Proof of Proposition 1
In any bubbleless steady state, we have bkt+1= bkt= bk and bRt+1= bRt= bR for all t: Substituting these into (4) and rearranging terms gives
Rb
Substituting the steady state conditions into (15) and rearranging terms gives (17). Note that the func-tion : R+! R+ de…ned in (47) is continuously di¤erentiable and satis…es (0) = 0. Straightforward di¤erentiation gives
0 R =b
1Rb1 1 1 + 1Rb1 1 1 + 1Rb1 1 2
> 0; for any > 0:
Hence, there exists a unique value of bR > 0 that solves (47). Using (17), one can obtain a unique value of bk > 0: This establishes the existence and uniqueness of bubbleless steady state.
Proof of Proposition 2
First, consider the case when = 1: Equations (14) and (15) now become
bkt+1= 1
Since 2 (0; 1) ; there exists a unique non-trivial steady state bk > 0 which is globally stable. The second equation in (48) can be rewritten as
Rbt= 1 + A
1
1+ bkt 1 bkt ;
where ( ) is a strictly decreasing function.
Next, consider the case when < 1: To prove that the bubbleless steady state is globally saddle-path stable, we will use the same “phase diagram” approach as in Tirole (1985) and Weil (1987). To start, de…ne a function F : R+! R+ according to
Note that the unique bubbleless steady state must satisfy bk = F bR : Taking the logarithm of both sides of (49) and di¤erentiating the resultant expression with respect to R gives
RF0(R)
(i) e 1 and (ii)e < 1: Since ( ) is strictly increasing and bounded above by one, in the …rst scenario we have F0(R) < 0 for all R 0; limR!0F (R) = +1 and limR!1F (R) = 0: In the second scenario, F ( ) is a U-shaped function. Figures B1 and B2 provide a graphical illustration of these two scenarios.
In both diagrams, the function F ( ) and the vertical line representing R = bR divide the (R; k)-space into four quadrants:
The rest of the proof is divided into a number of intermediate steps. These steps are valid both when e 1 and whene < 1:
Step 1 For any initial value RbT; bkT > 0; there exists a unique sequencen
RbT +1; bkT +1; bRT +2; bkT +2; :::o that solves the dynamical system in (14)-(15). Whether this is part of a non-stationary bubbleless equilib-rium depends on the location of RbT; bkT on the (R; k)-space. A solutionn
RbT +1; bkT +1; bRT +2; bkT +2; :::o is said to originate from Qn if RbT; bkT 2 Qn; for n 2 f1; 2; 3; 4g : In the …rst step of the proof, it is
shown that any solution that originates from Q1 or Q3 cannot be part of a bubbleless equilibrium.
which implies bRt+1 < bRt R : Recall that the functionb ( ) de…ned in (4) is strictly increasing when
< 1. Then it follows from (14) that
bkt+1 = 1 following: Any solution that originates from Q1 is a strictly decreasing sequence and is con…ned in Q1; i.e., Rbt; bkt 2 Q1 for all t T: Since both bkt and bRt are strictly decreasing over time, in the long run we will have either bkt= 0 or bRt= 0, which cannot happen in equilibrium.
Using a similar argument, we can show that any solution that originates from Q3 is a strictly in-creasing sequence and is con…ned in Q3: Using the young consumer’s budget constraint and the capital market clearing condition, we can obtain the following condition
bst= bkt+1
1 + n <wbtblt wbt= (1 ) Rbt
1
:
Obviously, this will be violated at some point if both bkt and bRtare strictly increasing over time. Hence, any solution that originates from Q3 cannot be part of a bubbleless equilibrium.
