lecture 4 modern growth theories - rutgers...
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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
Lecture 4ModernGrowthTheories
Prof.Paczkowski
ReadingAssignments Part I
Reading Assignments
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 2 / 116
Lecture 4ModernGrowthTheories
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 3 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Introduction Part II
Introduction
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 4 / 116
Lecture 4ModernGrowthTheories
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 5 / 116
Lecture 4ModernGrowthTheories
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 5 / 116
Lecture 4ModernGrowthTheories
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 5 / 116
Lecture 4ModernGrowthTheories
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 5 / 116
Lecture 4ModernGrowthTheories
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 5 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Part III
Harrod-Domar
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 6 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 7 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 7 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 8 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 8 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 8 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 8 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 8 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 8 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 8 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 9 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 9 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 9 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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. . .
Muscle
EducationHealthWork ethicEtc.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 9 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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. . .
MuscleEducation
HealthWork ethicEtc.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 9 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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. . .
MuscleEducationHealth
Work ethicEtc.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 9 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 ethic
Etc.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 9 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 9 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 10 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 10 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 10 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 10 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 10 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 10 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 10 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 10 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 10 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 11 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 11 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 11 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 11 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 11 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 11 / 116
E
Kuv
Q2
Q1
E0
K0
Isoquant Map
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 13 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 13 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 13 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 13 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 13 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 13 / 116
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
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 16 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 16 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 16 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 17 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 17 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 17 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 17 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 17 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 17 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Notation: The single dot over I indicates the rate of growth inI over time
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 18 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 19 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 19 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 19 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 20 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 20 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 21 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Conjecture
Q =s
v
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 22 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 23 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 23 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 23 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 23 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 23 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 23 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 23 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 23 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 24 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 24 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 24 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 24 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 24 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 24 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 25 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 25 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 25 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 25 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 25 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 26 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 27 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 27 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 27 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 27 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 27 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 27 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 28 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Conjecture
C = sv
Let C ′t = cQt be consumption net of Ap
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 28 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Conjecture
C = sv
Let C ′t = cQt be consumption net of Ap
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 28 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 28 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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
v
1
C ′dC ′
dt=
s
v
⇒ C =s
v
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 28 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 28 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 28 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 29 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Conjecture
S = sv
Let S ′t = sQt be saving net of Ap
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 29 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Conjecture
S = sv
Let S ′t = sQt be saving net of Ap
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 29 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 29 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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
v
1
S
dS
dt=
s
v
⇒ S =s
v
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 29 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 29 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 29 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 30 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 30 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 30 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 30 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 30 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 30 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 30 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Finally, by the production function, if real output grows at therate, Q, then capital must also grow at the same rate
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 31 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 32 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 32 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 32 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 32 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 32 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 32 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 32 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
This results in a ”balanced growth” equilibrium. . .
Q = C = S = K = I =s
v
Everything must grow at the warranted rate, sv
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 33 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
This results in a ”balanced growth” equilibrium. . .
Q
= C = S = K = I =s
v
Everything must grow at the warranted rate, sv
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 33 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
This results in a ”balanced growth” equilibrium. . .
Q = C
= S = K = I =s
v
Everything must grow at the warranted rate, sv
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 33 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
This results in a ”balanced growth” equilibrium. . .
Q = C = S
= K = I =s
v
Everything must grow at the warranted rate, sv
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 33 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
This results in a ”balanced growth” equilibrium. . .
Q = C = S = K
= I =s
v
Everything must grow at the warranted rate, sv
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 33 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
This results in a ”balanced growth” equilibrium. . .
Q = C = S = K = I
=s
v
Everything must grow at the warranted rate, sv
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 33 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
This results in a ”balanced growth” equilibrium. . .
Q = C = S = K = I =s
v
Everything must grow at the warranted rate, sv
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 33 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
This results in a ”balanced growth” equilibrium. . .
