Chapter 7－Equilibrium Analysis

Chapter 7－Equilibrium Analysis

( I ) Demand Shock：G↑ (政府提振內需)

Crusoe world analysis

c , y

l 0

G

1 E

F

*( )

y 總產量

n*

c*

∆G

D

生產可能線整個下移 G∆

^{w E}

( )

^{=MPL E}

( )

^{=MPL D}

( )

^{>MPL F}

( )

^{=w F}

( )

G↑by G∆ ⇒ IE(income effect)

* * * * *

c ↓ n ↑ ⇒ y ↑ w =MPL ↓

w

n

(

( ) ( )

)

s ,

n w d T

+ −−

(

( ),( ),( )

)

nd w A k

+

− +

w*

n*

Excess supply , ^s

IE n ↑

S F E

w

y w*

Excess demand E

( )

( ) ( ), ^* yd w a G

+ + +

(

( ),( ),( )

)

ys w A k

− + +

*

*

y c G

= +

∆G

cd G

∆ < ∆

A

Demand and supply analysis

Labor market Good market

(1) Direct impact( at the original w ) ^*

G↑ ⇒ n → , ^d y → (因這兩項只跟 w, A, k 有關) ^s y ↑ by d ∆ G

(2) Income effect( at the original w ) ^* G↑ ⇒ T ↑ by G∆ ⇒IE,c ↓ ^d n ↑ ^s

∆c^d < ∆ ：因為G c^d +wl^d = +w (d−T) ∆c^d + ∆ = −∆ = −∆ w l^d T G ∴ ∆c^d < ∆ G

由圖可知當 w 不變情況下 labor market 有 excess supply，good market 有 excess demand，因此 w 會改變直到兩市場均衡。

(3) Wage effect

labor market 有 excess supply ⇒ w ↓

對個人來說有 SE(替代效果)：c ↓ ^d n ↑ 由 S 點移至 F 點 ^s 對廠商來說n ↑ ^d y ↑ ^s

Net equilibrium at point F：w ↓ ^* n ↑ ^* y ↑ ^* c ↓ ^*

G↑ ⇒T ↑⇒IE c: ^d ↓ n^s ↑ G↑ ⇒ w↓⇒SE c: ^d ↓ n^s ↓

只看n 無法看出到底式上升還下降，但因平衡時^s n^s =n^d，且n 線 ^d 不動，所以由 w ↓ 可知n 一定上升，因此^d n ↑ 。 ^*

(4) Business cycle implications

Pro-cyclical(正向循環)：Corr( x, GDP)>0 GDP= 產量 = y

y

n ( ^, )

AF k n

Couter-cyclical(負向循環)：Corr( x, GDP)<0

由前面推導可知改 G∆ ，Corr( c, y)<0、Corr( w, y)<0、Corr( n, y)>0，

但此結果與實際數據不合，原因是因為除 G∆ 外還有其他效果。

( II ) Supply Shock：A↑

Crusoe world analysis

生產可能線向上移動 MPL↑

c , y

l 0

G

1 F

E

*( )

y 總產量

n*

c*

D

A↑ ⇒ IE：c ↑ , ^d n ↓ (D → F) ^s

MPL↑，SE：c ↑ , ^d n ↑ (E → D) ^s

Net equilibrium：w^*=MPL^* ↑ , c ↑ , ^* y^*↑(∵y^* =c^*+G), n^*? 證明w 上升：^*

( ) ( ) ( ) ( ) ( ) ( ) ( )

w F =MPL F >MPL D =MRS D >MRS E =MPL E =w E

w

n

(

( ) ( )

)

s ,

n w d T

+ −−

(

( ),( ),( )

)

nd w A k

+

− +

w*

n*

Excess demand E

w

y w*

Excess supply E

*

*

y c G

= +

( )

( ) ( ), ^* yd w a G

+ + +

(

( ),( ),( )