Step 2 We now show that any solution that originates from Q2will never enter Q4; i.e., RbT; bkT 2 Q2
implies Rbt; bkt 2 Q= 4, for all t > T ; likewise, any solution that originates from Q4 will never enter Q2:
Suppose Rbt; bkt is in Q2 for some t T: Then we have
The second line uses the fact that ( ) is strictly increasing and bRt+1 > bR : The last equality follows from the steady-state condition in (16). Since > 1; we also have bRt+11 > bRt 1: This, together with (15) and (50), implies
which gives rise to a contradiction. Hence, any solution that originates from Q2 will never enter Q4: Using similar arguments, we can show that any solution that originates from Q4 will never enter Q2:
Step 3 Consider a solution that originates from Q2: As shown in Step 2, RbT; bkT 2 Q2 implies RbT +1 > bRT: If bRT +1 R ; then the economy is in Qb 3 at time T + 1 and by the results in Step 1, we
know that bRt will diverge to in…nity in the long run. If bRT +1 < bR ; then using (14) we can obtain Q1 at time T + 1: By the results in Step 1, we know that all subsequent values of bRt will be strictly less than bR : Second, if bRT +1< bR and F bRT +1 < bkT +1; then that means the economy remains in Q2 at time T + 1: In addition, we have bRT +1 > bRT and bkT > bkT +1 which means the economy is now getting closer to the steady state R ; bb k : Thus, any solution that originates from Q2 has three possible fates:
(i) It will enter Q3 at some point and bRtwill then diverge to in…nity. (ii) It will enter Q1 at some point and bRt will be strictly less than bR afterward. (iii) It will converge to the bubbleless steady state. For reasons explained above, the …rst two types of solutions cannot be part of an equilibrium. Hence, a solution originating from Q2 is an equilibrium path only if it converges to the steady state R ; bb k : The above argument also shows that, along the convergent path, bktis decreasing towards bk while bRt is increasing towards bR :
Using a similar argument, we can show that any solution originating from Q4 is an equilibrium path only if it converges to the steady state R ; bb k ; and that along the convergent path, bkt is increasing towards bk while bRt is decreasing towards bR :
Step 4 We now establish the uniqueness of saddle path. Fix bkT > 0: Suppose the contrary that there exists two saddle paths, denoted by n
Rb0t; bkt0o1
which implies bR0T +1 > bRT +100 > 0: Using (14), we can get Hence, we can rule out the possibility of multiple saddle paths.
In sum, we have shown that any equilibrium path that originates from a given value of bkT > 0 must be unique and converge to the bubbleless steady state. Hence, the dynamical system in (14)-(15) is globally saddle-path stable. The one-to-one relationship between bRT and bkT can be captured by a function : R+ ! R+: Since the saddle path is downward sloping in the (R; k)-space, ( ) must be strictly decreasing. This completes the proof of Proposition 2.
Proof of Proposition 3
In the post-crash economy, optimal labor supply is determined by (3). Setting = 1 gives blt= 1+A
1 1+
for all t: In the pre-crash economy, optimal labor supply is determined by
Alt + = (wt)1
which is equation (41) in Appendix A, where
t+1
When = 1; the right-hand side (RHS) of the above equation becomes
Hence, we have Al +1t = 1 + for all t: The desired result follows immediately from this expression.
This completes the proof of Proposition 3.
Proof of Proposition 4
The main ideas of the proof are as follows. In any conditional bubbly steady state, we have a > 0 which is equivalent to > 1: This, together with < 1 and R R ; implies two things: k > bb k and Rb0 (k ) > bR : But as we have seen in Proposition 2, these two results cannot be both true which means we have reached a contradiction. Hence, it must be the case that R > bR :
The main task of the proof is to verify the following two claims:
Claim #1 Suppose < 1 and > 1: Then R R implies l > bb l and k > bk :
When evaluated in a recurring bubbly equilibrium, equation (41) becomes
A (l ) + = (w )1
On the other hand, the value of bl in the bubbleless steady state is determined by of bR is determined by (16), which can be rewritten as
1 b = 1 + 1 (1 + n)1 1 b 1
1
: (51)
On the other hand, the relationship between R and bR0 is characterized by (18), which is derived from (44) in Appendix A. The latter can be rewritten as
1 = 1 + ( 1) + [ q (1 + n)] 1 Rb0
= 1 + ( 1) + 1q 1 (1 + n)1 1 b0: (52)
Combining (51) and (52) gives
1 b = ( 1) + 1 (1 + n)1 1 q 1 b0 b 1 1 : (53)
, q 1 Rb0< Rb 1
1
(1 + n)1
, bR
1 1
< [q (1 + n)]1 Rb0 : This establishes Claim #1.
