g c ) exp ² 2 c = 2 (1 + ³ ) ³ ² = (1 + ´ ) 1 and rearranging we have r t = ´ + ³g c ³ (1 + ³ ) 2 c = 2 The &rst two terms are consistent with the Keynes-Ramsey condition: if agents dis- count rate increases, they must be compensated for foregoing current consumption so interest rate increases; if consumption growth increases, then current consumption is 1
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relatively more scarce so interest rates increase. If uncertainty over future consump- tion increases, this reduces the certainty equivalent of future consumption so current consumption is relatively more abundant. Consequently, interest rates are lower. With regard to the risk free rate puzzle, the above expression implies a larger risk
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prelims_Macro Prelim ANSWERS July 2007 - Answers to the...

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