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Lecture 4ModernGrowthTheories

Prof.Paczkowski

Lecture 4Modern Growth Theories

Prof. Paczkowski

Rutgers University

Spring Semester, 2009

Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 1 / 116


Prof.Paczkowski

ReadingAssignments Part I

Reading Assignments



Prof.Paczkowski

ReadingAssignments

Reading Assignments

Solows Noble Announcementhttp:www.nobel.seeconomicslaureates1987press.html

”Financing growth and development in the transitioneconomies: the role of domestic savings” Economic AnalysisDivision, UN/ECE”http:www.unece.orgpressspecialevent00rcmfd documentssavinv.pdf

”The ghost of financing gap: testing the growth model used inthe international financial institutions”William EasterlyJ. of Development EconomicsVolume 60, Issue 2 (December 1999) Pages 423-438



Prof.Paczkowski

Introduction Part II

Introduction



Prof.Paczkowski

Introduction

Introduction

We will study two modern growth theories. . .

1 Harrod-Domar Growth Model

2 Solow Neoclassical Growth Theory

Later, we will discuss endogenous growth theory.



Prof.Paczkowski

Introduction

Introduction







Prof.Paczkowski

Introduction

Introduction







Prof.Paczkowski

Introduction

Introduction







Prof.Paczkowski

Introduction

Introduction







Prof.Paczkowski

Harrod-Domar

Warranted Rateof Growth

Equilibrium

Part III

Harrod-Domar



Prof.Paczkowski

Harrod-Domar


Equilibrium

Harrod-Domar

Harrod-Domar model developed during the early days of thepost-World War II Keynesian Revolution in the 1940s

Harrod (1939) and Domar (1947)

Extension of Keynesian theory to growth



Prof.Paczkowski

Harrod-Domar


Equilibrium

Harrod-Domar

Harrod-Domar model developed during the early days of thepost-World War II Keynesian Revolution in the 1940s

Harrod (1939) and Domar (1947)

Extension of Keynesian theory to growth



Prof.Paczkowski

Harrod-Domar


Equilibrium

Harrod-Domar(Continued)

The model assumes a fixed proportions production function

An aggregate production function with fixed technicalcoefficients

Qt = min

(Kt

v,

Et

u

)

where

Qt = output net of depreciation of capital at time t

Kt = real physical capital stock at time t

v = utilized capital − output ratio (constant)

Et = effective labor force (not body count) at time t

u = employed effective labor − output ratio (constant)



Prof.Paczkowski

Harrod-Domar


Equilibrium




Qt = min

(Kt

v,

Et

u

)

where








Prof.Paczkowski

Harrod-Domar


Equilibrium




Qt = min

(Kt

v,

Et

u

)where








Prof.Paczkowski

Harrod-Domar


Equilibrium




Qt = min

(Kt

v,

Et

u

)where








Prof.Paczkowski

Harrod-Domar


Equilibrium




Qt = min

(Kt

v,

Et

u

)where








Prof.Paczkowski

Harrod-Domar


Equilibrium




Qt = min

(Kt

v,

Et

u

)where








Prof.Paczkowski

Harrod-Domar


Equilibrium




Qt = min

(Kt

v,

Et

u

)where








Prof.Paczkowski

Harrod-Domar


Equilibrium


We will use effective labor, E, rather than a body count, L

This modification results from viewing labor as acomposite of skills, not just brute force

Labor per se brings more to production than just muscle

Effective labor embodies. . .

MuscleEducationHealthWork ethicEtc.



Prof.Paczkowski

Harrod-Domar


Equilibrium









Prof.Paczkowski

Harrod-Domar


Equilibrium









Prof.Paczkowski

Harrod-Domar


Equilibrium






Muscle

EducationHealthWork ethicEtc.



Prof.Paczkowski

Harrod-Domar


Equilibrium






MuscleEducation

HealthWork ethicEtc.



Prof.Paczkowski

Harrod-Domar


Equilibrium






MuscleEducationHealth

Work ethicEtc.



Prof.Paczkowski

Harrod-Domar


Equilibrium






MuscleEducationHealthWork ethic

Etc.



Prof.Paczkowski

Harrod-Domar


Equilibrium









Prof.Paczkowski

Harrod-Domar


Equilibrium


v is utilized capital-output ratio, a constant

This is a technical coefficient given by the currenttechnology

It shows the amount of capital required per unit of outputproduced

v =K

Q

The inverse is the average product of capital:

1

v=

Q

K= APK

Multiplying K by 1v gives Q

Similarly, 1u is the average product of effective labor



Prof.Paczkowski

Harrod-Domar


Equilibrium





v =K

Q


1

v=

Q

K= APK





Prof.Paczkowski

Harrod-Domar


Equilibrium





v =K

Q


1

v=

Q

K= APK





Prof.Paczkowski

Harrod-Domar


Equilibrium





v =K

Q


1

v=

Q

K= APK





Prof.Paczkowski

Harrod-Domar


Equilibrium





v =K

Q


1

v=

Q

K= APK





Prof.Paczkowski

Harrod-Domar


Equilibrium





v =K

Q


1

v=

Q

K= APK





Prof.Paczkowski

Harrod-Domar


Equilibrium





v =K

Q


1

v=

Q

K= APK





Prof.Paczkowski

Harrod-Domar


Equilibrium





v =K

Q


1

v=

Q

K= APK





Prof.Paczkowski

Harrod-Domar


Equilibrium





v =K

Q


1

v=

Q

K= APK





Prof.Paczkowski

Harrod-Domar


Equilibrium


ConjectureKE = v

u

Proof:

K

E=

K

Q

Q

E

= v1

usince v =

K

Qand

Q

E=

1

u

=v

u

This is a ratio of technical coefficients.



Prof.Paczkowski

Harrod-Domar


Equilibrium


ConjectureKE = v

u

Proof:

K

E=

K

Q

Q

E

= v1

usince v =

K

Qand

Q

E=

1

u

=v

u




Prof.Paczkowski

Harrod-Domar


Equilibrium


ConjectureKE = v

u

Proof:

K

E=

K

Q

Q

E

= v1

usince v =

K

Qand

Q

E=

1

u

=v

u




Prof.Paczkowski

Harrod-Domar


Equilibrium


ConjectureKE = v

u

Proof:

K

E=

K

Q

Q

E

= v1

usince v =

K

Qand

Q

E=

1

u

=v

u




Prof.Paczkowski

Harrod-Domar


Equilibrium


ConjectureKE = v

u

Proof:

K

E=

K

Q

Q

E

= v1

usince v =

K

Qand

Q

E=

1

u

=v

u




Prof.Paczkowski

Harrod-Domar


Equilibrium


ConjectureKE = v

u

Proof:

K

E=

K

Q

Q

E

= v1

usince v =

K

Qand

Q

E=

1

u

=v

u



E

Kuv

Q2

Q1

E0

K0

Isoquant Map


Prof.Paczkowski

Harrod-Domar


Equilibrium


In a Keynesian world, there is unemployment so effective laboris not an issue

We’ll return to this effective labor issue later

Harrod and Domar assumed that effective labor is not anissue since they wrote during this period

Since effective labor is plentiful, output, Qt , is a functionof capital only

Qt =Kt

v



Prof.Paczkowski

Harrod-Domar


Equilibrium






Qt =Kt

v



Prof.Paczkowski

Harrod-Domar


Equilibrium






Qt =Kt

v



Prof.Paczkowski

Harrod-Domar


Equilibrium






Qt =Kt

v



Prof.Paczkowski

Harrod-Domar


Equilibrium






Qt =Kt

v



Prof.Paczkowski

Harrod-Domar


Equilibrium






Qt =Kt

v


E

Kuv

Q1

Q0

E0 = Excess Labor

K0

No Output Change

Isoquant Map

E

Kuv

Q1

Q0

E0 = Excess Labor

K1

No Output Change

Output Change

Isoquant Map

K0


Prof.Paczkowski

Harrod-Domar


Equilibrium


In macroequilibrium, the resources available for adding to thecapital stock, St , equal the actual additions, dKt

dt , which is netinvestment, so. . .

