上财金融学院高宏课件(谭继军) (10)
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上财金融学院高宏课件(谭继军) (2)
— Again, we are interested in the properties of s. This time we use implicit function theorem to find
∂st ∂st ∂w1 , ∂w2
rief review of implicit function theorem (see Appendix ) — Define H (st , w1 , w2 , rt ) ≡ −U1 (w1 − st ) + rt U2 (wt + rt st ) = 0, where st is endogenous variable, and w1 , w2 and rt are parameters. If U is additive separable, which ensures c1 and c2 are normal goods, then 1> U11 ∂st 1 = − ∂w = > 0, 2U ∂H ∂w1 U11 + rt 22 ∂s
∂H rU2 ∂s ∂τ = − ∂H = < 0. 2 ∂τ U11 + r (1 − τ )U22 ∂s
From
∂s ∂τ ,
only substitute effect exists, no income effect.
— Because τ only changes gross real interest rate, real income doesn’t change at all (rsi (1 − τ ) + w2 + a and τ rsi = a ⇒ rsi + w2 ). Income is compensated. — Another word: Though people’s real income doesn’t change, they still response to the tax by decreasing saving. Another word: No neutral tax on price (interest rate), even if real income doesn’t change.
高宏课件CHAPTER(1)
8.6 不确定性
8.7 金融市场的不完全性
h
11
f)
pK
(t)
其中: f为公司所得税减税比率
h
(8.5)
5
4.模型的不足之处
(1)假定资本存量调整是完全有弹性的，无成本 的，无限制的，实际上投资受限于产出规模， 因此投资和资本不可能无限连续(infinite)变 动；
(2)未考虑预期因素: 当预期需求上升或资本成 本下降时，K↑；
(3)对模型的修正: 引入资本存量的调整成本
1）内部调整成本：如安装新机器设备的成本， 训练工人操纵新机器设备的成本，利率变动 的成本等。
2）外部成本：资本h品价格的变化。
6
8.2 Q Theory Model of Investment
with Adjustment Cost
1.假定：
一产业有N个同质厂商
(1)一个典型厂商的实际利润与其资本存量(k(t))成 正比（若规模报酬不变、产品市场完全竞争、
K(t) I(t) 2. 厂商的利润函数(8.6) 即利润=收益－I－调整成本 若用离散时间，则(8.7)
h
8
预 算 约 束 : kt1 kt It
最 优 化 : (8.8) t : 资 本 的 边 际 收 益
令 q t (1 r ) t , 变 成 (8 .9 )
L : (8.10) I
为总收益, Xs:产品价格和要素成本等 假定:K 0,KK0
2.合意的资本存量的确定
K P K (K ,X 1 ,X h2 ,X 3 , ,X n ) r K 0 3
K()rK (8.1)
厂 商 合 意 的 资 本 存 量 Kf(rK,X's)
对 (8.1)式 对 rk求 导 :
上财金融学院高宏课件(谭继军) (9)
— Introduce fiat money, a constant nominal money stock M . — Denote Pt as the price level at t, $/goods. — Two–period agents with endowment (w1 , w2 ) when young and old. — When young: Pt (w1 − c1t ) = mt , where mt is quantity of money acquired by a young agent. So w1 − c1t = which is called real money balance. Note that $/($/goods)=goods, so real money balance means how many goods are held. When old: Pt+1 c2t+1 = Pt+1 w2 + mt , or c2t+1 = w2 + mt m t Pt = w2 + · . Pt+1 Pt Pt+1 mt , Pt
which means real money balance demand/capita equals real money balance supply/capita. ⇒ Pt M = f (w1 , w2 , ), Pt N t Pt+1 where both M and Nt are exogenous. — Define zt =
— The agent max U (c1t , c2t+1 ), s.t. w1 − c1t = c2t+1 ⇔ max U (w1 −
which means real money balance demand/capita equals real money balance supply/capita. ⇒ Pt M = f (w1 , w2 , ), Pt N t Pt+1 where both M and Nt are exogenous. — Define zt =
— The agent max U (c1t , c2t+1 ), s.t. w1 − c1t = c2t+1 ⇔ max U (w1 −
上财金融学院高宏课件(谭继军) (3)
— This is less than the maximum sustainable production.
Graphically (insert a graph) How can the government solve this problem?
— Using less centralized methods, short of taking over the entire economy as done in the G.R. administration.
— Transfers from young to old, which is called social security.
—
Look at
U1(w1−k) U2(w2+f (k))
.
Such transfers mean
U1(w1−k−τ ) U2(w2+f (k)+σ)
.