)

ys w A k

+

− +

Demand and supply analysis

Labor market Good market

(1) Direct impact( at the original w ) ^* A↑ ⇒ y ↑ , ^s n ↑ ^d

(2) Income effect( at the original w ) ^* ∆ = ∆ − ∆d y^s w n^d y=AF(k,n)

(

^,

) (

^,

)

s d

n MPL

y F k n A AF k n n

∆ = ∆ + ∆ 偏微分

(

^,

)

n

(

^,

)

^d

(

^,

)

d F k n A AF k n w n F k n A

∆ = ∆ + − ∆ = ∆

 

∴∆ ↑ IE：d c ↑ , ^d n ↓ ^s

(3) Wage effect w ↑

Example：

^{u c l}

( )

^{, =lnc+lnl y= Anα 0<α < 1}

Assume G=qy 0< < q 1 y c G= + ^c=

(

^{1−q y}

)

Equilibrium condition： ^l

c

MRS u w MPL

=u = =

(

¹

)

c= −q y

(

¹

)

1 1

c c q y y

l ⁼ n^{= −} n ⁼MPL⁼α n

− − (解是 A, K, G 的函數)

w

n ns

nd

w*

n*

E A

w

y

w* A E

*

*

y c G

= +

( )

( )

cd τ G

− +

ys

^*

( )

^0,1

n 1

q α

= α ∈

+ − 此時n 和 A 無關，因為 IE 和 SE 抵銷。 ^* y^* =An^*α

*

*

*

w y α n

= c^* =y^*− G

( III ) Wage Income Tax↑

Demand and supply analysis

Given budget constrain G T= +τwn

G→, τ↑ ⇒ T↓ no IE only SE ⇒c ↓ ^d n ↓ ^s

Market clearing conditions

( )

( ) ( )

(

( )

)

, , ,

d s

c w τ G y w A k

+ − + = −

( )

( ) ( )( ) ( ) ( )

, , ,

d s

n w A k n w τ

− + + + −

 

 =

 

Labor market Good market

Net equilibrium：τ↑ ⇒ w ↑ ^* n ↓ ^* y ↓ ^* c ↓ ^*

複數均衡

假設政府所收 tax revenue R=τwn lnR=lnτ +lnn w fix 所以不考慮

ln ln

ln 1 ln

R n

τ ^{= +} τ

τ 0% 10% 50% 60%

(1-τ)w w 0.9w 0.5w 0.4w

change 下降 10% 下降 20%

做出 R 對τ關係圖如下

R

τ 1

τ

_h

τ

_l

0

R

τ=0：沒課稅所以 R=0

τ=1：所賺的錢全課稅，沒有人要工作，R=0 若要收 tax revenue= R 有τ_l和τ_h兩種選擇

Example：

( )

^{, ln ln}

u c l = c+ l y=n^α 0<α < G=0 c G y1 + = ⇒ c= y

(

¹

) (

¹

)

1

l c

u c y

MRS MPL

u n τ α τ n

= = = − = −

−

( )

( )

* 1 *

1 1

n α τ n

α τ τ

= − ⇒ ↑⇒ ↓

+ −

( ) ( )

( )

1

1 1

R wn y n

n

α τ α

τ τ τ α ατ

α τ

 − 

 

= =   =  + − 

(出自毛老師書) 有兩個平衡點：a, b 為高稅率、c, d 為低稅率

( IV ) 消費券(Consumption Voucher)

均衡式子分析

Consumer：

( )

maxu c+v l, c c v= + (total consumption) st ^{c= −}

(

^{1 τ}

)

^wn+

(

^d−T

)

l+ = n 1

FOC：

( )

( ) ( )

,

, 1

,

l l c

c

u c l

MRS w

u c l τ

=  = −

 ───①

Firms：

( )

max d= f n −wn

FOC：^{MPL= f '}

( )

