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policy-it

# policy-it - Short notes on a simple global solution method1...

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Short notes on a simple global solution method 1 Consider the following recursive maximization problem: V ( a, e ) = max c,a 0 ( u ( c ) + β P e 0 { e 1 ,e 2 } π ( e 0 | e ) V ( a 0 , e 0 ) ) subject to c e + (1 + r ) a a 0 a 0 ≥ − φ The Euler equation (substituting in the budget constraint) is u 0 ( e + (1 + r ) a a 0 ) β X e 0 { e 1 ,e 2 } π ( e 0 | e ) V a ( a 0 , e 0 ) with equality if a 0 > 0 . The Envelope condition is V a ( a, e ) = u 0 ( e + (1 + r ) a a 0 )(1 + r ) We are looking for a decision rule a ( a, e ) for savings (given values for r and φ ) . 1. Construct a grid on a = { a 1 , a 2 ...a n } . Set a 1 = φ. 2. Guess a vector of values ˆ V a for all combinations for a and e on the grid: ˆ V a = n ˆ V a ( a 1 , e 1 ) , ˆ V a ( a 2 , e 1 ) , ..., ˆ V a ( a n , e 1 ) , ˆ V a ( a 1 , e 2 ) , ..., ˆ V a ( a n , e 2 ) o (e.g. ˆ V a = 1 ) 3. Take the fi rst point on the grid ( a 1 , e 1 ) . 4. Check whether u 0 ( e 1 + (1 + r ) a 1 + φ ) β X e 0 { e 1 ,e 2 } π ( e 0 | e 1 ) ˆ V a ( a 1 , e 0 ) > 0 . 1 Jonathan Heathcote, May 13 2000, last updated April 2006.

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If it is then a ( a 1 , e 1 ) = φ If it is not then a ( a 1 , e 1 ) > φ. Solve for a ( a 1 , e 1 ) as follows: 1. Note that by concavity
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