1 Consumption and Risky Assets

Transcription

1 Soluions o Problem Se 8 Econ 0A - nd Half - Fall 011 Prof David Romer, GSI: Vicoria Vanasco 1 Consumpion and Risky Asses Consumer's lifeime uiliy: U = u(c 1 )+E[u(c )] Income: Y 1 = Ȳ cerain and Y F ( Ȳ,σ y) is random variable Iniial wealh A0 is zero a Wih no nancial asses and a soring echnology available, he problem is given by: U = u(c 1 )+E[u(c )] c 1 +s = Ȳ c = Y +s FOC wr s (by choosing how much o save he consumer pins down (c 1,c ) : u (c 1 ) = E[u (c )] Now, we add a risky asse o his economy, wih reurn r ( r,σ r), wih E(rY ) = 0 b The consumer will buy a sricly posiive amoun i r > 0, ie, if he expeced reurn on his invesmen exceeds he risk free rae (in our case zero), i will be opimal for he consumer o inves a leas an ε > 0 amoun of is income in he risky asse This is rue because for a small ε, consumpion is period is "almos" uncorrelaed wih he reurn on his asse (since C = Y +ε(1+r)), so he covariance is zero and i has posiive expeced reurn he consumer should inves some posiive amoun Also, noice ha for very small amouns of risk, he uiliy funcion can be aproximaed by a linear funcion (hink of a aylor expansion around Y ), ie agens are 'almos' risk neural when i comes o small amouns of risk This are boh reasons why he agen always akes some risk (as long as he risk is small, he linear aproximaion works) c This saemen is no rue because he consumer is risk averse (u < 0), and hus won' be willing o leverage up o inniy o go long on he risky asse There is an opimal level (inerior soluion) ha will deermine he opimal amoun o be invesed in he risky asse The dierence wih ou answer o par b is ha for no small amouns of invesmen in he risky asse, consumpion and he reurn on he asse become highly correlaed, and hus he asse becomes less and less aracive since remember he agen ulimaely cares abou correlaion beween reurns and marginal uiliy of consumpion! 1

2 d Consumer now has o choose how much o inves in he risk free and how much o inves in risky asse Le α be he fracion of income invesed in risky asse and β he fracion invesed in risk free The consumer needs o choose α and β o max expeced uiliy: c 1 c max u(c 1)+E[u(c )] {α,β} = Ȳ (1 α β) = Y +[(1+r)α+β]Ȳ FOC (α) u (c 1 )Ȳ +E[u (c )(1+r)]Ȳ = 0 (β) u (c 1 )Ȳ +E[u (c )]Ȳ = 0 Can be re-wrien as follows: u (c 1 ) = E[u (c )(1+r)] (1) u (c 1 ) = E[u (c )] () Where (1) is he Euler equaion for he risky asse and () he Euler equaion for he risk-free asse (equal o he ne derived in par a) The Risk Free Rae Puzzle We solved his in secion a long ime ago! The quickes way is o use he resul: where CRRA = C()U (C()) U (C()) and consumpion growh, we obain Really, really high! Ċ() C() = r() ρ C()U (C()) U (C()) is he coecien of relaive risk aversion By pugging in he calibraion for CRRA,ρ, 15 = r 5 4 = r = 11

3 3 Romer, Problem 810 (in he end) 4 Asse Pricing I Our saring poin is: which implies: [ k=1 P 1 = E 1 [ k= ( ) ] k 1 U (C k ) 1+ρ U (C ) Z k ( ) ] k 1 1 U (C k ) 1+ρ U (C 1 ) Z k Going back o our iniial equaion: [ k=1 [ ( ) 1 U (C 1 ) 1+ρ U (C ) Z 1 + ( ) ] k 1 U (C k ) 1+ρ U (C ) Z k k= [ ( ) [ ( 1 U ) (C 1 ) 1 U 1+ρ U (C ) Z (C 1 ) 1 +E 1 1+ρ U (C ) ( ) ] k 1 U (C k ) 1+ρ U (C ) Z k k= ( ) ]] k 1 1 U (C k ) 1+ρ U (C 1 ) Z k [( ) 1 U ( ) (C 1 ) 1 U ] [( ) 1+ρ U (C ) Z (C 1 ) 1 U ρ U (C ) P (C 1 ) 1 = E 1+ρ U (C ) EXTRA PROBLEMS 5 Romer, Problem 914 (in he end) 6 Asse Pricing II If he expeced reurn on Asse A exceeds he expeced reurn on Asse B: A Financial markes canno be in equilibrium B Financial markes can be in equilibrium, bu only if here are noise raders ] (Z 1 +P 1 ) C A raional invesor would choose o hold Asse B only if he covariance of Asse B's reurn wih he marginal uiliy of consumpion is greaer han he covariance of Asse A's reurn wih he marginal uiliy of consumpion D A raional invesor would choose o hold Asse B only if he covariance of Asse B's reurn wih he marginal uiliy of consumpion is less han he covariance of Asse A's reurn wih he marginal uiliy of consumpion 3

