中級宏觀經(jīng)濟學(xué)-lecture07經(jīng)濟增長之一

IntermediatemacroeconomicsDai

Guang2005

SpringDepartmentof

EconomicsNanjing

University

ofFinance&

EconomicsLECTURE

SEVENEconomic

Growth

Ilearning

objectives學(xué)習(xí)封

濟的Solow

模型。Learn

theclosed

economy

Solow

model理解一個國家的生活水平如何依賴于它的儲蓄率和人口增長率。See

how

a

country’sstandard

of

living

depends

on

its

savingand

population

growth

rates黃金法則。Learn

how

to

use

the“GoldenRule”to

find

the

optimal

savings

rate

andcapital

stockEconomic

GrowthILECTURE

7slide

1模型The

Solow

ModelRobert

Solow,

MITwon

Nobel

Prize

for

contributions

tothe

study

of

economic

growth已經(jīng)成為增長理論的一個范式paradigm:widely

used

in

policy

makingben ark

against

whi

ostrecent

growth

theories

are

compared尋找經(jīng)濟在長期內(nèi)增長的決定因素Economic

GrowthILECTURE

7slide

2Economic

GrowthILECTURE

7slide

3Solow

model

和Lecture

3’s

model的不同K

不再固定is

no

longer

fixed:投資使它增加investment

causes

it

to

grow,折舊使它減少depreciation

causes

it

to

shrink.L

不再固定is

no

longer

fixed:人口增長使它增大population

growth

causes

itto

grow.消費函數(shù)更簡單The

consumption

function

issimpler.How

Solow

model

is

different

from

Chapter

3’s

model

Economic

GrowthILECTURE

7slide

4沒有G

和T

No

G

or

T(only

to

simplify

presentation;we

canstill

do

fiscalpolicy

experiments)外表不同Cosmeticdifferences.Economic

GrowthILECTURE

7slide

5生產(chǎn)函數(shù)The

production

function加總形式In

aggregate

terms:Y

=

F

(K,

L

)定義Define:y=Y/L

=人均產(chǎn)出k

=K/L

=人均資本假設(shè)規(guī)模收益不變constant

returns

to

scale:zY

=

F

(zK,

zL

)

for

any

z

>

0提出L,有：Y/L

=

F

(K/L

,

1)y

=

F

(k,

1)y

=

f(k)

其中

f(k)

=

F

(k,

1)The

production

function人均產(chǎn)出,y人均資本,k注意：這一生產(chǎn)函數(shù)的資本邊際產(chǎn)量是遞減的。Note:

thisproduction

function

exhibitsdiminishing

MPK.f(k)MPK

=f(k

+1)

–

f(k)1Economic

GrowthILECTURE

7slide

6國民收入恒等式The

national

e

identity

Y

=

C

+

I

(記住

假設(shè)沒有G

)以“人均per

worker”形式就是:y

=

c

+

i其中

c

=

C/L

且

i

=

I/LEconomic

GrowthILECTURE

7slide

7消費函數(shù)The

consumption

functions

=儲蓄率the

saving

rate,假設(shè)s

是一個外生變量is

anexogenous

parameterNote:

s

是唯一的一個不等于它的大寫形除以L

的小寫字母the

only

lowercasevariable

that

is

not

equal

toits

uppercase

version

divided

by

L消費函數(shù)Consumption

function:c

=

(1–s)y

(per

worker)Economic

GrowthILECTURE

7slide

8Economic

GrowthILECTURE

7slide

9儲蓄和投資Saving

and

investment儲蓄（人均）

=sy國民收入恒等式y(tǒng)

=c

+i就是:

i

=

y

–

c

=

sy

(投資

=

儲蓄,

likein

chap.

