200a_f08_ps_9_ak2

200a_f08_ps_9_ak2 - 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 9 ANSWER KEY 1. 1.1 Figure 1 CDFs for Lotteries L & and L && 1.2 L & does not FOSD L && : Analytically, recall part 3 of PS8, Constance, who likes money ( u C ( x ) > ), prefers L && to L & ; i.e., not everyone who likes money considers L & at least as good as L && : Graphically, from Figure 1, CDF graph of L & is not completely to the right of that of L && (they cross with each other). 1.3 L & does not SOSD L && : Analytically, recall part 3 of PS8, Constance, who is risk averse ( u 00 C ( x ) < ), prefers L && to L & ; i.e., not everyone who is risk averse considers L & at least as good as L && : Graphically, from Figure 1, the area under CDF graph of L & up to 100 is : 84 & 90 = 75 : 6 and it is greater than the area under CDF graph of L && up to 100 which is : 2. 2.1 Since u B ;u G ;u D display CRRA, their wealth expansion paths are rays through the origin. The coe cients of ARA and RRA of u B ;u G ;u D and u R ;u Q are as follows, 1 r A ( x;u ) r R ( x;u ) u B ( x ) = ln x 1 =x 1 u G ( x ) = p x 1 = (2 x ) 1 = 2 u D ( x ) = & x & 1 2 =x 2 u R ( x ) = & e & x 1 x u Q ( x ) = & ( a & x ) 2 ;x < a 1 = ( a & x ) a= ( a & x ) Let & p 1 p 2 1 & & & > 1 : Then the optimal solution in general could be characterized by the equation u i ( x i 1 ) u i ( x i 2 ) = &;i = B;G;D;R;Q: The wealth expansion path for u B is de&ned by & = u B & x B 1 u B ( x B 2 ) = x B 2 x B 1 ;i:e:; ~ x B 2 ( x 1 ) = &x 1 : (1) The wealth expansion path for u G is de&ned by & = u G & x G 1 u G ( x G 2 ) = & x G 1 & 1 = 2 ( x G 2 ) & 1 = 2 ;i:e:; ~ x G 2 ( x 1 ) = & 2 x 1 : (2) The wealth expansion path for u D is de&ned by & = u D & x D 1 u D ( x D 2 ) = & x D 1 & 2 ( x D 2 ) & 2 ;i:e:; ~ x D 2 ( x 1 ) = & 1 = 2 x 1 : (3) The wealth expansion path for u R is de&ned by & = u R & x R 1 u R ( x R 2 ) = e & x R 1 e & x R 2 ;i:e:; ~ x R 2 ( x 1 ) = ln & + x 1 : (4) The wealth expansion path for...
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200a_f08_ps_9_ak2 - University of California, Davis ARE/ECN...

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