Supplementary Materials for Asset Bubbles, Collateral, and Policy Analysis

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1 Supplemenary Maerials for Asse Bubbles, Collaeral, and Policy Analysis Jianjun Miao Pengfei Wang y Jing hou z Augus 20, 205 Secion A provides proofs of all proposiions in he main ex. Secion B provides proofs of resuls in Secion 5.3 of he main ex. Secion C analyzes he impac of foreign purchases of domesic bonds. Secion D describes daa sources for Figure in he main ex. A Proofs of all proposiions Proof of Proposiion : equaion and wrie i as We subsiue he conjecured value funcion ino he Bellman v ( j )K j p ( j )H j ' ( j )B j = k K j P H j B j j I j P H j B j H j ;I j ;B j v () K j f () d ' ()B j f () d f p () H j f () d subjec o K j = ( ) K j I j (A.) B j f P H j ; (A.2) H j!h j ; (A.3) Deparmen of Economics, Boson Universiy, 270 Bay Sae oad, Boson, MA Tel.: miaoj@bu.edu. Homepage: hp://people.bu.edu/miaoj. y Deparmen of Economics, Hong Kong Universiy of Science and Technology, Clear Waer Bay, Hong Kong. Tel: (852) pfwang@us.hk z Deparmen of Economics, Hong Kong Universiy of Science and Technology, Clear Waer Bay, Hong Kong. Tel: (852) jzhoubb@us.hk.

2 and 0 j I j k K j B j f B j P (H j H j ): (A.4) De ning Q = v () f () d and plugging (A.) ino he Bellman equaion above, we obain (0) in he main ex. More speci cally, if P > p () f () d; hen each enrepreneur j wans o sell land so ha H j =!H j all all j: Then he land marke canno clear. Bu if P < p () f () d; hen each enrepreneur j wans o buy as much land as possible. Thus we mus have P = p () f () d: Similarly we mus have = f = ' () f () d: Now he Bellman equaion becomes v ( j )K j p ( j )H j ' ( j )B j = H j ;I j ;B j k K j P H j B j j I j Q I j Q ( )K j : (A.5) (i) By (A.5) and (A.4), when j Q ; we mus have j I j = k K j B j f B j P (H j H j ): (A.6) In addiion, i follows from (A.2) and (A.3) ha boh he borrowing and resaleabiliy consrains bind. When j > Q ; i follows ha I j = 0: Because B j and H j are canceled ou in he objecive of (A.5), he enrepreneur is indi eren among he feasible choices of B j and H j : (ii) Subsiuing he decision rules in par (i) ino (A.5) and maching coe ciens, we obain ( Q v ( j ) = j k ( )Q if j Q ; ; (A.7) k ( )Q if j > Q p ( j ) = ( P (!!) Q j P if j Q ; ; (A.8) P if j > Q ( Q ' ( j ) = j if j Q ; : (A.9) if j > Q Using equaion (0) in he main ex of he paper, and he de niion of Q ; we can derive equaions ()-(3) in he main ex. The ransversaliy condiions follow from he in niehorizon dynamic opimizaion problem, e.g., Ekeland and Scheinkman (986). Q.E.D. 2

3 Proof of Proposiion 2: By par (i) of Proposiion, we can derive aggregae invesmen I = [ k K j (!!) P H j B j ] dj: j Q j Since j is independenly and idenically disribued and since K j ; B j ; and H j are predeermined, j is independen of hese variables. By a law of large numbers, we obain I = j Q j [ k K j (!!) P H j B j ] dj j Q dj j Q dj = E [ k K j (!!) P H j B j ] j j Q j = dj k K j dj (!!) P H j dj j Q j = [ k K (!!) P ] f () d; Q j Q dj where we have used he marke-clearing condiions o derive he las equaliy. By equaion () in he main ex and he labor marke-clearing condiion, = N = N j dj = W K j dj = W K : B j dj From his equaion, we can derive oher equaions in he proposiion. Q.E.D. Proof of Proposiion 3: The righ hand side of equaion (20) in he main ex is sricly decreasing in. When approaches he lower suppor of he disribuion for j, he righhand side approaches in nie. When approaches he upper suppor of he disribuion, he righ-hand side approaches < ( ): Thus, by he Inermediae Value Theorem, here is a unique soluion 2 ( min ; ) o equaion (20) in he main ex. Condiion (2) in he main ex ensures ha C f > 0: Q.E.D. Proof of Proposiions 4 and 5: seady-sae Tobin s Q, denoed by, sais es he equaion,!( ) = By he Inermediae Value Theorem, if Equaion () in he main ex implies ha he bubbly!( ) < 3 : (A.0) ; (A.)

