Economic Growth Core hypothesis: Economic growth doesn’t just happen; rather, it is endogenous, and depends on the choices society makes about political.

Economic GrowthEconomic GrowthCore hypothesis: Economic growth doesn’t just happen; rather, it is endogenous, and depends on the choices society makes about political and economic organization, policies, and history.

In this module, we look at

•History of thought on growth

•Stylized facts of growth

•Early models of growth, from Malthus to Solow

•Current models of endogenous growth

History:The first revolution: Adam Smith (1723-

1790)

History:The first revolution: Adam Smith (1723-

1790)Theory of wealth

creation, public policy, and economic growth

Theory of wealth creation, public policy, and economic growth SavingSaving and and

investmentinvestment are are by-products by-products and precursors and precursors of domestic of domestic and foreign and foreign tradetrade

SavingSaving and and investmentinvestment are are by-products by-products and precursors and precursors of domestic of domestic and foreign and foreign tradetrade

size of the marketsize of the market

division of labourdivision of labour

efficiencyefficiency

The first revolution:Adam Smith


Saving and investment stimulate growthdirect effects through accumulationof capitalindirect effects through labour productivityfurther indirect effects through interaction with exchange and trade, through foreign investmentdomestic market can take the place of foreign markets

Smith’s reference to ‘private misconduct’ and the ‘publick extravagance of government’

Problem of public corruption and what economists now call “regulatory capture”

Distinction between quantity and quality Quality enhances the productivity of workers and

other technological inputs to production, and permits further technical innovation to occur

Mutual advantages of trade and growth, links to geography

First recognition of the concept of comparative advantage



Benefits from division of labourBenefits from division of labour

If If specializationspecialization increases efficiency increases efficiencyand wealth and, thereby, economicand wealth and, thereby, economicgrowth, then ...growth, then ...

... just about ... just about anythinganything that increases that increases efficiencyefficiency by the same amount, other things by the same amount, other things being equal, should be expected to have the being equal, should be expected to have the same effect on growth.same effect on growth.



Implications for growth

... all... all other equivalent means of other equivalent means of increasing the increasing the efficiency efficiency oror quality quality of labour, capital, and land should be of labour, capital, and land should be expected to affect economic growth in the expected to affect economic growth in the same way.same way.

If If foreign tradeforeign trade enlarges the enlarges themarket and thus facilitates further market and thus facilitates further division of labour à la Smith, thereby division of labour à la Smith, thereby increasing wealth and growth, then ...increasing wealth and growth, then ...



Smith on education, efficiency, and growthDistinction between the quantity and quality of labour

education, by increasing labour productivity, also increases efficiency and growth

Smith feared the economic, political, and social consequences of inferior education among the masses

He favoured public support for educationFirst recognition of the external economic benefit to society of

mandatory universal education



The first revolution:Adam Smith - Summing up

The first revolution:Adam Smith - Summing up

Economic growth = increase in the quantity and quality of the three main factors of production: labor, capital, and land

Growth accounting is based on this classification

Two shortcomings:Fixed quantity of land – diminishing returnsIncrease in the labour force does not really

count as a source of economic growth

Adam Smith’s followersAdam Smith’s followers

Impact of the distribution of wealth Impact of the distribution of wealth and of foreign tradeand of foreign trade

David RicardoDavid Ricardo

Thomas MalthusThomas Malthus

Question of population growth and its effectQuestion of population growth and its effecton economic growthon economic growth

Malthus: A Formal Model

Real Wage

Real Wagew*

CBR

CDR

CBR, CDR

Labor(Pop.)

Ld=labor demand

Ls=labor supply

NRI=0

Effects of CharityA Malthusian Perspective

Real Wage

Real Wagew*

CBR

CDR

CBR, CDR

Labor(Pop.)

Ld

Ls

(w*+c)

CBR>CDR=> Growth

Ls2

Growth shiftsLs curve up thusreducing the effective wage.

(w*-c)

The effectivewage falls untilCBR=CDR, leavingthe level of living as was prior to charity.

1

2

3

Worker receivesw*-c from laborand c in charity.

Malthus: The Plague

Real Wage

Real Wagew*

CBR

CDR

CBR, CDR

Labor(Pop.)