4 To see his, check ou he Euler equaion for risky asses You should have wo Euler equaions, one for asse A and one for asse B Compare hem o ge he answer 7 Asymeric Informaion and Invesmen Consider he model of invesmen under asymmeric informaion in Secion 89 of Romer Suppose ha iniially he enrepreneur is underaking he projec, and ha (1+r)(1 W) is sricly less han R MAX Describe how each of he following aecs D: a A small increase in W: An increase in W implies an increase in he downpaymen done by he enrepreneur, and a decrease in he loan made by he lender This decreases he probabiliy of being in a sae ha has o be moniored, and hus should decrease D b A small increase in r: An increase in r implies an increase in he ouside opion lenders have, and hus D would have o be higher o saisfy he paricipaion consrain of he lenders (o coninue o leave he lender indieren beween invesing in he risky projec of saving a he risk free rae) c A small increase in c: An increase in c has wo eecs Firs, i decreases he incenives o monior, since i is now more expensive o do so, and so his pus preasure on D o decrease Second, by increasing he monioring coss, i decreases he expeced payo of he lenders, and hus D would have o increase o saisfy he paricipaion consrain I don' hink we can ell wha happens o D (or a leas i is no obvious o me wihou doind he mah) d Insead of being disribued uniformly on [0, γ], he oupu of he projec is disribued uniformly on [γ b,γ+b], and here is a small increase in b In his case, he increase in b is an increase in he dispersion of he reurns Since for he lender his only means an increase in he dispersion of he bad paymens and no he good ones, D has o increase o compensae him e Insead of being disribued uniformly on [0, γ], he oupu of he projec is disribued uniformly on [b,γ+b], and here is a small increase in b An increase in b here should be inerpreed as an upward shif in he suppor of reurns, which implies an upward shif in he suppor of paymens o he bank for he non-monioring zone To saisfy he paricipaion consrain wih equaliy (do no leave any surplus o he lender), D would have o decrease 4

5 Exercises from he Book Romer (4 h Ed) Problem 810 (a) Suppose he individual reduces her consumpion by a small (formally infiniesimal) amoun dc in period The uiliy cos of doing his equals he marginal uiliy of consumpion in period, 1/C, imes dc Thus we have (1) uiliy cos = dc/c This reducion in consumpion allows he individual o purchase dc/p rees in period In period + 1, he individual receives he exra oupu from her addiional holdings of rees She ges o consume an exra [dc/p ]Y 1 The individual hen sells her addiional holdings of rees for [dc/p ]P 1 and consumes he proceeds Thus her oal exra consumpion in period + 1 is given by [dc/p ]Y 1 + [dc/p ]P 1 The marginal uiliy of consumpion in period + 1 is 1/C 1 Thus he expeced discouned uiliy benefi from his acion is 1 1 dc () expeced uiliy benefi = E C P Y dc P P ρ 1 If he individual is opimizing, a marginal change of his ype mus leave expeced uiliy unchanged This means ha he uiliy cos mus equal he expeced uiliy benefi, or dc 1 1 dc (3) E ( ) C C P Y P = ρ 1 Canceling he dc's (which is somewha informal) gives us (4) = E ( 1 1) C 1 C P Y P ρ 1 We can now solve equaion (4) for P in erms of Y and expecaions involving Y 1, P 1 and C 1 Noe ha we can replace C wih Y and ha P is no uncerain a ime Using hese facs, equaion (4) can be rewrien as (5) ( 1 1) Y P E = Y + P 1+ ρ C 1 Solving equaion (5) for he price of a ree in period gives us Y Y + P (6) P = E 1 + ρ C 1 (b) Since C s = Y s for all s 0, equaion (6) can be wrien as Y P Y Y P 1 1 (7) P = E 1+ = + E 1 + ρ Y ρ 1 + ρ Y 1 Equaion (7) holds for all periods and so we can wrie he price of a ree in period + 1 as Y Y P 1 1 (8) P 1= + E ρ 1 + ρ Y Subsiuing equaion (8) ino equaion (7) yields Y Y 1 1 P (9) P = + E + E ρ 1 + ρ 1 + ρ 1 + ρ Y Now use he law of ieraed projecions ha saes ha for any variable x, E E 1 x = E x, o obain

