中级宏观经济学-lecture07经济增长之一

IntermediatemacroeconomicsDai

Guang2005

SpringDepartmentof

EconomicsNanjing

University

ofFinance&

EconomicsLECTURE

SEVENEconomic

Growth

Ilearning

objectives学习封

济的Solow

模型。Learn

theclosed

economy

Solow

model理解一个国家的生活水平如何依赖于它的储蓄率和人口增长率。See

how

a

country’sstandard

of

living

depends

on

its

savingand

population

growth

rates黄金法则。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已经成为增长理论的一个范式paradigm:widely

used

in

policy

makingben ark

against

whi

ostrecent

growth

theories

are

compared寻找经济在长期内增长的决定因素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:投资使它增加investment

causes

it

to

grow,折旧使它减少depreciation

causes

it

to

shrink.L

不再固定is

no

longer

fixed:人口增长使它增大population

growth

causes

itto

grow.消费函数更简单The

consumption

function

issimpler.How

Solow

model

is

different

from

Chapter

3’s

model

Economic

GrowthILECTURE

7slide

4没有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生产函数The

production

function加总形式In

aggregate

terms:Y

=

F

(K,

L

)定义Define:y=Y/L

=人均产出k

=K/L

=人均资本假设规模收益不变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人均产出,y人均资本,k注意：这一生产函数的资本边际产量是递减的。Note:

thisproduction

function

exhibitsdiminishing

MPK.f(k)MPK

=f(k

+1)

–

f(k)1Economic

GrowthILECTURE

7slide

6国民收入恒等式The

national

e

identity

Y

=

C

+

I

(记住

假设没有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消费函数The

consumption

functions

=储蓄率the

saving

rate,假设s

是一个外生变量is

anexogenous

parameterNote:

s

是唯一的一个不等于它的大写形除以L

的小写字母the

only

lowercasevariable

that

is

not

equal

toits

uppercase

version

divided

by

L消费函数Consumption

function:c

=

(1–s)y

(per

worker)Economic

GrowthILECTURE

7slide

8Economic

GrowthILECTURE

7slide

9储蓄和投资Saving

and

investment储蓄（人均）

=sy国民收入恒等式y

=c

+i就是:

i

=

y

–

c

=

sy

(投资

=

储蓄,

likein

chap.

3!)于是有：i

=

sy

=

sf(k)产出，消费和投资Output,

consumption, and

investment

人均产出,y人均资本,kf(k)sf(k)k1y1i1c1Economic

GrowthILECTURE

7slide

10折旧Depreciation人均折旧,k人均资本,k

=折旧率=每期磨损掉的资本比率k1Economic

GrowthILECTURE

7slide

11资本积累Capital

accumulationThe

basic

idea基本思想:投资使资本存量增大，折旧使资本存量减少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)

,

这就是:k

=

s

f(k)

–

kK

的变动方程式k

=

s

f(k)

–

kSolow

模型的

方程式决定了资本随时间变化的行为…这样，它也就决定了所有其它内生变量的行为，因为它们都依赖于k.比如.,人均收入:y=f(k)人均消费:c=(1–s)

f(k)Economic

GrowthILECTURE

7slide

14Economic

GrowthILECTURE

7slide

15稳态The

steady

statek

=

s

f(k)

–

k如果投资正好可以补偿折旧If

investment

is

just

enough

to

coverdepreciation

即[sf(k)

=

k

],那么人均资本就保持不变:k

=

0.这一不变的值k*

就是稳态资本存量。Thisconstant

value,denoted

k*,is

called

the

steadystate

capital

stock.The

steady

state投资和折旧人均资本,ksf(k)kk*Economic

GrowthILECTURE

7slide

16向稳态的移动Moving

toward

the

steadystateInvestmentanddepreciationsf(k)kk

=

sf(k)

k折旧depreciationk投资investmentk1k*Capital

perworker,

kEconomic

GrowthILECTURE

7slide

17Moving

toward

the

steady

stateInvestmentanddepreciationsf(k)kk

=

sf(k)

kk1k*Capital

perworker,

kEconomic

GrowthILECTURE

7slide

18k向稳态的移动Moving

toward

the

steadystateInvestmentanddepreciationsf(k)kk

=

sf(k)

kk1

k2k*Capital

perworker,

kEconomic

GrowthILECTURE

7slide

19k向稳态的移动Moving

toward

the

steadystateInvestmentanddepreciationsf(k)kk

=

sf(k)

k投资investment折旧depreciationk2k*Capital

perworker,

kEconomic

GrowthILECTURE

7slide

20k向稳态的移动Moving

toward

the

steadystateInvestmentanddepreciationCapital

perworker,

ksf(k)kk

=

sf(k)

kEconomic

GrowthILECTURE

7slide

21kk2

k3

k*向稳态的移动Moving

toward

the

steadystateInvestmentanddepreciationCapital

perworker,

ksf(k)kk

=

sf(k)

kk3

k*总结:只要k

<k*,投资就会超过折旧，且k

继续向k*.增长。Economic

GrowthILECTURE

7slide

22Economic

GrowthILECTURE

7slide

23Have

a

try!画出SOLOW模型，并标出稳态时的k*.大的经济初始在水平轴上，找出一个比k*资本存量，记为k1.用刚才的方法看看k会随时间怎么变化。k

会向稳态移动还是其它方向移动呢？例子A

numericalexampleK

L

K

1/

2L1

/

2生产函数Y

F

(K

,L)