Ld

Ls

CDR2

Ls2

w2*

1

2

Technological Advances

Real Wage

Real Wagew*

CBR

CDR

CBR, CDR

Labor(Pop.)

Ld

Ls

NRI=0

Ld2

w2

1

2

3 population grows

Ls24

5: wage returns to w*

Adam Smith’s followers

John Stuart Mill

rejected Malthus’s prediction that population would outgrow productive capacity

more and better education would restrain population growth

distribution a different matter than production but can be changed through policy

Karl Marx

Economic mechanisms driving production and distribution are closely related

Anticipates Henry Ford’s comment on the importance of income as a determinant of aggregate demand

General equilibrium effects are important

The limits to growth observed by Malthus are inescapable ‘technological unemployment’


Alfred Marshall

organization as a fourth factor of production

made explicit the connection between education and growth

distribution of income and wealth matters for efficiency and growth

‘Knowledge is our most powerful engine of production ... Organization aids knowledge’


Joseph Schumpetertechnology through invention, innovation, and entrepreneurship

rent-seekers motivated by monopoly profits

perfectly competitive markets may not be very conducive to economic growth

no rent to capture under perfect competition


John Maynard Keynes

Accumulation of capital

‘Science and technical inventions’

‘I draw the conclusion that, assuming no important wars and no important increase in population, the economic problem may be solved, or be at least within sight of solution, within a hundred years.’


Modern Models of Growth

Modern Models of Growth

Stylized Facts of GrowthStylized Facts of GrowthPer capita growth rate

Stylized Facts of GrowthStylized Facts of GrowthReturn to Capital

Stylized Facts of GrowthStylized Facts of GrowthWhy is the rate of depreciation

increasing?

Stylized Facts of GrowthStylized Facts of GrowthCapital-Output Ratio

Stylized Facts of GrowthStylized Facts of GrowthInvestment rates

Stylized Facts of GrowthStylized Facts of GrowthConsumption and income

Time-series data 1929-82, in 1982 $$

Enter mathematics: Harrod and DomarEnter mathematics: Harrod and Domar

Paul Samuelson’s Foundations of Economic Analysis (1948)

laid the basis for mathematical economics, including the modelling of dynamic interactions among macroeconomic variables


Net investment equals the increase in the capital stockNet investment equals the increase in the capital stock

… … net of depreciation due to net of depreciation due to physicalphysical or or economiceconomic wear and tear wear and tear

High level of investment entails High level of investment entails an increasing level of the capital an increasing level of the capital stockstockHigh levels of saving and investment are good for High levels of saving and investment are good for growth even if they are stationary, that is, not growth even if they are stationary, that is, not increasingincreasingBy continuously augmenting the capital By continuously augmenting the capital stock ...stock ...

… … even even stationarystationary levels of saving and levels of saving and investment relative to output drive output investment relative to output drive output higher and higherhigher and higher, thus generating , thus generating economic growtheconomic growth

•FlowsFlows of investment add to the of investment add to the stock stock of of capitalcapital


Efficiency is crucial for growthEfficiency is crucial for growth

High level of efficiency stimulates growth High level of efficiency stimulates growth by ...by ...

… … amplifying the effects of a given level of amplifying the effects of a given level of saving and investment on the rate of growth of saving and investment on the rate of growth of outputoutput

All that is required is a steady All that is required is a steady accumulation of capital through accumulation of capital through saving and investmentsaving and investment

A given level of efficiency, including the A given level of efficiency, including the state of technology will, then translate state of technology will, then translate the capital accumulation into economic the capital accumulation into economic growthgrowth


Harrod and Domar expressed the dynamic relationship between saving, efficiency, and growth in a simple equation

The Harrod-Domar model The Harrod-Domar model

So, Samuelson’s work neatly formalized, So, Samuelson’s work neatly formalized, simplified, and summarized the essence of simplified, and summarized the essence of

almost 200 years’ theorizing about almost 200 years’ theorizing about economic growtheconomic growth

The Harrod-Domar modelThe Harrod-Domar model

Economic growth depends on three factors:A. the saving rateB. the capital/output ratioC. the depreciation rate