St = It ≡dKt

dt

This is a Keynesian concept but with a dynamic twist given bythe time subscript



Prof.Paczkowski

Harrod-Domar


Equilibrium




St = It ≡dKt

dt




Prof.Paczkowski

Harrod-Domar


Equilibrium




St = It ≡dKt

dt




Prof.Paczkowski

Harrod-Domar


Equilibrium


Note that the change in investment over time is the secondderivative

dItdt

=dKtdt

dt

=dK 2

t

dt2

or

I =1

It

dItdt

=1

It

dKtdt

dt

=1

dKtdt

dKtdt

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium



dItdt

=dKtdt

dt

=dK 2

t

dt2

or

I =1

It

dItdt

=1

It

dKtdt

dt

=1

dKtdt

dKtdt

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium



dItdt

=dKtdt

dt

=dK 2

t

dt2

or

I =1

It

dItdt

=1

It

dKtdt

dt

=1

dKtdt

dKtdt

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium



dItdt

=dKtdt

dt

=dK 2

t

dt2

or

I =1

It

dItdt

=1

It

dKtdt

dt

=1

dKtdt

dKtdt

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium



dItdt

=dKtdt

dt

=dK 2

t

dt2

or

I =1

It

dItdt

=1

It

dKtdt

dt

=1

dKtdt

dKtdt

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium



dItdt

=dKtdt

dt

=dK 2

t

dt2

or

I =1

It

dItdt

=1

It

dKtdt

dt

=1

dKtdt

dKtdt

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium


Notation: The single dot over I indicates the rate of growth inI over time



Prof.Paczkowski

Harrod-Domar


Equilibrium


The amount of resources for growth made available throughsaving at time t, St , is given by1. . .

St = sQt

dSt

dQt= s =

St

Qt

so both the marginal and average propensity to save equal thesaving ratio, s = 1− c .

1Net of that needed to replace depreciated capital



Prof.Paczkowski

Harrod-Domar


Equilibrium



St = sQt

dSt

dQt= s =

St

Qt





Prof.Paczkowski

Harrod-Domar


Equilibrium



St = sQt

dSt

dQt= s =

St

Qt





Prof.Paczkowski

Harrod-Domar


Equilibrium


But note that. . .

dSt

dt= s

dQt

dt

since everything is a function of time, t2

2This assume that autonomous spending is independent of time, so. . .

dAP

dt= 0



Prof.Paczkowski

Harrod-Domar


Equilibrium


But note that. . .

dSt

dt= s

dQt

dt

since everything is a function of time, t2

2This assume that autonomous spending is independent of time, so. . .

dAP

dt= 0



Prof.Paczkowski

Harrod-Domar


Equilibrium


We have five key concepts. . .

1 Fixed proportions production function: Q = Kv

2 Keynesian consumption function: C = cQ

3 Keynesian saving function: S = sQ = Q − cQ

4 Keynesian Identity: S = I

5 Investment function: I = dKdt



Prof.Paczkowski

Harrod-Domar


Equilibrium



1 Fixed proportions production function:

Q = Kv







Prof.Paczkowski

Harrod-Domar


Equilibrium



1 Fixed proportions production function:

Q = Kv







Prof.Paczkowski

Harrod-Domar


Equilibrium










Prof.Paczkowski

Harrod-Domar


Equilibrium




2 Keynesian consumption function:

C = cQ






Prof.Paczkowski

Harrod-Domar


Equilibrium




2 Keynesian consumption function:

C = cQ






Prof.Paczkowski

Harrod-Domar


Equilibrium










Prof.Paczkowski

Harrod-Domar


Equilibrium





3 Keynesian saving function:

S = sQ = Q − cQ





Prof.Paczkowski

Harrod-Domar


Equilibrium





3 Keynesian saving function:

S = sQ = Q − cQ





Prof.Paczkowski

Harrod-Domar


Equilibrium










Prof.Paczkowski

Harrod-Domar


Equilibrium






4 Keynesian Identity:

S = I




Prof.Paczkowski

Harrod-Domar


Equilibrium






4 Keynesian Identity:

S = I




Prof.Paczkowski

Harrod-Domar


Equilibrium










Prof.Paczkowski

Harrod-Domar


Equilibrium







5 Investment function:

I = dKdt



Prof.Paczkowski

Harrod-Domar


Equilibrium







5 Investment function:

I = dKdt



Prof.Paczkowski

Harrod-Domar


Equilibrium










Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

Q =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Proof:

Dropping the t notation to simplify. . .

dQ

dt=

1

v

dK

dtsince Q =

K

v

=1

vI since I =

dK

dt

=1

vS since I = S

=s

vQ since S = sQ

1

Q

dQ

dt≡ Q =

s

v

Notation: The single dot over Q indicates the rate of growthin Q



Prof.Paczkowski

Harrod-Domar


Equilibrium


Proof:


dQ

dt=

1

v

dK

dtsince Q =

K

v

=1

vI since I =

dK

dt

=1

vS since I = S

=s

vQ since S = sQ

1

Q

dQ

dt≡ Q =

s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium


Proof:


dQ

dt=

1

v

dK

dtsince Q =

K

v

=1

vI since I =

dK

dt

=1

vS since I = S

=s

vQ since S = sQ

1

Q

dQ

dt≡ Q =

s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium


Proof:


dQ

dt=

1

v

dK

dtsince Q =

K

v

=1

vI since I =

dK

dt

=1

vS since I = S

=s

vQ since S = sQ

1

Q

dQ

dt≡ Q =

s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium


Proof:


dQ

dt=

1

v

dK

dtsince Q =

K

v

=1

vI since I =

dK

dt

=1

vS since I = S

=s

vQ since S = sQ

1

Q

dQ

dt≡ Q =

s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium


Proof:


dQ

dt=

1

v

dK

dtsince Q =

K

v

=1

vI since I =

dK

dt

=1

vS since I = S

=s

vQ since S = sQ

1

Q

dQ

dt≡ Q =

s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium


Proof:


dQ

dt=

1

v

dK

dtsince Q =

K

v

=1

vI since I =

dK

dt

=1

vS since I = S

=s

vQ since S = sQ

1

Q

dQ

dt≡ Q =

s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium


Proof:


dQ

dt=

1

v

dK

dtsince Q =

K

v

=1

vI since I =

dK

dt

=1

vS since I = S

=s

vQ since S = sQ

1

Q

dQ

dt≡ Q =

s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium


The growth in real output is a function of the saving ratio, s,and the capital-output ratio, v, which is technologicallydetermined

If the saving ratio, s, increases, growth rate increases

∂Q

∂s> 0

If the capital-output ratio, v, decreases (or output per unitof capital (APK ) increases; i.e., capital is moreproductive), growth rate increases

∂Q

∂v< 0

[or

∂Q

∂(

1v

) > 0 so ↑ APK ⇒↑ Q

]



Prof.Paczkowski

Harrod-Domar


Equilibrium




∂Q

∂s> 0


∂Q

∂v< 0

[or

∂Q

∂(

1v

) > 0 so ↑ APK ⇒↑ Q

]



Prof.Paczkowski

Harrod-Domar


Equilibrium




∂Q

∂s> 0


∂Q

∂v< 0

[or

∂Q

∂(

1v

) > 0 so ↑ APK ⇒↑ Q

]



Prof.Paczkowski

Harrod-Domar


Equilibrium




∂Q

∂s> 0


∂Q

∂v< 0

[or

∂Q

∂(

1v

) > 0 so ↑ APK ⇒↑ Q

]



Prof.Paczkowski

Harrod-Domar


Equilibrium




∂Q

∂s> 0


∂Q

∂v< 0

[or

∂Q

∂(

1v

) > 0 so ↑ APK ⇒↑ Q

]



Prof.Paczkowski

Harrod-Domar


Equilibrium




∂Q

∂s> 0


∂Q

∂v< 0

[or

∂Q

∂(

1v

) > 0 so ↑ APK ⇒↑ Q

]



Prof.Paczkowski

Harrod-Domar


Equilibrium


The result that. . .