— For a given k, these transfers increase c2 and lower c1.
— The C.E. tells us the best that individuals can do for a given budget.
— It may not be the best possible allocation. The key to understanding these models is to distinguish between the decision of individuals and those of the social planner.
A social planner’s problem: maximizing utility of future generations, ignoring the initial old (Golden Rule of Phelps (AER, 1965)).
Graphically (insert a graph) How can the government solve this problem?
— Using less centralized methods, short of taking over the entire economy as done in the G.R. administration.
— Transfers from young to old, which is called social security.
—
Look at
U1(w1−k) U2(w2+f (k))
.
Such transfers mean
U1(w1−k−τ ) U2(w2+f (k)+σ)
.
— For a given k, these transfers increase c2 and lower c1.
— The C.E. tells us the best that individuals can do for a given budget.
— It may not be the best possible allocation. The key to understanding these models is to distinguish between the decision of individuals and those of the social planner.
A social planner’s problem: maximizing utility of future generations, ignoring the initial old (Golden Rule of Phelps (AER, 1965)).
上财罗大庆高宏课件 (9)
• Since all periods are identical, the competitive equilibrium prices and consumption allocation are
∞ {r t }∞ t=1 = {0.01}t=1 , ∞ {c 1 t , c 2 t }∞ t=1 = {2.2, 2}t=1 .
• Optimization problem solved by an individual born in period t is to maximize u(c1t , c2t+1 ) subject to the Present Value budget constraint. • Logarithmic utility example: – Maximize u(c1t , c2t+1 ) = ln c1t + β ln c2t+1 subject to c 1t + c2t+1 y2t+1 = y 1t + . 1 + rt 1 + rt
(II) Heterogeneity within Cohorts
• Heterogeneity creates incentives for trade. • Assume there are two types of agents born in each period. • Type 1 (group 1) – Nt individuals of type 1 are born in period t – they have utility function u(c1t , c2t+1 ) – and an endowment stream {y1t , y2t+1 }. • Type 2 (group 2) – Nt∗ individuals of type 2 are born in period t
高宏(08)2014年11月17日§2.1§2.2(纯白)
•第三次作业：第117页：•2.8 —2.11•11月24日交第3次作业2014-11-17 高宏（8）《高宏》讲义，张延著。
版权所有2•四、家庭的最大化问题•1、家庭最大化问题的一阶条件•家庭的问题是，在预算约束条件下选择c(t)的路径以最大化一生效用。
尽管这涉及选择每一时点上的c(而非像标准的最大化问题那样，仅选择有限的一组变量)，传统的最大化方法仍可使用。
2014-11-17 高宏（8）《高宏》讲义，张延著。
版权所有3•由于消费的边际效用总为正，所以家庭满足其预算约束的等号形式。
因此，我们可用目标函数(2.14)和预算约束(2.7)来构造拉格朗日函数：•目标函数：∞[ e -βt c(t)1-θ／(1-θ) ] dt (2.14)•U ≡ B∫t=02014-11-17 高宏（8）《高宏》讲义，张延著。
版权所有4•约束条件：•∫t=0∞e -R(t) c(t) e(n+g)t dt•≤ k(0) + ∫t=0∞e -R(t) w(t) e(n+g)t dt (2.7)•L= B∫t=0∞[e -βt c(t) 1-θ／(1-θ)]dt +λ[ k(0) +•∫t=0∞e -R(t) e(n+g)t w(t)dt -∫t=0∞e -R(t) e(n+g)t c(t)dt ]•(2.15)2014-11-17 高宏（8）《高宏》讲义，张延著。
版权所有5•c(t)：选择变量，政策可控制。
如何选择，以最大化一生的效用。
•k(t)：状态变量。
•t ：时间变量。
2014-11-17 高宏（8）《高宏》讲义，张延著。
版权所有6•即以某一时期t为例，写出某一时期t的拉格朗日函数：•L = B e -βt c(t)1-θ／(1-θ) +•λ [ k(0) +e -R(t)e(n+g)t w(t) -e -R(t)e(n+g)t c(t) ]•存在极值的一阶条件为：•∂L ／∂c=B e-βt c(t)-θ-λe -R(t)e(n + g)t= 0 2014-11-17 高宏（8）《高宏》讲义，张延著。
高宏课件3 lecture_OK
7
5 Stationary paths,interest rates and golden rules
• Pro 20.E.2 r>0 or r<0; • Golden Rule
8
6 Dynamics
• Forms of Dynamics (see figure20.F1-4) • Shocks:Transitory or Permanent (fig 20.F.7)
3
1 Introduction
• Structure of Time • Dynamic Aspects • Dynamics Equilibrium
4
2 Intertemporal Utility
• Comsumption stream c=(c0,…ct…),Bounded,||ct||<infinite
第三章 宏观经济学中的微观经济 学基础——时间与均衡
掌握动态一般均衡的基础知识； 了解动态宏观经济学的微观基础；