4 hen (A.0) has a unique soluion 2 ( min ; ) : We can hen derive he seady-sae renal rae of capial kb using equaion (2) in he main ex, kb = ( ) ; : (A.2) We hen use (8) in he main ex o deermine he seady-sae capial sock K b : The bubbly seady-sae invesmen, oupu, and consumpion are given by I b = K b, Y b = K b ; and C b = Y b land price P; I b ; respecively. We use equaion (5) in he main ex o deermine he seady-sae P Y b = 2 4!( ) ( ) ; We need P > 0 and C b > 0 for he exisence of a bubbly seady sae. The proof consiss of wo pars. 3 5 : (A.3) Par I. Suppose ha he bubbly and bubbly seady saes coexis. We hen prove Proposiion 5 and he necessiy of condiion (24) in he main ex. Sep. We prove < : By equaion (20) in he main ex, ( ) = Equaion (A.3) and P > 0 imply ha ; : (A.4) kb < : Combining equaion (A.2) and he preceding inequaliy, we can derive ha ( ) = kb ; " # F ( ) < : Combining he preceding inequaliy wih (A.4) yields ; < " This inequaliy is equivalen o he following inequaliy: # F ( ) : F ( ) > F ( ) : 4

5 Thus < : Sep 2. We prove kf > kb : The seady-sae version of equaion (2) in he main ex is given by = ( ) k The equaion above implies ha kf > kb since < : ; : (A.5) Q Sep 3. Because kb = K b < kf = K f, we have K b > K f. Hence, Y b = K b > Y f = K b ; I b = K b > I f = K f. In addiion, equaion (3) in he main ex implies ha = fb = ( ; ) < ( ; ) = = ff, i.e., fb > ff : Sep 4. In he bubbly seady sae, equaion () in he main ex implies ha = (!!) : (A.6) Since < ; condiion (24) in he main ex mus hold. This proves he necessiy of (24) in he main ex as well as Proposiion 5. Par II. Now,we suppose ha condiions (22), (23), and (24) in he main ex hold. We hen prove ha he bubbly and bubbleless seady saes coexis. Sep. The righ-hand side of (A.0) is sricly increasing in : I is equal o 0 when = min and equal o when = : If condiion (A.) holds, hen (A.0) has a unique soluion 2 ( min ; ) by he Inermediae Value Theorem. Sep 2. By (A.6) and condiion (24) in he main ex, < ; where is given by equaion (20) in he main ex. By Sep 2 of Par I, kb < kf : Condiion (23) in he main ex implies ha kf > kb > : By Proposiion 3, a bubbleless seady sae exiss. Since Sep 3. To show he exisence of a bubbly seady sae, we mus show C b > 0 and P > 0: C b = Y b I b = Y b K b = K b K b = = K b ( kb = ) ; condiion (23) in he main ex ensures ha C b > 0 holds. We now check P > 0: By (A.3), P Y b = 2 4!( ) ( ) 2 4!( ) ( )!( ) ( ) ; ; 3 5 > 5 ( ) = = 0; (A.7) 5 3

6 where he rs inequaliy follows from > by Sep 2 of Par II and he second equaliy follows from equaion (20) in he main ex. Q.E.D. Proof of Proposiion 6: J(Q ) = Q min Denoe by F he cumulaive disribuion funcion of and de ne. We can use Proposiion 2 o show ha he equilibrium sysem can be described by he following four di erence equaions: C K ( )K = K ; (A.8) Q = ( )Q K C C [Q J(Q ) F (Q )] ; (A.9) P C = P C f (!!)[Q J(Q ) F (Q )]g ; (A.20) K ( )K = [K (!!)P ]J(Q ); (A.2) for four unknowns fk ; C ; Q ; P g. Only K is predeermined. The oher hree variables are nonpredeermined. Linearizing P around zero and log-linearizing Q, K and C around heir bubbleless seady sae values, we obain C f Kf ^C K f K f ^K ( )K f K f ^K = ^K ; ^Q ^C = ^C ( )[ ( )] ^K ^Q ; P = f (!!)[ J( ) F ( )]g P ; (A.22) ^K ( ) ^K = f( ) J( ) ^Q ^K (!!) J( ) K f P ; where ^C, ^K ; and ^Q denoe log-deviaion from he seady sae. We rewrie he sysem in he following marix form: 2 B 6 4 ^C ^K ^Q ^C 7 5 = G 6 ^K 7 4 ^Q 5 ; where P 2 3 K 0 f K 0 0 f B = 6 ( )( ) B 34 5 ; P 6