Q =s

v

is what Harrod, writing before Domar, called the warrantedrate of growth

The ratio, sv , is the warranted rate

Since 1v is fixed by technology and s is fixed by saving

habits of people, then sv is fixed



Prof.Paczkowski

Harrod-Domar


Equilibrium



Q =s

v







Prof.Paczkowski

Harrod-Domar


Equilibrium



Q =s

v







Prof.Paczkowski

Harrod-Domar


Equilibrium



Q =s

v







Prof.Paczkowski

Harrod-Domar


Equilibrium



Q =s

v







Prof.Paczkowski

Harrod-Domar


Equilibrium


A higher saving ratio, which gives the proportion of resourcesavailable to increase the capital stock and, hence, output, or alower capital-output ratio (i.e., higher average product ofcapital) , which gives the magnitude of the effect of an increasein capital on increasing output, result in a higher warrantedrate of growth.



Prof.Paczkowski

Harrod-Domar


Equilibrium


We can make a bold statement. . .

The following all grow at the warranted rate of growth sinceoutput drives or determines all of these from simplemacrotheory which states that all markets must be inequilibrium for the economy to be in equilibrium.

Saving and consumption

Investment

The capital stock

Effective labor employed



Prof.Paczkowski

Harrod-Domar


Equilibrium





Investment

The capital stock




Prof.Paczkowski

Harrod-Domar


Equilibrium





Investment

The capital stock




Prof.Paczkowski

Harrod-Domar


Equilibrium





Investment

The capital stock




Prof.Paczkowski

Harrod-Domar


Equilibrium





Investment

The capital stock




Prof.Paczkowski

Harrod-Domar


Equilibrium





Investment

The capital stock




Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

C = sv

Let C ′t = cQt be consumption net of Ap

dC ′

dt= c

dQ

dt1

c

dC ′

dt=

dQ

dt1

Q

1

c

dC ′

dt=

1

Q

dQ

dt=

s

v1

C ′dC ′

dt=

s

v

⇒ C =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

C = sv


dC ′

dt= c

dQ

dt

1

c

dC ′

dt=

dQ

dt1

Q

1

c

dC ′

dt=

1

Q

dQ

dt=

s

v1

C ′dC ′

dt=

s

v

⇒ C =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

C = sv


dC ′

dt= c

dQ

dt1

c

dC ′

dt=

dQ

dt

1

Q

1

c

dC ′

dt=

1

Q

dQ

dt=

s

v1

C ′dC ′

dt=

s

v

⇒ C =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

C = sv


dC ′

dt= c

dQ

dt1

c

dC ′

dt=

dQ

dt1

Q

1

c

dC ′

dt=

1

Q

dQ

dt

=s

v1

C ′dC ′

dt=

s

v

⇒ C =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

C = sv


dC ′

dt= c

dQ

dt1

c

dC ′

dt=

dQ

dt1

Q

1

c

dC ′

dt=

1

Q

dQ

dt=

s

v

1

C ′dC ′

dt=

s

v

⇒ C =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

C = sv


dC ′

dt= c

dQ

dt1

c

dC ′

dt=

dQ

dt1

Q

1

c

dC ′

dt=

1

Q

dQ

dt=

s

v1

C ′dC ′

dt=

s

v

⇒ C =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

C = sv


dC ′

dt= c

dQ

dt1

c

dC ′

dt=

dQ

dt1

Q

1

c

dC ′

dt=

1

Q

dQ

dt=

s

v1

C ′dC ′

dt=

s

v

⇒ C =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = sv

Let S ′t = sQt be saving net of Ap

dS

dt= s

dQ

dt1

s

dS

dt=

dQ

dt1

Q

1

s

dS

dt=

1

Q

dQ

dt=

s

v1

S

dS

dt=

s

v

⇒ S =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = sv


dS

dt= s

dQ

dt

1

s

dS

dt=

dQ

dt1

Q

1

s

dS

dt=

1

Q

dQ

dt=

s

v1

S

dS

dt=

s

v

⇒ S =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = sv


dS

dt= s

dQ

dt1

s

dS

dt=

dQ

dt

1

Q

1

s

dS

dt=

1

Q

dQ

dt=

s

v1

S

dS

dt=

s

v

⇒ S =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = sv


dS

dt= s

dQ

dt1

s

dS

dt=

dQ

dt1

Q

1

s

dS

dt=

1

Q

dQ

dt

=s

v1

S

dS

dt=

s

v

⇒ S =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = sv


dS

dt= s

dQ

dt1

s

dS

dt=

dQ

dt1

Q

1

s

dS

dt=

1

Q

dQ

dt=

s

v

1

S

dS

dt=

s

v

⇒ S =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = sv


dS

dt= s

dQ

dt1

s

dS

dt=

dQ

dt1

Q

1

s

dS

dt=

1

Q

dQ

dt=

s

v1

S

dS

dt=

s

v

⇒ S =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = sv


dS

dt= s

dQ

dt1

s

dS

dt=

dQ

dt1

Q

1

s

dS

dt=

1

Q

dQ

dt=

s

v1

S

dS

dt=

s

v

⇒ S =s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = I = sv

Since St = It in Keynesian equilibrium, we must have. . .

S =1

S

dS

dt

=1

S

dI

dt

=1

I

dI

dt

Using. . .

S = I ⇒ dS

dt=

dI

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = I = sv


S =1

S

dS

dt

=1

S

dI

dt

=1

I

dI

dt

Using. . .

S = I ⇒ dS

dt=

dI

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = I = sv


S =1

S

dS

dt

=1

S

dI

dt

=1

I

dI

dt

Using. . .

S = I ⇒ dS

dt=

dI

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = I = sv


S =1

S

dS

dt

=1

S

dI

dt

=1

I

dI

dt

Using. . .

S = I ⇒ dS

dt=

dI

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = I = sv


S =1

S

dS

dt

=1

S

dI

dt

=1

I

dI

dt

Using. . .

S = I ⇒ dS

dt=

dI

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = I = sv


S =1

S

dS

dt

=1

S

dI

dt

=1

I

dI

dt

Using. . .

S = I ⇒ dS

dt=

dI

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

S = I = sv


S =1

S

dS

dt

=1

S

dI

dt

=1

I

dI

dt

Using. . .

S = I ⇒ dS

dt=

dI

dt



Prof.Paczkowski

Harrod-Domar


Equilibrium


Finally, by the production function, if real output grows at therate, Q, then capital must also grow at the same rate



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

Q = K = sv

Q = min

(K

v,

E

u

)dQ

dt=

1

v

dK

dt1

K

dQ

dt=

1

v

(1

K

dK

dt

)v

K

dQ

dt= K

1

Q

dQ

dt=

s

v= K

Using K = vQ



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

Q = K = sv

Q = min

(K

v,

E

u

)

dQ

dt=

1

v

dK

dt1

K

dQ

dt=

1

v

(1

K

dK

dt

)v

K

dQ

dt= K

1

Q

dQ

dt=

s

v= K

Using K = vQ



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

Q = K = sv

Q = min

(K

v,

E

u

)dQ

dt=

1

v

dK

dt

1

K

dQ

dt=

1

v

(1

K

dK

dt

)v

K

dQ

dt= K

1

Q

dQ

dt=

s

v= K

Using K = vQ



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

Q = K = sv

Q = min

(K

v,

E

u

)dQ

dt=

1

v

dK

dt1

K

dQ

dt=

1

v

(1

K

dK

dt

)

v

K

dQ

dt= K

1

Q

dQ

dt=

s

v= K

Using K = vQ



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

Q = K = sv

Q = min

(K

v,

E

u

)dQ

dt=

1

v

dK

dt1

K

dQ

dt=

1

v

(1

K

dK

dt

)v

K

dQ

dt= K

1

Q

dQ

dt=

s

v= K

Using K = vQ



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

Q = K = sv

Q = min

(K

v,

E

u

)dQ

dt=

1

v

dK

dt1

K

dQ

dt=

1

v

(1

K

dK

dt

)v

K

dQ

dt= K

1

Q

dQ

dt=

s

v

= K

Using K = vQ



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

Q = K = sv

Q = min

(K

v,

E

u

)dQ

dt=

1

v

dK

dt1

K

dQ

dt=

1

v

(1

K

dK

dt

)v

K

dQ

dt= K

1

Q

dQ

dt=

s

v= K

Using K = vQ



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

Q = K = sv

Q = min

(K

v,

E

u

)dQ

dt=

1

v

dK

dt1

K

dQ

dt=

1

v

(1

K

dK

dt

)v

K

dQ

dt= K

1

Q

dQ

dt=

s

v= K

Using K = vQProf. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 32 / 116


Prof.Paczkowski

Harrod-Domar


Equilibrium


This results in a ”balanced growth” equilibrium. . .