Proof of Claim #2 First, note that > 1 is true if and only if
q (1 + n) >
2
4q + (1 q) Rb0 R
!1 3 5 R
, q (1 )
> (1 q) b0!1
: (55)
Next, rewrite (54) as
q 1
"
q (1 ) (1 q) b0
#1
b0 < b 1
1
, 1 < (1 q) b0 b
!1
: (56)
Using (55) and (56), and the assumptions of b and < 1; we can get
q (1 )
> (1 q) b0!1
(1 q) b0 b
!1
> 1 ;
which implies q > : Using (56) and q > , we can get
(1 q) b0 b
!1
> 1 > 1 q ) b0 > b :
This establishes Claim #2.
0 2,000 4,000 6,000 8,000 10,000 12,000
Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03
Dow Jones Index
0 400 800 1,200 1,600
S&P 500
S&P 500 Dow Jones
Figure 1: Dow Jones Industrial Average and S&P 500, 1995-2003.
120 140 160 180 200 220
Jun-03 Jun-05 Jun-07 Jun-09
Home Price Index
Figure 2: Case-Shiller 20-City Home Price Index, June 2003 to June 2010.
0 2,000 4,000 6,000 8,000 10,000 12,000
Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03
Dow Jones Index
72,000 76,000 80,000 84,000 88,000 92,000
Total Employment (thousands)
DJI Employment
Figure 3: Total Employment and Dow Jones Index, 1995-2003.
0 2,000 4,000 6,000 8,000 10,000 12,000
Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03
Dow Jones Index
80 85 90 95 100 105
Index of Aggregate Hours
DJI Aggregate Hours
Figure 4: Aggregate Hours and Dow Jones Index, 1995-2003.
2,000 4,000 6,000 8,000 10,000 12,000
1995 1996 1997 1998 1999 2000 2001 2002 2003
Dow Jones Index
800 900 1,000 1,100 1,200 1,300 1,400 1,500
Nonresidential Investment (billions)
Dow Jones
Nonresidential Investment
Figure 5: Private Nonresidential Fixed Investment and Dow Jones Index, 1995Q1 to 2003Q4.
120 140 160 180 200 220
Jun-03 Jun-05 Jun-07 Jun-09
Home Price Index
80,000 85,000 90,000 95,000 100,000
Total Employment (thousands)
Home Price Index
Employment
Figure 6: Total Employment and Home Price Index, June 2003 to June 2010.
120 140 160 180 200 220
Jun-03 Jun-05 Jun-07 Jun-09
Home Price Index
90 100 110 120
Index of Aggregate Hours
Home Price Index
Index of Aggregate Hours
Figure 7: Aggregate Hours and Home Price Index, June 2003 to June 2010.
0.9 1.0 1.1 1.2 1.3 1.4 1.5
2003 2004 2005 2006 2007 2008 2009
Nonresidential Fixed Investment
Home Price Index
Figure 8: Private Nonresidential Fixed Investment and Home Price Index, 2003Q3 to 2010Q3.
Figure 9: Probability Tree Diagram of the Asset Price Shock.
1.1 1.2 1.3 1.4 1.5
0 1 2 3 4 5 6 7 8 9 10 11 12
sigma = 0.10 sigma = 0.15
sigma= 0.2 sigma = 0.3
Time
FIgure 10: Time Paths of Interest Rate under Di¤erent Values of :
0.45 0.55 0.65 0.75
0 1 2 3 4 5 6 7 8 9 10 11 12
sigma = 0.10 sigma = 0.15 sigma= 0.2 sigma = 0.3
Time
Figure 11: Time Paths of Labor Supply under Di¤erent Values of :
0.05 0.06 0.07 0.08
0 1 2 3 4 5 6 7 8 9 10 11 12
sigma = 0.10 sigma = 0.15 sigma= 0.2 sigma = 0.3
Time
Figure 12: Time Paths of Capital under Di¤erent Values of :
Figure B1: Phase Diagram for the case whene 1:
Figure B2: Phase Diagram for the case whene < 1:
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