^{n =w───②}

Goods market equilibrium：

Walra’s law 使 labor market 自動平衡

( )

c= + = =c v y f n ^───③ Government Constrain(G=0)：

v= +T τwn───④

給定v τ 政府決定後，T 自動固定，若 v ↑ 對其他變數影響 求

{

^{c n y T w}^{, , , ,}

}

^其中

{

^{c n y}^{, ,}

}

^都和v無關，因為①②③式中沒有v

圖形分析

Labor market Good market

v ↑直接效果

y w*

*

*

y c G

= +

d d

c =c +v

ys

w

替代效果

課稅T ↑效果

w*

w 替代效果

課稅T ↑效果

n ns

nd

n*

因為要發消費券要課稅，所以 v ↑ ⇒T ↑⇒c^d ↓ , n ↑ ^s

但此時勞動市場 w↑，商品市場 w↓，不可能達到平衡，所以若要

達到平衡，則所有效果必須抵銷，使c 及^d n 回到原處。 ^s

Net equilibrium： v ↑ c , ^d n , ^* y , ^* w 都不變。 ^*

Chapter 8－Consumption and Saving

( I ) The Model Economy (Endowment economy)

取自毛慶生老師課本 Ch8 Model 中包含：

(1) 2 agents：consumers & government

(2) 2 competitive market：goods & credit (bond) market

Assumption：

(1) 所有人都為全知 (2) 只存活 2 期

(3) 沒有廠商，所有商品都是天上掉下來的

( II ) Fisher two period models

Budget constrain

(1) 定義變數

b

0

b

₁

b

₂

Period 1 Period 2

1 1 1

a = y − T a

₂

= y

₂

− T

₂

c

1

c

₂

Endowment product：y 、₁ y (天上掉下來的) ₂ Lump-sum tax：T 、₁ T ₂

Disposable income：a 、₁ a ₂ Consumption：c 、₁ c ₂

b ：Initial bond holding at t=1 (原始擁有的債券，一出生就決定，0

自己不可以決定) b >0 ⇒ assent (lender) 0

b <0 ⇒ liberty (borrower) 0

b ：Bond holding at the end of t=1 ₁

(2) Budget constrain

t=1 t=1 t=1 t=1

1

(1 )

0 1 1

a + + r b = c + b

1 (1 ) 0

a + +r b ：Source of found

1 1

c + ：Use of found b rb ：利息 0

Saving：S1 =

(

a1+rb0

)

−c1 =b1−b0 (Disposable income which is not consume)

t=

t=

t=

t=2 2 2 2

2

(1 )

1 2 2

a + + r b = c + b

2 2 2

1 1

c b a

b r

+ −

= + 代入 t=1 時的式子可得終身預算限制式

Life time budget constrain Life time budget constrain Life time budget constrain Life time budget constrain

( )

2 2 2

1 1 1 0

1 1 1

c b a

c a r b

r r r

 

+ + = + + +

+ +  + 

( )

2

1 1 0

1

a a r b x

r

 

+ + + =

 + 

  ：終身財富

終身預算中有除以(1+r)的可以想成第 2 期折算回第 1 期時的價 格，因第 1 期不消費而拿來儲蓄，第二期會多(1+r)，所以折算回來 時要除以(1+r)。

t=1 t=2

1 → 1+r

1

1 r+ ← 1

(3) Reason of b =0 2

b ＞0 ⇒ leave bequest after death ⇒ not optimal ₂

b ＜0 ⇒ in debt after death (大家都想要₂ b = −∞ ) ⇒ market won’t ₂ allow it ⇒ not feasible

∴optimal b = 0 ₂

圖形

Assume b₂ =b₀ = 0

Life time budget constrain： ₁ ² ₁ ²

1 1

c a

c a x

r r

+ = + =

+ +

Ex：

A：

(

a a1, 2

) (

= 0,105

)

r=5% x=100 B：

(

a a1, 2

) (

= 100, 0

)

r=5% x=100

c ₂

c₁ borrowing ( )

11 a +r

1 x a a2

( ) ( )

