Lecture 3 Economic Growth - Lakehead Universityflash.lakeheadu.ca/~kyu/E5118/M3.pdf · 1 increases in capital stock 2 increases in human resources: population, immigration, participation

Lecture 3 Economic GrowthEconomics 5118 Macroeconomic Theory

Kam Yu

Winter 2013

Outline

1 Introduction

2 Modelling Economic Growth

3 The Solow-Swan ModelTheoryGrowth and DevelopmentBalanced Growth

4 Theory of Optimal GrowthThe ModelSteady StateComparing Models

5 Endogenous GrowthThe AK ModelHuman Capital ModelObservations

Kam Yu (LU) Lecture 3 Economic Growth Winter 2013 2 / 39

Introduction

Industrial Revolution and Capitalism

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Figure !.! World economic history in one picture. Incomes rose sharply in many countries after 1800 but declined in others.

typical English worker of 1800, even though the English table by then included such exotics as tea, pepper, and sugar.

And hunter-gatherer societies are egalitarian. Material consumption varies little across the members. In contrast, inequality was pervasive in the agrarian economies that dominated the world in 1800. The riches of a few dwarfed the pinched allocations of the masses. Jane Austen may have written about re-fined conversations over tea served in china cups. But for the majority of the English as late as 1813 conditions were no better than for their naked ancestors of the African savannah. The Darcys were few, the poor plentiful.

So, even according to the broadest measures of material life, average welfare, if anything, declined from the Stone Age to 1800. The poor of 1800, those who lived by their unskilled labor alone, would have been better off if transferred to a hunter-gatherer band.

The Industrial Revolution, a mere two hundred years ago, changed for-ever the possibilities for material consumption. Incomes per person began to undergo sustained growth in a favored group of countries. The richest mod-ern economies are now ten to twenty times wealthier than the 1800 average. Moreover the biggest beneficiary of the Industrial Revolution has so far been

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Source: Clark (2007)


Introduction

And Growing Faster and Faster


Introduction

Sources of Economic Growth

1 increases in capital stock

2 increases in human resources: population, immigration, participationrate, education

3 technological progress: new methods of production, more efficientmachinery and structures.

4 Openness to trade


Modelling Economic Growth

Basic Set-Up of the Model

Capital letters for total quantities:

Variable Symbol Growth Rate

Population Nt nCapital Kt γTechnology – µOutput Yt –Consumption Ct –Investment It –Depreciation – −δ

Cobb-Douglas production function:

Yt = Ft(Kt ,Nt , t) = (1 + µ)tKαt N

1−αt .

Lowercase letters for per capita quantities:

yt = (1 + µ)t(Kαt N

1−αt

Nt

)= (1 + µ)tkαt .


Modelling Economic Growth

Identities and Dynamics

National income identity:Yt = Ct + It .

Capital accumulation:∆Kt+1 = It − δKt .

Population dynamics:Nt = (1 + n)tN0.


The Solow-Swan Model

Assumptions of the Solow-Swan Model

1 Growth rates of population, n, and technological progress, µ areexogenous.

2 The saving rate,

st = 1− Ct

Yt= 1− ct

yt,

is also exogenous. That is, st = s.

3 All savings are invested, that is,

st =ItYt

= it .

4 The objective is to maximize output per capita, or equivalently capitalper capita.


The Solow-Swan Model Theory

Capital Accumulation

From the capital accumulation equation,

∆Kt+1

Kt=

ItKt− δ =

It/Yt

Kt/Yt− δ

= sYt/Nt

Kt/Nt− δ = s

ytkt− δ.

In per capita term,

∆kt+1

kt' ∆Kt+1

Kt− ∆Nt+1

Nt(exercise)

= sytkt− (δ + n),

or∆kt+1 = syt − (δ + n)kt . (3.1)



Graphical Solution

Per capita output and saving

-

6

k

y

��

��

��

(δ + n)kt

syt

yt

Capital accumulation

-

6

k

∆kt+1

��

γkt

syt − (δ + n)kt = ∆kt+1

k∗

γk∗



Per Capita Growth Rate of Capital

From (3.1) the growth rate of capital per capita is

γ =∆kt+1

kt= s

ytkt− (δ + n). (3.1a)

The growth rate γ is decreasing in k since

dγ

dkt= s

[kt(dyt/dkt)− yt

k2t

]= −syt

k2t

[1− kt

yt

dytdkt

]< 0,

and the capital elasticity of output is always less that one, i.e.,

ktyt

dytdkt

< 1. (Exercise)

Conclusion: The larger the capital stock per capita, the lower the growthrate.