6 Y Y Y (10) P E P = ρ ( 1+ ( 1+ Y Afer repeaed subsiuions, we will have Y Y Y Y (11) P E P s = + + Κ + + s s 1 + ρ ( 1+ ( 1+ ( 1+ Y s Imposing he no-bubbles condiion ha lim s E [(P s /Y s )/(1 + s ] = 0, he price of a ree in period can be wrien as 1 (1) P = Y Κ 1 ρ ( 1+ Since 1/(1 + < 1, he sum converges and we can wrie 1 ( 1+ 1 ( 1+ (13) P = Y = Y 1 [ 1 ( 1+ ] ρ ( 1+ ρ Thus, finally, he price of a ree in period is (14) P = Y /ρ (c) There are wo effecs of an increase in he expeced value of dividends a some fuure dae Firs, a a given marginal uiliy of consumpion, he higher expeced dividends increase he araciveness of owning rees This ends o raise he curren price of a ree Since consumpion equals dividends in his model, however, higher expeced dividends in ha fuure period mean higher consumpion and lower marginal uiliy of consumpion in ha fuure period This ends o reduce he araciveness of owning rees he ree is going o pay off more in a ime when marginal uiliy is expeced o be low and hus ends o lower he curren price of a ree In he case of logarihmic uiliy, hese wo forces exacly offse each oher, leaving he curren price of a ree unchanged in he face of a rise in expeced fuure dividends (d) The pah of consumpion is equivalen o he pah of oupu Thus if oupu follows a random walk, so does consumpion Bu if oupu does no follow a random walk, hen consumpion does no eiher Problem 914 (a) Consider he value of a uni of deb I pays off one uni of oupu a ime +, for all 0 The consumer values his payoff according o he marginal uiliy of consumpion a each ime + Thus he value of having one uni of oupu a ime + raher han a is equal o he discouned marginal uiliy of consumpion a ime + relaive o he marginal uiliy of consumpion a ime, which is given by e -ρ u '(C( + ))/u '(C()) Thus he value of a uni of deb a ime is simply he appropriaely discouned "sum" of all he fuure payoffs, or (1) P e E u C ( + ) ( ) = ρ d Equiy holders are he residual claiman and hus a ime +, 0, hey receive he addiional profi generaed by he marginal uni of capial, π(k( + )), minus he oal amoun paid o bond holders, which is b (he oal number of ousanding bonds) Again, individuals value his payoff a ime + according o he discouned marginal uiliy of consumpion a ime + relaive o he marginal uiliy of consumpion a ime Thus he value of he equiy in he marginal uni of capial is () V e E u C ( + ) ( ) = ρ ( π( K( ) ) b) d

7 (b) Adding equaion () o b imes equaion (1) gives us he following marke value of he claim on he marginal uni of capial: ( ) (3) P b V e be u C ( + ) d e E u C ( + ) ρ ρ ( ) + ( ) = + ( π( K( ) ) b) d Combining he inegrals yields (4) P b V e E u C ( + ) ( ) + ( ) = ρ ( b+ π( K( ) ) b) d, and hus (5) P b V e E u C ( + ) ( ) + ( ) = ρ π( K( ) ) d The division of financing beween bonds and equiy as capured by b, he number of ousanding bonds, does no affec he marke value of he claims on he marginal uni of capial The presen discouned value of ha uni of capial is deermined by is expeced effec on he pah of profis Since he division of π(k( + )) beween bonds and equiy does no affec he size of π(k( + )), i does no affec he marke value of he claim on he uni of capial (c) The marke value of each of he n asses is given by (6) V e E u C ( + ) i ( ) = ρ d i ( ) d There will be n equaions of he form of (6) Adding hese n equaions ogeher gives us he following oal value of he n financial insrumens: (7) V V e E u C ( + ) 1( ) + Κ + n ( ) = ρ ( d1( ) + Κ + d n ( ) ) d Since d 1 ( + ) + + d n ( + ) = π(k( + )), we can rewrie equaion (7) as (8) V V e E u C ( + ) 1( ) + Κ + n ( ) = ρ π( K( ) ) d The oal marke value of he n financial insrumens is deermined by he expeced effec on he pah of profis of he marginal uni of capial I does no depend upon he individual payoffs o he asses (d) The value of a uni of deb coninues o be given by equaion (1) The value of a uni of equiy is now (9) V e E u C ( + ) ( ) = ρ {( 1 θ) [ π( K( ) ) b] } d Adding equaion (9) o b imes equaion (1) gives he following marke value of he claims on he marginal uni of capial: ( ) (10) P b V e be u C ( + ) d e E u C ( + ) ρ ρ ( ) + ( ) = + [ ( ) ] ( ) ( ) {( 1 θ) π K( ) b } d C( ) C( ) Combining he inegrals yields (11) P b V e E u C ( + ) ( ) + ( ) = ρ [( 1 θ) π( K( ) ) + θb] d Now he division of he financing beween bonds and equiy does maer The number of bonds issued, b, does affec he marke value of he claim on he marginal uni of capial The division of he addiional profis beween bonds and capial does affec he size of hose profis Specifically, a swich oward deb financing increases profis since ineres paymens are ax deducible

8