= wt , b
wt
=
(1 − α ) Yt
lt Nt
⇒
ln
⎢⎡(1
⎣
−
st
)
Yt Nt
⎤ ⎥ ⎦
−
ln(1
−
lt
)
=
ln
⎢⎡(1
⎣
−
α
)
Yt lt Nt
⎤ ⎥ ⎦
−
ln
b
( ) ⇒ ln lt − ln(1− lt ) = ln(1−α )− ln 1− s∗ − ln b
( ) l∗
=
1−α
(1−α )+ b 1−
consumption in next period will not change the expected utility.
The cost of this change:
e−ρt ⎜⎜⎝⎛
1 ct
⎟⎟⎠⎞ ×
∆c
3
The benefit of this change:
Et
⎨⎧e− ⎩
ρ
(t
+1)
U
=
∞
∑e
t =0
−
ρtu(ct
,1
−
lt
)
Nt H
ln Nt = N + nt, where N = ln N0
ut = ln ct + b ln(1− lt ) b > 0
z Shocks: First autoregressive process with white-noise disturbances ε 1) ln At = A + gt + A~t A~t = ρ A A~t−1 + ε A,t −1 < ρ A < 1
2. Theories of Fluctuations
z In the framework of the Ramsey model z Two extensions:
i) Including a source of disturbances: real shock. ii) Including variations in employment
2.4 The General Case
z Log-linearizing the model around the balanced growth path
C~t = aCK K~t + aCA A~t + aCGG~t
L~t
=
aLK K~t
+
aLA
~ At
+
aLG
~ Gt
z The intratemporal first-order condition
wt = (1−α )Ktα (At Lt )−α At rt = αKtα −1(At Lt )1−α − δ
z Utility function:
U
=
∞
∑ (1 +
t=0
ρ )−t u(ct ,1− lt
)
Nt H
Nt = N0 (1+ n)t n < ρ
Other expression by using the fact ln(1+ x) ≈ x :
period and has no initial wealth.
L = ln c + b ln(1− l)+ λ(wl − c)
FOC: Budget constraint:
1−λ =0 c − b + λw = 0 1−l c = wl
⇒ l* = 1 1+ b
Intuition:
z
Two periods:
ቤተ መጻሕፍቲ ባይዱ
( ) ⇒
aCK K~t
+
aCA A~t
+
aCGG~t
+
⎜⎜⎝⎛
1
l∗ − l∗
+α
⎟⎟⎠⎞
aLK K~t
+
aLA A~t
+
aLGG~t
= (1−α )A~t + αK~t
Because it must hold for all values of K~t , A~t ,G~t , we have:
⎜⎜⎝⎛
1 ct +1
⎟⎟⎠⎞
×
(1
+
rt
+1
)⎬⎫∆c
⎭
Optimization requires:
e − ρt
⎜⎜⎝⎛
1 ct
⎟⎟⎠⎞∆c
=
Et
⎨⎧e− ⎩
ρ
(t
+1)
⎜⎜⎝⎛
1 ct +1
⎟⎟⎠⎞(1 +
rt +1 )⎬⎫∆c
⎭
⇔
1 ct
=
e−
ρ
Et
⎜⎜⎝⎛
1
+ ct
rt +1
+1
⎟⎟⎠⎞
i.e.
s∗
Intuition: here fluctuations don’t reflect any market failures, RBC models imply that observed aggregate output movements represent the time-varying Pareto optimum.
Real Business Cycle Theory
1. Introduction: some facts about economic fluctuations
z 我们关注两个重要的指标:
why?
产出(GDP)和失业率(Unemployment rate)
z 我们关注经济的短期波动
why?
z 有哪些波动特质我们可以观测到? 1) 这些波动没有简单的规律可循 – modern macroeconomics has turned away from attempts to interpret fluctuations as combinations of deterministic cycles of different lengths, e.g., Kitchin(3 years), Juglar(10 years), Kuznets(20 years), and Kondratiev(50 years). 2) 产 出 的 各 个 组 成 部 分 的 波 动 相 当 不 同 – inventories vs. nondurable consumption 3) 波动常常是”牛长熊短” 4) 产业升级和政府干预经济并没有改变波动本身 5) Okun’s law: 3 percent shortfall in GDP relates to 1 percent rise in the unemployment rate.
z The evolution function of capital:
Kt +1 = Kt + It − δKt = Kt + Yt − Ct − Gt − δKt
z The lump-sum taxation z Competitive markets for labor, capital, and output
c1
+1 1+ r
c2
=
w1l1
+1 1+ r
w2l2
L
=
ln
c1
+
b
ln(1 −
l1 ) +
e−ρ
[ln
c2
+
b
ln(1 −
l2
)]+
λ⎜⎛ ⎝
w1l1
+
1 1+
r
w2l2
−
c1
−
1
1 +
r
c2
⎟⎞ ⎠
FOC:
1 − λ = 0 e−ρ − λ = 0
c1
c2 1+ r
−
b 1− l1
+
λw1
=
0
2) ln Gt = G + (n + g )t + G~t
G~t = ρGG~t−1 + εG,t −1 < ρG < 1
2
2.2 HH Behavior z For simplicity: the household has only one member and he lives only for one
rt
= α ⎜⎜⎝⎛
At Lt Kt
⎟⎟⎠⎞1−α
−1
Solution: (s,l)
ct
=
(1
−
st
)
Yt Nt
1+ rt
=α
Yt Kt
z
1 ct
=
e−
ρ
Et
⎜⎜⎝⎛
1
+ ct
rt +1
+1
⎟⎟⎠⎞
⇒
−
ln
⎢⎡(1
⎣
−
st
)
Yt Nt
⎤ ⎥ ⎦
=
−ρ
+
ln
Et
⎡ ⎢ ⎣
(1
αYt +1 /
− st +1)Yt
ρ
Et
⎜⎜⎝⎛
1
+ ct
rt +1
+1
⎟⎟⎠⎞
( ) rt+1 = αKtα+−11 At+1Lt+1 1−α − δ