7 wih 2 G = 6 4 C f Kf ( )K f K f f( ) J( ) (!!) J( ) K f B 34 f (!!)[ J( ) F ( )]g: ; I is sraighforward o check ha G is inverible. To sudy he local dynamics around he bubbleless seady sae, we sudy he eigenvalues of he marix M G B. Firs, we check ha 0 mus be an eigenvalue of marix M. Noe ha marix B is singular because is columns and 3 are linearly dependen. Thus de(m) = de(m 0 I) = 0, implying ha 0 is an eigenvalue. Second, noe ha de(m B 34 I) = de[g B B 34 G G] = de(g ) de(b B 34 G) = 0: Thus B 34 = f(!!)[ J( ) F ( )]g is an eigenvalue of marix M. Le B 34 denoe his eigenvalue. Third, we can show ha he oher wo eigenvalues are posiive real numbers, wih one greaer han and he oher smaller han. Le 2 and 3 denoe hese wo eigenvalues. We can hen wrie de(m I) = ( )( 2 b c); where b d ( [ ( )] C f K f (2 )f( ) f( ) J( ) c d " C f K f ( )( ) f( ) J( ) ; K f # f( ) ; J( ) C f Kf d ( ) C f K f ( )f( ) f( ) J( ) > 0: 7

8 Since c < 0, i follows ha 2 3 > 0. We can also show ha b c = ( ) C f d Y f ( ) ( ) f( ) J( ) > 0: Thus he quadraic equaion always has wo real soluions, wih one larger han and he oher smaller han. Wihou loss of generaliy, we suppose ha 2 < < 3. We hen have wo eigenvalues (0 and 2 ) inside he uni circle and one ( 3 ) ouside he uni circle. Wheher he local dynamic around he bubbleless seady sae is deerminae depends on wheher = f (!!) [ J( ) F ( )]g is smaller han. By Proposiion 4, when boh he bubbly and bubbleless seady saes exis, condiion (24) in he main ex mus hold, i.e., = f (!!) [ J( ) F ( )]g > ; implying ha he marix M has wo eigenvalues ouside he uni circle and wo eigenvalues inside he uni circle. Since here are hree nonpredeermined variables, his means ha he bubbleless seady sae is a saddle wih indeerminacy of degree. When only he bubbleless seady sae exiss, we mus have = f (!!) [ J( ) F ( )]g : If < ; hen he marix M has hree eigenvalues inside he uni circles and one eigenvalue ouside he uni circle, implying ha he local dynamic is deerminae. If = ; hen (A.22) implies ha P = P : Since lim! P = 0, i follows ha P = 0 for all : In boh cases, here is a unique equilibrium, which is bubbleless. Q.E.D. Proof of Proposiion 7: See he proofs of Proposiion 8 and Lemma 2. Q.E.D. Proof of Proposiion 8: Consider he equilibrium wihou bubble rs. Le = and! ( ) = 0: Equaion (2) in he main ex becomes Q Y = ( s )Y ( s )Y s Y We can furher reduce he above equaion o Q = s s s Equaion (5) in he main ex implies ha K = I = Y Q Q ; : Q = s Y ; 8 ; : (A.23)

9 or s = Q : (A.24) Hence he sysem of wo di erence equaions (A.23) and (A.24) deermine he bubbleless equilibrium rajecories for Q and s. Clearly a consan seady sae is a soluion o he sysem. The following lemma shows ha his is he unique local soluion. Lemma There is a unique local soluion o he sysem of wo equaions (A.23) and (A.24), which is he bubbleless seady sae. Proof: We use F o denoe he cumulaive disribuion funcion of and de ne H (Q ). In he seady sae, equaions (A.23) and (A.24) imply ha Q = s f [ H ( ) F ( )] ; s f = H ( ) : Then log-linearizing he sysem around his seady sae, we obain ^Q We rewrie he above wo equaions as where s f s f ^s = s f s f ^s ^s ^Q ; ^s = f ( ) H ( ) ^Q : ^s ^s B f = G ^ ; ^Q B f = 2 G f = 4 # s f ; " sf s f f(q f) H() In order o undersand he local dynamics around his seady sae, we need o sudy he wo eigenvalues of he marix G f B f. They are 0 and f where f = 5 : s f f() s f H() : f() s f H() 9

