ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 2

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1 ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER Seion Eerise -: Coninuiy of he uiliy funion Le λ ( ) be he monooni uiliy funion defined in he proof of eisene of uiliy funion If his funion is oninuous y hen for ny ε > 0 here eiss some δ > 0 suh h y δ λ( ) λ( y) < ε Thus if he funion is no oninuous hen for some ˆε nd ny deresing sequene { δ } = wih limi zero, here eiss some bundle y in δ neighborhood of y suh h λ( y ) λ( y) > ˆ ε Le { } y + = be he subsequene of { y } = suh h λ ( y + ) > λ ( y ) + ˆ ε nd le { y } =, be he remining erms (for whih λ( y ) λ( y) < ˆ ε ) Suppose h he firs sequene is infinie Th is here is n infinie sequene + λ λ( y) + ˆ ε y,whih, in he limi pprohes y Sine preferenes re oninuous i follows h ( y) y λ + ε Sine preferenes re monooni, y ( y) λ Bu his is impossible sine ( y) λ y = Finlly, if he sequene { y + } = is finie we n mke essenilly he sme rgumen o obin onrdiion for he infinie subsequene { y } = for whih λ( y ) < λ( y) ε Eerise -4: Suffiien Condiions for Conve Preferenes u ( ) is onve so for ny λ λ λ λ,, X, u ( ) ( ) u ( ) + u ( ) Summing m over i follows hu ( ) = fu ( ( )) = u( ) is onve From he nswer o he = previous eerise i follows h preferenes re onve Eerise -6: Srily onve preferenes nd sri qusi-onve uiliy Seion 9 pge

2 () Suppose h 0 y nd 0 y Then U( ) U( y) nd U( ) U( y) If U is λ qusi-onve, i follows h U( ) > Min{ U( ), U( )} U( y) Therefore λ y nd so preferenes re srily onve (b) Suppose h preferenes re onve on X nd n be represened by uiliy funion U For ny Then 0 y nd, X, hoose y X so h U( y) = Min{ U( ), U( )} y Sine preferenes re srily onve, for ny onve ombinion λ, λ λ y Therefore U( ) > U( y) = Min{ U( ), U( )} Thus if preferenes re srily onve, he uiliy funion is srily qusi-onve Consumer Choie wih ommodiies Eerise -: Elsiiy Rules Firs we show h he hree definiions re equivlen θ ln = Hene θ ln = = E [, θ ] θ Also lnθ = Hene θ ln θ θ = = = E [, θ ] lnθ θ Appeling o hese definiions yield he following resuls ln (ln + ln ) ln (i) Eb [, θ ] = = = = E [, θ ] ln bθ (ln b+ ln θ) lnθ (ii) E[ y, θ ] = θ ln y = θ (ln + ln y) = θ ln + θ ln y = E[, θ] + E[ y, θ] ln ln (iv) E[, ] = = = E[, θ ] θ lnθ lnθ Seion 9 pge

3 Eerise -4: CES Preferenes ( < ) () U = + Define u = U Then round n indifferene urve indifferene urve hrough (, ), u = + Hene on he + = () If = his beomes + = Rerrnging, = Noe h s, () We n lso wrie he indifferene urve in he form = g( ) = Arguing symmerilly, s, Thus, s depied, he inerseion poin of he wo sympoes is (, ) B A Seion 9 pge 3

4 (b) If = he indifferene urve hrough (,) is + = Tking he limi n n n n s, n Wih n = 4 > 3 () In he limi Thus he poin B onverges o A Sine he indifferene n urve lies in he shded region i mus beome more nd more nerly L-shped s n inreses 3 Consumer Choie wih n ommodiies Eerise 3: Qusi-liner onve preferenes nd muli-vlued demnd Sge : Bev solves, M{ p + p y} This is speil se of he problem emined in he previous eerise Thus y mimized uiliy is u( p, y) = ( ) ( ) y = p p ( pp) Sge : Bev solves M{( u p, y) + y + p I} y, Noe h he indifferene urves re liner U 3 MRS( 3, y) = = ( pp) U y nd he prie rio is p 3 Thus Bev onsumes only ommodiy 3 if ( p p ) > p 3 nd only he firs ommodiies if he inequliy is reversed In he borderline se Bev is indifferen beween onsuming only ommodiy 3 nd only he firs ommodiies or ny onve ombinion of hese wo bundles Seion 9 pge 4

5 Eerise 3-4: Slusky equion wih endowmens () The onsumer s opimizion problem is U = M{ U( ) p p I} The dul problem sks how muh sh he needs o be ompensed s he prie veor hnges M ( pu, ) = Min{ p p U( ) U } Le ( pu, ) be ompensed demnd nd le ( pi, ) be unompensed demnd Then I follows h ( p, U) = ( p, M( p, U )) i i i M = + p p I p Wriing down he Lgrngin he opimum,, M ( pu, ) = L= p ( ) + λ( U( ) U ) By he Envelope Theorem, M L = = ( ) p p i i i Hene = + ( ) p p I We herefore hve he Slusky equion, ( pi, ) ( pu, ) ( ) ( pi, ) p p I i = i i () We inrodue he supersrip h o denoe individul h h h h i i (, ) = i (, ) ( ) p I p U p p I Sine every individul hs he sme mrginl propensiy o onsume ou of inome, we n sum over individuls s follows Seion 9 pge 5

6 p H H H h h i pi = i pu α h= p h= h= (, ) (, ) ( ) In equilibrium supply = demnd so he finl summion mus be zero Then he mrke level he unompensed effe is he sme s he ompensed effe () Le he number of hours vilble be 0 nd le 0 be hours worked he wge w= p 0 The budge onsrin is hen Rerrnging, p p + p ( ) p00 + p p00 + p Appeling o he Slusky equion, = + ( 0 0) p p I 0 0 The own ompensed effe on he demnd for leisure is negive However he inome effe is posiive Thus demnd for leisure my rise or fll Hene he supply of lbor,, my fll or rise 0 0 Seion 5 Choie Over Time Eerise 5-: All-or-nohing onsumer () For Cobb-Dougls uiliy funion U = he indire uiliy funion is α α α α V( p, I) = ki, where k = ( ) ( ) p p α α Thus he indire uiliy funion is U () = k + k (b) Given welh of W, he onsumer hs budge onsrin + = W Thus here + r is orner soluion for ll ineres res Moreover, if k/ k = + r, he onsumer is indifferen beween ll poins on he budge line () Wih sri qusi-onviy here n never be wo opiml poins on he budge line For if here were, ll onve ombinions would be srily preferred Sine he budge Seion 9 pge 6

7 se is onve, hese onve ombinions re lso fesible Bu hen neiher of he originl wo poins ould hve been opiml fer ll Eerise 5-4: Fuures nd fuure spo pries () For he sndrd Cobb-Dougls uiliy mimizion problem n M{ U ( ) = α ln p p ω} = The onsumer s ependiure shre on ommodiy is Le p = n α α i= i pi be he prie of ommodiy i in period Then p ω ( p, p, p, p = (,, δ, δ ) 3( + δ ) (b) Tol spending in period is p ω p ω p = p + p = (+ ) = 3( + δ ) ( + δ ) () Sving is he differene beween he vlue of he firs period endowmen nd period spending Th is S = p ω p = p ω p ω + p ω + δ ( ) = δ ( ) p ω ( ) p ω + δ + δ Therefore he onsumer sves if nd only if p ω < δ p ω Seion 9 pge 7