从微um and Time， Chapter 20；in Mas-Colell, Whinston, Green, Microeconomics Theory,Oxford University Press,1995
2
Outline
• Introduction • Intertemporal Utility • Intertemporal Production and Effeciency • Equilibrium:One-Comsumer Case • Stationary Paths,Interest Rates and Golden Rules • Dynamics • Equilibrium:Several Consumers • Overlapping Generations • Remarks on Nonequilibrium Dynamics
5 Stationary paths,interest rates and golden rules
• Pro 20.E.2 r>0 or r<0; • Golden Rule
8
6 Dynamics
• Forms of Dynamics (see figure20.F1-4) • Shocks:Transitory or Permanent (fig 20.F.7)
3
1 Introduction
• Structure of Time • Dynamic Aspects • Dynamics Equilibrium
4
2 Intertemporal Utility
• Comsumption stream c=(c0,…ct…),Bounded,||ct||<infinite
第三章 宏观经济学中的微观经济 学基础——时间与均衡
掌握动态一般均衡的基础知识； 了解动态宏观经济学的微观基础；
从微um and Time， Chapter 20；in Mas-Colell, Whinston, Green, Microeconomics Theory,Oxford University Press,1995
2
Outline
• Introduction • Intertemporal Utility • Intertemporal Production and Effeciency • Equilibrium:One-Comsumer Case • Stationary Paths,Interest Rates and Golden Rules • Dynamics • Equilibrium:Several Consumers • Overlapping Generations • Remarks on Nonequilibrium Dynamics
上财金融学院高宏课件(谭继军) (5)
Case 2: r < n, government debt/GNP falls over time. (insert a graph)
— The government’s budget constraint: total government expenses equal total government revenues.
or
r
1
பைடு நூலகம்gt + n bt−1 = τ1 + n τ2 + bt.
— Consider a cut in taxes of 5 units per young person in period t but no
change in g.
So there must be an increase of 5 goods in government debt per young
Now in equilibrium,
U1 U2
= r = f (k∗).
So if r = n, the golden rule
conditions
are
meet
(
U1 U2
=
n
and
f
(k)
=
n).
All the government needs to know to implement this is the long run
3
3 The Effect of the National Debt on Capital and Saving
Assume a tax of τ1 goods on each young person and τ2 goods on each old person and f (k) = x = r. Incorporating these taxes into individuals’ budget constraint:
— The government’s budget constraint: total government expenses equal total government revenues.
or
r
1
பைடு நூலகம்gt + n bt−1 = τ1 + n τ2 + bt.
— Consider a cut in taxes of 5 units per young person in period t but no
change in g.
So there must be an increase of 5 goods in government debt per young
Now in equilibrium,
U1 U2
= r = f (k∗).
So if r = n, the golden rule
conditions
are
meet
(
U1 U2
=
n
and
f
(k)
=
n).
All the government needs to know to implement this is the long run
3
3 The Effect of the National Debt on Capital and Saving
Assume a tax of τ1 goods on each young person and τ2 goods on each old person and f (k) = x = r. Incorporating these taxes into individuals’ budget constraint:
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zt+1 = 1 − δ + f (kt+1) · · · (∗∗).
zt
1+n
— Steady state:
1. k = 0 and z = 0 (trivial steady state).
2.
z
= 0 ⇒ kt+1
=
1 1+n
s(kt,
kt+1
).
So
any
non–trivial
steady–state
λ
c2
:
U2
=
1
+
. n
⇒ U1 = 1 + n. U2
So R¯GR = 1 + n = U1 . U2
While the C.E.:
U1 = R = 1 − δ + f (k∗). U2
Normally, k∗ > kGR, f (k∗) < f (kGR). People save two much. ⇒ Dynamic
— There are firms with production function, F (K, L), (CRS and F =
K FK
+ LFL).
Denote
k
=
K L
,
so
f (k)
=
F (k, 1).
— Firms: F.O.C.:
max F (Kt, Lt) − rtKt − wtLt.
By the first order condition of the problem,
st = s(wt, 1 − δ + rt+1).
— Saving market clearing condition:
M Kt+1 + Pt = Nts(w(kt), 1 − δ + f (kt+1)).