1 1 2 1

a +r +a = +r x

lending Slope = 1+r

(^{1 2} )

a +r

Ex：lending rate ＜ borrow rate c ₂

c₁ borrowing

a1

a2

lending

Ex：卡奴(不能再借了) c ₂

c₁ borrowing

a1

a2

lending

Preference

U c c

(

1, 2

)

=u c

( )

1 +βu c

( )

2 1

0 1

β 1

< = ρ <

+

(1) U c c1

(

1, 2

)

>0，U2

(

c c1, 2

)

>0，U11

(

c c1, 2

)

<0，U22

(

c c1, 2

)

<0 (2) Prefer diversity

( )

( )

( ) ( )

1 1 2 1

1,2

2 1 2 2

, '

, '

u c c u c

MRS ⁼u c c ⁼βu c ^{↓ as c ↑ 1} (3) c &₁ c are normal goods ₂

(4) U is time separable or time additively⇒ MU of c is independent of MU 1

of c and vice versa ₂

(5) U is time preference：β < 表示在 t=2 消費時的效用較低 1

( )

( )

¹2 ^{1 2}

' 1

1 ( )

'

MRS u c when c c

u c ρ

β β

= = = + =

c ₂

c₁ c1=c

c2 =c

45 line0

slope= +1 ρ

無異曲線

Decision problem

Given

{

a a r1, 2,

}

{ }

( ) ( )

1 2 1

1 2

, ,

max

c c b

u c + β u c

St： c₁+b₁=a₁ (b = ) ───₀ 0 ①

c2 =a2 + +

(

1 r b

)

1 (b = ) ───₂ 0 ② Or st： ₁ ² ₁ ²

1 1

c a

c a

r r

+ = +

+ + 決定關鍵為b ₁

FOC：

When a fix ₁ ∆ =b₁ 1⇒ ∆ = ⇒ ∆ = +c1 1 c2

(

1 r

)

Marginal cost of saving in current utility：u c'

( )

1

Marginal gain of saving in current utility：βu c'

( )(

2 1+r

) ( )

1

( )(

2

)

' ' 1

u c =βu c +r ───③

( ) ( )

¹

1,2

2

' 1

'

MRS u c r

βu c

= = +

由上面①②③可解

{

c c b1, 2, 1

} c

₂

c

₁

c

1

c

2

( ) 1 MRS E = + r

e

a

1

a

2

E

B(lending) A(borrowing)

( ) 1 MRS B < + r ( ) 1

MRS A > + r

無論初始稟賦在 A 還是 B，最終選擇都在 E 點，只是b 正負號不同而已。 ₁

初始在 A：b ＜0 ₁ 初始在 B：b ＞0 ₁

Shocks

(1) r fix，a ↑，1 a fix (temporary increase in income) ₂ a ↑₁ ⇒ ↑ Wealth effect(WE)：x c₁^d ↑,c₂^d ↑,S₁ =b₁↑

S₁ =b₁↑ 是因為希望各期消費都可以上升(consumption smoothing)

c

₂

c

₁

c

1

c

2

e

a

1

a

2

F

f c

1

∆ ∆ = ∆ b

₁

S

₁

因為 r 不變，所以斜率不變。

S₁ =a₁− c₁ ∆ = ∆ − ∆ > S₁ a₁ c₁ 0 ( ₁ ² ₁

1

c c x a

r

∆ + ∆ = ∆ = ∆

∵ + ⇒ ∆a1 > ∆c1 )

(2) r fix，a fix，1 a ↑ (temporary increase in income) ₂ a ↑₂ ⇒ ↑ Wealth effect(WE)：x c₁^d ↑,c₂^d ↑,S₁=b₁↓

(3) r fix，a ↑，1 a ↑ (permanent increase in income) ₂ a ↑，₁ a ↑₂ ⇒ ↑ Wealth effect(WE)：x c₁^d ↑,c₂^d ↑ ∆ 要看哪個上升較多，如果上升量差不多S₁ ∆ 不大 S₁

MPC ( marginal propensity to consume) MPS ( marginal propensity to save)

¹

1

MPC c a

= ∆

∆ ^{1 1 1}

1 1

S a c 1

MPS MPC

a a

∆ ∆ − ∆

= = = −

∆ ∆

temporary increase in income：

MPC is relatively small，MPS is relatively large because of consumption smoothing

permanent increase in income：

MPC ≈ 1 MPS ≈ 0

(4) Simple Keyne’s consumption function c_t = +a by_t 0< < b 1

b is constant which is independent of temporarily or permanently increase of income.

