Brad De Long è Maury Obstfeld, Petra Geraats è Galina Hale-Borisova. Econ 202B, Fall 1998

Transcription

1 uggested olutions to oblem et 4 Bad De Long Mauy Obstfeld, eta Geaats Galina Hale-Boisova Econ 22B, Fall Moal hazad and asset-pice bubbles. a The epesentative entepeneu boows B on date 1 and invests it in X isky assets and X safe assets such that B = X + X : 1 On date 2, the isky and safe assets payoæx and X,espectively. In addition, the entepeneu has to pay the bank B if X +X ç b and X +X othewise. o, the date 2 payoæ to the entepeneu equals æ = ubstituting 1, æ = 8 é : X + X, B if X + X ç B othewise 8 é : X, X if X, X ç othewise = max fx, X ; g : 2 The payoæ function æ is gaphed in ægue 1. b The expected payoæ E ëæë inceases as the vaiance of ises, given and X. The eason is that æ isconvex. Intuitively, a highe vaiance inceases the pobabilityofvalues of close to and MAX. In the latte case, the payoæ to the entepeneu inceases, but in the fome case the entepeneu simply defaults and the payoæ is not aæected. As a esult, the expected payoæ inceases. : 1

2 Π Π() MAX Figue 1: The entepeneu's payoæ function c Let UX denote the entepeneu's expected net beneæts of investing in X isky assets. Then, substituting 2 yields UX = E ëæ ë, cx = = Z dh+ X, X dh, cx X, X dh, cx : The æst-ode optimality condition with espect to X is U X = eaanging gives, dh, c X =: 3 c X = = ç, Z dh, Z dh, dh, ë1, H ë : dh d Besides the optimality condition 4, thee ae thee othe equilibium conditions. Fist, equilibium in asset makets equies that demand 2 4

3 and supply of the isky asset ae equal: X =1: 5 econd, the entepeneu's budget constaint 1 has to hold. ubstituting 5 gives X + = B: 6 Thid, the isk-fee goss etun is detemined by the goss maginal poduct of the safe poduction technology at the aggegate safe investment level X,so = f X : 7 We willnow show that a unique equilibium exists when ç éc 1. Combining 4 and 5, ç, c 1 = Z dh+ë1, H ë : The ight-hand side of this equation is a function of which we will denote by G. Notice that G = and that G é if and only if é. o, in equilibium, is stictly positive when ç éc 1. Futhemoe, applying Leibniz' ule, G =H +ë1, H ë, H =1, H so that ég é 1foé é MAX. Hence, G is a oneto-one function of and so 8 uniquely detemines the value of ; to be pecise, = G,1 ç ç, c 1 ç.thus, fo ç éc 1, 8 deænes an invese elationship between and which is depicted by the hypebola in ægue 2. Combining the emaining two equilibium conditions, 6 and 7, yields 8 = f B, : 9 =,f B, é, 9 deænes a positive elationship between and which is depicted by the schedule in ægue 2. The intesection of the and cuves detemines the equilibium values of and. Clealy, thee is a unique equilibium when éc ç 1. Concening the banks, they eceive B if X + X ç B, o equivalently, using 5 and 6, if ç. But if é, they only get 3

4 Figue 2: Equilibium inteest ates and asset pices X + X éb. In equilibium, é and the expected payoæ to the banks equals V = é Z Z X + X dh+ BdH+ BdH BdH =B: 1 As a consequence, banks ean an expected etun stictly below on thei loans B. e ecall that the downwad-sloping schedule in ægue 2 is deæned by 3 and 5, so dh, f ç g, c 1 = : eaanging and solving fo the asset pice gives = 1 " MAX dh, c 1 f ç g : 11 f If entepeneus ænanced asset puchases entiely out of thei own wealth B, thei payoæ would be æ æ =X + X : 4

5 Taking into account the oppotunity costs B, the expected net beneæts equal U æ X = E ëæ æ ë, B, cx = X dh, X, cx ; using the budget constaint 6. Then, the æst-ode optimality condition fo the epesentative entepeneu is U æ X = dh,, c X =: ubstituting 5 and eaanging gives the fundamental asset pice in equilibium æ = 1 " dh, c 1 = 1 h i ç, c 1 : 12 In wods, the fundamental pice of the isky asset is the pesent discounted value of the diæeence between the expected etun ç and the maginal cost of investment c 1. o, æ equals the expected pesent discounted value of the net maginal beneæts fom isky assets. g eaanging the equilibium condition fo the isky asset 8, = ç, c 1 + H, Z dh é ç, c 1 = æ : 13 h i The inequality follows fom the fact that é dh =H, i.e. the conditional expectation of fom to must be less then. The last equality in 13 uses 12 and the assumption that the inteest ate in equations 11 and 12 is the same. Theefoe, é æ. h In an economy whee entepeneus ænance investment out of thei own wealth B, the schedule is no longe applicable. Instead, equilibium in the maket fo isky assets is descibed by 12, depicted by the æ æ schedule in ægue 3. ince é æ fo a given level of the inteest ate, the æ æ locus lies to the lowe left of the cuve. The schedule descibing the maket fo safe assets is not aæected. o, in the case of self-ænanced investment, the equilibium is given by theintesection of the æ æ and schedules. Clealy, the equilibium isk-fee inteest ate æ and asset pice æ ae both lowe than thei values in the case of bank-ænancing. 5