The Solow-Swan Model Growth and Development

Implications of the Solow-Swan Model

1 Since dγ/dk < 0, developing countries have higher growth rates thandeveloped countries.

2 Since dγ/ds = y/k > 0, and dγ/dδ = dγ/dn < 0, a higher savingrate, lower depreciation rate, or lower population growth rate wouldincrease γ.

3 Technical progress in each period increases yt/kt and therefore raisesγ.

Developing countries have higher growth rates due to diminishing marginalproduct of capital. Does empirical evidence supports this claim?



Empirical evidence on MPK — Developed Countries

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The B.E. Journal of Macroeconomics, Vol. 9 [2009], Iss. 1 (Topics), Art. 16

http://www.bepress.com/bejm/vol9/iss1/art16

Source: Mello (2009)



Empirical evidence on MPK — Developing Countries !"#$%&'!()!*+*,!#-!./0/-1!23#45'&6/-71!89,91!/-7!8929!

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The B.E. Journal of Macroeconomics, Vol. 9 [2009], Iss. 1 (Topics), Art. 16

http://www.bepress.com/bejm/vol9/iss1/art16

Source: Mello (2009)


The Solow-Swan Model Balanced Growth

Balanced Growth

Recall that yt = (1 + µ)tkαt . 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 sameas that of output. The idea of balanced growth is for y , c, and k havingthe same growth rate. This requires µ+ αγ = γ or γ = µ/(1− α).


Theory of Optimal Growth The Model

Optimal Growth

In the Solow-Swan model the objective is to maximize output percapita in every period, which is similar to the golden rule. Now weshift our objective to maximization of the present value ofintertemporal welfare.

A useful math trick: rewrite the Cobb-Douglas production function as

Yt = (1 + µ)tKαt N

1−αt

= Kαt [(1 + µ)t/(1−α)Nt ]

1−α

= Kαt (N#

t )1−α,

where N#t = (1 + µ)t/(1−α)Nt is called effective labour.



Effective Labour

Since Nt = (1 + n)tN0,

N#t = (1 + µ)t/(1−α)Nt

= [(1 + µ)1/(1−α)(1 + n)]tN0

= (1 + η)tN0,

where 1 + η = (1 + µ)1/(1−α)(1 + n) or, using log approximation,

η ' n +µ

1− α.

Output and capital stock per unit of effective labour are

y#t =Yt

N#t

=Yt

(1 + η)tN0,

k#t =Kt

N#t

=Kt

(1 + η)tN0.



National Income and Capital Accumulation

Consumption and investment per unit of effective labour are

c#t =Ct

N#t

=Ct

(1 + η)tN0,

i#t =It

N#t

=It

(1 + η)tN0.

The production function becomes

y#t = (k#t )α.

The national income identity becomes

y#t = c#t + i#t .

The capital accumulation equation becomes

(1 + η)k#t+1 = i#t + (1− δ)k#t .



Resource Constraints and Utility

The last three equations gives the resource constraint

(k#t )α = c#t + (1 + η)k#t+1 − (1− δ)k#t .

Instantaneous utility function has the function form of constantrelative risk aversion:

U(Ct) =C 1−σt

1− σ

=

[(1 + η)tN0c

#t

]1−σ1− σ

=

[(c#t )1−σ

1− σ

](1 + η)(1−σ)t ,

where N0 is normalized to 1.



Optimization

The optimization problem is

maxc#t+s ,k

#t+s+1

∞∑s=0

β̃s

[(c#t+s)1−σ

1− σ

](1 + η)(1−σ)t

subject to (k#t )α = c#t + (1 + η)k#t+1 − (1− δ)k#t ,

where β̃ = β(1 + η)1−σ. The Lagrangian is

Lt =∞∑s=0

{β̃s

[(c#t+s)1−σ

1− σ

](1 + η)(1−σ)t

+ λt+s

[(k#t+s)α − c#t+s − (1 + η)k#t+s+1 + (1− δ)k#t+s

]}.



First-Order Conditions

∂Lt∂c#t+s

= β̃s(c#t+s)−σ(1 + η)(1−σ)t − λt+s = 0, s ≥ 0,

∂Lt∂k#t+s

= λt+s

[α(k#t+s)α−1 + 1− δ

]− λt+s−1(1 + η) = 0, s ≥ 1.

The Euler equation is

β̃

(c#t+1

c#t

)−σ [α(k#t+1)α−1 + 1− δ

]= 1 + η.

Note that the Euler equation is the same as in Chapter 2 if we set η = 0.


Theory of Optimal Growth Steady State

Steady State

Since k#t and c#t are kt and ct adjusted for technological andpopulation growth, in the steady state ∆k#t+1 = ∆c#t+1 = 0.