3!)于是有：i

=

sy

=

sf(k)產(chǎn)出，消費和投資Output,

consumption, and

investment

人均產(chǎn)出,y人均資本,kf(k)sf(k)k1y1i1c1Economic

GrowthILECTURE

7slide

10折舊Depreciation人均折舊,k人均資本,k

=折舊率=每期磨損掉的資本比率k1Economic

GrowthILECTURE

7slide

11資本積累Capital

accumulationThe

basic

idea基本思想:投資使資本存量增大，折舊使資本存量減少Investment

makesthe

capital

stock

bigger,depreciation

makes

it

smaller.Economic

GrowthILECTURE

7slide

12Economic

GrowthILECTURE

7slide

13Capital

accumulationChangein

capital

stock

= investment

–

depreciationk

=

i

–

k由

i

=

sf(k)

,

這就是:k

=

s

f(k)

–

kK

的變動方程式k

=

s

f(k)

–

kSolow

模型的

方程式?jīng)Q定了資本隨時間變化的行為…這樣，它也就決定了所有其它內(nèi)生變量的行為，因為它們都依賴于k.比如.,人均收入:y=f(k)人均消費:c=(1–s)

f(k)Economic

GrowthILECTURE

7slide

14Economic

GrowthILECTURE

7slide

15穩(wěn)態(tài)The

steady

statek

=

s

f(k)

–

k如果投資正好可以補償折舊If

investment

is

just

enough

to

coverdepreciation

即[sf(k)

=

k

],那么人均資本就保持不變:k

=

0.這一不變的值k*

就是穩(wěn)態(tài)資本存量。Thisconstant

value,denoted

k*,is

called

the

steadystate

capital

stock.The

steady

state投資和折舊人均資本,ksf(k)kk*Economic

GrowthILECTURE

7slide

16向穩(wěn)態(tài)的移動Moving

toward

the

steadystateInvestmentanddepreciationsf(k)kk

=

sf(k)

k折舊depreciationk投資investmentk1k*Capital

perworker,

kEconomic

GrowthILECTURE

7slide

17Moving

toward

the

steady

stateInvestmentanddepreciationsf(k)kk

=

sf(k)

kk1k*Capital

perworker,

kEconomic

GrowthILECTURE

7slide

18k向穩(wěn)態(tài)的移動Moving

toward

the

steadystateInvestmentanddepreciationsf(k)kk

=

sf(k)

kk1

k2k*Capital

perworker,

kEconomic

GrowthILECTURE

7slide

19k向穩(wěn)態(tài)的移動Moving

toward

the

steadystateInvestmentanddepreciationsf(k)kk

=

sf(k)

k投資investment折舊depreciationk2k*Capital

perworker,

kEconomic

GrowthILECTURE

7slide

20k向穩(wěn)態(tài)的移動Moving

toward

the

steadystateInvestmentanddepreciationCapital

perworker,

ksf(k)kk

=

sf(k)

kEconomic

GrowthILECTURE

7slide

21kk2

k3

k*向穩(wěn)態(tài)的移動Moving

toward

the

steadystateInvestmentanddepreciationCapital

perworker,

ksf(k)kk

=

sf(k)

kk3

k*總結(jié):只要k

<k*,投資就會超過折舊，且k

繼續(xù)向k*.增長。Economic

GrowthILECTURE

7slide

22Economic

GrowthILECTURE

7slide

23Have

a

try!畫出SOLOW模型，并標(biāo)出穩(wěn)態(tài)時的k*.大的經(jīng)濟初始在水平軸上，找出一個比k*資本存量，記為k1.用剛才的方法看看k會隨時間怎么變化。k

會向穩(wěn)態(tài)移動還是其它方向移動呢？例子A

numericalexampleK

L

K

1/

2L1

/

2生產(chǎn)函數(shù)Y

F

(K

,L)

集約形式:YEconomic

GrowthILECTURE

7slide

24

K

1/

2L

K

1/

2L1/

2L

L

將

y

=

Y/L

and

k

=

K/L

代入得到：y

f

(k

)

k

1/

2Economic

GrowthILECTURE

7slide

25A

numerical

example,

cont.假設(shè):s

=

0.3

=0.1初始資本存量k

=4.0slide

2向穩(wěn)態(tài)靠攏Approaching

the

SteadyStateAssumptions:

y

k

;

s

0.3;

