高级宏观经济学课件(厦门大学,龚敏) chap 1

Lecture 1 The Solow Growth Model Romer(2001), Advanced Macroeconomics, Chapter 1 Blanchard, Olivier J., 2nd edition, Macroeconomics, Prentice Hall

曼昆，《宏观经济学》，人民大学，2000年巴罗，萨拉伊马丁，《经济增长》，中国社会科学出版社，2000年

Some Basic Facts about Economic Growth Assumptions of One-Sector Growth Models The Dynamics of the Model

The Impact of a Change in the Saving Rate The Speed of Convergence

The Solow Model and the Central Questions of Growth Theory

Empirical Applications

一、Some Basic Facts about Economic

Growth: the stylized facts of growth

1. Economic growth through deep time(F2)

2. The stylized facts

? 0,>>y L Y （F3）

? o k L K >> ,(F4)

? t cons Y K Y K tan ,== (F5)

? t cons Y

P Y W t cons Y

P t cons K P tan 1,tan ,tan =-====ρ

3. international differences in the

standard of living (F6)

4. The Solow growth model

The ultimate objective of research on economic growth is to determine whether there are possibilities for raising overall growth or bringing standards of living in poor countries closer to those in the world

leaders.

The Solow model is the starting point for almost all analyses of growth.

The principal conclusion of the Solow model is that the accumulation of

physical capital cannot account for either the vast growth over time in output per person or the cast geographic differences in output per person. The model treats other potential sources of differences in real incomes as either exogenous and thus not explained by the model.( in the case of technological progress) or absent altogether(in the case of positive externalities from capital).

二、 Basic assumptions of one-sector

growth models

1． Input and Output

),(t t t t L A K F Y

1) Labor-augmenting or

Harrod-neutral

),(t t t t L K A F Y = capital

augmenting

),(t t t t L K F A Y = Hicks-ueutral

2) The ratio of capital to output, K/Y, eventually settles down

3) Constant returns to scale

(first-degree homogeneity)

? The economy is big enough that the gains from specialization

have been exhausted.

? Inputs other than capital, labor, and knowledge are relatively

unimportant.

4) intensive form :

? The amount of output per unit of effective labor depends only

on the quantity of capital per

unit of effective labor, and not

on the overall size of the

economy.

? output per worker

)(k Af AL

Y A L Y == 5) Assumptions

0)0(=f

0,0<''>'f f

0)(lim )(lim 0

='∞='∞→→k f k f k k ( Inada conditions)

? The path of the economy does

not diverge

? Cobb-Douglas function

αα==k AL

K k f )()( 2． the evolution of the input into production

1) Labor and knowledge grow at

constant rates:

nt L L nL L e

L L t t t

nt t +=?=?=00ln )ln( gt L A gA A e

A A t t

t gt t +=?=?=00ln )ln(

2) Investment and capital formation t t t t t sY S I dt

dK dt dK I S G

I T S =====+=+ t t t K sY K δ-=

三、 The simple dynamics of growth model

1． The Harrod-Domar model

1) The fixed-coefficients production

function

??????αυ=t t t L K Y ,min

2) Saving, investment, and the warranted rate of growth

υ

==s I Y 3) Labor force growth:g n Y +=

4) The Harrod-Domar condition:υ=+s g n

2． The Solow model

1) Cobb-Douglas function(F7)

k

k y AL

K AL Y

K Y k k MPk k MPk k AL

Y y AL K Y )(10)1()(0

)(211===υ<-αα=??>α====-α-αα

α

-α

2) The dynamics of k

g y L

Y k f Le

K f Le Y E Y y E K F Y e L E gt gt t t t t

g n t +=======+ )()()()

,()(0

O ne way: actual and break-even investment

k g n k sf k dt

dk t t )()(δ++-== 0)()(>?δ++>k k g n k sf t

, k is rising 0)()(>?δ++

k g n k sf t , k is falling 0)()(>?δ++=k

k g n k sf t , k is constant *)(*)(k g n k sf δ++=

T he second way: speed

k g n k sf k dt

dk t t )()(δ++-== )()()ln(δ++-=g n k

k sf dt k d t t )(*

*)(δ++=g n k k sf )()ln(1δ++-=-αg n sk dt

k d t t )]([)ln()()ln(1δ++-?α=α==-ααg n sk dt

k d dt k dlin dt y d t t t t

3) the balanced growth path

*****

****

**

**)

(*k A L K y A L Y y k Y K k L A K Y L A Y k f y t t t t t

t t t t t t t t t ======

**)(k s

g n k f ++δ=

)]()([)()()),((t t t t t t

t t t t t t t k f k k f A k f L K k f A dL L A K F d w '-='-== )()),((t t t t t k f dK

L A K F d r '== t t t t t Y K r L w =+

The Solow model implies that,

regardless of its starting point, the

economy converges to a balanced

growth path --- a situation where each

variable of the model is growing at a

constant rate. On the balanced

growth path, the growth rate of

output per worker is determined

solely by the rate of technological

progress.

四、The impact of a change in the saving

rate

The division of the government’s

purchanses between consumption and

investment goods, the division of its

revenues between taxed and borrowing,

and its tax treatments of saving and

investment are all likely to affect the fraction of output that is invested.

1. the impact on output(F24,F25)

A change in the saving rate has a level effect but not a growth effec t: it changes the economy’s balanced growt h path, and thus the level of output per worker at any point in time, but it does not affect the growth rate of output per worker in the balanced growth path.

In the Solow model only changes in the rate of technological progress have growth effects; all other changes have only level effects .