The Euler equation becomes

β̃[α(k#∗)α−1 + 1− δ

]= 1 + η.

Solving for k#∗ (exercise),

k#∗ '(σ(n + (µ/(1− α))) + δ + θ

α

)−1/(1−α).


Theory of Optimal Growth Steady State

Capital, Output, and Consumption

Although k#t is unchanged in steady state, capital stock per capita,kt = Kt/Nt , is growing due to technological progress:

kt = k#∗[(1 + µ)1/(1−α)

]t,

which means that kt grows at a rate of approximately µ/(1− α).

Similarly, since y#t =(k#t

)αand

yt = y#∗t

[(1 + µ)1/(1−α)

]t,

ct = c#∗t

[(1 + µ)1/(1−α)

]t,

output and consumption per capita grow at the same rate ofµ/(1− α) (balanced growth).


Theory of Optimal Growth Comparing Models

Saving Rate

The saving rate along the optimal growth path is

st = 1− Ct/Yt = 1− c#t /y#t .

Since

y#t = (k#t )α,

c#t = (k#t )α − (η + δ)k#t ,

k#t =

(ση + δ + θ

α

)−1/(1−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) Lecture 3 Economic Growth Winter 2013 24 / 39


Empirical Observations — U.S. Time Series

Is saving rate constant?



Empirical Observations — Cross-Sectional


Endogenous Growth

New Growth Theory

In the previous models technical progress is exogenous.

For most developing countries that is a good assumption. Newtechnologies are usually embodied in imported goods and services andfrom foreign direct investment.

For developed countries to maintain their edge, they have to invest inR&D using resources. Technical progress becomes an endogenousdecision.

The subject is often called new growth theory.


Endogenous Growth

The Technological Race


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, notdecreasing in kt .

From (3.1a), the growth rate of capital is

γ = stytkt− (δ + n)

= stA− (δ + n).

Therefore the capital growth is independent of the level of capital.



The AK Model






= stA− (δ + n).




The AK Model






= stA− (δ + n).




The AK Model






= stA− (δ + n).




The AK Model






= stA− (δ + n).



Endogenous Growth Human Capital Model

Human Capital Model

Separation of human capital, ht , and physical capital, kt , bothexpressed in per capita form.

Production function:

yt = Akαt h1−αt , 0 ≤ α ≤ 1.

Assuming both types of capital depreciate at the same rate δ, thecapital accumulation equations are

∆kt+1 = ikt − δkt ,∆ht+1 = iht − δht ,



Human Capital Model



yt = Akαt h1−αt , 0 ≤ α ≤ 1.





Human Capital Model



yt = Akαt h1−αt , 0 ≤ α ≤ 1.





Optimization Problem

max∞∑s=0

βsc1−σt+s

1− σ

subject to

Akαt+sh1−αt+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 differenttechnologies in producing physical and human capital.



Optimization

The Lagrangian is

Lt =∞∑s=0

{βs

c1−σt+s

1− σ+ λt+s

[Akαt+sh

1−αt+s − ct+s

− (kt+s+1 + ht+s+1) + (1− δ)(kt+s + ht+s)]}

The first-Order Conditions are

∂Lt∂ct+s

= βsc−σt+s − λt+s = 0, s ≥ 0,

∂Lt∂kt+s

= λt+s

[αAkα−1t+s h

1−αt+s + 1− δ

]− λt+s−1 = 0, s ≥ 1,

∂Lt∂ht+s

= λt+s

[(1− α)Akαt+sh

−αt+s + 1− δ

]− λt+s−1 = 0, s ≥ 1.



Optimization

The Lagrangian is

Lt =∞∑s=0

{βs

c1−σt+s

1− σ+ λt+s

[Akαt+sh

1−αt+s − ct+s

− (kt+s+1 + ht+s+1) + (1− δ)(kt+s + ht+s)]}

The first-Order Conditions are

∂Lt∂ct+s

= βsc−σt+s − λt+s = 0, s ≥ 0,

∂Lt∂kt+s

= λt+s

[αAkα−1t+s h

1−αt+s + 1− δ

]− λt+s−1 = 0, s ≥ 1,

∂Lt∂ht+s

= λt+s

[(1− α)Akαt+sh

−αt+s + 1− δ

]− λt+s−1 = 0, s ≥ 1.



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,

kt+1

ht+1=

α

1− α,

which is a constant. Substituting this into the Euler equation, we get thegrowth rate of consumption as (exercise)

log

(ct+1

ct

)=

1

σ

[Aαα(1− α)(1−α) − δ − θ

].



