复旦大学博士生宏观经济学讲义(一)

=0
.
⎟
c (t )
=
ce (t) ce (t )
=
f
'(ke (t))(n + ™ ⌠
+g
+©)
=
0
.
k e (t )
=
f
(ke (t)) (n + ™
+
g
+ © )ke (t)
=
0
f '(ke (t)) = n + ™ + g + © ce (t) = f (ke (t)) (n + ™ + g)ke (t)
1/⎝ = s* SolowSwan 1/⎝ > s* s*
dt = + wte
dt + a = w + a = W
0
0
214
ct
= c e(1/⎝ )[rt〉 ]t 0
219 218
+ c(0) = W /( e dt) [rt (1⎝ )/⎝ 〉 /⎝ +n]t 0
5
^
Yt = F (Kt, AtLt) = F (Kt, L t)
^
A(t) = e©t L t = AtLt
f '(ke (t)) (n + ™ + g + © ) ⎝
⎟c (t) t
2(ce , ke )
.
⎟
c(t )
= ce (t) = ce (t )
f '(ke (t)) (n + ™ + g + © ) ⎝
.
k e (t )
=
f
(ke (t)) (n + ™
+
g)ke (t) ce (t)
n>0 t = 0 1t L(t) = ent
A(t) = egt
Y (t) = F(K (t), A(t)L(t))
.
K
(t
)
=
F
(K
(t
),
A(t
)L(t ))

™
K
(t )

C (t )
c(t) = C(t)
k(t) = K (t) c (t) =
L(t )
L(t )
(1.1)
(1.2) C(t ) e A(t)L(t)
^
lim[k e
0
t
+ ] = 0 t
(
^
f
'(
k
v
)™
©
n
) dv
^^
227 230 (c, k t)
.
.
^
^
^
^
4ct) /(ct) = 0 (k t) /(k t) = 0
^
f '(kt) = ™ + 〉 +⎝©
9
223 224 225 226
227 228 229 230 231
232
^
kt
)

™
〉
⎝©
⎝
.
kt
= wt + rkt ct nkt
kt
=
^
k
e©
t
t
225 226 229
.
^
kt
= w e©t t
+
(r

n

^
©
)
k
t^
ct
.
^
kt
=
f
(k
t)

^
(©
+
n+™
^
)k
t
^ ct
3k t
=
ke©
^ t
^
f '(kt) = rt + ™ 216TVC
cs / ct 6 210
⌠
=
[ u
'(
cs / cs)
ct /u
'(
ct
d[u '(cs) ) d (cs
/ /
u '(ct ct )
)] ]
1
.
r= + c ⌠c
.
c= ⌠ (r 〉) c
CIES
u(c)
=
c1⎝ 1⎝
1
⎝
>
0,⎝
⎯ 1u(c)
=
log c
⎝
=1
⎝ ⌠ = 1/⎝ 212
e
1.14
.
ce
(t
)
c
(et)U ''(ce=(t))c U '(ce (t))
(t)[ ef
'(k
(t)) (n + ™e + g + © )]
⎝
=
c e(t)U ''(ce (t))⌠ = 1/⎝ U '(ce (t))
(1.6) (1.7) (1.8)
(1.9) (1.10)
(1.11)
®
∝
=
u
'(c)e(〉 n)t
H
∝
® ∝ = (r n)∝
. = a
.
lim[∝tat] = 0
t
26
27 28
3 〉〉=〉
n
Benthamite felicity function
4
5
.
At
=
wtLt
+
rtAt