但台灣實際 saving rate

1993 t

t t

S y

1990

2009 25%

30%

可見上述模型錯誤。

(5) r↑，a fix，1 a fix ₂

r↑ ⇒ opportunity cost of c ↑₁ ⇒c₁↓ ，c ↑ ，₂ S ↑ ₁ Intertemporal substitution effect (ISE) (跨期替代效果) 下圖平衡點由 E→E’

考慮是否有財富效果：

₁ ² ₁ ²

1 1

c a

c a x

r r

+ = + =

+ + ：r ↑ ⇒ x ↓

分三種情況考慮

a. Lender

c

₂

c

₁

c

1

c

2

slope = + 1 r

e

a

1

a

2

E

x F

E’

1 ' slope = + r

1 1 1 0

S =a −c >

WE：(E’→F) c ↑ ，₁ c ↑ ，₂ S ↓ ₁

On net：c₁?，c ↑ ，₂ S₁?

b. Borrower

c

₂

c

₁

c

1

c

2

slope = + 1 r e

a

1

a

2

E

x F

E’

1 ' slope = + r

1 1 1 0

S =a −c >

ISE：r↑⇒c₁↓ ，c ↑ ，₂ S ↑ (E→E’) ₁

WE：r↑⇒c₁↓ ，c ↓ ，₂ S ↑ (E’→F) ₁

On net：c ↓ ，₁ c₂?，S ↑ ₁

c. Representative consumer

c

₂

c

₁

1 1

a = c

2 2

a = c e=E

E’

1 0

S =

只有 ISE(E→E’)

d. Example

^{u c}

( )

^{=lnc u c'}

( )

¹

= c

FOC：u c'

( )

1 =βu c'

( )(

2 1+r

)

BC： ₁ ² ₁ ²

1 1

c a

c a x

r r

+ = + =

+ + 求c c S ₁, ₂, ₁ FOC：c2 =β

(

1+r c

)

1

₁ 1 ₁ ²

1 2 1

a

c x a

r ρ

β ρ

 +  

= + = +  + +  1 β 1

= ρ +

( ) ( )

1 2

2

1 1

1 2

r a a

c β r x

β ρ

+ +

= + =

+ +

_{1 1 1} 1 ^{1 2}

2 1 1

a a

S a c

r ρ

ρ ρ

 

= − = ++  + − + 

Temporary increase in income：∆ = ，a₁ 1 ∆a₂ = 0 ₁ ¹

1

1 2 MPC c

a

ρ ρ

∆ +

= =

∆ + ， ₁ 1

MPS 2

= ρ +

Permanent increase in income：∆ = ，a₁ 1 ∆a₂ = 1 ₁ ¹

1

2 1

2 1 1

c r

MPC a r

ρ ρ

 

∆ +  + 

= ∆ = +  + ≈ ，MPS ≈ ₁ 0

If r= ρ MPC = ，₁ 1 MPS = ₁ 0

由 FOC： ²

( )

1

ln c ln ln 1 ln

r r

c β β

 

= + + ≈ +

 

  r↑by 1% ²

1

c c

⇒ ↑ by 1%

跨期替代彈性：

( )

2 1

ln ln 1 d c

c

d r

 

 

  +

c

₂

c

₁

A

B

1

1

slope = + r

2 2 1

1 ( )

slope = + r r > r

2 1 B

slope c c

 

=  

 

2 1 A

slope c c

 

=  

 

若無異曲線圖形為以下

則 r ↑ 時 ²

1

ln c c

 