Q = C = S = K = I =s

v

Everything must grow at the warranted rate, sv



Prof.Paczkowski

Harrod-Domar


Equilibrium



Q

= C = S = K = I =s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C

= S = K = I =s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C = S

= K = I =s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C = S = K

= I =s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C = S = K = I

=s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C = S = K = I =s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C = S = K = I =s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium


The key Harrod-Domar result

Q =s

v

demonstrates the existence of a steady growth path for theeconomy


The stability of the growth path, however, is in doubt



Prof.Paczkowski

Harrod-Domar


Equilibrium


The key Harrod-Domar result

Q =s

v

demonstrates the existence of a steady growth path for theeconomy


The stability of the growth path, however, is in doubt



Prof.Paczkowski

Harrod-Domar


Equilibrium


The stability of a system is just as important as the existenceof an equilibrium in the system

We don’t want a system to explode


D

SP

QStable Equilibrium Unstable Equilibrium

D

P

Q

S

Initially assume price too low


Prof.Paczkowski

Harrod-Domar


Equilibrium


If the economy is growing at the warranted rate, theinvestment expenditures of firms are ”warranted”, i.e., justified,by their profitability, since the growth in aggregate demandequals the growth in capacity output they provide.

sv can be interpreted as the growth in real capacity

But what if the economy should temporarily diverge fromthe fixed warranted rate?



Prof.Paczkowski

Harrod-Domar


Equilibrium


If the economy is growing at the warranted rate, theinvestment expenditures of firms are ”warranted”, i.e., justified,by their profitability, since the growth in aggregate demandequals the growth in capacity output they provide.

sv can be interpreted as the growth in real capacity

But what if the economy should temporarily diverge fromthe fixed warranted rate?



Prof.Paczkowski

Harrod-Domar


Equilibrium


If aggregate demand temporarily falls because desiredinvestment is less than saving (I < S in a traditional Keynesianframework), then output grows slower than the warranted rate



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

I < S ⇒ Q <s

v

Proof:3

I < S ⇒ 1

vI <

s

vQ

⇒ 1

Q

dQ

dt<

s

v

⇒ Q <s

v

The economy can grow faster.

3Use: Q = Kv⇒ dQ

dt= 1

vdKdt

= 1vI



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

I < S ⇒ Q <s

v

Proof:3

I < S ⇒ 1

vI <

s

vQ

⇒ 1

Q

dQ

dt<

s

v

⇒ Q <s

v


3Use: Q = Kv⇒ dQ

dt= 1

vdKdt

= 1vI



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

I < S ⇒ Q <s

v

Proof:3

I < S ⇒ 1

vI <

s

vQ

⇒ 1

Q

dQ

dt<

s

v

⇒ Q <s

v


3Use: Q = Kv⇒ dQ

dt= 1

vdKdt

= 1vI



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

I < S ⇒ Q <s

v

Proof:3

I < S ⇒ 1

vI <

s

vQ

⇒ 1

Q

dQ

dt<

s

v

⇒ Q <s

v


3Use: Q = Kv⇒ dQ

dt= 1

vdKdt

= 1vI



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

I < S ⇒ Q <s

v

Proof:3

I < S ⇒ 1

vI <

s

vQ

⇒ 1

Q

dQ

dt<

s

v

⇒ Q <s

v


3Use: Q = Kv⇒ dQ

dt= 1

vdKdt

= 1vI



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

I < S ⇒ Q <s

v

Proof:3

I < S ⇒ 1

vI <

s

vQ

⇒ 1

Q

dQ

dt<

s

v

⇒ Q <s

v


3Use: Q = Kv⇒ dQ

dt= 1

vdKdt

= 1vI



Prof.Paczkowski

Harrod-Domar


Equilibrium


This slower real growth (less than warranted or justified)induces a smaller increase in investment in the next periodbecause profitability falls since consumption is down because ofthe slower real income growth: Q ↓⇒ C ↓

There is also a smaller growth in output because less wasadded to capacity due to the lower real investment

But less capacity means less long-term growth, and soon. . .

There will be cumulative movements away from the fixedwarranted rate of growth, s

v



Prof.Paczkowski

Harrod-Domar


Equilibrium






v



Prof.Paczkowski

Harrod-Domar


Equilibrium






v



Prof.Paczkowski

Harrod-Domar


Equilibrium






v


The economy falls off a knife-edge

t

Q

But…A Knife-edge Path

Time whenI<S

Time whenI=S

Real GrowthPath Given by

vs


Prof.Paczkowski

Harrod-Domar


Equilibrium


The problem is that there is only one simple equationdescribing the economy.

An equation is missing to fully specify equilibrium

The missing equation represents the effective labor market

In the long-run, bond and money markets are irrelevant inthis frameworkThe capital markets, and their importance, will beintroduced later



Prof.Paczkowski

Harrod-Domar


Equilibrium








Prof.Paczkowski

Harrod-Domar


Equilibrium








Prof.Paczkowski

Harrod-Domar


Equilibrium





In the long-run, bond and money markets are irrelevant inthis framework

The capital markets, and their importance, will beintroduced later



Prof.Paczkowski

Harrod-Domar


Equilibrium








Prof.Paczkowski

Harrod-Domar


Equilibrium


Recall that real growth, Q, is given by. . .

Q

= C = S = K = I =s

v= E

so the economy must also grow at the same rate as effectivelabor, E.



Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C

= S = K = I =s

v= E




Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C = S

= K = I =s

v= E




Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C = S = K

= I =s

v= E




Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C = S = K = I

=s

v= E




Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C = S = K = I =s

v

= E




Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C = S = K = I =s

v= E




Prof.Paczkowski

Harrod-Domar


Equilibrium



Q = C = S = K = I =s

v= E




Prof.Paczkowski

Harrod-Domar


Equilibrium


Just for argument, let’s say 1,000,000 people enter the laborforce each year. Then the real economy must generate thismany new jobs just to keep even.



Prof.Paczkowski

Harrod-Domar


Equilibrium


Now consider the effective labor force, Et

Assume an exponential growth in the effective labor force

Et = E0e(n+λ)t

where

n = rate of growth of the labor force, not the size

λ = the rate of labor augmenting technical change



Prof.Paczkowski

Harrod-Domar


Equilibrium


Now consider the effective labor force, Et

Assume an exponential growth in the effective labor force

Et = E0e(n+λ)t

where

n = rate of growth of the labor force, not the size

λ = the rate of labor augmenting technical change



Prof.Paczkowski

Harrod-Domar


Equilibrium


The exponential, e, is used to reflect the continuouscompounding convenience in mathematical formulations butalso, it is realistic for a dynamic economy which does not growby fits and starts



Prof.Paczkowski

Harrod-Domar


Equilibrium


Basically, labor grows by two factors. . .

1 Body-count growth, n : ent

2 Productivity growth, λ : eλt

The productivity growth is due to technical change that has amultiplicative effect. Therefore, the total growth in labor is. . .

enteλt = e(n+λ)t



Prof.Paczkowski

Harrod-Domar


Equilibrium






enteλt = e(n+λ)t



Prof.Paczkowski

Harrod-Domar


Equilibrium






enteλt = e(n+λ)t



Prof.Paczkowski

Harrod-Domar


Equilibrium






enteλt = e(n+λ)t



Prof.Paczkowski

Harrod-Domar


Equilibrium






enteλt = e(n+λ)t



Prof.Paczkowski

Harrod-Domar


Equilibrium






enteλt = e(n+λ)t



Prof.Paczkowski

Harrod-Domar


Equilibrium


Harrod-neutral technical change

Definition

Technical change specified this way is Harrod-neutral since anincrease in the efficiency or productivity of the labor force atthe rate λ (due to better training or education) has the sameconsequence for the effective labor supply as if the labor forceitself (i.e., a body count) increased at the rate n. . .

