200a_f08_ps_7_ak

# 200a_f08_ps_7_ak - University of California Davis ARE/ECN...

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University of California, Davis ARE/ECN 200A Fall 2008 PROBLEM SET 7 ANSWER KEY 1. A Tale of Twelve Cities As a prerequisite for application of either Theorem 1 or Theorem 2, all individuals must have preference relations that can be represented by indirect utility functions in the Gorman form, v i ( p; w i ) = a i ( p ) + b i ( p ) w i . City 1: No. The two types of citizens have Gorman forms with a A ( p ) = 0 ; b A ( p ) = 1 3 p 1 ± 1 = 3 2 3 p 2 ± 2 = 3 ; a B ( p ) = 0 ; b B ( p ) = 2 3 p 1 ± 2 = 3 1 3 p 2 ± 1 = 3 : Since b A ( p ) 6 = b B ( p ) ; Theorem 1 does not apply. We do have the Gorman form restriction for Theorem 2 with a A ( p ) = a B ( p ) = 0 (indicating that both preference relations are homothetic). However, the wealth domain is unrestricted so that Theorem 2 also does not apply. Therefore, City 1 does not have a positive representative consumer (by either Theorem 1 or 2). City 2: YES, by Theorem 2. Beacuse a A ( p ) = a B ( p ) = a C ( p ) = 0 and wealth vectors are restricted. City 3: YES, by Theorem 1. The two types of citizens have Gorman forms with a A ( p ) = 0 ; b A ( p ) = 1 3 p 1 ± 1 = 3 2 3 p 2 ± 2 = 3 ; a F ( p ) = ( p 1 + 2 p 2 ) 1 3 p 1 ± 1 = 3 2 3 p 2 ± 2 = 3 ; b F ( p ) = 1 3 p 1 ± 1 = 3 2 3 p 2 ± 2 = 3 : Since b A ( p ) = b F ( p ) ; Theorem 1 does apply this time. However, with a F ( p ) 6 = 0 ; we do not have the Gorman form restriction for Theorem 2. Therefore, City 3 does have a positive representative consumer by Theorem 1, but not by Theorem 2. City 4: YES, by Theorem 1. Same reasoning as in City 3. 1

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City 5: No. The two types of citizens have Gorman forms with a A ( p ) = 0 ; b A ( p ) = 1 3 p 1 ± 1 = 3 2 3 p 2 ± 2 = 3 ; a G ( p ) = ( p 1 + 2 p 2 ) 2 3 p 1 ± 2 = 3 1 3 p 2 ± 1 = 3 ; b G ( p ) = 2 3 p 1 ± 2 = 3 1 3 p 2 ± 1 = 3 : Since b A ( p ) 6 = b G ( p ) ; Theorem 1 does not apply, and with a G ( p ) 6 = 0 ; Theorem 2 also does not apply. (Theorem 2 would not apply anyway since the domain of wealth vectors is not appropriately restricted.) Therefore, City 5 does not have a positive representative consumer (by either Theorem 1 or 2). City 6: No. The two types of citizens have Gorman forms with a D ( p ) = ln p 1 p 2 ± 1 ; b D ( p ) = 1 p 1 ; a E ( p ) = ln p 2 p 1 ± 1 ; b E ( p ) = 1 p 2 : Since b D ( p ) 6 = b E ( p ) ; Theorem 1 does not apply, and with a D ( p ) 6 = 0 and a E ( p ) 6 = 0 ; Theorem 2 also does not apply. (Theorem 2 would not apply anyway since the domain of wealth vectors is not appropriately restricted.) Therefore, City 6 does not have a positive representative
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200a_f08_ps_7_ak - University of California Davis ARE/ECN...

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