 

 ^不變

Chapter 9－Dynamic family (Infinite life)

( I ) Budget Constrain t=1,2,3…

∞

Preferences

( )

( ) ( ) ( )

( )

1 2

2

1 2 3

1 1

,

0 1

t t t

U c c

u c u c u c

u c

β β

β β

∞

−

=

= + + +

=

∑

< <

Lifetime budget constrain

(1) Variables

t 1

b

₋

b

_t

b

_t+1

Period t Period t+1

t t t

a = y − T a

_t+1

= y

_t+1

− T

_t+1

c

t

c

_t+1

b_t−1：Initial bond holding at time t b ：End bond holding at time t _t

(2) Lifetime budget constrain c_t+b_t =a_t + +

(

1 r_t−1

)

b_t−1 ∀t St =

(

at +r bt₋1 t₋1

)

−ct =bt−bt₋1 ∀t

Let r_t =r ∀ ， t b = ，Consider the equations from t=1 to t=T< ∞ ₀ 0

1 1 1

2 2 2 1

1

1:

2 : (1 )

:

_{T T T}

(1 )

_T

t c b a

t c b a r b

t T c b a r b

₋

= + =

= + = + +

= + = + +

 

用 t=1 和 t=2 式子來消去b 可得_{1 1} ^{2 2} ₁ ²

1 1 1

c b a

c a

r r r

+ + = +

+ + +

用上式和 t=3 來消去b 可得 ₂

( ) ( ) ( )

3 3 3

2 2

1 2 2 1 2

1 1 1 1 1

c b a

c a

c a

r r r r r

+ + + = + +

+ + + + + 重複上述過程可得

( )

¹

( )

¹

( )

¹

1

1 1

1

1

T T

t T t

t T t

t t

c b a

r

⁻

r

⁻

r

⁻

= =

+ =

+ + +

∑ ∑

期時上式右邊

( ) ( )

3 2

1 2

1 1

a a

a r r

+ + +

+ +
就是各期稟賦折算到今天的

價值總和。

Assume：

(

¹

)

^{1 0}

T T

b r ⁻ + =

b >_T 0⇒ not optimal b <_T 0⇒ not feasible ∴b_T = 0

T→ ∞

( )

¹

( )

¹

1

1

1

1

t t

t t

t t

c a

x

r r

∞ ∞

− −

= =

= =

+ +

∑ ∑

( II ) Decision Problem

Conditions

Given

{

at = yt−T rt^, t

}

t^∞₌₁

{ } ¹

( )

, 1

max

t t

t c b t

t

β u c

∞ −

=

∑

St c_t+b_t =a_t + +

(

1 r_t−1

)

b_t−1 ∀t b = ₀ 0

FOC

For any time t consider the trade off between c and _t c_t−1 holding other fixed

t 1

b

₋

b

t

t 1

b

₊

Period t Period t+1

c

t

c

t₊1

a

t

a

_t+1

fixed

( )

( )

1 1

1 1 1

1 1

t t t t t

t t t t t

c b a r b

c b a r b

− −

+ + +

+ = + +

+ = + +

1 1 1 1

t t t t t

c b S c₊ r

∆ = − ⇒ ∆ = ∆ = ⇒ ∆ = + Marginal cost of saving at t：u c^'

( )

t

Marginal gain of saving at t：βu c'

(

_t+1

)(

1+r_t

)

FOC：u c'

( )

_t =βu c'

( )(

_t+1 1+r_t

)

∀t

( )

1

(

2

)(

1

)

' _t ' _t 1 _t

u c₊ =βu c₊ +r₊

( )

²

(

2

) ( )(

1

)

' _t ' _t 1 _t 1 _t

u c =β u c₊ ^ +r +r₊ ^

Marginal gain：若今天存一塊，明天不花，後天才花，則到後天會有

(

1+r_t

)(

1+r_t+1

)