Q = C = S = K = I =s
v
Everything must grow at the warranted rate, sv
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 33 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
The key Harrod-Domar result
Q =s
v
demonstrates the existence of a steady growth path for theeconomy
Everything must grow at the warranted rate, sv
The stability of the growth path, however, is in doubt
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 34 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
The key Harrod-Domar result
Q =s
v
demonstrates the existence of a steady growth path for theeconomy
Everything must grow at the warranted rate, sv
The stability of the growth path, however, is in doubt
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 34 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 35 / 116
D
SP
QStable Equilibrium Unstable Equilibrium
D
P
Q
S
Initially assume price too low
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 37 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 37 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 38 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 39 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 39 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 39 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 39 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 39 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 39 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 40 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 40 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 40 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 40 / 116
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
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 42 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 42 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 42 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 framework
The capital markets, and their importance, will beintroduced later
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 42 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 42 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 43 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 43 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 43 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 43 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 43 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 43 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 43 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 43 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 44 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 45 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 45 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 46 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 47 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 47 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 47 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 47 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 47 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 47 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 48 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 49 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 49 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 49 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Note that. . .
ln (Et) = ln (E0) + (n + λ) t
so. . .
1
E
dE
dt≡ E = (n + λ)
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 50 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Note that. . .
ln (Et) = ln (E0) + (n + λ) t
so. . .
1
E
dE
dt≡ E = (n + λ)
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 50 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Note that. . .
ln (Et) = ln (E0) + (n + λ) t
so. . .
1
E
dE
dt≡ E = (n + λ)
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 50 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 51 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 52 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 52 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 52 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 52 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 52 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 52 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 53 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 53 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 53 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 53 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 53 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 54 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 54 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 54 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 54 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 55 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 55 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Subsequent work specified. . .
the determinants of the four variables
their interrelations
the way they adjust to satisfy the equality sv = (n + λ)
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 56 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Subsequent work specified. . .
the determinants of the four variables
their interrelations
the way they adjust to satisfy the equality sv = (n + λ)
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 56 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Subsequent work specified. . .
the determinants of the four variables
their interrelations
the way they adjust to satisfy the equality sv = (n + λ)
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 56 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Harrod-Domar
Warranted Rateof Growth
Equilibrium
Harrod-Domar(Continued)
Subsequent work specified. . .
the determinants of the four variables
their interrelations
the way they adjust to satisfy the equality sv = (n + λ)
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 56 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Part IV
Solow’s Neoclassical Growth Model
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 57 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 58 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 58 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 58 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
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 assumed
Stabilization policy is successful
2 Fixed technical coefficients production function is dropped
Substitution between factors of production is now assumed
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 58 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 58 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 58 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 58 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 59 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 59 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 59 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 60 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 60 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 60 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 60 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 60 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
If we divide by λE , we obtain. . .
Q
E= Q
(K
E, 1
)= Q
(K
E
)
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 61 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
If we divide by λE , we obtain. . .
Q
E= Q
(K
E, 1
)= Q
(K
E
)
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 61 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Or. . .
q = q (k)
where
q =Q
E
k =K
Edq
dk> 0
d2q
dk2< 0
Note: ”k” is called ”capital intensity”
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 62 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Or. . .
q = q (k)
where
q =Q
E
k =K
Edq
dk> 0
d2q
dk2< 0
Note: ”k” is called ”capital intensity”
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 62 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Or. . .
q = q (k)
where
q =Q
E
k =K
Edq
dk> 0
d2q
dk2< 0
Note: ”k” is called ”capital intensity”
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 62 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 63 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 63 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 63 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 growth
We have apples-to-apples comparison
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 63 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 63 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 64 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 64 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 64 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 65 / 116
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
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 67 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 67 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 67 / 116
k
q
q(k)
sq(k), 0<s<1
A
k*
q*
B
Note:Recall how the Keynesian Cross diagram was divided into parts
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 69 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 69 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 69 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 69 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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
E
I
E= sq (k)
where k = KE
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 69 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 69 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 69 / 116
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
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 71 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 71 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 71 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 71 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 71 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 72 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 72 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 72 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 73 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 73 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 73 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 73 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 73 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 74 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 74 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 74 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 = 0
But, 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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 74 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 changing
With E growing at the same rate, QE is unchanged: the
economy produces just enough jobs to meet the neweffective workers
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 74 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 74 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 75 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 75 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 76 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 76 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 76 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 76 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 76 / 116
k
q
q(k)
sq(k), 0<s<1
k*
q*For this to be in steady-state, then…
this must be fixed
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 78 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 78 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 78 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 78 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 78 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Must know how effective labor behaves
Assume that effective labor grows at the proportional raten + λ; that is,
E = n + λ
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 79 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Must know how effective labor behaves
Assume that effective labor grows at the proportional raten + λ; that is,
E = n + λ
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 79 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 80 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 80 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 80 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 80 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 80 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 81 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 81 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 81 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 82 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 82 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 82 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 82 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 82 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 82 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 82 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 83 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 83 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 83 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 83 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 83 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 84 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 84 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 84 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 84 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 84 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 85 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 86 / 116
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
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 88 / 116
k
q
q*
k*
q(k)
sq(k)
)k(n
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 90 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 90 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 90 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 90 / 116
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
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 92 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 92 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 92 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 92 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 92 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 92 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 92 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 92 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Proof (Continued):
Or. . .