Ct
a
=
.
d(
A) t / Lt
dt
=
=
.
A tLt

At
L
2 t
.
H (ce, ke, ⎣;t) ke
3TVC
lim
t
⎣
(t
)
ke
(t
)
=
0
e©tU '(ce (t)) = ⎣(t)
.
⎣(t) / ⎣(t) =
f '(ke (t)) (n + ™ + g)
lim
t
⎣
(t
)
ke
(t
)
=
0
1(1.9)(1.10)⎣ (t )
(1.9)t :
.
⎣ (t )
=
e© tU
^ ^^
^
Yt = Lt yt = Lt f (k t)
f
(k
t)
=
^
F
(Kt
/
AtLt ,1)
7
217 218 219 220 221 222
7 LYKKYL
8
Yt
^
Kt = f '(k)
Yt = [(Lf t
f (0) = 0 f '(0) = f '( ) = 0 Inada
^
^
^
k t) k tf '(k t)]e©t
.
c=
r〉
c⎝
27
∝ (t) = ∝(0)et(rtn) ∝ (0) > 0
28
6=
〉r ct +=1[ ct
(1+ r) ]⎝ ⎝1 = 1/ ⌠ (1 + 〉 )
7
29 210
211
212 213 214 215 u '(ct) 1+ r u '(ct + 1) 1+ 〉
lim[atet(rtn) ] = 0
334 k * < k gold
〉 e = 〉 + ⎝© > n + ©
^
f '(k t) = © + n + ™
^
^
8©
10
. ^
k =0
^
^
k **
k
334 335
〉 + ⎝© 〉 ⎝
^
^
c' 0 232 c 0
^
c'' 0 TVC
^
1DD c'' 0 k ,
n+©
e e rtt (n+© )t rt
lim
[ate
+
t
( rv n
) dv
]
ε
0
0
t
rt n
+ rt rt =
lim[a(t)et(rtn)] ε 0 t
3 U0
1
t
rvdv 24 t
0
H = u(x)e(〉n) + ∝[w + (r n)a c]
22
23 24 24’ 25
∝
27 28 26
H c
=
0
(1.12) (1.13)
(1.14)
(1.15)
cs / ct
⌠
=
[ u
'(
cs / cs)
ct /u
'(
ct
)
d[u
'(cs ) d (cs
/ /
u '(ct ct )
)]
]
1
(1.16)
3
s t ⌠ = u '(c) cu ''(c)
1.15
.
⎟
c(t )
= ce (t) = ce (t )
Blanchard Fischer1989Barro Sala-I-Martin1995Zilibotti Dirk,kruger .
Ramsey (1928)Cass1965Koopmas1965
1
k (t) =
k (t )
e
A(t)
.
k e (t) = f (ke (t)) ce (t) (n + ™ + g)ke (t)
c(t )1⎝ 1⎝
= e〉t
(ce (t)egt 1 ⎝
1)⎝
=
e(〉 (1⎝
) g )tU
(c
(t))
e
© = (〉 (1⎝ )g)
(1.3) (1.4)
© >0
+ max
ce (t ),ke (t ) 0
e
e tU (c (t))dt
.
s.t.k e (t)
=
f(ke (t)) ce (t) (n + ™
^
^
kt0BBkt
.
^
^
ct 212 = ⌠ (r 〉)
c
^
c
^
k t 0 r 212
212
7 ⎝ ⎝
11
high⎝
low⎝
^
⎝k* high
^
k * ⎝ low
22
⎝
8 CD f (k) = A k〈
^
^
s* = 〈 (© + n + ™ ) /(〉 +⎝ x + ™ )
BarroSala-I-Martin 1995P89, AppendixB
216
t
at < 0 21624
216rt n
rt n rt n 24
4
d [ae ( r n ) t
]
/
dt
=
a
e
. (r

n )t
a(r n)e(rn)t
a
t

(r

.
n)at
= wt ct
e ( r n )t
T
(
[ (rtn)t ] /
)
T
( rt n )t
L
t=
.
=A t Lt
.
=L t Lt
A=t w + ra c na Lt
6
.
∝ = uc''(
) c e(〉n)
.