乘上 MU 後在折算回今天，就是上是右方。

1

(1 )

t t

b r

∆

+

= +

t t+1

t

1

∆ = − c

1

0

c

t₊

∆ =

t+2

2

(1 )(1

1

)

t

t t

b

r r

+

+

∆

= + +

In general：u c'

( )

_t =β^ju c'

( )

_t+j ^

⁽

1+r_t

⁾⁽

1+r_t+1

⁾

(

1+r_{t+ −j} 1

)

^ j=1, 2,

( ) ( )

,

^{( )(}

1

⁾ (

1

)

' 1 1 1

'

t

t t j t t t j

j t j

u c MRS r r r

β u c₊ <sup>= + = + + +
+ + −</sup>

Discuss

上面說今天存一塊，明天不花，後天才花。但若改成明天花一些，

後天在花一些，會不會又什麼不一樣?

Plane1：∆ = − 全留到 t+2 c_t 1

Plane2：∆ = − c_t 1 ∆c_t+1 = ，r_t ∆b_t+1= ，1 ∆c_t+2 = +1 r_t+1 每期都花一些

FOC：

Plane1：u c'

( )

_t =β²u c'

(

_t+2

) (

^ 1+r_t

)(

1+r_t+1

)

^ Plane2：u c'

( )

_t =βu c'

( )

_t+1 r_t +β²u c'

(

_t+2

)(

1+r_t+1

)

Pf：兩者一樣

( )

1

(

2

)(

1

)

' _t ' _t 1 _t

u c₊ =βu c₊ +r₊

∵

Plane2：u c'

( )

_t =βu c'

( )

_t+1 r_t +β²u c'

(

_t+2

)(

1+r_t+1

)

=β²u c'

(

_t+2

)(

1+r_t+1

)

r_t+β²u c'

(

_t+2

)(

1+r_t+1

)

=β²u c'

(

_t+2

) (

^1+r_t

)(

1+r_t+1

)

^ = Plane1

意義：不論如何分配資源的效果都一樣，因為終身效用已經極大化。

Summary

ISE：

(1) r ↑ known at time t _t ⇒c c_t^d, _t^d₊₁,c_t^d₊₂
,c_t^d_+j

r ↑ 表示儲蓄有利_t ⇒c_t^d ↓,c_t^d₊₁↑,c_t^d₊₂ ↑
,c_t^d_+j ↑

(2) r_t+j ↑ known at time t ⇒c c_t^d, _t^d₊₁,c_t^d₊₂
,c_t^d_+j

只要今天知道，今天行為就會改，不用等到 t=t+j

1 2 1 1

, , , , ,

d d d d d d

t t t t j t j t j

c c₊ c₊ c_{+ −} c₊ c_{+ +}

⇒ ↓ ↓ ↓
↓ ↓ ↑

(3) r_t−1↑ suddenly：分 lender & borrower 考慮 (和 Ch8 同)

c ₂

c₁ c1=c

c2=c

45 line0

1 slope= +r

1 slope= +ρ r=ρ

WE：

(1) a ↑ temporarily：_t c ↑ for all t _t^d S ↑ _t^d

(2) a ↑ permanently：_t c ↑ for all t _t^d S → _t^d Summary：

1 1

( ) ( ) ( ) ( )

, , ; , ,

d d

t t t t t

c c r r

₊

a a

₊

− − + +

 

=  

 

ISE WE

1 1 2

( ) ( ) ( ) ( ) ( )

, , ; , , ,

d d

t t t t t t

S c r r

₊

a a

₊

a

₊

+ + + − −

 

=  

 

ISE WE

( III ) Optimal Consumption Profile (Assume

r_t =r ∀t

)

Relation of r &ρ ( )

( ) ( )

1

' 1

' 1 1

t t

u c r

r t

u c β

+ ρ

= + = + ∀

+ ,r ρ為外生 (1) r =ρ

u c'

( )

_t =u c'

( )

_t+1 ∀t c_t = constant tc ∀

用 FOC + lifetime budget constrain 可以解所有變數 所以用 lifetime budget constrain 解 c

( )