10 I is sraighforward ha 0 < f < since 0 < f() s f H() s f < f() s f H() and 0 < <. This means he wo eigenvalues are boh inside he uni circle. Therefore here is a unique local soluion o he sysem since boh s and Q are nonpredeermined. The lemma above shows ha he seady sae is he unique soluion o he sysem of wo equaions (A.23) and (A.24) for s and Q in he neighborhood of he bubbleless seady sae. We hen use hese wo equaions o deermine by = F ( ) : (A.25) F (Q Since lim f ) Qf! = and lim F ( ) min! = 0, by he Inermediae Value Theorem, here is a unique soluion in ( min ; ). Once is deermined, Q f hen he saving rae is given by from equaion (A.23). s f = ; : (A.26) We need s f 2 (0; ) for a bubbleless equilibrium o exis. This condiion is equivalen o (2) in he main ex. This is because ha equaion implies ha >. By equaion Q f (A.24), s f = f () d <. We compue he life-ime uiliy as U f (K 0 ) = X =0 where s f is given by equaion (A.26) and [ln( s f ) ln(y )] = ln( s f ) ln(k ) = ln (s f Y ) = ln(s f ) ln(k ): Hence, he welfare for any given K 0 is given by U f (K 0 ) = ln( s f ) X ln(k ); =0 ln(s f ) ln(k 0 ) : (A.27) Nex, consider he equilibrium pah o he bubbly seady sae. Equaion (A.23) sill holds. Equaion (5) in he main ex becomes K = I = (Y P ) Q = s Y : 0

11 Dividing by Y on he wo sides of his equaion yields s = ( p ) Q ; where p = P =Y. Using C = ( s ) Y ; we rewrie equaion () in he main ex as p = p Q ; : s s (A.28) (A.29) Therefore, he sysem of hree di erence equaions (A.23), (A.28) and (A.29) deermine hree sequences for s ; p ; and Q : Clearly, he seady sae is a soluion o his sysem. The following lemma shows ha i is a unique local soluion. Lemma 2 There exiss a unique soluion s = s b, p = p b and Q = for all o he sysem of hree equaions (A.23), (A.28) and (A.29) in he neighborhood of he bubbly seady sae. Proof: Subsiuing (A.28) ino (A.23) and (A.29), we obain Q ( p ) Q ; f () d = Q f () d ( p ) Q f () d ( p ) ; Q f () d p ( p ) p Q ; f () d = Q f () d ( p ) : Q f () d As before, denoe J (Q ) = Q f () d. Then Q ; f () d = Q J (Q ) F (Q ). A he bubbly seady sae, he wo equaions above imply ha = [ J ( ) F ( )] ; = ( p b ) J ( ) : We log-linearize hese wo di erence equaions around he bubbly seady sae and obain s b p b ^p s b f( ) s b p b s b J( ) ^Q = s b p b ^p s b f( ) ^Q ; s b p b p b s b J( ) = p b ^p f( ) ^Q s b p b s b J( ) s b p b ^p s b f( ) s b p b p b s b J( ) ^Q :

12 We rewrie he wo equaions above in he following form ^p ^p B b = G ^ ; ^Q where and B b = G b = " s b p b s b p b p b s b f( ) s b J( ) s b f( ) s b J( ) s b s b p b p b p b " s b s b p b p b s b f( ) s b J( ) s b p b p b s b f( ) J( ) As before, we need o check he eigenvalues of he marix G b B b. The characerisic funcion of he marix G b B b is 2 b c where hen b " 2 d p b s b s b p b p b f( ) J( ) s b f( ) J( ) c s b f( ) > 0; d p b s b J( ) d f( ) s b J( ) s b p b > 0: s b p b We hen prove he following wo facs: () 0 < c < ; (2) b c > 0. () Claim 0 < c < : Since which implies 0 < c <. (2) Claim b c > 0. 0 s b f( ) s b J( ) 0 < p b < ; # f( ) s b J( ) ; s b f( ) 0 < s b J( ) p b < f( ) s b J( ) < f( ) s b J( ) s b p b ; s b p b : # s b s b p b p b # < 0; 2