Et = E0e(n+λ)t



Prof.Paczkowski

Harrod-Domar


Equilibrium


The rate of growth of the effective labor force underHarrod-neutral technical change is. . .

E = (n + λ)

Harrod called this the natural rate of growth



Prof.Paczkowski

Harrod-Domar


Equilibrium



E = (n + λ)




Prof.Paczkowski

Harrod-Domar


Equilibrium



E = (n + λ)




Prof.Paczkowski

Harrod-Domar


Equilibrium


Note that. . .

ln (Et) = ln (E0) + (n + λ) t

so. . .

1

E

dE

dt≡ E = (n + λ)



Prof.Paczkowski

Harrod-Domar


Equilibrium


Note that. . .

ln (Et) = ln (E0) + (n + λ) t

so. . .

1

E

dE

dt≡ E = (n + λ)



Prof.Paczkowski

Harrod-Domar


Equilibrium


Note that. . .

ln (Et) = ln (E0) + (n + λ) t

so. . .

1

E

dE

dt≡ E = (n + λ)



Prof.Paczkowski

Harrod-Domar


Equilibrium


With a fixed coefficients production function, the natural rateof growth is also the maximum sustainable rate of growth ofoutput: cannot grow faster than effective labor since Q = E



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

E =s

v

Proof:

If Q =E

u

ThendQ

dt=

1

u

dE

dt1

Eu

dQ

dt=

1

E

dE

dt1

uQu

dQ

dt=

1

E

dE

dt

Q = E



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

E =s

v

Proof:

If Q =E

u

ThendQ

dt=

1

u

dE

dt1

Eu

dQ

dt=

1

E

dE

dt1

uQu

dQ

dt=

1

E

dE

dt

Q = E



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

E =s

v

Proof:

If Q =E

u

ThendQ

dt=

1

u

dE

dt1

Eu

dQ

dt=

1

E

dE

dt1

uQu

dQ

dt=

1

E

dE

dt

Q = E



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

E =s

v

Proof:

If Q =E

u

ThendQ

dt=

1

u

dE

dt

1

Eu

dQ

dt=

1

E

dE

dt1

uQu

dQ

dt=

1

E

dE

dt

Q = E



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

E =s

v

Proof:

If Q =E

u

ThendQ

dt=

1

u

dE

dt1

Eu

dQ

dt=

1

E

dE

dt

1

uQu

dQ

dt=

1

E

dE

dt

Q = E



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

E =s

v

Proof:

If Q =E

u

ThendQ

dt=

1

u

dE

dt1

Eu

dQ

dt=

1

E

dE

dt1

uQu

dQ

dt=

1

E

dE

dt

Q = E



Prof.Paczkowski

Harrod-Domar


Equilibrium


Conjecture

E =s

v

Proof:

If Q =E

u

ThendQ

dt=

1

u

dE

dt1

Eu

dQ

dt=

1

E

dE

dt1

uQu

dQ

dt=

1

E

dE

dt

Q = EProf. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 52 / 116


Prof.Paczkowski

Harrod-Domar


Equilibrium


Since E = (n + λ), then. . .

To maintain full employment (a Keynesian concept), thewarranted rate must equal the natural rate

s

v= (n + λ) using Q =

s

v

That is, the labor market growth must equal the goodsmarket or capacity growth, a definite Keynesian concept



Prof.Paczkowski

Harrod-Domar


Equilibrium




s


s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium




s


s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium




s


s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium




s


s

v




Prof.Paczkowski

Harrod-Domar


Equilibrium


The Harrod-Domar model assumes that all four variables arefixed

Everything is exogenous

Then. . .

s

v= (n + λ)

would prevail only by accident, and perhaps at a highunemployment rate

But the warranted rate is unstable anyway!



Prof.Paczkowski

Harrod-Domar


Equilibrium




Then. . .

s

v= (n + λ)





Prof.Paczkowski

Harrod-Domar


Equilibrium




Then. . .

s

v= (n + λ)





Prof.Paczkowski

Harrod-Domar


Equilibrium




Then. . .

s

v= (n + λ)





Prof.Paczkowski

Harrod-Domar


Equilibrium


If the economy is in equilibrium (and this is a big IF) with thewarranted rate equal to the natural rate, this equilibrium wouldbe unstable.

A slight movement away from equilibrium would lead toimplosion or explosion.



Prof.Paczkowski

Harrod-Domar


Equilibrium


If the economy is in equilibrium (and this is a big IF) with thewarranted rate equal to the natural rate, this equilibrium wouldbe unstable.

A slight movement away from equilibrium would lead toimplosion or explosion.



Prof.Paczkowski

Harrod-Domar


Equilibrium


Subsequent work specified. . .

the determinants of the four variables

their interrelations

the way they adjust to satisfy the equality sv = (n + λ)



Prof.Paczkowski

Harrod-Domar


Equilibrium








Prof.Paczkowski

Harrod-Domar


Equilibrium








Prof.Paczkowski

Harrod-Domar


Equilibrium








Prof.Paczkowski

Solow’sNeoclassicalGrowth Model

Part IV

Solow’s Neoclassical Growth Model



Prof.Paczkowski


Solow’s Model

Solows model differs from the Harrod-Domar model in tworespects

1 Keynesian concern with fluctuations in aggregate demand(i.e., Business Cycle) is dropped

Full employment of labor and capital is assumedStabilization policy is successful

2 Fixed technical coefficients production function is dropped

Substitution between factors of production is now assumed



Prof.Paczkowski


Solow’s Model








Prof.Paczkowski


Solow’s Model








Prof.Paczkowski


Solow’s Model



Full employment of labor and capital is assumed

Stabilization policy is successful





Prof.Paczkowski


Solow’s Model








Prof.Paczkowski


Solow’s Model








Prof.Paczkowski


Solow’s Model








Prof.Paczkowski


Solow’s Model(Continued)

Changes in the variables λ, n, and s are analyzed within thismodel

Harrod-Domar assumed them to be constant

These will be held constant here for a while and thenchanged later



Prof.Paczkowski








Prof.Paczkowski








Prof.Paczkowski



Write the production function as. . .

Q = Q (K ,E )

where we assume the time subscript, t, can be dropped.

Assume this is homogeneous-of-degree-one

This means that if all factors are multiplied by a constant,λ, then output is multiplied by the same constant:

λQ = Q (λK , λE )

as shown earlier



Prof.Paczkowski




Q = Q (K ,E )





as shown earlier



Prof.Paczkowski




Q = Q (K ,E )





as shown earlier



Prof.Paczkowski




Q = Q (K ,E )





as shown earlier



Prof.Paczkowski




Q = Q (K ,E )





as shown earlier



Prof.Paczkowski



If we divide by λE , we obtain. . .

Q

E= Q

(K

E, 1

)= Q

(K

E

)



Prof.Paczkowski



If we divide by λE , we obtain. . .