Conclusion

1 Optimal consumption grows at a constant rate given preferences andtechnology.

2 Ratios of physical to human capital are constant through time andtherefore have the same growth rate. As balanced growth they areequal to the consumption growth rate.

3 The production function can be written as

yt = Akαt h1−αt = A

(ktht

)−(1−α)kt

= A

(α

1− α

)−(1−α)kt = A∗kt .

Therefore the model is effectively the AK model.


Endogenous Growth Observations

Income and Growth Rates of 112 CountriesVOL. 1, NO. 1 3LUCAS: TRADE AND THE DIFFUSION OF THE INDUSTRIAL REVOLUTION

Sachs and Warner are explicit about their de!nition of openness, but it is a com-plicated de!nition. To be classi!ed as open, an economy must pass !ve tests. It must (a) have effective protection rates less than 40 percent, (b) have quotas on less than 40 percent of imports, (c) have no currency controls or black markets in currency, (d) have no export marketing boards, and (e) not be socialist (using the de!nition in Janos Kornai 1992). Clearly, these standards do not hold an economy to a Smithian ideal of laissez-faire; there is plenty of room for Japanese or Korean mercantilism. The currency control test is, I think, just a way of tagging governments that can-not keep their hands to themselves. The export marketing boards are an African device (carried over from colonial times) requiring farmers to sell export crops to the government at a low price set by the government, which then resells them abroad at world prices. Kornai’s “socialist” countries are the communist dictatorships. The focus of the Sachs-Warner classi!cation is on the abilities of individuals to engage freely in international trade. High trade volumes—think of the oil exporters or bar-ter deals within the old Soviet bloc (the Soviet Union and countries it controlled)—are not accepted as proof of openness.1

Sachs and Warner provide a detailed, country-by-country appendix describing the way their criteria are applied over the 20-year period covered in their study. An evident limitation of their de!nition of openness is its zero-one character (a country is labelled either open or closed for the entire period). The problems this raises are even more serious in my application, which covers the 40-year period up to 2000. Thus, my Figure 2 classi!es all of Eastern Europe as closed, even though most of

1 Ellen R. McGrattan and Prescott (2007) propose a de!nition of “openness” based on receptivity to foreign direct investment. It would be useful to incorporate this criterion into the Sachs-Warner classi!cation scheme, but my guess is that few countries would be reclassi!ed if this were done.

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Source: Lucas (2009)



The Importance of Being Open4 AMERICAN ECONOMIC JOURNAL: MACROECONOMICS JANUARY 2009

these countries opened after 1990 and many are now members of the European Union. Many other countries have undertaken major policy reforms. A replication that reclassi!ed all of the countries based on applying the criteria (a)–(e) to the entire period would be an important improvement.2

There is controversy over whether the superior growth performance of the coun-tries classi!ed as open by Sachs and Warner arises from differences in trade policies or from other factors. This is unavoidable. Figure 3 looks at 25 European countries, open and closed, from Figure 2. The open economies are simply western Europe; the closed ones are the former communist countries of eastern Europe. The information in the !gure is not enough to let us separate the effects of trade policy from the effects of central planning, followed in many countries by the chaotic transitions of the 1990s.

The other striking feature of Figure 3, and the feature I will emphasize in this study, is the regularity of the behavior of the open western economies. These points on the graph trace a downward sloping curve that illustrates the equalizing forces operating within the set of market economies. The poorer a western European coun-try was in 1960, the faster it grew between 1960 and 2000. This equalizing, which could have taken place in the !rst half of the century but did not, is widely attributed to the formation and gradual expansion of the European Union over these 40 years.3 Morever, going back to Figure 2, we can see that a curve !t to the open European countries will also !t the fast growing Asian economies.

2 See Romain Wazciarg and Karen Horn Welch (2003) for an interesting paper that also uses post-1990 evi-dence and exploits the panel character of the data set to examine the growth effects of within-period changes in trade policies.

3 Dan Ben-David (1993) documented the role of the Eupean Economic Community in equalizing incomes among the original six members. His conclusions would certainly be strengthened by including Spain and other later entrants.

F"#$%& 2. I'()*& +', G%)-./ R+.&0, 112 C)$'.%"&0

Source: Lucas (2009)



A Subset of Developing Countries



Growth History Comparison



References

Clark, Gregory (2007) A Farewell to Alms: A Brief Economic History of theWorld, 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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Lecture 3 Economic Growth - Lakehead Universityflash.lakeheadu.ca/~kyu/E5118/M3.pdf · 1 increases in capital stock 2 increases in human resources: population, immigration, participation