13 We use he de niion of b and c o compue b c = p b > 0: d p b p b Given hese four facs, if he wo roos (denoed by and 2 ) are real numbers, hey mus boh be posiive because 0 < 2 = c < and 2 = b > 0. Since b c > 0, he wo roos mus be smaller han, oherwise 2 >. If he wo eigenvalues are complex numbers, i follows from 0 < c = 2 < ha hey mus be inside he uni circle. We conclude ha, in boh cases, here is a unique local soluion for p and Q since boh are nonpredeermined variables. The soluion is he bubbly seady sae. We hen use (A.28) o deermine he soluion for s ; which is also he seady sae value. Now we compue he bubbly equilibrium welfare for any given iniial non-seady-sae capial sock K 0 : U b (K 0 ) = ln( s b) ln(s b) ln(k 0 ) : (A.30) We hen compare U f (K 0 ) and U b (K 0 ). Noe ha he life-ime uiliy in equaion (25) of he main ex as a funcion of he saving rae s is concave and has a imum a s =. Using (9) of he main ex and =, s f = = kf = f () d: By assumpion, s f >. By Proposiion 5, kb < kf : Thus s b = = kb > = kf = s f > >. Hence, U f (K 0 ) > U b (K 0 ). Anoher su cien condiion for U f (K 0 ) > U b (K 0 ) is 2 [0; ]. Under his condiion, ; f () d > ; so ha s b > s f = ; f () d >, where he equaion for s f follows from (A.26). Finally, we compare U f (K f ) and U b (K b ). In any seady sae, K = I = sk : I follows ha he seady-sae capial sock sais es K = s =( hen compue he seady-sae welfare as U = ln ( s) ln (s) ) : Using (25) in he main ex, we can I is a concave funcion of s and is imized a s =. Since s b > s f >, i follows ha U f (K f ) > U b (K b ). Q.E.D. : 3

14 B Proof of resuls in Secion 5.3 We rs prove he following resul. Proposiion The equilibrium sysem for he economy wih ransacion axes is given by he following equaions: ( ) P = P Q ( Q Q <Q [! (!)( )] Q (B.) ) 2 (!) ; I = k K Q f () d P H P H [! (!) ( )] Q<Q Q f () d; (B.2) f () d and equaions (2), (3), (6), (7), and (8) in he main ex for nine variables fc ; I ; Y ; K ; W ; k ; f ; Q ; P g. The usual ransversaliy condiions hold. Proof: We conjecure ha enrepreneur j s value funcion akes he form, V ( j ; K j ; H j ; B j ) = v ( j )K j p ( j )H j saisfy ' ( j )B j, where v ; p and ' are funcions o be deermined and ( )P = p ( j )dj; (B.3) = ' f ( j )dj: (B.4) Denoe Q = v ( j )dj as Tobin s marginal Q. Given he preceding conjecure, we can rewrie he Bellman equaion as subjec o (A.2), (A.3), I j 0, and v ( j )K j p ( j )H j ' ( j )B j (B.5) = I j ;H j k K j B j (Q j ) I j Q ( )K j P (H j H j ) P jh j H j j ( )P H j ; k K j j I j P (H j H j ) B j f B j P jh j H j j 0: (B.6) 4

15 Noe ha erms relaed o B j are canceled ou in he Bellman equaion. We rs consider a low- enrepreneur wih j Q. This enrepreneur would like o inves as much as possible unil (B.6) and (A.2) bind. Thus, and j I j = k K j B j f B j P (H j H j ) P jh j H j j ; B j f = P H j : Subsiuing his invesmen rule ino he preceding Bellman equaion yields v ( j )K j p ( j )H j ' ( j )B j (B.7) = k K j B j P (H j H j ) P jh j H j j Q ( )K j H j 0 Q [ k K j B j P H j P (H j H j ) P jh j H j j] j ( )P H j ; subjec o (A.3). Now consider he choice of H j. We claim ha enrepreneur j will never buy land (i.e. H j > H j ) because his would imply ha he marginal bene of holding one more uni of land is negaive, i.e., ( ) Q P < 0. I mus be he case ha H j H j. We j can hen compue he marginal bene of holding one more uni of land as Q 2 ( ) P : j This expression is posiive when j > Q. In his case, enrepreneur j will keep buying unil H j = H j : However, when j < Q, he marginal bene of holding one more uni of land is negaive so ha enrepreneur j prefers o sell as much land as possible unil H j =!H j. In sum, opimal land holdings are given by ( Hj when H j = Q < j Q!H j when j Q : Subsiuing he decision rule for H j above ino he Bellman equaion in (B.7), we can simplify he Bellman equaion. In paricular, for j Q, he value funcion sais es v ( j )K j p ( j )H j ' ( j )B j (B.8) = Q ( k K j B j ) Q ( )K j j Q ( ) (!) [! (!) ( )]! ( ) P H j ; j 5

