Lecture 3 Economic Growth Economics 5118 Macroeconomic Theory
Kam Yu
Winter 2013
Outline 1
Introduction
2
Modelling Economic Growth
3
The Solow-Swan Model Theory Growth and Development Balanced Growth
4
Theory of Optimal Growth The Model Steady State Comparing Models
5
Endogenous Growth The AK Model Human Capital Model Observations Kam Yu (LU)
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Introduction
Industrial Revolution and Capitalism 12
Income per person (1800= 1)
10
Great Div ergence 8
6
Indus trial Rev olution
4
2
0 BC 1000
Malthus ian Trap
-500
0
500
1000
1500
AD 2000
Figure . World economic history in one picture. Incomes rose sharply in many countries after 1800 but declined in others.
Source: Clark (2007) Kam Yu (LU)
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Introduction
And Growing Faster and Faster
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Introduction
Sources of Economic Growth
1
increases in capital stock
2
increases in human resources: population, immigration, participation rate, education
3
technological progress: new methods of production, more efficient machinery and structures.
4
Openness to trade
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Modelling Economic Growth
Basic Set-Up of the Model Capital letters for total quantities: Variable Population Capital Technology Output Consumption Investment Depreciation
Symbol Nt Kt – Yt Ct It –
Growth Rate n γ µ – – – −δ
Cobb-Douglas production function: Yt = Ft (Kt , Nt , t) = (1 + µ)t Ktα Nt1−α . Lowercase letters for per capita quantities: α 1−α K t Nt t yt = (1 + µ) = (1 + µ)t ktα . Nt Kam Yu (LU)
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Modelling Economic Growth
Identities and Dynamics
National income identity: Yt = Ct + It . Capital accumulation: ∆Kt+1 = It − δKt . Population dynamics: Nt = (1 + n)t N0 .
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The Solow-Swan Model
Assumptions of the Solow-Swan Model 1
Growth rates of population, n, and technological progress, µ are exogenous.
2
The saving rate, st = 1 −
Ct ct =1− , Yt yt
is also exogenous. That is, st = s. 3
All savings are invested, that is, st =
4
It = it . Yt
The objective is to maximize output per capita, or equivalently capital per capita.
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The Solow-Swan Model
Theory
Capital Accumulation From the capital accumulation equation, ∆Kt+1 It It /Yt = −δ = −δ Kt Kt Kt /Yt yt Yt /Nt − δ = s − δ. =s Kt /Nt kt In per capita term, ∆kt+1 kt
∆Kt+1 ∆Nt+1 − Kt Nt yt = s − (δ + n), kt
'
(exercise)
or ∆kt+1 = syt − (δ + n)kt . Kam Yu (LU)
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The Solow-Swan Model
Theory
Graphical Solution
y
∆kt+1
6
γkt
6 yt (δ + n)kt
-k
Per capita output and saving
Kam Yu (LU)
γk ∗
syt k
syt − (δ + n)kt = ∆kt+1 -k ∗
Capital accumulation
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The Solow-Swan Model
Theory
Per Capita Growth Rate of Capital From (3.1) the growth rate of capital per capita is γ=
∆kt+1 yt = s − (δ + n). kt kt
(3.1a)
The growth rate γ is decreasing in k since dγ kt (dyt /dkt ) − yt syt kt dyt =s = − 1 − < 0, dkt yt dkt kt2 kt2 and the capital elasticity of output is always less that one, i.e., kt dyt < 1. (Exercise) yt dkt Conclusion: The larger the capital stock per capita, the lower the growth rate. Kam Yu (LU)
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The Solow-Swan Model
Growth and Development
Implications of the Solow-Swan Model
1
Since dγ/dk < 0, developing countries have higher growth rates than developed countries.
2
Since dγ/ds = y /k > 0, and dγ/dδ = dγ/dn < 0, a higher saving rate, lower depreciation rate, or lower population growth rate would increase γ.
3
Technical progress in each period increases yt /kt and therefore raises γ.
Developing countries have higher growth rates due to diminishing marginal product of capital. Does empirical evidence supports this claim?
