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# homework9_sol - ECO 475 HW9 Solution Joon Song 1 Computing...

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Unformatted text preview: ECO 475 HW9 Solution Joon Song Nov 18, 2009 1. Computing the Equilibrium To solve the problem, we need to specify a within-period timing assumption. There are two cases. Case1 ADE: Decision is made before realization of time state. SME: Decision at time t is made before realization of time t state. Case2 ADE: Decision is made after realization of time state. SME: Decision at time t is made after realization of time t state. We are going to solve in detail for case 1, and to mention brie&y how the result will change if the timing assumption comes from case2. (1) ADE The ADE consists of price f p t ( s t ) g 1 t =0 ;s t 2 S t and allocation & c i t ( s t ) ; i = 1 ; 2 ¡ 1 t =0 ;s t 2 S t such that 1) Given f p t ( s t ) g 1 t =0 ;s t 2 S t , for i = 1 ; 2 , & c i t ( s t ) ¡ 1 t =0 ;s t 2 S t solves the following: max f c i t ( s t ) g 1 P t =0 P s t 2 S t & t ¡ ¢ s t £ ln ¢ c i t ¢ s t ££ subject to 1 P t =0 P s t 2 S t p t ¢ s t £ c i t ¢ s t £ & 1 P t =0 P s t 2 S t p t ¢ s t £ e i t ¢ s t £ (1) c i t ¢ s t £ ¡ ; 8 t; 8 s t . 2) Market clears: 2 P i =1 c i t ¢ s t £ = 2 P i =1 e i t ¢ s t £ ; 8 t; 8 s t . (2) The FOCs from consumer¡s problem is that & t ¡ ( s t ) c i t ( s t ) = ¢ i p t ¢ s t £ ; 8 i; t; s t (3) where ¢ i is a Lagrange multiplier for constraint (1) . Thus & t +1 ¡ ( s t ; s t +1 ) c i t +1 ( s t ; s t +1 ) = ¢ i p t +1 ¢ s t ; s t +1 £ ; 8 i; t; s t ; s t +1 . (4) 1 By (3) and (4) , 1 c i t ( s t ) = &p t ( s t ) ¡ ( s t +1 ) p t +1 ( s t ; s t +1 ) 1 c i t +1 ( s t ; s t +1 ) ; 8 i; t; s t ; s t +1 (5) where we use ¡ ( s t +1 ) = ¡ & s t +1 j s t ¡ (iid-over-time) = ¡ ( s t ; s t +1 ) ¡ ( s t ) . Combining this with the market clearing condition, we have &p t ( s t ) ¡ ( s t +1 ) p t +1 ( s t ; s t +1 ) = 1 ; 8 t; s t ; s t +1 . (6) Therefore, by (5) , 8 i; t; s t , c i t & s t ¡ = c i ;A if s = A c i t & s t ¡ = c i ;B if s = B (7) where c i ;s = c i ( s = s ) ( s = A; B ) . In addition, from (3) , we have p ;B c i ;B = 1 & ¡ ¡ p ;A c i ;A ; 8 i; t; s t . (8) Combining (1) , (6) , and (7) , we have p ;A c 1 ;A + p ;B c 1 ;B 1 & & = p ;A ¡ 2 + &¡ 1 & & ( p ;A ¡ 2 + p ;B ¡ 2) p ;A c 2 ;A + p ;B c 2 ;B 1 & & = p ;B ¡ 2 + + & (1 & ¡ ) 1 & & ( p ;A ¡ 2 + p ;B ¡ 2) . (9) Plugging (8) into (9) , we have c 1 ;A = 2 (1 & & ) ¡ + 2 &¡ 2 ¢ 1 + p ;B p ;A £ c 2 ;A = 2 (1 & & ) ¡ p ;B p ;A + 2 &¡ (1 & ¡ ) ¢ 1 + p ;B p ;A £ . (10) Combining this with the market clearing conditions, we have p ;B p ;A = 1 & ¡ ¡ . (11) 2 Normalize p ;A = & . Then, p ;B = 1 & & and by (10) , c 1 ;A = 2 & c 2 ;A = 2 (1 & & ) . (12) Moreover, by (8) and (11) , c 1 ;B = 2 & c 2 ;B = 2 (1 & & ) . (13) Therefore, by (7) , c 1 t & s t ¡ = 2 &; 8 t; s t c 2 t & s t ¡ = 2 (1 & & ) ; 8 t; s t . (14) and by (6) , p t & s t ¡ = ¡ t & j (1 & & ) t +1 & j ; 8 t; s t . (15) where j is the number of occurrences of state A in history s t ....
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homework9_sol - ECO 475 HW9 Solution Joon Song 1 Computing...

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