The Harrod-Domar model:Mathematics


Notation:Y denotes national incomeK denotes capital stockS denotes savingY denotes national income



Assumptions:Saving is proportional to income: S=sYCapital-output ratio is constant: K=vYInvestment (newly produced capital goods) must be

allocated between increasing the stock of capital and replacing depreciated capital: I=K+K

At equilibrium S=I (desired saving =desired investment)



Harrod-Domar equationFrom the capital-output ratio assumption, we can

write K=v Y.Substituting into the expression for investment, we

have I=v Y+vYUsing the equilibrium condition, we then have

sY= v Y+vY or Y/Y=s/v-Example: s=0.2, v=3, yields a growth rate of

roughly 3%.

The Harrod-Domar modelThe Harrod-Domar model

Shortcomings: Neither theory nor empirical evidence seemed to provide

much support for the capital/output ratio as an exogenous behavioural parameter in the model

a more elaborate formulation of the link between capital and output was called for

The model did not leave much room for the other crucial factor of production, labor

population or labor-force growth is absent from the formula, which explains output growth solely by saving and efficiency

The second revolution: The neoclassical model

The second revolution: The neoclassical model

Even so, saving and efficiency play an important role for growth over long periods, that is, the medium term

Economic growth was considered immune to economic policy, good or bad

According to Solow, saving behaviour was no longer relevant for long-run growth, nor was efficiency in a broad sense, except insofar as it mattered for technology

Since population growth is basically a demographic phenomenon and, hence, exogenous from an economic point of view, it must follow that economic growth is also

exogenous

The second revolution:The neoclassical modelThe second revolution:The neoclassical model

… but also with a constant rate of growth of output per capita, a constant rate of interest, and a constant distribution of national income between labour and capital, all of which seemed to apply to the real world

Once attained, the long-run equilibrium is consistent with not only a constant capital/output ratio

… is better viewed as an endogenous variable, which moves over time and ultimately reaches long-run equilibrium

Solow showed how the capital/output ratio, rather than being exogenously fixed as in the Harrod-Domar

model,

The Neoclassical ModelMathematics


Output is produced via a production function which uses capital and labor as inputs

where the parameter a is between 0 and 1.Taking logs on both sides and differentiating yields

Here, g is the rate of growth of output in percentage terms, n is the exogenously given rate of growth of the labor force (or equivalently, of population), and is the rate of growth of the capital stock.

aaKLY 1

K

Kaang

Y

Y 1

K

K



From empirical work by Kuznets, it is plausible to assume that the long-run capital-output ratio is constant, which implies that

Plugging into the growth equation, then, we have

gK

K

ng

anag

gaanK

Kaang

)1(1



Thus, in the Solow model, the long-run rate of growth is determined entirely by the exogenously given rate of population growth.It also follows that in the long-run, there can be no growth in

per capita outputSince we obviously have seen significant increases in

standard of living since the onset of industrialization in the early 1800’s, the model must be modified if it is to explain this.



We can explain the observed growth in per capita output by assuming that technological change makes the labor input more productive over time, due to factors such as better technology or better education of the workforce. With this assumption, the production function becomes

B represents some initial state of technologye is the base of the natural logarithmLabor productivity grows at the rate qWe refer to as the efficiency unit equivalent of the labor input

aaqt KLeBY 1

Leqt



Log differentiating the production function now gives

As before, taking the long-run capital-output ratio as constant yields

So, we now have that growth is exogenous, being driven by productivity improvements, but per capital growth is now positive and equal to q.

K

Kaqnag

)1(

qng



Comparing the growth equation for the Solow model with that of the Harrod-Domar model, we see that we must now have

If all the parameters n,q,s,v, and are exogenously given, then we would generally not expect the equality above to hold. Mathematically, the Harrod-Domar model is now over-identified.Solow resolved this over-identification by assuming that the capital-output

ratio, rather than being exogenously specified, was a function of the other parameters of the model:

v

sqng

qn

s

g

sv



How do we know the capital-output ratio is the right parameter to make endogenous?Consider the original definition of investment:This can be re-written as

Since saving must equal investment in the long-run, I/Y=s, and we may then solve the equation above for the capital-output ratio as

Hence, changes in any of the right-hand parameters will affect the value of the capital-output ratio.