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The Solow-Swan Model
Growth and Development
Empirical evidence on MPK — Developed Countries The B.E. Journal of Macroeconomics, Vol. 9 [2009], Iss. 1 (Topics), Art. 16
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Lecture 3 Economic Growth
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The Solow-Swan Model
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The Solow-Swan Model
Balanced Growth
Balanced Growth Recall that yt = (1 + µ)t ktα . Then the per capita growth rate of output is ∆yt+1 yt
' log yt+1 − log yt = (t + 1) log(1 + µ) + α log kt+1 −t log(1 + µ) − α log kt = log(1 + µ) + α(log kt+1 − log kt ) ' µ + αγ.
Since ct = (1 − s)yt , the per capita consumption growth rate is the same as that of output. The idea of balanced growth is for y , c, and k having the same growth rate. This requires µ + αγ = γ or γ = µ/(1 − α).
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Theory of Optimal Growth
The Model
Optimal Growth
In the Solow-Swan model the objective is to maximize output per capita in every period, which is similar to the golden rule. Now we shift our objective to maximization of the present value of intertemporal welfare. A useful math trick: rewrite the Cobb-Douglas production function as Yt
= (1 + µ)t Ktα Nt1−α = Ktα [(1 + µ)t/(1−α) Nt ]1−α = Ktα (Nt# )1−α ,
where Nt# = (1 + µ)t/(1−α) Nt is called effective labour.
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Theory of Optimal Growth
The Model
Effective Labour Since Nt = (1 + n)t N0 , Nt# = (1 + µ)t/(1−α) Nt = [(1 + µ)1/(1−α) (1 + n)]t N0 = (1 + η)t N0 , where 1 + η = (1 + µ)1/(1−α) (1 + n) or, using log approximation, µ η 'n+ . 1−α Output and capital stock per unit of effective labour are yt# = kt# = Kam Yu (LU)
Yt Nt# Kt Nt#
= =
Yt , (1 + η)t N0 Kt . (1 + η)t N0
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Theory of Optimal Growth
The Model
National Income and Capital Accumulation Consumption and investment per unit of effective labour are ct# = it# =
Ct Nt# It Nt#
= =
Ct , (1 + η)t N0 It . (1 + η)t N0
The production function becomes yt# = (kt# )α . The national income identity becomes yt# = ct# + it# . The capital accumulation equation becomes # (1 + η)kt+1 = it# + (1 − δ)kt# . Kam Yu (LU)
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Theory of Optimal Growth
The Model
Resource Constraints and Utility The last three equations gives the resource constraint # (kt# )α = ct# + (1 + η)kt+1 − (1 − δ)kt# .
Instantaneous utility function has the function form of constant relative risk aversion: U(Ct ) =
=
Ct1−σ 1−σ h i1−σ (1 + η)t N0 ct# "
=
1 −#σ
(ct# )1−σ 1−σ
(1 + η)(1−σ)t ,
where N0 is normalized to 1. Kam Yu (LU)
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Theory of Optimal Growth
The Model
Optimization The optimization problem is " # ∞ # 1−σ X (c ) t+s s max β˜ (1 + η)(1−σ)t # # 1 − σ ct+s ,kt+s+1 s=0
# subject to (kt# )α = ct# + (1 + η)kt+1 − (1 − δ)kt# ,
where β˜ = β(1 + η)1−σ . The Lagrangian is # ( " ∞ # 1−σ X (c ) t+s s (1 + η)(1−σ)t Lt = β˜ 1−σ s=0
) h i # α # # # + λt+s (kt+s ) − ct+s − (1 + η)kt+s+1 + (1 − δ)kt+s .
Kam Yu (LU)
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Theory of Optimal Growth
The Model
First-Order Conditions
∂Lt
# −σ = β˜s (ct+s ) (1 + η)(1−σ)t − λt+s = 0,
# ∂ct+s
∂Lt
s ≥ 0,
h i # α−1 = λt+s α(kt+s ) +1−δ
# ∂kt+s
− λt+s−1 (1 + η) = 0,
s ≥ 1.
The Euler equation is β˜
# ct+1
ct#
!−σ
h i # α−1 ) + 1 − δ = 1 + η. α(kt+1
Note that the Euler equation is the same as in Chapter 2 if we set η = 0. Kam Yu (LU)
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Theory of Optimal Growth
Steady State
Steady State
Since kt# and ct# are kt and ct adjusted for technological and # # population growth, in the steady state ∆kt+1 = ∆ct+1 = 0. The Euler equation becomes h i β˜ α(k #∗ )α−1 + 1 − δ = 1 + η. Solving for k #∗ (exercise), k
Kam Yu (LU)
#∗
'
σ(n + (µ/(1 − α))) + δ + θ α
Lecture 3 Economic Growth
−1/(1−α) .