16 h i where ( ) (!) Q j (!) ( ) P H j is he invesmen nanced by selling a fracion (!) of he curren land holdings ne of ransacion ax, Q!P H j is he j invesmen nanced by borrowing using a fracion! of he curren land holdings as collaeral, and! ( ) P H j is he shadow value of he land. For Q < j Q, he value funcion sais es where Q j v ( j )K j p ( j )H j ' ( j )B j (B.9) = Q Q ( k K j B j ) Q ( )K j ( ) P H j ; j j P H j is he invesmen nanced by borrowing wih he curren land holdings as collaeral, and ( ) P H j is he shadow value of he land. Nex, consider a high- enrepreneur wih j > Q. In his case, invesing is unpro able so ha I j = 0. The Bellman equaion in (B.5) becomes v ( j )K j p ( j )H j ' ( j )B j (B.0) = H j 0 kk j B j P (H j H j ) P jh j H j j Q ( )K j ( )P H j ; subjec o (A.3). If H j H j, hen he marginal bene of holding one more uni of land is 2P > 0. In his case, he enrepreneur will increase land holdings unil H j = H j : If H j H j ; hen he erms relaed o H j are canceled ou in he preceding Bellman equaion. This means ha he enrepreneur is indi eren among any feasible choices of H j H j. We can hen rewrie (B.0) as v ( j )K j p ( j )H j ' ( j )B j = k K j ( )P H j B j Q ( )K j : (B.) Maching coe ciens of K j ; H j and B j on he wo sides of equaions (B.8), (B.9), and (B.), respecively, we can derive expressions for v ( j ); p ( j ); and ' ( j ): Subsiuing hese expressions ino (B.3), (B.4) and using he de niion of Q ; we obain equaions (B.) and (2) and (3) in he main ex afer some manipulaion. 6

17 Using a law of large numbers, we compue aggregae invesmen as h I = k K j B j [! (!) ( )] P H j idj j Q j h k K j B j P H j idj Q< jq j = dj k K j dj [! (!) ( )] P H j dj j Q j dj k K j dj P H j dj B j dj Q< jq j = k K f () d Q P "[! (!) ( )] f () d Q Q<Q B j dj f () d where we have used he marke-clearing condiion H = : We can also derive equaions (6), (7), and (8) in he main ex as before. I is known from he lieraure ha he ransversaliy condiions are par of he necessary and su cien condiions for opimaliy in in nie-horizon problems. Q.E.D. By adaping he proof of Proposiion 4, we can show ha here is a unique bubbly seady sae in which P = P > 0 for all if and only if ( < < ( ) [! (!)( )] Q ( ) f ) 2 (!) ; where is deermined by (20) in he main ex. Since he righ-hand side of he above inequaliy is decreasing in ; when is su cienly high he inequaliy is violaed. As a resul, a land bubble canno exis. # ; C Foreign purchases of bonds We now use our model o sudy he impac of capial in ow hrough foreign purchases of domesic privae bonds on asse bubbles. I is ofen argued ha he increased capial ows as a resul he global saving glus were an imporan reason for he US housing bubbles (see e.g., Bernanke s celebraed speech on he global saving glu and he U.S. curren accoun de ci on March 0, 2005, and Greenspan s esimony a he Financial Crisis Inquiry Commission 7