Q

E= Q

(K

E, 1

)= Q

(K

E

)



Prof.Paczkowski



Or. . .

q = q (k)

where

q =Q

E

k =K

Edq

dk> 0

d2q

dk2< 0

Note: ”k” is called ”capital intensity”



Prof.Paczkowski



Or. . .

q = q (k)

where

q =Q

E

k =K

Edq

dk> 0

d2q

dk2< 0




Prof.Paczkowski



Or. . .

q = q (k)

where

q =Q

E

k =K

Edq

dk> 0

d2q

dk2< 0




Prof.Paczkowski



Looking at the problem from the output per effective workerperspective has an advantage

The data are ”normalized” for the size of the population

Large economies (in people terms) and smaller economiescan be compared

Otherwise, the pure size would distort the effects of themain drivers of growthWe have apples-to-apples comparison



Prof.Paczkowski









Prof.Paczkowski









Prof.Paczkowski






Otherwise, the pure size would distort the effects of themain drivers of growth

We have apples-to-apples comparison



Prof.Paczkowski









Prof.Paczkowski



Note the difference from Harrod-Domar

The Harrod-Domar Model had output as a function of thecapital stock only

The Neoclassical Model has output as a function of capitaland labor through the amount of capital per effectiveworker



Prof.Paczkowski








Prof.Paczkowski








Prof.Paczkowski



This means that output per effective worker is a function ofcapital per effective worker

This is shown as the schedule q(k) in next chart


k

q

q(k)

Capital per Effective Worker

Rea

l Out

put p

er E

ffec

tive

Wor

ker

Total Real Output Production Function

EK k ,

EQ q

0 dk

qd ,0dkdq q(k), q 2

2


Prof.Paczkowski



Assume that the saving ratio is constant

Further assume 0 < s < 1, which should hold since theMarginal Propensity to Consume (MPC) is 0 < c < 1 ands = 1− c

Saving per effective worker is sq(k) and the saving pereffective worker curve lies below the curve q(k)



Prof.Paczkowski








Prof.Paczkowski







k

q

q(k)

sq(k), 0<s<1

A

k*

q*

B

Note:Recall how the Keynesian Cross diagram was divided into parts


Prof.Paczkowski



Note that. . .

Since Saving = Investment in equilibrium (S = I), thensq(k) also measures investment per effective worker, or thechange in the capital stock per effective worker since. . .

I

E=

S

E

=sQ

EI

E= sq (k)

where k = KE



Prof.Paczkowski



Note that. . .


I

E=

S

E

=sQ

EI

E= sq (k)

where k = KE



Prof.Paczkowski



Note that. . .


I

E=

S

E

=sQ

EI

E= sq (k)

where k = KE



Prof.Paczkowski



Note that. . .


I

E=

S

E

=sQ

EI

E= sq (k)

where k = KE



Prof.Paczkowski



Note that. . .


I

E=

S

E

=sQ

E

I

E= sq (k)

where k = KE



Prof.Paczkowski



Note that. . .


I

E=

S

E

=sQ

EI

E= sq (k)

where k = KE



Prof.Paczkowski



Note that. . .


I

E=

S

E

=sQ

EI

E= sq (k)

where k = KE


k

q

q(k)

Consumption/Effective Worker

sq(k), 0<s<1

Saving/Effective Worker =Investment/Effective Worker

A

k*

q*

B

Note:Recall how the Keynesian Cross diagram was divided into parts


Prof.Paczkowski



If capital per effective worker is an arbitrary k∗, then. . .

Saving per effective worker equals investment per effectiveworker (equals k*B)

Consumption per effective worker is AB

Saving per effective worker plus consumption per effectiveworker equals output per effective worker

This follows since Q = C + S from Keynesian macrotheoryand National Income Accounting



Prof.Paczkowski










Prof.Paczkowski










Prof.Paczkowski










Prof.Paczkowski










Prof.Paczkowski



We do not know how k will change over time or if it willapproach a steady-state equilibrium value

The k we looked at, k∗, is just an arbitrary k used toillustrate the decomposition of output per effective worker,q

A select k is needed for steady-state equilibrium



Prof.Paczkowski








Prof.Paczkowski








Prof.Paczkowski



The steady-state is defined by the requirements that. . .

1 Output and employment grow at some constantproportional rate, g (recall warranted and natural rates)

2 Net saving and investment are a constant fraction ofoutput

3 Net investment must be growing at the same proportionalrate as output - and so must the stock of capital since it isthe sum of past net investments

This implies that the capital/output ratio will be constant



Prof.Paczkowski










Prof.Paczkowski










Prof.Paczkowski










Prof.Paczkowski










Prof.Paczkowski



Steady-state equilibrium occurs when q does not change overtime; that is, q = 0

The economy may perhaps be growing, but itsrate of growth per effective worker is not changing

There is a distinction between the level of the ratio,q = Q

E , and the level and change in the level of real GDP,Q

If Q and E rise at the same rate, then q is a constantnumber and q = 0But, the economy is clearly growing since Q is changingWith E growing at the same rate, Q

E is unchanged: theeconomy produces just enough jobs to meet the neweffective workers



Prof.Paczkowski











Prof.Paczkowski











Prof.Paczkowski







If Q and E rise at the same rate, then q is a constantnumber and q = 0

But, the economy is clearly growing since Q is changingWith E growing at the same rate, Q




Prof.Paczkowski







If Q and E rise at the same rate, then q is a constantnumber and q = 0But, the economy is clearly growing since Q is changing

With E growing at the same rate, QE is unchanged: the

economy produces just enough jobs to meet the neweffective workers



Prof.Paczkowski











Prof.Paczkowski



Something is needed to keep q fixed and the something is kremaining fixed

That is, for steady-state equilibrium, we need. . .

q = k = 0



Prof.Paczkowski



Something is needed to keep q fixed and the something is kremaining fixed

That is, for steady-state equilibrium, we need. . .

q = k = 0



Prof.Paczkowski



But this requires that K and E grow at the same rate since. . .

q =Q

E

= Q

(K

E

)

There is an equation missing . . . again!

We need at least two equations (i.e., curves) to identifythe optimal k



Prof.Paczkowski




q =Q

E

= Q

(K

E

)





Prof.Paczkowski




q =Q

E

= Q

(K

E

)





Prof.Paczkowski




q =Q

E

= Q

(K

E

)





Prof.Paczkowski




q =Q

E

= Q

(K

E

)




k

q

q(k)

sq(k), 0<s<1

k*

q*For this to be in steady-state, then…

this must be fixed


Prof.Paczkowski



The missing equation needed for steady-state is for effectivelabor (i.e., the labor market) . . . again!

Labor was really ignored in Harrod-Domar because therewas excess labor at that time

Labor was not an issue – capital (or lack of it) was theissue because of the Keynesian influence

But we saw that the labor market was necessary tocomplete the economy

However, in Harrod-Domar, the real and labor sides wouldbe in equilibrium by chance only



Prof.Paczkowski










Prof.Paczkowski










Prof.Paczkowski










Prof.Paczkowski










Prof.Paczkowski



Must know how effective labor behaves

Assume that effective labor grows at the proportional raten + λ; that is,

E = n + λ



Prof.Paczkowski



Must know how effective labor behaves

Assume that effective labor grows at the proportional raten + λ; that is,

E = n + λ



Prof.Paczkowski



Recall that

k =K

E

so that. . .

ln(k) = ln(K )− ln(E )

which is a function of time, as are all our terms



Prof.Paczkowski



Recall that

k =K

E

so that. . .





Prof.Paczkowski



Recall that

k =K

E

so that. . .





Prof.Paczkowski



Recall that

k =K

E

so that. . .





Prof.Paczkowski



Recall that

k =K

E

so that. . .





Prof.Paczkowski



The condition to leave k unchanged over time, required forsteady-state, that is q = k = 0, is. . .

1

k

dk

dt=

1

K

dK

dt− 1

E

dE

dt= 0



Prof.Paczkowski




1

k

dk

dt=

1

K

dK

dt− 1

E

dE

dt

= 0



Prof.Paczkowski




1

k

dk

dt=

1

K

dK

dt− 1

E

dE

dt= 0



Prof.Paczkowski



Or. . .

k = K − E

= 0

Or. . .

K = E

This should not be surprising to have k constant over time

The capital stock must grow at the same rate as effectivelabor, which makes sense from the ratio being fixed. Butwe saw this before!



Prof.Paczkowski



Or. . .

k = K − E

= 0

Or. . .

K = E





Prof.Paczkowski



Or. . .

k = K − E

= 0

Or. . .

K = E





Prof.Paczkowski



Or. . .

k = K − E

= 0

Or. . .

K = E





Prof.Paczkowski



Or. . .

k = K − E

= 0

Or. . .

K = E





Prof.Paczkowski



Or. . .

k = K − E

= 0

Or. . .

K = E





Prof.Paczkowski



Or. . .

k = K − E

= 0

Or. . .