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Theory of Optimal Growth
Steady State
Capital, Output, and Consumption Although kt# is unchanged in steady state, capital stock per capita, kt = Kt /Nt , is growing due to technological progress: h it kt = k #∗ (1 + µ)1/(1−α) , which means that kt grows at a rate of approximately µ/(1 − α). α Similarly, since yt# = kt# and yt ct
h it = yt#∗ (1 + µ)1/(1−α) , h it = ct#∗ (1 + µ)1/(1−α) ,
output and consumption per capita grow at the same rate of µ/(1 − α) (balanced growth). Kam Yu (LU)
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Theory of Optimal Growth
Comparing Models
Saving Rate The saving rate along the optimal growth path is st = 1 − Ct /Yt = 1 − ct# /yt# . Since yt# = (kt# )α , ct# = (kt# )α − (η + δ)kt# , ση + δ + θ −1/(1−α) # , kt = α the optimal saving rate is st =
α(η + δ) . ση + δ + θ
Therefore the saving rate is constant as in the Solow-Swan model. Kam Yu (LU)
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Theory of Optimal Growth
Comparing Models
Empirical Observations — U.S. Time Series Is saving rate constant?
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Theory of Optimal Growth
Comparing Models
Empirical Observations — Cross-Sectional
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Endogenous Growth
New Growth Theory
In the previous models technical progress is exogenous. For most developing countries that is a good assumption. New technologies are usually embodied in imported goods and services and from foreign direct investment. For developed countries to maintain their edge, they have to invest in R&D using resources. Technical progress becomes an endogenous decision. The subject is often called new growth theory.
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Endogenous Growth
The Technological Race
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Endogenous Growth
The AK Model
The AK Model Production function: Yt = AKt , A > 0, or in per capita form, yt = Akt .
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Endogenous Growth
The AK Model
The AK Model Production function: Yt = AKt , A > 0, or in per capita form, yt = Akt . Kt can be an aggregate form of capital such as physical, human, intellectual properties, etc.
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Endogenous Growth
The AK Model
The AK Model Production function: Yt = AKt , A > 0, or in per capita form, yt = Akt . Kt can be an aggregate form of capital such as physical, human, intellectual properties, etc. The key point is production exhibits constant returns to scale in Kt . The average product of capital, yt /kt = A, is constant, not decreasing in kt .
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Endogenous Growth
The AK Model
The AK Model Production function: Yt = AKt , A > 0, or in per capita form, yt = Akt . Kt can be an aggregate form of capital such as physical, human, intellectual properties, etc. The key point is production exhibits constant returns to scale in Kt . The average product of capital, yt /kt = A, is constant, not decreasing in kt . From (3.1a), the growth rate of capital is yt − (δ + n) kt = st A − (δ + n).
γ = st
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Endogenous Growth
The AK Model
The AK Model Production function: Yt = AKt , A > 0, or in per capita form, yt = Akt . Kt can be an aggregate form of capital such as physical, human, intellectual properties, etc. The key point is production exhibits constant returns to scale in Kt . The average product of capital, yt /kt = A, is constant, not decreasing in kt . From (3.1a), the growth rate of capital is yt − (δ + n) kt = st A − (δ + n).
γ = st
Therefore the capital growth is independent of the level of capital. Kam Yu (LU)
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Endogenous Growth
Human Capital Model
Human Capital Model Separation of human capital, ht , and physical capital, kt , both expressed in per capita form.
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Endogenous Growth
Human Capital Model
Human Capital Model Separation of human capital, ht , and physical capital, kt , both expressed in per capita form. Production function: yt = Aktα ht1−α ,
Kam Yu (LU)
0 ≤ α ≤ 1.
Lecture 3 Economic Growth
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Endogenous Growth
Human Capital Model
Human Capital Model Separation of human capital, ht , and physical capital, kt , both expressed in per capita form. Production function: yt = Aktα ht1−α ,
0 ≤ α ≤ 1.
