notes on the solow modelpfwang.people.ust.hk/advanced macro lecture note 2-solow...notes on the...

Notes on the Solow Model

Pengfei Wang

Hong Kong University of Science and Technology

2010

pfwang (Institute) Notes on the Solow Model 03/09 1 / 32

Introduction: Basic Facts about Economic Growth

Small di¤erence in growth rate can make large di¤erence in income inthe long run.


Introduction: Basic Facts about Economic Growth(continued)


Questions to be Addressed

How important is the rate of capital accumulation to a nationseconomic growth?

What are the economic forces that ultimately allow poor countries tocatch up with the richest countries in living standards?

How does a nations growth rate evolve over time? Is there a limit? Ifso, what determines the limit the rate of saving? the rate ofpopulation growth? or the amount of natural resources?


Assumption

There is only a single good, so relative prices do not play any role;

There is no money and no government spending, so government doesnot play any role;

Full employment at all time, so unemployment does not matter;

Except inputs, the production technology does not change over time;

There are only two types of inputs: capital and labor;

The rates of saving, depreciation, population growth, and technologyprogress are constant.

These features are both defects and virtues of the model.


Assumption (continued)

More specically and mathematically, we have

Production functionY = F (Kt ,AtLt ) (1)

where K denotes capital stock, A denotes labor productivity, Ldenotes labor units.

Properties:

F01 > 0,F

02 > 0,F

0012 > 0,F

0021 > 0,F

001 < 0,F

002 < 0 (2)

F (λK ,AλL) = λF (K , L) for λ > 0

limx!0F 0(x , y) = ∞; limy!0F 0(x , y) = ∞ (3)limx!∞F 0(x , y) = 0; limy!∞F 0(x , y) = 0


Assumption (continued)

More specically and mathematically, we have

Resource Allocation

St = It = sYt (4)

Ct + It = Yt

Evolution of Inputs

K̇t = It � δKt (5)Ȧt = gAtL̇t = nLt


Model Dynamics

Per-e¤ective labor economy: Dene x = XAL then

y =F (K ,AL)AL

= F (KAL, 1) = f (k) (6)

c + i = y (7)

k̇ + (g + n)k = i � δk (8)where

k̇ =d( KAL )dt

=K̇AL� K (ȦL+ AL̇)

(AL)2=K̇AL� (g + n)k (9)

andK̇AL

= i � δk (10)


Model Dynamics (continued)

the property of f (k) = F ( KAL , 1)

f 0(k) = F01(K ,AL) (11)

f 0(k) =∂hF (K ,AL)AL

i∂( KAL )

=∂F (K ,AL)

∂K= F1(K ,AL) > 0 (12)

f00(k) =

∂F1(K ,AL)

∂( KAL )= ALF11(K ,AL) < 0 (13)


Steady-State

A steady state of a dynamic system is a situation where all the variables inthe system are constant, xt = x̄ . Assuming that a steady state exists inthe transformed Solow model, then

k̇ = ċ = ẏ = i̇ = 0, (14)

i.eXAL

= const for X = K ,Y ,C , I (15)

soK̇K=ẎY=ĊC=İI= g + n (16)

this is so called balanced growth path.


Stabability of Steady-State

In S-S, (8) impliesı̄ = (δ+ g + n)k̄ (17)

andk̇t = sf (kt )� (g + n+ δ)kt = π(kt ) (18)

the steady-state k̄ is given by

sf (k̄) = (g + n+ δ)k̄ (19)

Noticeπ0(k) = sf 0(k)� (g + n+ δ);π00(k) = sf 00(k) < 0 (20)

so we havek̇t > 0 if kt < k̄ ;k̇t < 0 if kt > k̄ (21)


Stabability of Steady-State

The phase diagram


E¤ects of Saving rate

In S-S we havesf (k̄) = (g + n+ δ)k̄. (22)

totally di¤erentiating this equation gives

f (k)ds + sf 0(k)dk = (g + n+ δ)dk, (23)

which implies

dkds=

f (k)(g + n+ δ)� sf (0k) =

f (k)

s f (k )k � sf 0(k)(24)

ordk/kds/s

=f (k)

f (k)� f 0(k)k =1

1� α > 0 (25)


E¤ects of Saving rate (continued)

Note 1: α is output elasticity of capital, we must have

0 < α < 1 (26)

Proof:

α =f 0(k)kf (k)

> 0 (27)

is easy to see.



To see α < 1, we have

α =F 01(K ,AL)KF (K ,AL)

(28)

Notice by constant return to scale, we have

F (λK ,λAL) = λF (K ,AL) (29)

totally di¤erentiating with respect to λ, and evaluated λ = 1

F 01(K ,AL)K + F02(K ,AL)AL = Y (30)

and F02 > 0, so we have

α < 1 (31)



E¤ect on outputdyds=dfdkdkds

(32)

we havedy/yds/s

= (dfdkky)(dkdssk) =

α

1� α > 0



E¤ect on consumption

c = (1� s)f (k) (33)

we have

dcds

= �f (k) + (1� s)f 0 dkds

= �f (k) + (1� s)(kff 0)fk(dkdssk)ks

= �f (k) + (1� s) α1� α

f (k)s

(34)

so we have

dc/cds/s

=

��f (k) + (1� s) α

1� αf (k)s

�s

(1� s)f (k)= � s

1� s +α

1� α (35)



The golden rule: The golden rule is the best S-S where S-Sconsumption is maximized. Since S-S consumption depends on thesaving rate, we can nd the golden rule rate of saving such that S-Sconsumption is maximized:

dcds= 0 (36)

ors = α (37)


