John Riley 6 December 200 NWER TO ODD NUMBERED EXERCIE IN CHPTER 7 ecion 7 Exercie 7-: m m uppoe ˆ, m=,, M (a For M = 2, i i eay o how ha I implie I From I, for any probabiliy vecor ( p, p 2, 2 2 ˆ ( p, p ˆ 2 :, ( p, p2:, 2 2 ˆ ( p, p ˆ ˆ ˆ 2 :, ( p, p2:, 2 2 Therefore 2 2 ˆ ˆ 2 ˆ ˆ 2 2 2 ( p, p :, ( p, p :, * p p * p p (b Define = (,, :,, ˆ = (,, : ˆ,, ˆ p p p p * * By hypohei he equivalence hold for M = Thu ˆ Nex noe ha * ( p, p :,, = ( p, p :, ( p, p :,, ( p, p :, * ˆ ˆ ˆ ˆ = ince I hold for M=2, i follow ha p p p p ˆ ˆ * * (, :, (, :, Exercie 7-3: (a Conider any pair of loerie (, c (, cˆ ˆ If he fir i (wealy preferred hen vc ( ˆ vc ( ˆ Hence + B v( c + B ˆ v( cˆ probabiliie um o, we can rewrie hi a follow ince he nwer o Chaper 7 page
John Riley 6 December 200 ( + Bv( c ˆ ( + Bv( cˆ (b ny increaing monoonic ranformaion of a uiliy funcion i an equivalen repreenaion of preference Conider he logarihmic ranformaion of U, ln U = ( c ln c ince hi i an expeced uiliy funcion, he conumer preference are conien wih he expeced uiliy axiom ecion 72 Exercie 72-: Ri averion rading in ae claim mare Define FOC x = arg Max{ v( x + v( x p x I} v ( c p = v ( c p x 2 2 2 2 2 p uppoe ha > o ha v ( c > v ( c2 hence ha c2 > c p 2 2 Conider he logarihm of he marginal rae of ubiuion Thi can be expreed a follow c2 c2 d d v ( c MR c c2 MR c c = MR c c dc = dc dc dc v ( c c c ln (, ln (, ln (, ln c2 c2 v ( c = dc = ( c dc v ( c c c Thu he higher he degree of abolue ri averion, he greaer he change in he marginal rae of ubiuion lo on he cerainy line he marginal rae of ubiuion i equal o he raio of probabiliie i herefore independen of he degree of ri averion We have herefore hown ha a all poin above he cerainy line he marginal rae of ubiuion i higher for an individual wih higher abolue averion o ri I nwer o Chaper 7 page 2
John Riley 6 December 200 follow ha he more ri avere individual will chooe a final conumpion budndle ( c, c cloer o he cerainy line 2 Exercie 72-3: Riy Choice wih 2 commodiie (a Wihin each ae he conumer ha a Cobb-Dougla uiliy funcion From Chaper 2, he indirec uiliy funcion i α+ β I α β V( p, I = α β + p x p y α β Thi i concave in income if only if he econd derivaive wih repec o income i non-poiive, ha i, α + β (b I i eaily checed ha he econd derivaive wih repec o px i poiive Thu, holding income conan, i follow from Jenen Inequaliy, ha uch individual ricly prefer an uncerain price o i expecaion In a mare economy price uncerainy i generaed by ome underlying (echnological hoc Thi hoc affec boh he income in each ae a well a he price Thu i mae lile ene o imply udying he effec of price uncerainy Exercie 72-5 Wealh Effec on ae hare (a If an individual inve a fracion z of hi wealh in he riy ae, he conumpion in each ae would be c = ( W Wz( + r + Wz( + r 2 Then he expeced uiliy can be wrien a: U( z = E{ V( W( + r + Wzθ }, where θ = r2 r The marginal uiliy of increaing hi hare of ae 2 i θ θ U ( z = Wv ( W ( + r + Wz We hen a wha i he effec of an increae in wealh on he marginal uiliy of inveing in he riy ae nwer o Chaper 7 page 3