¹

( )

²

1

1 1 1

1 1

1 1

t t t

c r

x c c

r r

r r

∞

−

=

  +

= =  + + + =

+  + + 

∑

1 c r x

∴ = r +

(2) r >ρ

u c'

( )

_t >u c'

( )

_t+1 ∀t

c_t <c_t+1 ∀ (邊際效用遞減) t

c

₂

c

₁

45 line

0

r > ρ r = ρ

r > ρ

slope = + 1 r

(3) r <ρ

u c'

( )

_t <u c'

( )

_t+1 ∀t

c_t >c_t+1 ∀ (邊際效用遞減) t

c

₂

c

₁

45 line

0

r < ρ

r=

ρ

r<

ρ

slope = + 1 r

r

_t

t r = ρ

1 10 20

c

_t

t 1

c r x

= r +

10 21

c

_t

t r > ρ

r = ρ

r < ρ

假設終身所得相同，則上方 3 條線積分面積相同。

(4) Example 1

t=1~9：r =ρ 所以各期消費相等但較少 c_t = < c c t=10~20：r >ρ c_t <c_t+1

t=21~ ∞ ：r =ρ 所以各期消費相等但較多 c_t =c'> c 因不考慮財富效果，所以右圖兩條線積分面積=終身財富=x 若考慮財富效果，右圖圖形形狀不變，只是上下移動而已。

(5) Example 2

^{u c}

( )

^=lnc

( )

( )

1 ¹

' 1

' 1

t t

t t

u c r c

u c ρ c

+ +

= + = +

消費成長率=lnc_t+1−lnc_t =ln 1

(

+r

)

−ln 1

(

+ρ

)

≈ −r ρ

( IV ) Permanent Income Hypothesis (Milton Friedman)

Review

{ } ¹

( )

, 1

max

t t

t c b t

t

β u c

∞ −

=

∑

St c_t+b_t =a_t + +

(

1 r_t−1

)

b_t−1 ∀t b = ₀ 0 Or ¹

(

¹

)

^{1 1}

(

¹

)

^{1 (1 ) 0 1}

t t

t t

t t

c a

r b x

r r

∞ ∞

− −

= =

= + + =

+ +

∑ ∑

^if rt = r

x ：Lifetime wealth at t=1 1

Milton Friedman

P T

t t t

y = y +y

y ：Observed (actual) income at time t t P

y ：Permanent income ex：工作所得 t T

y ：transitory income ex：中獎 t P

y 和t y 是無法分別觀測到的，Milton 認為消費只和_t^T y 有關，和_t^P y 無^T_t 關。

P

t t

c =φy φ=some constant (close to one)

t 1990

y

t P

y

t

c

t

P

y

t T

y

t

因為消費只和y 有關，所以_t^P y 都被儲蓄，_t^T y 有正有負都被儲蓄_t^T

⇒ consumption smoothing

Define：A person’s permanent income from a perspective of time t is an imputed or imaginary constant flow of income from t on, of which the discounted sum equals the person’s lifetime wealth at time t.

2 period case (

b =0 0

)

2

1 1

1 x a a

= + r

+ let y =permanent income

1 1

y y x

+ r =

+ 1 ₁

2

y r x

r +

 

=  + 

2

1 1

1 x a a

= + r

+ 代入 1 ₁ 1 ₂

2 2

y r a a

r r

+

   

= +  + +  真實所得的加權平均

c

₂

c

₁

1

y = y

P

a

2

45 line

0

slope = + 1 r B

2

y = y

P

A

a

1

1^T

0 y >

2

0

y <

T

每一期 y 相同 ⇒ 在45⁰線上 1 1

y y x

+ r =

+ ⇒ 在預算限制線上

所以可知所求在兩線交點

FOC：u c'

( )

1 =βu c'

( )(

2 1+r1

)

¹

β 1

= ρ

+ if r= ρ

1 2

c =c = y ∴ = φ 1