Growth E¤ects of Saving rate (continued)

Growth E¤ect:k̇ = sf (k)� (g + n+ δ)k (38)

dk̇ds

= f (k) + sf 0(k)dkds� (g + n+ δ)dk

ds

= f (k) + [f 0(k)k � (g + n+ δ)ks

]dkdssk

= f (k) + (f 0(k)k � f (k)) 11� α

= f (k) + f (k)α� 11� α

= 0 (39)


Transitional Dynamics

Algebraic analysis:

k̇t = sf (kt )� (g + n+ δ)kt' (sf 0(k̄)� (g + n+ δ))(kt � k̄) (40)

since sf 0(k̄) = sf 0(k̄) k̄f̄f̄k̄ = α

s f̄k̄ = α(g + n+ δ), we have

k̇t ' �(1� α)(g + n+ δ)(kt � k̄) (41)� �λ(kt � k̄)

this says that near S-S, the growth rate is approximately constant andequal to �λ. Hence λ = (1� α)(g + n+ δ) is called the speed ofconvergence.solving this equation implies

ln(kt � k̄) + const = �λt (42)notice that k0 is given so we have

ln(k0 � k̄) + const = 0 (43)hence

ln(kt � k̄) = ln(k0 � k̄)� λt (44)or

kt � k̄ = (k0 � k̄)e�λt (45)


Transitional Dynamics

solving this equation implies

ln(kt � k̄) + const = �λt (46)

notice that k0 is given so we have

ln(k0 � k̄) + const = 0 (47)

henceln(kt � k̄) = ln(k0 � k̄)� λt (48)

orkt � k̄ = (k0 � k̄)e�λt (49)


Transitional Dynamics (continued)

Now consider a small change of s on kt

dktds

=dk̄ds(1� e�λt ) � 0 (50)

which is zero at t = 0 and increases in a diminishing manner towardsd k̄ds with t.

growth e¤ect on k̇t , by k̇t = �λ(kt � k̄) we have

dk̇tds

' �λ(dktds� dk̄ds) (51)

� λe�λt dk̄ds

which is positive at t = 0 and decreases exponentially towards zerowith t.



The e¤ect on outputyt = f (kt )

henceyt ' f (k̄) + f 0(k̄)(kt � k̄) (52)

so we have

ẏt ' f 0(k̄)k̇t� �λf 0(k̄)(kt � k̄)' �λ(yt � ȳ) (53)



Since we haveẏt ' �λ(yt � ȳ) (54)

so we can solveyt � ȳ = (y0 � ȳ)e�λt . (55)

so we havedytds

= (1� e�λt )dȳds, (56)

which is similar to the analysis to capital.



The e¤ect on consumption

ct = yt � (g + n+ δ)kt � k̇t (57)

so we have

dctds

=dytds� (g + n+ δ)dkt

ds� dk̇tds

= (1� e�λt )dȳds� (g + n+ δ)dk̄

ds(1� e�λt )� λe�λt dk̄

ds(58)

which is always negative at t = 0.


A micro-foundation of aggregate production function

rms: Suppose there is a continuum of rms measured by i 2 [0, 1].Its production function is

y(i) = ε(i)min [k(i),An(i)] (59)

the total output is

Y =Z 10y(i)di (60)

Where k(i) is the capital used by rm i and n(i) is the labor input insuch a rm, the aggregate technology level is A.

rm specic technology level ε(i) drawn from a common distributionfunction. Pr [ε(i) � ε] = ε�σ with the density function equal tof (ε) = σε�σ�1.



timing : Before the realization of ε(i), rm needs to install capital.After that ε(i) realizes, and rm i decides whether to produce or not.

Let r be the interest rate, and w be the real wage and the goodsprice be unit, the rms problem can be solved by backwardreduction. First consider the labor decision, its prot is

π(ε(i)) = ε(i)An(i)� wn(i) (61)

s.t An(i) � k(i) (62)



In this case, the labor decision and output is

n(i) =

(k (i )A if ε(i) �

wA

0 otherwise

)(63)

y(i) =�

ε(i)k(i) if ε(i) � wA0 otherwise

�(64)



The rms prot is

π(i) =

(ε(i)k(i)� k (i )A w if ε(i) �

wA

0 otherwise

)(65)

Hence its expected prot when its install capital is :

�rk(i) + k(i)∞ZwA

�ε� w

A

�f (ε)dε



This implies

r =

∞ZwA

�ε� w

A

�f (ε)dε

=

∞ZwA

εf (ε)dε��wA

�1�σ=

1σ� 1

�wA

�1�σ(66)

and the total production:

Y = K

∞ZwA

εf (ε)dε =σ

σ� 1�wA

�1�σK (67)



Interest rate isrK =

1σY (68)

and the real wage is �wA

�1�σ=

σ� 1σ

YK

the total wage payment

wN = wKA

Z ∞wA

f (ε)dε = K�wA

�σ�1(69)

or we have :

wN =σ� 1

σY (70)



Finally use w = A�

σ�1σ

YK

� 11�σ to compute the aggregate ouput. we

have

A�

σ� 1σ

YK

� 11�σN =

σ� 1σ

Y (71)

or we have

ANK1

σ�1σ� 1

σ

σ1�σ= Y

σσ�1 (72)

This yields

Y =σ

σ� 1K1σ [AN ]

σ�1σ (73)


Solow-ModelIntroduction

The micro-foundation of Aggregate production function

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