savingproblem - ECON 714 MACROECONOMIC THEORY II TA TIM LEE...

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Unformatted text preview: ECON 714: MACROECONOMIC THEORY II TA: TIM LEE FEBRUARY 26, 2010 Some advice on Ljungqvist and Sargent ( 2004 ): It’s a great book, but hard to learn from be- cause it’s not really written in a sequential fashion but rather grouped by topic. (This handout is basically a copy of it.) It doesn’t say much about dynamic programming ( Stokey and Lucas ( 1989 ) is the standard for that), but it does touch most (if not all) topics we will cover from now (including Noah’s part). Here’s my two cents if you want to study macro a bit more: First study Chapter 12 (Recursive CE) , you should be able to understand the jargon by now. Then Chapter 8 (Complete Markets) and skip to Part IV (Incomplete Markets) . Noah will cover Asset Pricing (Chapter 13) and probably some Money (Chapters 24 and 25) and Search (Chapters 6 and 26) . Arguably, the book should have been written in this order anyway... Permanent Income Hypothesis Solve the deterministic individual problem max { c t , a t + 1 } ∞ ∑ t = β t u ( c t ) s.t. c t + a t + 1 = y t + ( 1 + r ) a t ∀ t F.O.C.’s are c t : u ( c t ) = λ t a t + 1 : λ t = β ( 1 + r ) λ t + 1 Assuming β ( 1 + r ) = 1, we obtain u ( c t ) = u ( c t + 1 ) . Hence if u ( · ) is strictly concave, c t + j = c t for all j . Then writing out the budget constraints, c t + a t + 1 = y t + ( 1 + r ) a t c t + a t + 2 = y t + 1 + ( 1 + r ) a t + 1 c t + a t + 3 = y t + 2 + ( 1 + r ) a t + 2 c t + a t + 4 = y t + 3 + ( 1 + r ) a t + 3 ··· Multiply iteratively by 1 1 + r to get c t + a t + 1 = y t + ( 1 + r ) a t 1 1 1 + r c t + 1 1 + r a t + 2 = 1 1 + r y t + 1 + a t + 1 ( 1 1 + r ) 2 c t + ( 1 1 + r ) 2 a t + 3 = ( 1 1 + r ) 2 y t + 2 + ( 1 1 + r ) a t + 2 ( 1 1 + r ) 3 c t + ( 1 1 + r ) 3 a t + 4 = ( 1 1 + r ) 3 y t + 3 + ( 1 1 + r ) 2 a t + 3 ··· Then adding up LHS’s and RHS’s, notice that the a ’s cancel out, so we obtain ∞ ∑ j = ( 1 1 + r ) j c t + lim J → ∞ ( 1 1 + r ) J a t + J + 1 = ∞ ∑ j = ( 1 1 + r ) j y t + j + ( 1 + r ) a t By the TVC or borrowing constraint, the last term of LHS is 0. HenceBy the TVC or borrowing constraint, the last term of LHS is 0....
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savingproblem - ECON 714 MACROECONOMIC THEORY II TA TIM LEE...

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