0.1;initial

k

4.0Year6123k4.0004.2004.395y

c

i

δk2.000

1.400

0.600

0.4002.049

1.435

0.615

0.4202.096

1.467

0.629

0.440k0.2000.1950.189slide

2Approaching

the

Steady

State:A

Numerical

ExampleAssumptions:y

k

;s

0.3;

0.1;initial

k

4.0Yearkyciδkk14.0002.0001.4000.6000.4000.20024.2002.0491.4350.6150.4200.19534.3952.0961.4670.6290.4400.18944.5842.1411.4990.6420.4580.184…105.6022.3671.6570.7100.5600.150…257.3512.7061.8940.8120.7320.080…1008.9622.9942.0960.8980.8960.002…∞9.0003.0002.1000.9000.9000.0007Economic

GrowthILECTURE

7slide

28練習(xí)：解出穩(wěn)態(tài)

=

0.1,

and

y

=

k

1/2繼續(xù)假設(shè)：s

=

0.3,使用動態(tài)方程k

=

sf(k)

k解出穩(wěn)態(tài)的k,y,和c.答案穩(wěn)態(tài)的定義當(dāng)

k

0時的方程由假設(shè)而來k

0s

f

(k

*)

k

*0.3

k

*

0.1k

*k

*3

k

*

k

*解出有:

k

*

9

and

y

*

k

*

3最后,Economic

GrowthILECTURE

7slide

29c

*

(1

s

)y

*

0.7

3

2.1一次儲蓄率的增加An

increase

in

thesaving

rate一次儲蓄率的增加提高了投資……這導(dǎo)致資本存量向新的穩(wěn)態(tài)調(diào)整:Investmentanddepreciationk1ks2

f(k)s1

f(k)k

*

k

*2Economic

GrowthILECTURE

7slide

30Prediction:Higher

s

higher

k*.And

since

y

=

f(k)

,higher

k*

higher

y*

.這樣，Solow模型

具有更高儲蓄率和投資率的國家在長期內(nèi)具有更高的資本水平和更高的人均收入。Thus,

the

Solow

model

predicts

that

countrieswith

higher

rates

of

saving

and

investment

willhave

higher

levels

of

capital

and e

perworker

in

the

longrun.Economic

GrowthILECTURE

7slide

31EgyptChadPakistanIndonesiaZimbabweKenyaCameroonUgandaMexicoIvoryCoastBrazilPeruU.K.U.S.IsraelGermanyCanadaDenmarkFranceItalyJapanFinlandSingapore10,0001,000e

perin

1992(logarithmic

scale)100,0001000510152025

30

35

40Investmentas

percentage

of

output(average

1960

–1992)投資率和人均收入的國際Economic

GrowthILECTURE

7slide

32黃金法則：介紹The

Golden

Rule:introduction已經(jīng)知道不同的s會導(dǎo)致不同的穩(wěn)態(tài)，但是怎樣才能知道哪個才是“最好”的穩(wěn)態(tài)呢？經(jīng)濟福利依賴于消費，因此“最好”的穩(wěn)態(tài)應(yīng)該具有最高c*= (1–s)

f(k*)的人均消費水平:而一次s

的提高：導(dǎo)致了更高的k*

and

y*,這將導(dǎo)致一個更高的c*另一方面，它又減少了收入中消費的份額(1–s),這將導(dǎo)致一個更低的c*那么,怎樣才能找到一個

s

和

k*

最大化

c*Economic

GrowthILECTURE

7slide

33黃金水平的資本存量goldk

*

the

Golden

Rule

levelofcapital,穩(wěn)態(tài)時k

的值，它最大化消費.為了找出這一黃金水平,首先將c*

表達成k*

的函數(shù):=c*

y*

i*i*=

f(k*)

=

f(k*)

k*一般地:i

=

k

+

k穩(wěn)態(tài)中有:i*=

k*因為k

=0.Economic

GrowthILECTURE

7slide

34The

Golden

Rule

Capital

Stock然后,畫出f(k*)和k*,并找出它們之間距離最大的那一點.穩(wěn)態(tài)時的產(chǎn)出和折舊穩(wěn)態(tài)時的人k*f(k*)goldk*goldc

