Econ 401A Three extra questions John Riley. Homework 3 Due Tuesday, Nov 28

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1 Econ 40 ree etr uestions Jon Riley Homework Due uesdy, Nov 8 Finncil engineering in coconut economy ere re two risky ssets Plnttion s gross stte contingent return of z (60,80) e mrket vlue of tis lnttion (finncil sset) is P 00 Plnttion s stte contingent return of z (0, 60) nd mrket vlue of P 00 e S&P terefore s vlue of 500 n emle of simle engineered sset is te S&P mutul fund It olds n eul sre of te two ssets If its mrket vlue is 50, ten its olding of te two ssets must (, ) (, ) S& P Finncil engineers wis to design riskless mutul fund tt s te sme return (0,0) in te two sttes ey terefore urcse sres (, ) suc tt en nd () Solve tese two eutions for te sres nd (One is times te oter) () Wt is te mrket vlue of tis riskless mutul fund? Net consider te design of mutul fund tt ys off 0 in stte nd zero in stte Wit suc n sset ville it is esy for n investor increse is wet in stte Remrk: One of te untities is negtive So te fund is selling tt sset sort (c) Solve for te sres ( ˆ ˆ, ) in tis new mutul fund (d) Wt is its mrket vlue? Net consider te design of mutul fund tt ys off zero in stte nd 0 in stte Wit suc n sset ville it is esy for n investor increse is wet in stte (e) Solve for te sres ( ˆ ˆ, ) in tis new mutul fund (f) Wt is its mrket vlue? Use tese results to solve for s, s,, were s is te mrket vlue of finncil instrument yong only unit only in stte s Remrk: e oint of tis ecercise is to sow tt it tere re t lest n mny finncil ssets s sttes, it my e ossile to relicte ll stte contingent trding (insurnce mrkets) y trding in sset mrkets

2 Econ 40 ree etr uestions Jon Riley etting on e Gme (08) In stte te rojns win In stte te ruins win ev ( ruin) tinks tt te roility of rojn victory is 04 nd ommy ( rojn) tinks tt te roility of of rojn victory is e ommy s utility function is v ( ) ln( ) ev s utility fucntion is v ( ) ln( ) 08 ot ev nd ommy ve welt of ŵ million ink of tem s reresenttive ruins nd rojns If ommy mkes et tt ys off in te event of rojn victory e ys te mrket odds of nd so must y te ookie ( ) if te ojns lose en is stte contingent welt is ˆ ˆ (, ) ( w, w ( ) ) () Sow tt tis is euvlent to te following stte contingent welt constrint wˆ wˆ We normlize so tt te sum of te rices is () Hence sow tt if ommy mkes is otiml et, ten is welt if te outcome is stte (rojns win) is wˆ (c) Sow lso tt ev s welt if te outcome is stte, ( wˆ ) stisfies (d) Suose tt 0 Sow tt te euilirium rice of stte clims is Hence sow tt te mrket odds re Hencefort consider te cse 0 (e) Solve for te new euirium rice of stte clim nd ence te new mrket odds How do te mket odds vry wit?

3 Econ 40 ree etr uestions Jon Riley (f) Wt is te intuition for te effect of iger rmeter? HIN Wt is te degree of reltive risk version of ec ettor? wo sset economy (te coconut lnttion economy) sset s return z (00,00) sset contingent welt is (400,00) roility of stte is s return z (00,0) us te totl stte Ec consumer s te sme utility function u( ) ln e Initilly suose tt le owns sset nd ev owns sset () Solve for te PE lloctions in tis economy () If tere re cometitive mrkets for stte clims (insurnce), elin wy te euilirium stte clims rice rtio is eul te endowment rtio nd ence elin wy te euilirium consumtion rtios must : (c) Comre te vlue of te two endowments (Coose eiter (or some oter metod) to solve for te euilirium lloction (,) or (, ) (d) Suose tt tere re no stte clims rices ut te mrket vlue of te ssets P ) Use tis eul to te vlue in te stte clims mrket euilirium Elin ow it is ossile for te two consumers to trde sres nd rec te sme outcome s in (c) Wt frction of ec sset would e trded? (e) Suose tt tis trde is comleted en stte contingent mrkets uneectedly oen Would tere e ny furter trde in tese mrkets? (f) Would ny of te ove results cnge if tere were tree investors le owns sset ev nd Crles ve 50% ownersi of sset nd P re 4 Wlrsin Euilirium nd Preto efficiency in tree commodity model Suose tt le nd ev ve te sme utility function U( ) ( ) ( ) ( ) / / / 6 le s endowment (50,80,0) nd ev s endowment (50, 0,780) so tt te ggregte endowment is (00, 400,900) () Sow tt te utility function is omotetic nd ence el to te reresnttive gent roc to sow tt (,,) is WE rice vector () Sow tt for n lloction to PE lloction 4 nd 9

4 Econ 40 ree etr uestions Jon Riley (c) Comre te mrket vlues of ec consumer s endowment Hence or oterwise solve for te euilirium consumtion vectors nd Oter eercises (Do not nd in nswers) 5 Stte clims euiliri nd sset mrket euiliri (do not nd in) () Reinterret te model of uestion 4 s one in wic tere re tree sttes e vector of roilities is (,, ) (,, ) Ec endowment is te stte contingent numer of 6 coconut trees surving tree different tyes of storm () Use te euilrium stte clims rice vector to solve for te vlue of te two coconut lnttions (c) Wt re te euilirium outcomes nd? (d) Suose tere re no stte clims mrkets ut tere is stock mrket were te lnttions cn e trded Eln wy it is ossile to cive te sme outcome s in (c) y ecnging sres (e) Given tese rices, is tere ny oter trde wic would mke one of te consumers etter off? 6 lmost omotetic economy Ec of H consumers s te sme utility function U(, ) v( ) v( ) were (, ) is n individul secific rmeter e endowment of individul is Define ˆ nd ˆ () Elin wy te following is n euivlent economy Ec of H consumers s te sme utility function u( ˆ ˆ ˆ ˆ, ) v( ) v( ) e endowment of individul is ˆ () Give two emles of functions v() (ecluding v( ) ln ) for wic te trnsformed economy is omotetic Hencefort ssume tt v( ) ln (c) Solve for WE rice rtio in tis second economy s function of ˆ ( ˆ ˆ, ) (d) s long s te rmeters stisfy rices in te originl economy re sufficiently smll, elin wy tese re lso euilirium 4

5 Econ 40 ree etr uestions Jon Riley 7 wo eriod economy wit roduction le nd ev ve te sme utility function U(, ) v( ) v( ) e endowments re (0, 0) nd (4,0) e first eriod commodity cn e trnsformed into eriod outut ccording to z / () Use te nswer to uestion 6 to trnsform te rolem into n euivlent economy wit omotetic references Hencefort ssume tt ( ˆ v ) ln ˆ () Sow tt in te trnsformed economy te otiml first eriod inut is te WE rices in tis economy? z * ˆ Wt re (c) Elin wy tese re lso WE rices in te originl economy 5