K = E





Prof.Paczkowski



Multiplying by k and substituting, so nothing changed, weget. . .

K k = E k

Or. . . (1

K

dK

dt

)k = (n + λ) k

where we used E = n + λ and the definition of K



Prof.Paczkowski




K k = E k

Or. . . (1

K

dK

dt

)k = (n + λ) k




Prof.Paczkowski




K k = E k

Or. . .

(1

K

dK

dt

)k = (n + λ) k




Prof.Paczkowski




K k = E k

Or. . . (1

K

dK

dt

)k = (n + λ) k




Prof.Paczkowski




K k = E k

Or. . . (1

K

dK

dt

)k = (n + λ) k




Prof.Paczkowski



But. . .

(1

K

dK

dt

)k = (n + λ) k(

1

K

)I

(K

E

)= (n + λ) k

I

E= (n + λ) k

Investment per effective worker equals the growth in effectivelabor times k



Prof.Paczkowski



But. . . (1

K

dK

dt

)k = (n + λ) k

(1

K

)I

(K

E

)= (n + λ) k

I

E= (n + λ) k




Prof.Paczkowski



But. . . (1

K

dK

dt

)k = (n + λ) k(

1

K

)I

(K

E

)= (n + λ) k

I

E= (n + λ) k




Prof.Paczkowski



But. . . (1

K

dK

dt

)k = (n + λ) k(

1

K

)I

(K

E

)= (n + λ) k

I

E= (n + λ) k




Prof.Paczkowski



But. . . (1

K

dK

dt

)k = (n + λ) k(

1

K

)I

(K

E

)= (n + λ) k

I

E= (n + λ) k




Prof.Paczkowski



This result shows the amount of investment per effectiveworker necessary at different levels of k to keep k constant – tomaintain a steady-state – given the rate of growth of effectivelabor

The ratio of capital per effective worker (i.e., capitalintensity) is fixed, even if the labor force grows



Prof.Paczkowski



Alternatively, it can be viewed as the amount of investmentnecessary to equip the addition to the effective labor force(measured by n + λ) with exactly the same amount of capitalequipment per effective worker as is used by the existingeffective labor force.

This process of capital accumulation with k constant isknown as capital widening


Capital

Capital Widening

Cap

ital

Dee

peni

ng

More capital but same amount per effective worker

Mor

e ca

pita

l for

eac

h ef

fect

ive

wor

ker

Note: Capital Deepening is discussed later

k constant

k ch

angi

ng


Prof.Paczkowski



The result. . .

I

E= (n + λ) k

is a linear function with constant slope (n + λ) and zerointercept

Since output and investment are denominated the same(and both are per effective worker), they can be plotted onthe same scale


k

q

q*

k*

q(k)

sq(k)

)k(n


Prof.Paczkowski



Conjecture

The point of intersection of the (n + λ) k schedule and the sq (k)schedule gives the equilibrium steady-state value of capital pereffective worker, k*

(n + λ) k shows the amount ofinvestment per effective worker necessary to keep kconstant: needed

sq (k) shows the actual saving per effective workerforthcoming: actual

Remember that S = I



Prof.Paczkowski



Conjecture




Remember that S = I



Prof.Paczkowski



Conjecture




Remember that S = I



Prof.Paczkowski



Conjecture




Remember that S = I


k

q

q*

k*

q(k)

sq(k)

)k(n

Actual saving per Eforthcoming from sq(k)

Investment per Enecessary to keep k constant from λ)k (n


Prof.Paczkowski



Proof:

Actual growth of capital, as shown before, is

K =1

K

dK

dt

=I

K

=sQ

K

E

E

=sq (k)

k

Substitute into k = K − E = 0 and using E = n + λ, we get

sq (k)

k= n + λ



Prof.Paczkowski



Proof:


K =1

K

dK

dt

=I

K

=sQ

K

E

E

=sq (k)

k


sq (k)

k= n + λ



Prof.Paczkowski



Proof:


K =1

K

dK

dt

=I

K

=sQ

K

E

E

=sq (k)

k


sq (k)

k= n + λ



Prof.Paczkowski



Proof:


K =1

K

dK

dt

=I

K

=sQ

K

E

E

=sq (k)

k


sq (k)

k= n + λ



Prof.Paczkowski



Proof:


K =1

K

dK

dt

=I

K

=sQ

K

E

E

=sq (k)

k


sq (k)

k= n + λ



Prof.Paczkowski



Proof:


K =1

K

dK

dt

=I

K

=sQ

K

E

E

=sq (k)

k


sq (k)

k= n + λ



Prof.Paczkowski



Proof:


K =1

K

dK

dt

=I

K

=sQ

K

E

E

=sq (k)

k


sq (k)

k= n + λ



Prof.Paczkowski



Proof:


K =1

K

dK

dt

=I

K

=sQ

K

E

E

=sq (k)

k


sq (k)

k= n + λ



Prof.Paczkowski



Proof (Continued):

Or. . .

sq (k) = (n + λ) k

Or. . .

Saving = Investment



Prof.Paczkowski



Proof (Continued):

Or. . .

sq (k) = (n + λ) k

Or. . .

Saving = Investment



Prof.Paczkowski



Proof (Continued):

Or. . .

sq (k) = (n + λ) k

Or. . .

Saving = Investment



Prof.Paczkowski



Proof (Continued):

Or. . .

sq (k) = (n + λ) k

Or. . .

Saving = Investment



Prof.Paczkowski



Proof (Continued):

Or. . .

sq (k) = (n + λ) k

Or. . .

Saving = Investment



Prof.Paczkowski



Now observe that. . .

sq (k)

k= s

(QE

)(KE

)=

sQ

K

But, we know that QK = APK = 1

v . Therefore, . . .

sq (k) = (n + λ) k

⇒ s

v= n + λ

Or, the warranted rate = the natural rate



Prof.Paczkowski




sq (k)

k= s

(QE

)(KE

)

=sQ

K



sq (k) = (n + λ) k

⇒ s

v= n + λ




Prof.Paczkowski




sq (k)

k= s

(QE

)(KE

)=

sQ

K



sq (k) = (n + λ) k

⇒ s

v= n + λ




Prof.Paczkowski




sq (k)

k= s

(QE

)(KE

)=

sQ

K



sq (k) = (n + λ) k

⇒ s

v= n + λ




Prof.Paczkowski




sq (k)

k= s

(QE

)(KE

)=

sQ

K



sq (k) = (n + λ) k

⇒ s

v= n + λ




Prof.Paczkowski




sq (k)

k= s

(QE

)(KE

)=

sQ

K



sq (k) = (n + λ) k

⇒ s

v= n + λ




Prof.Paczkowski




sq (k)

k= s

(QE

)(KE

)=

sQ

K



sq (k) = (n + λ) k

⇒ s

v= n + λ




Prof.Paczkowski



Equilibrium is attained when the warranted rate of growthequals the natural rate of growth.

Same condition as before, but this time the factor v isdetermined by k, whereas previously v was unknown

v =k

q (k)



Prof.Paczkowski



Equilibrium is attained when the warranted rate of growthequals the natural rate of growth.

Same condition as before, but this time the factor v isdetermined by k, whereas previously v was unknown

v =k

q (k)



Prof.Paczkowski



But this is all worthless if we have another unstable equilibrium.We must show stability. Is the economy stable at k∗ and q∗?

We just have to show what happens out of equilibrium



Prof.Paczkowski



But this is all worthless if we have another unstable equilibrium.We must show stability. Is the economy stable at k∗ and q∗?

We just have to show what happens out of equilibrium



Prof.Paczkowski



Assume k < k∗

This implies that sq (k) > (n + λ) k

Actual investment is greater than needed so k rises: newworkers each get more capital

Note: We can interpret sq(k) as the actual investment and(n + λ) k as the amount of investment needed to equip thelabor force.So more is forthcoming than needed, so the stock pereffective worker must rise.

Assume k > k∗

This implies that sq (k) < (n + λ) k

Actual investment is less than needed so k falls: newworkers each get less capital



Prof.Paczkowski



Assume k < k∗




Assume k > k∗





Prof.Paczkowski



Assume k < k∗




Assume k > k∗





Prof.Paczkowski



Assume k < k∗



Note: We can interpret sq(k) as the actual investment and(n + λ) k as the amount of investment needed to equip thelabor force.