18 in April 200). We now exend our model o invesigae his hypohesis by assuming ha invesors from he res of worlds buy B domesic bonds. We will focus he bubbly equilibrium only. Wih capial in ow, he bond marke-clearing condiion is given by 0 B j dj = B ; (C.) By Proposiion in he main ex and he bond marke-clearing condiion (C.), we can derive aggregae invesmen The resource consrain becomes I = [ k K (!!)P B ] Q : (C.2) C I B = Y B f : (C.3) The oher equilibrium condiions described in Proposiion 2 in he main ex remain unchanged. Comparing he equilibrium sysem wih ha in benchmark model in he main ex, only equaions (C.2) and (C.3) are di eren. Therefore, in he bubbly seady sae, fi b ; Y b ; K b ; W b ; kb ; fb ; g are all he same as in he benchmark model in he main ex. We can derive he bubbly seadysae land price P wih capial in ow as P =!( ) 20 4@ ( ) ; 3 A Y b B 5 : (C.4) We can see ha he land price P increases wih B. In addiion, a dollar capial in ow increases he land price by!( ) > dollars. This means ha here is an muliplier e ec of capial in ow on land bubbles in he seady sae. Finally we use he resource consrain o derive C b = Y b I b B Y b I b (C.5) fb Since fb and he inequaliy is sric for! > 0, he households receive a ne income ransfer in he bubbly seady sae. This implies capial in ow will be welfare improving in he bubbly seady sae. In summary, we have he following proposiion characerizing he bubbly equilibrium wih capial in ow. 8

19 % % % % % % 5 Land Price 0 Ne Ineres ae Oupu Consumpion Invesmen Welfare Figure : Transiion pahs of he bubbly equilibrium in response o a gradual increase in foreign purchases of bonds. Parameers values are given by = 0:3; = 0:99; = 0:025; = 5:7;! = 0:2; = 0:75; B = ; and = 0:6: Proposiion 2 Suppose ha he assumpions in Proposiion 4 in he baseline model wihou capial in ow hold. Then here always exiss a bubbly equilibrium in which he values of fi b ; Y b ; K b ; W b ; kb ; fb ; g are he same as hose in Secion 3.4 of he main ex and C b and P > 0 are given by equaion (C.5) and (C.4). We now sudy he ransiion pah. We assume ha he economy is iniially in he bubbly seady sae wihou capial ows unil period 0. In period, he capial accoun is opened. We se he same parameers as in Secion 4.2 of he main ex. The seady-sae capial in ow is se o B =, which corresponds abou =3 of he iniial level of bubbly seady-sae oupu. Along he ransiion pah, B = B where 2 (0; ) : Figure plos he ne ineres rae f and he percenage deviaions of he land price P, oupu Y, consumpion C ; invesmen I ; and capial in ows B from heir bubbly seady-sae values wihou capial ows. The increase in he capial ow raises he demand for domesic bonds, hereby raising he bond price and lowering he ne ineres rae. The decline in he 9

20 ineres rae allows he household o subsiue bonds for land, hereby raising he land price dramaically on impac. Alernaively, he decline in he ineres rae lowers he discoun rae for land and hence raises he land price. As he capial in ow gradually rises o is seady sae level B, he ineres rae gradually rises back o is seady-sae value. This induces he household o gradually shif invesmens in land o invesmens in bonds. Thus he land price gradually falls back o is seady-sae value and he land price overshoos is long-run value on impac. The new seady-sae land price is higher han is old seady sae value due o he capial in ow. Wih capial in ow, he economy is able o nance higher consumpion and invesmen simulaneously because he ne ineres rae is negaive. The welfare as measured by discouned fuure uiliy increases accordingly. In he long run, oupu and invesmen reurn o is bubbly seady-sae levels wihou capial in ow, bu consumpion reaches a permanenly higher level. In sum, Figure con rms he hypohesis of he global saving glu ha foreign capial in ow will reduce ineres rae and push up asse prices. Bu unlike he convenional wisdom, capial in ow is bene cial o he recipien economy. D Daa descripion for Figure in he main ex We download he daa from he Deparmen of Economics of Queen s Universiy via he link les/oher/house_price_indices%20(oecd).xls. All series are quarerly and seasonally adjused. The daa are de ned as follows.. The nominal house price index of he US is he all-ransacion index (esimaed using sales price and appraisal daa) from Federal Housing Finance Agency (FHFA). 2. The nominal house price index of Japan is he naionwide urban land price index from he Japan eal Esae Insiue. 3. The nominal house price index of Spain is he average price per square meer of privae housing (more han one year old) from he Bank of Spain. 4. The nominal house price index of Greece is he price per square meer of residenial properies (all as) in urban areas from he Bank of Greece. 5. The real house price index used in Figure is he above nominal house price index de aed by he privae consumpion de aor. The average real index in 2000 is normalized o The price-income raio used in Figure is he raio of he nominal house price index o he nominal per capia disposable income. The sample average is normalized o

21 7. The price-renal raio used in Figure is he raio of he nominal house price index o he ren componen of he consumer price index. The sample average is normalized o 00. 2