Divided by Ntd model are still steady–state values here.
3. z = 0 ⇒ zt+1 = zt = z. From (**),
(kt)
= lim f (kt) − lim f (kt) = 0(Inada)
kt→∞
kt→∞
So we get kt+1 − kt graph as follows. (insert a graph) There exists at least a non–trivial steady state. It is not certain how many there are. If getting a steady state below, it must get one steady state upper.
⇒
max L
c1t ,c2t+1 ,zt ,it
= U (c1t, c2t+1) + λ1(wt − it −
mt Pt
− c1t)
+
λ2
(
mt Pt
Pt Pt+1
+
it(1
−
δ
+
rt+1)
−
c2t+1).
zt(
mt Pt
)
:
−λ1
+
λ2
Pt Pt+1
= 0,
it : −λ1 + λ2(1 − δ + rt+1) = 0.
— We still have wt = w(kt) and rt = f (kt). No change.
— But young:
max U (c1t, c2t+1),
5
s.t.
Ptc1t ≤ Ptwt − Ptit − mt, Pt+1c2t+1 ≤ mt + it(1 − δ + rt+1)Pt+1. v.s.
or
c1
+
1
c2 +
n
+
(1
+
n)kt+1
≤
f (kt)
+
(1
−
δ)kt.
⇔
max
U (c1,
c2)
+
λ(f (kt)
+
(1
−
δ)kt
−
c1
−
1
c2 +
n
−
kt+1(1
+
n)).
F.O.C.:
k : f (kGR) = n + δ, R¯GR = 1 − δ + f (kGR) = 1 + n,
c1 : U1 = λ,
c1t ≤ wt − st, (st = it)
c2t+1 ≤ stRt.
⇒
c1t
≤
wt
−
it
−
mt , Pt
c2t+1
≤
mt Pt
Pt Pt+1
+ it(1 − δ
+ rt+1),
where
Pt Pt+1
is
real
gross
rate
of
return
of
real
money
balance
and
1−
δ + rt+1 is rate of investment.
st
=
mt Pt
+
it,
⇒
Rtst
=
Pt ( mt Pt+1 Pt
+ it)
=
(1 − δ
+
rt+1)(
mt Pt
+ it).
So the budget constraints become:
c1t ≤ wt − st, c2t+1 ≤ Rtst.
6
So max U (wt − st, (1 − δ + rt+1)st).
growth), less capital/capita.
(insert a graph)
3.1 Steady state welfare
Social planner:
max U (c1, c2),
c1,c2,k
4
s.t. Ntc1 + Nt−1c2 + Kt+1 ≤ F (Kt, Nt) + (1 − δ)Kt,
)
dkt+1 dkt
)
⇒
dkt+1 (1 − s2f (kt+1) ) = s1w (kt) ,
dkt
1+n
1+n
where s1 > 0, s2 > 0, w (kt) > 0 and f (kt+1) < 0.
So dkt+1 > 0. dkt
3
–
Claim:
kt+1 kt
→ 0 as kt → ∞.
— F.O.C.:
s = s(w(kt), 1 − δ + f (kt+1)).
So Kt+1 = Nts(w(kt), 1 − δ + f (kt+1)),
⇒
Kt+1 Nt+1
=
Nt Nt+1
s(w(kt),
1
−
δ
+
f
(kt+1)),
⇒ 1
kt+1 = 1 + n s(w(kt), 1 − δ + f (kt+1)).
kt+1 +
M PtNt
·
Nt Nt+1
=
Nt Nt+1
s(w(kt),
1
−
δ
+
f
(kt+1)),
or
1
1
kt+1 + zt · 1 + n = 1 + n s(w(kt), 1 − δ + f (kt+1)) · · · (∗).
Also, ⇒
zt+1 = Pt ·
1 .
zt Pt+1 1 + n
If n ↑, look at the graph, mathematically keep kt+1, then w(k(kt)) need
to increase ⇒ kt need to increase. So the curve shifts right and the
steady state capital decrease. That is, higher 1 + n (higher population
0 ≤ kt+1 = s(wt, Rt) ≤ w(kt) kt (1 + n)kt (1 + n)kt
If w → constant, done; if not,
lim
kt→∞
w(kt) kt
=
lim ( f (kt) kt→∞ kt
−
f
(kt))
=
lim
kt→∞
f (kt) kt
−
lim f
kt→∞
— Then, from good market clearing,
Nt(w(kt) − st) + (1 − δ + rt)Kt + Kt+1 = F (Kt, Lt) + (1 − δ)Kt.