2. saving and consumption (welfare) )()()

()1(t t t t k sf k f k f s c -=-=

*

)(*)(*)(*)(*)

()1(*);

(**k g n k f k sf k f k f s c s k k δ++-=-=-== s

g n s k g n g n s k f s c ?δ?δ++-δ'=??),,,(*)](),,,(*([*if )(),,,(*(δ++>δ'g n g n s k f ,

0*>??s c if )(),,,(*(δ++<δ'g n g n s k f ,

0*

c the golden-rule level of the capital stock: Consumption is at its maximum possible level among balance

d growth paths . δ++=δ'=g n g n s k f MPK )),,,(*(

Quantitative Implications

1． The effect of s on output in the long run s

g n s k k f s y ?δ?'=??),,,(**)(*

),,,(*)()),,,(*(δδ++=δg n s k g n g n s k sf 0*)

()(*)(*>'-δ++=??k f s g n k f s k

?

*)](/*)(*[1*)(/*)(***k f k f k k f k f k s y y s '-'=???

*)(/*)(*k f k f k 'is the elasticity of output with respect to capital at *k k =. Denoting this by *)(k K α

*)(1*)(**k k s y y s K K α-α=???

If markets are competitive and there are no externalities, *)(/*)(*k f k f k ' is the share of total income that goes to capital on the balanced growth path.

3

1*)(≈αk K 21**=????s y y s A 10% increase in the saving rate (from

20% of output to 22%) raises output per worker in the long run by about 5% relative to the path it would have followed.

Significant changes in saving have only moderate effects on the level of output on the balanced growth path.

2． The speed of convergence

1) One way

)()(δ++-==γg n k

k sf k k k k k f )(资本的平均产品。

资本报酬递减+Inada 条件使0))((

()(：所有者的人均收入占人均总收入的份额

c

k y k g n sk γ=αγ=γδ++-=γα--)

()1( 2) The second way

k g n k sf k k

)()()(δ++-=

When k equals k *, 0=k

. A first-order Taylor-series approximation of k

around k =k * yields:

*])([*)()(*k t k k k k k k k k k -λ-≈-???

????????= ? I n the vicinity of the balanced growth path, k moves toward k * at a speed approximately proportional to its distance from k *. ? In addition, one can show that y approaches y * at the same rate that k approaches k *.

*][*0k k e

k

k t t -+≈λ-

*][*0y y e y y t t -≈-λ- ? T he half-life

The half-life is roughly equal to 70 divided by its growth rate in percent. More exactly,

5.0*=λ-t e

λ-=/)5.0ln(*t

%4%4~3%;2~1%;2~1≈λ?≈δ≈=≈y

y g n therefore k and y move 4% of the remaining distance toward k * and y * each year, and take approximately 18 years to get halfway to their balanced-growth-path values.

Cobb-Douglas 函数k, y 的收敛速度： )()1(δ++?α-=λg n

? N ot only is the overall impact of a substantial change in the saving rate modest, but it does not occur very quickly

3． D

4．

1)

2) 绝对收敛与条件收敛（F42）

2. 内生增长理论

1) AK model

c

y k g n sA A dK dY

K Y

AK

Y γ=γ=δ++-=γ===)((F43)：s ，

A 影响人均增长率；没有趋同。

2) 规模报酬递增：(F44)

)

(/)(0lim 1δ+-=γ>=+=∞→α

-αn k k sf A MPK L BK AK Y k k (F45)

3) CES function

linear

Douglas

Cobb L b bK Y L b a bK a A Y ∞→?-→?-→?-→?-=→?--∞→?-?-+??=?

??11

1111

0]

)1(,min[011

}])1[()1()({1 )(*)

(/)(1δ+-=γδ+-=γ?n sAba n k k sf k (F46)

? Case 1 δ

+>

+<

? Case 3 δ+<

3.贫困陷阱（F47）

五、The Solow Model and the Central

Questions of Growth Theory

The Solow model identifies two possible sources of variation---either over time or across parts of the world---in output per worker: differences in capital per worker and differences in the effectiveness of labor. We have seen, however, that only growth in the effectiveness of labor can lead to permanent growth in output per worker, and that for reasonable cases the impact of changes in capital per worker on output per worker is modest. As a result, only differences in the effectiveness of labor have any reasonable hope of accounting for the vast differences in wealth across time and space.

If output per worker differs by a factor of X, and the elasticity of output per worker with respect to capital per worker is K , log

capital per worker must differ by K X αln . That is, capital per worker differs by a factor of K X e )(ln or K

X α1. 100031,10?=α=K X

In sum, differences in capital per worker are far smaller than those needed to account for the differences in capital per worker that we are trying to understand.

The other potential source of variation on output per worker in the Solow model is the effectiveness of labor.

Empirical Applications

1. Growth accounting

A A L L K K Y Y ?+?α-+?α=?)1((F38)

◆ Solow residual ：L

L K K Y Y A A ?α--?α-?=?

)1(（F48,39）

It reflects all sources of growth other than the contribution of capital accumulation vir its private return.

◆ Denison(1967): different types of

capital and labor

◆Young(1995): the exceptionally rapid growth of the NICs

◆Hsieh(1998a): to exam the behavior of factor returns rather than quantities ◆Oliner and Sichel(2000), Jorgenson and Stiroh(2000), Whelan(2000): U.S. productivity growth rebound

2.Convergence

Growth is whether poor countries tend to grow faster than rich countries.

1)The Solow model predicts countries converge to their balanced growth paths

2)The Solow model implies that the rate of return on capital is lower in countries with more capital per worker 3)If there are lags in the diffusion of knowledge, income differences might tend to shrink as poorer countries gain access to state-of-the-art methods.

◆Baumol(1986):