John Riley 6 December 200 d U ( z = θ v ( c + θ W ( + r + z θ v ( c dw (D7- where c = W( + r + θwz Tha i, d U ( z = θ v ( c + θ c v ( c dw Le = θv ( c θr( c v ( c (D7-2 * z be he opimal fracional holding of he riy ae If d U * ( z 0 dw >, increaing wealh raie he marginal uiliy of inveing in he riy ae Then, a he higher wealh level, he individual inve more in he riy ae If relaive ri averion i d conan we can argue exacly a in he previou exercie ha U ( z = 0 I follow dw ha a change in wealh ha no effec on he opimal invemen (b The ric o analyzing i o mae ue of our aumpion ha relaive ri averion i increaing (IRR ince R( c i increaing Hence θr( c v ( c θr( W( + r v ( c (D7-3 ( θ R( c v ( c ( θ R( W( + r v ( c θr( c v ( c θr( W( + r v ( c dding inequaliy (D7-3, θr( c v ( c θr( W( + r v ( c Hence, from (D7-2, d U ( z 0 dw nwer o Chaper 7 page 4
John Riley 6 December 200 ecion 73 Exercie 73-: Porfolio Choice (a Final conumpion i c= ( w x + x( + θ = w+ x θ o expeced uiliy i U( x = E{ u xθ } (b Differeniaing by x, U x = E θu w+ x θ = * ( { ( } 0 (c u = θv xθ hence 2 u v xθ = v xθ + θxv xθ = v xθ( + θx x θ v xθ v ( c v ( c = v (( c + c w = v (( c R( c + ( c v ( c v ( c u (d We can appeal o Propoiion 73-6 if we can how ha ( x, θ i a concave funcion of θ ppealing o (c u = v (( c R( c + ( c + v (( c xr + x θ θ The righ h ide i negaive if R, R ( c 0 ( c > 0 Exercie 73-3 Mean uiliy preerving increae in ri (a θ = hu ( i he invere funcion, ha i u = v xh( u Differeniaing by u, = v xh( u xh ( u (b Gu ( = Pr{ u u} = Pr{ vw ( + x θ vw ( + xθ} = Pr{ θ θ} = F( θ (c vw ( + xhu ( = huv ( xhu ( Therefore vw ( + xhu ( = h ( uv xhu ( + huh ( ( uv xhu ( u nwer o Chaper 7 page 5
John Riley 6 December 200 v xh = ( + xh x v xh v xh v xh = ( + xh w x v xh v xh = ( + ( w R( w + xh w( w + xh x (d I follow ha vw ( + xhu ( i a concave funcion of if relaive ri averion i increaing abolue ri averion i decreaing (e U( x = E{ v xh( u} i a ricly concave funcion of x By Propoiion 73-6, if repreen a mean uiliy preerving in ri over B hen U ( x U * * > ( x o x < x B B ecion 74 Exercie 74-: Increaing lielihood raio ochaic dominance ( x ( x ( x (a hence ( x < ( ( x,,, ( x ( x ( x umming hee inequaliie, Hence ( x ( ( ( x x ( x (b Define ( x ( x ( x ( x Π ( x = ( x Π( x ( x ( x ( x hence Π ( x ( x Π ( x Π ( x Π ( x = ( x From (a, ppealing o Propoiion 73-5, he oupu diribuion under x exhibi condiional ochaic dominance over ha for acion x nwer o Chaper 7 page 6
John Riley 6 December 200 Exercie 74-3: Principal virual co funcion (a The agen receive hi reervaion uiliy Therefore U = v( w C( x Differeniaing by x, he exra wage needed o induce a higher x aifie dw v ( w( x = C ( x dx Therefore dw C ( x = dx v ( w( x The exra co of inducing he higher acion i herefore x2 x2 dw C ( x wx ( 2 wx ( = dx= dx dx v ( w( x x x (b Noe ha vwx ( ( = U + Cx ( Therefore he higher he reervaion uiliy, he higher i he wage wxu (, needed o induce acion x Then he marginal uiliy, v ( w( x, U, i lower ppealing o (a, wx ( 2 wx ( i higher nwer o Chaper 7 page 7