6 * * * * Figue 3: elf-ænanced vesus bank-ænanced investment i A ise in the amount ofbankloansb does not aæect the schedule. Howeve, the locus given by 9 shifts down and to the ight, as a highe value fo B educes fo a given level of. The new equilibium at the intesection of the and '' schedules featues a lowe inteest ate and a highe asset pice as illustated in ægue 4. Intuitively, the inceased supply of funds B boosts the demand fo the isky asset, theeby inæating the asset pice, and aises the amount invested in the safe asset, which depesses the isk-fee inteest ate. j The fact that the loans B ae supplied by foeign instead of domestic banks is ielevant fo the deivation of the equilibium elationships 8 and 9. o, fo a given capital inæow B, thevalues of and ae detemined as in pat d. The level of B, howeve, is no longe exogenous; foeign banks lend up to the point whee the expected payoæ fom domestic loans, V, equals W B. ubstituting the equilibium conditions 5 and 6 into 1 we obtain V = Z dh, H + B = W B: 6

7 Figue 4: The eæect of an incease in bank loans eaanging gives = W + 1 B " H, Z dh : 14 ince é h dh i =H, the domestic inteest ate exceeds the wold inteest ate: é W. Fomally, we have thee equations, 14, 9 and 8, which can be solved fo the thee endogenous vaiables, B and. Notice that 14 detemines a negative elationship between and B fo a given level of. This is depicted in ægue 5 by thehypebola FF with hoizontal asymptote W. In addition, we can wite 9 as = f B,! ; 15 =,1= ë= 2, 1=f B, =ë é, given. This also deænes a negative elationship between and B fo a given level of, which is depicted by the schedule. The cuve is convex and bounded below bythehypebola = =B, which is indicated by the dashed line. The open-economy equilibium is 7

8 F F W W F F B B B Figue 5: The eæect of an decease in the wold inteest ate detemined by the intesection of the FF and schedules, given the level of detemined by 8. A fall in the wold inteest ate shifts the FF schedule downwad as shown in ægue 5. As a esult, capital inæows B incease, the domestic inteest ate falls, and since is not aæected, the pice of the isky asset ises. 2. Compaing optimal consumption with complete and incomplete makets. a Ignoing nonnegativity, wite the unconstained maximization as max B2 ë1 + B1, B2 + Y1ë, a 2 ë1 + B 1, B2 + Y1ë E 1 The æst-ode condition fo B2 is: ç ë1 + B2 + Y2së, a 2 ë1 + B 2 + Y2së 2ç : C1 =E 1 fc2sg : The + 1 budget constaints in the poblem imply that E 1 C1 + C 2s =E B1 + Y1 + Y 2s : 8

9 Thus by substitution of the æst-ode condition, ç 1+ 1 that is, ç C1 =E B1 + Y1 + Y 2s C1 = 2+ E B1 + Y1 + Y 2s : 16 The implied values of C2s ae given by C2s = 1 + B2 + Y2s: ubstituting fo B2 =B1 + Y1, C1; whee C1 is given by 16 above, we obtain C2s = ç 1 + B1 + Y1, 2+ ç 1 + B1 + Y1 + E 1Y2 ; çç +Y2s: Fo the 1-hoizon case, we just get the usual ëpemanent-income" fomula, essentially eq. 32 of Chapte 2 suitably adapted. b The consumption fomula of pat a will be geneally valid if the nonnegativity constaint on consumption neve binds, that is, if, even when output hits its minimal date 2 value in state s = 1, C2 ç : This last inequality will hold if and only if C21 ç, which is equivalent to 1 + B1 + Y Y 21 ç E 1 Y2: If this inequality does not hold, then the nonnegativity constaint on C2 binds in at least one state of natue on date 2, so we cannot ignoe the associated Kuhn-Tucke multiplie see supplement Ato Chapte 2. In that case, the Kuhn-Tucke theoem pedicts that date 1 consumption must be lowe than E 1 fc2sg to make C21= in state 1 of date 2 when output is minimal. As a esult, the bond Eule equation no longe holds. Futhemoe, since C21= ë1 + B1 + Y1, C1ë+Y21 = we see that C1 =B1 + Y1 + Y21=1 + : 9

10 c The state-by-state Eule equations ae ps 1, a C1 = çs ë1, a C2së ; which educe to C1 = C2s; 8s; because we've assumed ps = çs: Thus consumption is constant acoss states and dates, equal to C,given ç by ç ç " X çc = 1 + B1 + Y s=1 ç ç " X = 1 + B1 + Y s=1 psy2s çsy2s The citical diæeence between the equation above and eq. 16 is that the peceding equation holds ex post as well as ex ante, i.e., it holds in evey state on date 2 as well as on date 1. Equation 16 above, in contast, implies that date 2 consumption vaies one-fo-one with the output ealization date 2 consumption is not insued in the ëbondsonly" asset egime. Thus the possibility of negative consumption is an issue in the bonds-only case, though not unde complete makets. : 1