(
〉

n)u
'(c)e(〉
n)t
26e(〉n)t = ∝ / u '(c) 29 27 29
.
r = 〉 [u ''(c)c](c ) u '(c) c
u ''(c)c u '(c)
T (rtn)t
+ da te
dt dt = + wte dt + cte dt
0
0
0
+ + aTe (rtn)T + T cte(rtn)dt t =
wte

(
rt
n )t
dt

T
a
(0)
0
0
T 216
(rt n)t
( rt n ) t
~
(0) (0) (0)
+ cte
+
g)ke (t)
ke (0) = ke 0
2
H (ce, ke, ⎣;t) = etU (c (te)) + ⎣(t)[ f (k (et)) c (et) (n + ™ + g)k e(t)] (1.5)
1ce (t)
2
2
H (ce, ke, ⎣;t) 0
ce
=
2
.
⎣(t) =
CRRA
⎝ 〉t e
+ max etU (c(t))dt
c(t ),k (t ) 0
.
s.t.k e (t)
=
f(ke (t)) ce (t) (n + ™
+
g)ke (t)
keห้องสมุดไป่ตู้(0) = ke 0
U
(c)
=
⎮〉⎮∫ 1cln1(⎝⎝c),,⎝⎝
⎯1 =1
⎮⌠
U((c t)) = e〉t
lim ertte(n+© )t kt = Blanchard Fischer1989
t
2Blanchard Fischer1989
e (ne+© )tkt
2DD
.
230k t
=
f
^
(k t)

(©
+
n
+
^
™)
k
t

ct
^
^
^
d2 k
t
dt 2
= ['f
.
.
.
^
^^
^
^
(k t) (© + n + ™ )]k t ct < 0 f '(k t) > (© + n + ™ ) ct > 0
复旦大学博士生宏观经济学讲义（一）
1
Frank Plumpton Ramsey
Frank Plumpton Ramsey (1903-1930), British mathematician and philosopher, best known for his work on the foundations of mathematics. But Ramsey also made remarkable contributions to epistemology, semantics, logic, philosophy of science, mathematics, statistics, probability and decision theory, economics and metaphysics.
^
f '(kt) = ™ + 〉 +⎝©
^
ct
=
f
(k
t
)

^
(©
+
n+™
^
)k
t
233
^
c
B
DD
. ^
c=0
^
c' 0
^
c0
^
c'' 0
^
k0
^
^
k*
k gold
12
© 232233
^
TVC231 r* = f '(k*) ™ > © + n 232
〉 > n + (1 ⎝ )©
〉e8 333
ce (t) , ke (t)
Lt = ent
5
^
^
c(t) , k(t)
©
ct , kt
21
3
+ U 0 = u(ct)e(〉n)t dt 0
〉 u '(0) = u '( ) = 0 〉 > n 4c
U0
rt wt At at = At / Lt rtat
2
.
at
= wt + rat ct nat5
''(c
(t)) c
(t)
© e©tU
'(c
.
(t ))
e
e
e
1.9
.
.
⎣(t) U ''(ce (t)) ce (t)
=
©
⎣(t) U '(ce (t))
1.10⎣ (t )
.
U (t )
''(ce (t)) ce © = [ f (k
(t)) (n + ™ + g)]
U '(ce (t))
∠t
^
=
L
t[
F
(
Kt
^
,
L
t
)

^
Rt
k
t

wt
e©
t
]
^
Rt Rt = rt + ™ L t
225 224
6 1c = ce© t ^
^
f '(kt) = rt + ™
[
f
^
(k
t)

^
k
tf
^
'(k t
)]e© t
= wt
2at = kt
.
^
(ct )
/(ct )
^
=
.
ct
/
ct

©
=
f
'(
(1.17)
(1.18) (1.19) (1.20) (1.21) (1.22) (1.23)
4
ce B ce' ce0 c0''
.
ce = 0
DD
.
ke = 0
ke0
ke*
k gold e
12
ke**
ke
g
c(t) , k(t)
1 Nt n Lt 1
Ct t c(t) = C(t) / L(t)