Assuming both types of capital depreciate at the same rate δ, the capital accumulation equations are ∆kt+1 = itk − δkt , ∆ht+1 = ith − δht ,
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Endogenous Growth
Human Capital Model
Optimization Problem
max
∞ X s=0
βs
1−σ ct+s 1−σ
subject to 1−α α ht+s Akt+s = ct+s + (kt+s+1 + ht+s+1 ) − (1 − δ)(kt+s + ht+s ).
Note: See section 3.5.2.2 of the textbook for a model that uses different technologies in producing physical and human capital.
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Endogenous Growth
Human Capital Model
Optimization The Lagrangian is ( ∞ h X c 1−σ 1−α α Lt = β s t+s + λt+s Akt+s ht+s − ct+s 1−σ s=0
− (kt+s+1 + ht+s+1 ) + (1 − δ)(kt+s
Kam Yu (LU)
Lecture 3 Economic Growth
i + ht+s )
Winter 2013
)
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Endogenous Growth
Human Capital Model
Optimization The Lagrangian is ( ∞ h X c 1−σ 1−α α Lt = β s t+s + λt+s Akt+s ht+s − ct+s 1−σ s=0
− (kt+s+1 + ht+s+1 ) + (1 − δ)(kt+s
i + ht+s )
)
The first-Order Conditions are ∂Lt ∂ct+s ∂Lt ∂kt+s ∂Lt ∂ht+s
−σ = β s ct+s − λt+s = 0,
s ≥ 0,
α−1 1−α = λt+s αAkt+s ht+s + 1 − δ − λt+s−1 = 0, s ≥ 1, −α α = λt+s (1 − α)Akt+s ht+s + 1 − δ − λt+s−1 = 0,
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Lecture 3 Economic Growth
s ≥ 1.
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Endogenous Growth
Human Capital Model
Euler equation The Euler equation is β
ct+1 ct
−σ "
αA
kt+1 ht+1
−(1−α)
# + 1 − δ = 1.
From the first-order conditions of k and h, α kt+1 = , ht+1 1−α which is a constant. Substituting this into the Euler equation, we get the growth rate of consumption as (exercise) i ct+1 1h α log = Aα (1 − α)(1−α) − δ − θ . ct σ Kam Yu (LU)
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Endogenous Growth
Human Capital Model
Conclusion 1
Optimal consumption grows at a constant rate given preferences and technology.
2
Ratios of physical to human capital are constant through time and therefore have the same growth rate. As balanced growth they are equal to the consumption growth rate.
3
The production function can be written as yt
=
Aktα ht1−α
= A
α 1−α
=A
kt ht
−(1−α)
−(1−α) kt kt = A∗ kt .
Therefore the model is effectively the AK model. Kam Yu (LU)
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Endogenous Growth
Observations
Income and Growth Rates of 112 Countries VOL. 1, NO. 1
LUCAS: TRADE AND THE DIFFUSION OF THE INDUSTRIAL REVOLUTION
3
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Figure 1. Income and Growth Rates, 112 Countries
Sachs and Warner are explicit about their definition of openness, but it is a comSource: Lucas (2009) plicated definition. To be classified as open, an economy must pass five tests. It must (a) have effective protection rates less3than 40 percent, than2013 Kam Yu (LU) Lecture Economic Growth (b) have quotas on less Winter
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Endogenous Growth
Observations
The Importance of Being Open 4
AMERICAN ECONOMIC JOURNAL: MACROECONOMICS
JANUARY 2009
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Figure 2. Income and Growth Rates, 112 Countries
these (2009) countries opened after 1990 and many are now members of the European Source: Lucas Union. Many other countries have undertaken major policy reforms. A replication Kam (LU) Lecture 3 Economic Growththe criteria (a)–(e) to the Winter 2013 thatYureclassified all of the countries based on applying entire
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Endogenous Growth
Observations
A Subset of Developing Countries
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Endogenous Growth
Observations
Growth History Comparison
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Endogenous Growth
Observations
References
Clark, Gregory (2007) A Farewell to Alms: A Brief Economic History of the World, Princeton: Princeton University Press. Lucas, Robert E. (2009) “Trade and the Diffusion of the Industrial Revolution,” American Economic Journal: Macroeconomics, 1(1), 1–25. Mello, Marcelo (2009) “Estimates of the Marginal Product of Capital, 1970–2000,” B.E. Journal of Macroeconomics, 9(1), Article 16.
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