Econ 40 Version John Riley Homeork Due uesdy, Nov 8 nsers nser to question () Double both sides of the second eqution nd subtrct the second eqution 60q 0q 0 60q 0q 0 b b 00q 0 hen q 0 (b) he vlue of the fund is Substituting, q b 0 P q b P qb 00 00 0 0 0 (c) 60 0 0 q qb 80 60 0 Solve to obtin 4 qˆ (, ) 0 0 (d) he vlue of the fund is P q b 4 P qb 00 00 0 0 0 (e) 60 0 0 q qb 80 60 0 Solve to obtin qˆ (, ) 0 0 (f) he vlue of this fund is 0 Oning the first fund costs 0 nd yields 0 in stte hus clim to unit in stte hs mrket vlue of Oning the second fund hs vlue of 0 nd yields 0 in stte hus clim to stte costs From (e) nd (f) the vlue of the mutul fund is 0 hus clim to unit in stte hs mrket vlue of
Econ 40 Version John Riley We could get this nother y sset ith return (60,80) is orth 00 60 80 00 he vlue of its stte clim is 60 0 the mrket vlue of its stte clims is 80 nd so 80 80 etting on he Gme () ˆ ˆ q, ( ) q ˆ q, ˆ q dding these equtions, ( ˆ ) (b) ommy s utility function is U ln ln ommy therefore solves the folloing roblem M{ U ( ) ( ) ˆ 0} FOC U U lying the Rtio Rule ( ) ˆ ˆ ˆ (c) For ev the FOC is U U b b eling to the Rtio Rule
Econ 40 Version John Riley b b ( b ) ( b ) b ˆ ˆ ( b ) b (d) If b 0 ˆ totl demnd for stte elth is ˆ ˆ otl suly is the sme of the elth of the to reresenttive bettors, ie For equilibrium suly = demnd so ˆ ˆ ˆ Rerrnging this eression 06 06 04 (e) ˆ ( b ) ˆ b, ˆ ˆ ( b ) b ˆ ˆ
Econ 40 Version John Riley ˆ ˆ ( b ) ˆ b ˆ ˆ ( b ) ˆ b he derivtive ith resect to b is negtive hus must increses the mrket odds decrese declines Since e hve normlized, (f) If you comute ev s reltive version to risk ( RR ( )) you ill find tht it is decreses s b Increses s ev becomes less risk verse she is illing to risk more nd therefore bet more hus the suly of bets on the ruins increses his loers the reltive rice *the mrket odds of rojn victory o sset economy () h h v( ) MRS ( ) h h h h h h v( ) For n lloction to be PE MRS MRS 4 hus the consumtion rtios re the sme s the ggregte endoment rtio (b) It follos from () tht MRS MRS for ny PE lloction y the first elfre theorem WE is PE In WE, it follos from the necessry conditions for utility mimiztion tht MRS MRS hen the WE rice rtio is hen choose the rice vector (,) (d) he vlue of the endoments re s follos:
Econ 40 Version John Riley P z (, ) (00,00) 400 P z (,) (00,0) 00 (d) le s endoment hs vlue equl to of the totl endoment thus in equilibrium he consumes of the totl endoment ie 800 00 (, ) ev s consumtion is equl to of the totl endoment thus in equilibrium she consumes of the totl endoment ie 400 00 (, ) P he sset rice rtio is P / P hus le sells of ev s lnttion ev is on the other side of this trde of his lnttion (nd so retins ) to urchse he resulting lloction is PE lloction he MRS re equl so neither consumer ill ish to mke ny dditionl trde (f) rguing s bove, in PE lloction ech consumer hs shre of the totl endoment if stte clims mrkets re oen hus ev nd Chrles tken together do ectly ht ev did bove So there is no chnge in ny of the conclusions 4 WE nd PE in three commodity model () h / h U ( ) U ( ) so utility is homothetic hen e cn consider the rer4esenttive gent His R endoment I (00,400,900) (b) His utility mimizing demnd must equte the mrginl utility er dollr U U U / / / o cler ll mrkets, 6 6 6 / / / 0 0 0 (c) Suose tht { ˆ, ˆ } is PE lloction Holding constnt the lloction of e cn consider the other to commodities If the MRS re not equl there re oortunities for mutul gin hen
Econ 40 Version John Riley MRS (, ) MRS (, ) is necessry condition for n lloction to be PE ( ) MRS (, ) ( ) nd / / / ( ) MRS(, ) ( ) / eling to the rtio rule Sme rgument for ny ir of commodities It follos tht ech consumes frction of the ggregte endoment he vlue of le s endoment is 50 nd the vlue of ev s is 050 hus le hs one qurter of the totl elth hen his consumtion is one qurter of the ggregte endoment