So more is forthcoming than needed, so the stock pereffective worker must rise.

Assume k > k∗





Prof.Paczkowski



Assume k < k∗




Assume k > k∗





Prof.Paczkowski



Assume k < k∗




Assume k > k∗





Prof.Paczkowski



Assume k < k∗




Assume k > k∗





Prof.Paczkowski



Assume k < k∗




Assume k > k∗





Prof.Paczkowski



Assume k < k∗




Assume k > k∗




k

q

q*

k1 k* k2

q(k)

sq(k)

)k(n

A

I > S I < S


Prof.Paczkowski



Now, let’s look at the rate of growth of the economy. Insteady-state equilibrium, output per effective worker isunchanged through time, which implies. . .

q =1

q

dq

dt= 0



Prof.Paczkowski




q =1

q

dq

dt

= 0



Prof.Paczkowski




q =1

q

dq

dt= 0



Prof.Paczkowski



Furthermore, from the homogeneous-of-degree-one productionfunction. . .

qt ⇒ Qt = Etq (kt)

= E0e(n+λ)tq (kt)



Prof.Paczkowski





= E0e(n+λ)tq (kt)



Prof.Paczkowski





= E0e(n+λ)tq (kt)



Prof.Paczkowski



Therefore, in steady-state equilibrium we have. . .

1

Q

dQ

dt≡ Q = (n + λ) +

1

q

dq

dt

Or. . .

Q = n + λ

The economy will grows at the natural rate.



Prof.Paczkowski




1

Q

dQ

dt≡ Q = (n + λ) +

1

q

dq

dt

Or. . .

Q = n + λ




Prof.Paczkowski




1

Q

dQ

dt≡ Q = (n + λ) +

1

q

dq

dt

Or. . .

Q = n + λ




Prof.Paczkowski




1

Q

dQ

dt≡ Q = (n + λ) +

1

q

dq

dt

Or. . .

Q = n + λ




Prof.Paczkowski




1

Q

dQ

dt≡ Q = (n + λ) +

1

q

dq

dt

Or. . .

Q = n + λ




Prof.Paczkowski



This result should be expected since q is constant insteady-state, q = Q

E . So we must have the numerator growingat the same rate as the denominator. . .

Q = n + λ



Prof.Paczkowski



Further, we can get. . .

Q = n + λ⇒ Qt = Q0e(n+λ)t

and, so, real output grows as a function of thelabor-augmenting technical change, λ

Note that if we takes logs, then. . .

ln (Qt) = ln (Q0) + (n + λ) t

which is a straight line, much like our investment pereffective worker function.



Prof.Paczkowski







ln (Qt) = ln (Q0) + (n + λ) t




Prof.Paczkowski







ln (Qt) = ln (Q0) + (n + λ) t



Time

ln(Qt)

ln(Q0)

tλnQln 0

tλn00 eQtλn Qln

:Note

Log of LEVEL

This shows that real output increases at a constant rate given by the exogenous factor, λn


Prof.Paczkowski



Let’s consider what happens when the key variables change. . .

1 saving ratio, s

2 natural rate of growth of labor, n

3 Labor-augmenting technical change, λ



Prof.Paczkowski




1 saving ratio, s





Prof.Paczkowski




1 saving ratio, s





Prof.Paczkowski



Consider an increase in s from s0 to s1

The first effect is to shift the saving schedules upward

The saving-per-effective worker schedule shifts up



Prof.Paczkowski








Prof.Paczkowski







k

q

q0*

k0*

q(k)

s0q(k)

)k(n A

B

At the old equilibrium rate, , we are no longer in equilibrium

k0*

s1q(k)


Prof.Paczkowski



The first effect is for the growth of capital proceeds at a fasterrate than the capital widening rate (at k0*) so. . .

sq (k)

k=

sQ

Kwhich is > n + λ

=I

Kwhich is > n + λ

=1

K

dK

dtwhich is > n + λ

= K which is > n + λ

at the old level k0*

Note: at k0*, capital is growing at the rate n + λ, so n + λ isalso the capital widening rate



Prof.Paczkowski




sq (k)

k=

sQ

Kwhich is > n + λ

=I

Kwhich is > n + λ

=1

K

dK

dtwhich is > n + λ






Prof.Paczkowski




sq (k)

k=

sQ

Kwhich is > n + λ

=I

Kwhich is > n + λ

=1

K

dK

dtwhich is > n + λ






Prof.Paczkowski




sq (k)

k=

sQ

Kwhich is > n + λ

=I

Kwhich is > n + λ

=1

K

dK

dtwhich is > n + λ






Prof.Paczkowski




sq (k)

k=

sQ

Kwhich is > n + λ

=I

Kwhich is > n + λ

=1

K

dK

dtwhich is > n + λ






Prof.Paczkowski




sq (k)

k=

sQ

Kwhich is > n + λ

=I

Kwhich is > n + λ

=1

K

dK

dtwhich is > n + λ






Prof.Paczkowski



The result is that k > 0 since k = K − (n + λ)

Capital deepening now takes place which means that. . .

q > 0

Economic growth is positive



Prof.Paczkowski





q > 0




Prof.Paczkowski





q > 0



Capital Widening

Cap

ital

Dee

peni

ng

More capital but same amount per effective worker

Mor

e ca

pita

l for

eac

h ef

fect

ive

wor

ker

Note: Capital Widening was discussed previously

k constant

k ch

angi

ng


Prof.Paczkowski



Growth continues, but because the production function exhibitsdiminishing returns, q rises proportionately less than k

The new saving schedule intersects the (n + λ)k scheduleat a new higher steady-state equilibrium capital pereffective worker, k1*, and, hence, a new output pereffective worker, q1*



Prof.Paczkowski



Growth continues, but because the production function exhibitsdiminishing returns, q rises proportionately less than k

The new saving schedule intersects the (n + λ)k scheduleat a new higher steady-state equilibrium capital pereffective worker, k1*, and, hence, a new output pereffective worker, q1*


k

q

q1*

q0*

k0* k1

*

q(k)

s1q(k)s0q(k)

)k(n A

B

This is a new, higher level of real output.


Prof.Paczkowski



Note what has happened. . .

The increase in the saving rate increased the warrantedrate, s

v , above the natural rate, n + λ at k0*

But. . .

The capital output ratio, q(k)k = 1

v , has fallen since outputhas risen proportionately less than capitalThus, the growth rate has only temporarily increased eventhough the level of output and capital per effective workerincreased

The steady-state growth rate is invariant with respect tothe saving rate

Growth is determined exclusively by the natural rate, n + λ



Prof.Paczkowski






But. . .







Prof.Paczkowski






But. . .







Prof.Paczkowski






But. . .


v , has fallen since outputhas risen proportionately less than capital

Thus, the growth rate has only temporarily increased eventhough the level of output and capital per effective workerincreased





Prof.Paczkowski






But. . .







Prof.Paczkowski






But. . .







Prof.Paczkowski






But. . .






Time

ln(Q)

ln(Q1)

ln(Q0)

t0

tλnQln 1

tλnQln 0

tλn11 eQtλn Qln

: thatNote

Log of LEVEL


Prof.Paczkowski



Real growth is given by. . .

Q = C = S = K = I = E = n + λ

The economy will only grow at the natural rate.



Prof.Paczkowski




Q

= C = S = K = I = E = n + λ




Prof.Paczkowski




Q = C

= S = K = I = E = n + λ




Prof.Paczkowski




Q = C = S

= K = I = E = n + λ




Prof.Paczkowski




Q = C = S = K

= I = E = n + λ




Prof.Paczkowski




Q = C = S = K = I

= E = n + λ




Prof.Paczkowski




Q = C = S = K = I = E

= n + λ




Prof.Paczkowski




Q = C = S = K = I = E = n + λ




Prof.Paczkowski




Q = C = S = K = I = E = n + λ




Prof.Paczkowski



Assignment

Show what happens when n and λ change


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