集约形式:YEconomic

GrowthILECTURE

7slide

24

K

1/

2L

K

1/

2L1/

2L

L

将

y

=

Y/L

and

k

=

K/L

代入得到：y

f

(k

)

k

1/

2Economic

GrowthILECTURE

7slide

25A

numerical

example,

cont.假设:s

=

0.3

=0.1初始资本存量k

=4.0slide

2向稳态靠拢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练习：解出稳态

=

0.1,

and

y

=

k

1/2继续假设：s

=

0.3,使用动态方程k

=

sf(k)

k解出稳态的k,y,和c.答案稳态的定义当

k

0时的方程由假设而来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

2.1一次储蓄率的增加An

increase

in

thesaving

rate一次储蓄率的增加提高了投资……这导致资本存量向新的稳态调整: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*

.这样，Solow模型

具有更高储蓄率和投资率的国家在长期内具有更高的资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)投资率和人均收入的国际Economic

GrowthILECTURE

7slide

32黄金法则：介绍The

Golden

Rule:introduction已经知道不同的s会导致不同的稳态，但是怎样才能知道哪个才是“最好”的稳态呢？经济福利依赖于消费，因此“最好”的稳态应该具有最高c*= (1–s)

f(k*)的人均消费水平:而一次s

的提高：导致了更高的k*

and

y*,这将导致一个更高的c*另一方面，它又减少了收入中消费的份额(1–s),这将导致一个更低的c*那么,怎样才能找到一个

s

和

k*

最大化

c*Economic

GrowthILECTURE

7slide

33黄金水平的资本存量goldk

*

the

Golden

Rule

levelofcapital,稳态时k

的值，它最大化消费.为了找出这一黄金水平,首先将c*

表达成k*

的函数:=c*

y*

i*i*=

f(k*)

=

f(k*)

k*一般地:i

=

k

+

k稳态中有:i*=

k*因为k

=0.Economic

GrowthILECTURE

7slide

34The

Golden

Rule

Capital

Stock然后,画出f(k*)和k*,并找出它们之间距离最大的那一点.稳态时的产出和折旧稳态时的人k*f(k*)goldk*goldc

*goldgoldi

*

k

*goldgoldy

*

f

(k

*

)均资本,k*slide

35Economic

GrowthILECTURE

7The

Golden

Rule

Capital

Stock稳态时的人k*f(k*)goldk

*gold均资本,k*slide

36Economic

GrowthILECTURE

7c

*当生产函数的斜率等于折旧曲线的斜率时，c*

=f(k*)

k*就最大化了:MPK

=

向黄金法则稳态水平的转移首先，经济并没有向黄金法则稳态水平靠拢的趋势。要达到这一黄金法则要求政策制定者调整s.这一调整必导致一个具有更高消费水平的新的稳态。.但是，在一调整过

，消费发生了什么样的变化呢？Economic

GrowthILECTURE

7slide

37以过多资本开始Starting

with

too

much

capital

那么提高c*要求降低s.在这一向黄金稳态的调整过,消费在所有的时点上都变得更高了.goldIf

k

*

k

*timet0ciyEconomic

GrowthILECTURE

7slide

38以资本过少开始Starting

with

too

little

capital

要提高

c*

就要求提高

s.未来的人口将享受更高的消费,但现在的人口必须经历一次初始消费水平的下降。goldIf

k

*

k

*t0timeiycEconomic

GrowthILECTURE

7slide

39小结这个简单的solow模型不能解释持续的经济

增长。

它仅仅是说更高的储蓄率会导致暂时的经济增长，但是经济最终会像稳态靠近。为了解释持续的经济增长，

试着考虑其它的增长来源：人口

and

技术进步。Economic

GrowthILECTURE

7slide

40人口增长Population

Growth假设人口，即劳动力，以不变的n

增长.

(n是外生变量)LLEconomic

GrowthILECTURE

7slide

41

n解释:

假设在第一年

L

=

1000

且增长率为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收支相抵的投资Break-even

investment(

+n)k

=收支相抵的投资break-eveninvestment,即使k

保持不变的投资水平。收支相抵的投资包括:k

替代资本的磨损部分nk

补充给新工人的资本动态方程The

equation

of

motion

for

k在有人口增长时,k的动态方程变为k

=

sf(k)

(

+

n)

k实际投资收支相抵的投资Economic

GrowthILECTURE

7slide

43模型The

Solow

Model

diagram收支相抵的投资水平人均资本,k(

+

n

)ksf(k)k*k

=

s

f(k)

(

+n)kEconomic

GrowthILECTURE

7slide

44人口增长的影响The

impact

of

population

growth

Investment,break-eveninvestmentCapital

perworker,

kk1*(

+n2)k(

+n1)ksf(k)k2*一次n

的增加导致了收支相抵投资水平的下降，这导致了更低的稳态资本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

具有更高人口增长率的国家在长期将会具有更低的资本水平和人均收入。Economic

GrowthILECTURE

7slide

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

perin

1992(logarithmic

scale)100,0003

4Population

growth

(percent

per

year)(average

1960

–1992)人口增长率和人均收入的国际Economic

GrowthILEC