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# homework6_sol - ECO 475 HW6 Solution Joon Song 1 Simple Job...

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Unformatted text preview: ECO 475 HW6 Solution Joon Song Oct 21, 2009 1. Simple Job Search Model A Let V ( w ) be a value of an unemployed worker with an o/er of w at hand, and W = [ w ; w ] be a support of F . Then V ( w ) = max & V E ( w ) ;V U ¡ , (1) where V E ( w ) = w + & ¢ (1 & ¡ ) V E ( w ) + ¡V U £ (2) V U = b + & Z V ( w ) dF ( w ) . (3) From (2) , V E ( w ) = w + &¡V U 1 & & (1 & ¡ ) , (4) and thus V ( w ) = max ¤ w + &¡V U 1 & & (1 & ¡ ) ;V U ¥ . (5) Since V E ( w ) is strictly increasing with w , an unemployed worker&s policy follows a reservation rule: w A R + &¡V U 1 & & (1 & ¡ ) = V U V U = w A R 1 & & . (6) Therefore, V ( w ) = max ¤ w + &¡V U 1 & & (1 & ¡ ) ;V U ¥ = max ¤ (1 & & ) w + &¡w A R (1 & & ) [1 & & (1 & ¡ )] ; w A R 1 & & ¥ = 1 (1 & & ) [1 & & (1 & ¡ )] max & (1 & & ) w + &¡w A R ; (1 & & ) w A R + &¡w A R ¡ = 1 1 & & (1 & ¡ ) max & w;w A R ¡ + &¡w A R (1 & & ) [1 & & (1 & ¡ )] . (7) 1 By (3) , (6) , and (7) , w A R 1 & & = b + & Z & 1 1 & & (1 & ¡ ) max ¡ w;w A R ¢ + &¡w A R (1 & & ) [1 & & (1 & ¡ )] £ dF ( w ) = b + & 1 & & (1 & ¡ ) Z max ¡ w;w A R ¢ dF ( w ) + & 2 ¡w A R (1 & & ) [1 & & (1 & ¡ )] ) w A R = 1 & & (1 & ¡ ) 1 + &¡ b + & 1 + &¡ Z max ¡ w;w A R ¢ dF ( w ) . (8) (1) T ( ¡ ) is a contraction. Let x;y 2 W . Without loss of generality (WLOG), x ¢ y . Then, j T ( y ) & T ( x ) j = & 1 + &¡ ¤ ¤ ¤ ¤ Z max f y;w g dF ( w ) & Z max f x;w g dF ( w ) ¤ ¤ ¤ ¤ = & 1 + &¡ ¤ ¤ ¤ ¤ Z w & x ( y & x ) dF ( w ) + Z x & w & y ( y & w ) dF ( w ) ¤ ¤ ¤ ¤ ¢ & 1 + &¡ ¤ ¤ ¤ ¤ Z ( y & x ) dF ( w ) ¤ ¤ ¤ ¤ = & 1 + &¡ j y & x j . Therefore, T ( ¡ ) is a contraction with modulus & 1+ &¡ < 1 (by assuming & 2 (0 ; 1) ). Note that we also need restrictions on parameters so that T ( W ) £ W , which amounts to T ( w ) 2 W and T ( w ) 2 W due to monotonicity of T : 1 & & (1 & ¡ ) 1 + &¡ b + &E ( w ) 1 + &¡ ¤ w 1 & & (1 & ¡ ) 1 + &¡ b + & 1 + &¡ w ¢ w i.e. w + & 1 & & (1 & ¡ ) [ w & E ( w )] ¢ b ¢ w . Since ( W ; j¡j ) is a complete metric space and T is a contraction, by the contraction mapping theorem, w A R exists and is unique in W . Model B Let V ( w ) be a value of an unemployed worker with an o/er of w at hand, and W = [ w ; w ] be a support of F . Then V ( w ) = max ¡ V E ( w ) ;V U ¢ , (9) where V E ( w ) = w + & ¥ (1 & ¡ ) Z max ¡ V E ( w ) ;V E ( w ) ¢ dF ( w ) + ¡V U ¦ (10) V U = b + & Z V ( w ) dF ( w ) . (11) 2 From (10) , we know that V E ( w ) is strictly increasing with w (Why?). Thus, an unemployed worker&s policy follows a reservation rule: V U = w B R + & & (1 & ¡ ) Z max ¡ V E ¢ w B R £ ;V E ( w ) ¤ dF ( w ) + ¡V U ¥ = w B R + & & (1 & ¡ ) Z max ¡ V U ;V E ( w ) ¤ dF ( w ) + ¡V U ¥ = w B R + & & (1 & ¡ ) Z V ( w ) dF ( w ) + ¡V U ¥ . (12) By (11) and (12) , V U = w B R & (1 & ¡ ) b (1 & & ) ¡ . (13) Moreover, by (10) and (12) , V E ( w ) & V U = w & w B R + & (1 & ¡ ) Z ¦ max ¡ V E ( w ) ;V E ( " ) ¤ & max ¡ V U ;V E ( " ) ¤§...
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homework6_sol - ECO 475 HW6 Solution Joon Song 1 Simple Job...

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