sq (k) = (n + λ) k
Or. . .
Saving = Investment
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 93 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Proof (Continued):
Or. . .
sq (k) = (n + λ) k
Or. . .
Saving = Investment
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 93 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Proof (Continued):
Or. . .
sq (k) = (n + λ) k
Or. . .
Saving = Investment
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 93 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Proof (Continued):
Or. . .
sq (k) = (n + λ) k
Or. . .
Saving = Investment
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 93 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Proof (Continued):
Or. . .
sq (k) = (n + λ) k
Or. . .
Saving = Investment
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 93 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 94 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 94 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 94 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 94 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 94 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 94 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 94 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 95 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 95 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 96 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 96 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 97 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 97 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 97 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 97 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 97 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 97 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 97 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 97 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 97 / 116
k
q
q*
k1 k* k2
q(k)
sq(k)
)k(n
A
I > S I < S
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 99 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 99 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 99 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Furthermore, from the homogeneous-of-degree-one productionfunction. . .
qt ⇒ Qt = Etq (kt)
= E0e(n+λ)tq (kt)
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 100 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Furthermore, from the homogeneous-of-degree-one productionfunction. . .
qt ⇒ Qt = Etq (kt)
= E0e(n+λ)tq (kt)
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 100 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Furthermore, from the homogeneous-of-degree-one productionfunction. . .
qt ⇒ Qt = Etq (kt)
= E0e(n+λ)tq (kt)
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 100 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 101 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 101 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 101 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 101 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 101 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 102 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 103 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 103 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 103 / 116
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
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 105 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 105 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 105 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 106 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 106 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 106 / 116
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)
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 108 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 108 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 108 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 108 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 108 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 108 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 109 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 109 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 109 / 116
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
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 111 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 111 / 116
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.
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 113 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 113 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 113 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 capital
Thus, 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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 113 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 113 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 113 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
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 (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 113 / 116
Time
ln(Q)
ln(Q1)
ln(Q0)
t0
tλnQln 1
tλnQln 0
tλn11 eQtλn Qln
: thatNote
Log of LEVEL
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Real growth is given by. . .
Q = C = S = K = I = E = n + λ
The economy will only grow at the natural rate.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 115 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Real growth is given by. . .
Q
= C = S = K = I = E = n + λ
The economy will only grow at the natural rate.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 115 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Real growth is given by. . .
Q = C
= S = K = I = E = n + λ
The economy will only grow at the natural rate.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 115 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Real growth is given by. . .
Q = C = S
= K = I = E = n + λ
The economy will only grow at the natural rate.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 115 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Real growth is given by. . .
Q = C = S = K
= I = E = n + λ
The economy will only grow at the natural rate.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 115 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Real growth is given by. . .
Q = C = S = K = I
= E = n + λ
The economy will only grow at the natural rate.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 115 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Real growth is given by. . .
Q = C = S = K = I = E
= n + λ
The economy will only grow at the natural rate.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 115 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Real growth is given by. . .
Q = C = S = K = I = E = n + λ
The economy will only grow at the natural rate.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 115 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Real growth is given by. . .
Q = C = S = K = I = E = n + λ
The economy will only grow at the natural rate.
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 115 / 116
Lecture 4ModernGrowthTheories
Prof.Paczkowski
Solow’sNeoclassicalGrowth Model
Solow’s Model(Continued)
Assignment
Show what happens when n and λ change
Prof. Paczkowski (Rutgers University) Lecture 4 Modern Growth Theories Spring Semester, 2009 116 / 116