*goldgoldi

*

k

*goldgoldy

*

f

(k

*

)均資本,k*slide

35Economic

GrowthILECTURE

7The

Golden

Rule

Capital

Stock穩(wěn)態(tài)時的人k*f(k*)goldk

*gold均資本,k*slide

36Economic

GrowthILECTURE

7c

*當(dāng)生產(chǎn)函數(shù)的斜率等于折舊曲線的斜率時，c*

=f(k*)

k*就最大化了:MPK

=

向黃金法則穩(wěn)態(tài)水平的轉(zhuǎn)移首先，經(jīng)濟并沒有向黃金法則穩(wěn)態(tài)水平靠攏的趨勢。要達到這一黃金法則要求政策制定者調(diào)整s.這一調(diào)整必導(dǎo)致一個具有更高消費水平的新的穩(wěn)態(tài)。.但是，在一調(diào)整過

，消費發(fā)生了什么樣的變化呢？Economic

GrowthILECTURE

7slide

37以過多資本開始Starting

with

too

much

capital

那么提高c*要求降低s.在這一向黃金穩(wěn)態(tài)的調(diào)整過,消費在所有的時點上都變得更高了.goldIf

k

*

k

*timet0ciyEconomic

GrowthILECTURE

7slide

38以資本過少開始Starting

with

too

little

capital

要提高

c*

就要求提高

s.未來的人口將享受更高的消費,但現(xiàn)在的人口必須經(jīng)歷一次初始消費水平的下降。goldIf

k

*

k

*t0timeiycEconomic

GrowthILECTURE

7slide

39小結(jié)這個簡單的solow模型不能解釋持續(xù)的經(jīng)濟

增長。

它僅僅是說更高的儲蓄率會導(dǎo)致暫時的經(jīng)濟增長，但是經(jīng)濟最終會像穩(wěn)態(tài)靠近。為了解釋持續(xù)的經(jīng)濟增長，

試著考慮其它的增長來源：人口

and

技術(shù)進步。Economic

GrowthILECTURE

7slide

40人口增長Population

Growth假設(shè)人口，即勞動力，以不變的n

增長.

(n是外生變量)LLEconomic

GrowthILECTURE

7slide

41

n解釋:

假設(shè)在第一年

L

=

1000

且增長率為2%/year

(n

=

0.02).那么

L

=

n

L

=

0.02

1000

=

20,因此第二年人口L

=1020。Economic

GrowthILECTURE

7slide

42收支相抵的投資Break-even

investment(

+n)k

=收支相抵的投資break-eveninvestment,即使k

保持不變的投資水平。收支相抵的投資包括:k

替代資本的磨損部分nk

補充給新工人的資本動態(tài)方程The

equation

of

motion

for

k在有人口增長時,k的動態(tài)方程變?yōu)閗

=

sf(k)

(

+

n)

k實際投資收支相抵的投資Economic

GrowthILECTURE

7slide

43模型The

Solow

Model

diagram收支相抵的投資水平人均資本,k(

+

n

)ksf(k)k*k

=

s

f(k)

(

+n)kEconomic

GrowthILECTURE

7slide

44人口增長的影響The

impact

of

population

growth

Investment,break-eveninvestmentCapital

perworker,

kk1*(

+n2)k(

+n1)ksf(k)k2*一次n

的增加導(dǎo)致了收支相抵投資水平的下降，這導(dǎo)致了更低的穩(wěn)態(tài)資本k.Economic

GrowthILECTURE

7slide

45Prediction:更高的n

更低的k*.由y

=f(k),更低的

k*

更低的

y*.于是，

模型

具有更高人口增長率的國家在長期將會具有更低的資本水平和人均收入。Economic

GrowthILECTURE

7slide

46ChadKenyaZimbabweCameroonPakistanUgandaIndonesiaIsraelMexicoBrazilPeruEgyptSingaporeU.S.U.K.CanadaFinlandJapanFranceIvoryCoastGermanyDenmarkItaly10,0001,000121000e

perin

1992(logarithmic

scale)100,0003

4Population

growth

(percent

per

year)(average

1960

–1992)人口增長率和人均收入的國際Economic

GrowthILEC