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200a_f08_ps_5_ak

200a_f08_ps_5_ak - University of California Davis ARE/ECN...

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Unformatted text preview: University of California, Davis ARE/ECN 200A Fall 2008 Joaquim Silvestre & Shaofeng Xu PROBLEM SET 5 ANSWER KEY 1. 1.1 By e ( p; v ( p; w )) = w; ln w = & + & L j =1 & j ln p j + 1 2 & L i =1 & L j =1 ¡ ij ln p i ln p j + v ( p; w ) ¡ j =1 p & j j ; i:e:;v ( p; w ) = ¡ L j =1 p & & j j (ln w & & & & L j =1 & j ln p j & 1 2 & L i =1 & L j =1 ¡ ij ln p i ln p j ) : 1.2 By Shephard¡s lemma, 8 k 2 f 1 ; :::;L g ; h k ( p; u ) = @e ( p; u ) @p k = e ( p; u ) @ ln e ( p; u ) @p k = e ( p; u ) & & k p k + 1 2 & L j =1 ¡ kj ln p j p k + 1 2 & L i =1 ¡ ik ln p i p k + u ¢ k p k ¡ j =1 p & j j ¡ = e ( p; u ) p k & & k + 1 2 & L i =1 ¡ kj ln p j + 1 2 & L i =1 ¡ ik ln p j + u¢ k ¡ j =1 p & j j ¡ = e ( p; u ) p k h & k + & L j =1 ¡ kj ln p j + u¢ k ¡ j =1 p & j j i ¢ ¡ ij = ¡ ji ; 8 i; j £ 1.3 8 k 2 f 1 ; :::;L g ; ~ x k ( p; w ) = h k ( p; v ( p; w )) = e ( p; v ( p; w )) p k h & k + & L i =1 ¡ kj ln p j + v ( p; w ) ¢ k ¡ j =1 p & j j i = w p k " & k + & L i =1 ¡ kj ln p j + ¤ ¡ L j =1 p & & j j (ln w & & & & L j =1 & j ln p j & 1 2 & L i =1 & L j =1 ¡ ij ln p i ln p j ) ¥ ¢ k ¡ j =1 p & j j # = w p k & & k + & L j =1 ¡ kj ln p j + (ln w & & & & L j =1 & j ln p j & 1 2 & L i =1 & L j =1 ¡ ij ln p i ln p j ) ¢ k ¡ 1 2. 2.1 By Roy&s identity, ~ x j ( p; w ) = & ( @v ( p; w ) [email protected] j ) = ( @v ( p; w ) [email protected] ) ; which if computed from (1) yields ~ x j ( p; w ) = & & h F ( p ) ln( w=C ( p )) + G ( p ) i & 2 & @F @p j ln ( w C ( p ) ) & C w ( & w C 2 ) @C @p j F ( ln w C ) 2 + @G @p j ¡ & h F ( p ) ln( w=C ( p )) + G ( p ) i & 2 & C w ( 1 C ) F ( ln w C ) 2 = @F @p j ln w C & C w ¢ & w C 2 £ @C @p j F + @G @p j ¢ ln w C £ 2 F w ; i:e:; ~ x j ( p; w ) = w F @F @p j ln w C + w C @C @p j + w F @G @p j ¤ ln w C ¥ 2 : This in turn yields b j ( p; w ) ¡ p j ~ x j w = p j F @F @p j ln w C + p j C @C @p j + p j F @G @p j ¤ ln w C ¥ 2 : 2.2 If G ( p ) = 0 ; 8 p; then b j = p j F @F @p j ln w C + p j C @C @p j ( ¢...
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