KKI

K

K

Y

K

Y

I

K

Ks

Y

K



We can also use this result to solve for the rate of growth of capital in terms of other parameters of the model:

Substituting for the rate of growth of capital in the Solow growth equation yields

This equation tells us that if we increase saving, then the economy will grow as long as the capital-output ratio remains constant.

We turn next to the question of whether this ratio will in fact remain constant.

K

Ys

K

K

K

Ysaqna

K

Kaqnag )1()1(



Dynamics of the capital-output ratioDefine the following ratios of capital and output per efficiency unit at time t:

The percentage rate of change of the first ratio is

where we use the relationship between the rate of change of capital to the

capital-output ratio to arrive at the right-hand side of the equation.

qt

qt

Le

Yy

Le

Kk

qnK

Ysqn

K

K

k

k



We can also write the production function in terms of the two ratios as

Now, since

substituting into the expression for the rate of growth of capital, we get a key equation:

akBk

y

K

Y

qnksBk

k a

aaaqtqt

aaqt

qtkBKLeB

Le

KLeB

Le

Yy

1111



The Solow differential equation

If k is small, so that is large, the the rate of change of k will be positive, so the capital stock will increase. On the other hand, if k is large, will be small, so that the rate of change will be negative.

This means that if we start at a low level of capital, the economy will accumulate capital, while if we somehow started with a large amount of capital, we will decumulate it.

Hence, independently of where the economy starts, it will evolve toward a steady-state at which

qnksBk

k a

ak

ak

0k



Steady-stateSet the time derivative of k to zero and solve for k

Using the definition of k, we can find the steady-state values of capital and output:

a

qn

sBk

1

ˆ

qta

a

aqtaqta

a

qta

Leqn

sBBLeLe

qn

sBBY

Leqn

sBK

1

1

1

1

ˆ

ˆ



Note that the steady-state capital stock and flow of output are actually growing, but in a balanced way, at the same rate, so that the capital output ratio remains constant at

Income distribution in the Solow modelStandard results from producer theory tell us that at the competitive equilibrium,

inputs are paid their marginal products. For the simple model with only capital and labor inputs, these are given by

qn

s

Y

K

wL

YaKaALMPL

rK

YaKALaMPK

aa

aa

11

11



Hence, for the Cobb-Douglas specification of technology, each factor of production is paid a constant share (a for labor, 1-a for capital) of output. This is consistent with data for modern industrial economies, where labor receives 2/3 of total output, while capital receives 1/3.

This also gives us a way to calibrate the model, since it says we should set a=2/3.

Since the capital-output ratio is constant, it also follows that along a balanced growth path, interest rates will remain constant.

For labor, the real wage will grow at the rate g-n=q, since labor productivity is growing over time.

Calibration spreadsheet



The third revolution:Endogenous growthThe third revolution:Endogenous growth

The neoclassical growth model seemed unable to answer some burning questions about economic growth

Is technological change exogenous from an economic point of view?

Do economists really have nothing to say about economic growth in the long run?

If output per capita grows at a rate that depends solely on - in fact, is equal to - the rate of technological progress, then why is it that the growth performance of different countries differs so radically over long periods?

What does the neoclassical model tell us about relative growth performance anyway?

The third revolution:Endogenous growthThe third revolution:Endogenous growth

Key idea Technology is not exogenous

Technology depends on economic factorsTechnological improvement depends on

Innovation – “Learning by doing”EducationBasic research

Technical innovation is external to firms’ decisionsBasic research generates new technologies available to allEducation and on-the-job learning spill over from one firm to

the next

Endogenous Growth:Mathematics


Human capital investmentFocus again on Cobb-Douglas production: We assume that some fraction h of the workforce is engaged in

innovation – basic research, fine-tuning technical processes within the firm, independent invention, or other educational pursuits. The remaining fraction (1-h) provides labor input for firms.

The effect of human capital accumulation on production is via A, which we now assume is an increasing function of the average amount of human capital accumulation

For specificity, we assume the production function is given by

aaKALY 1

Lh

aaaKLLhY 1



Increasing returns propertyThe inclusion of human capital accumulation effects on productivity

implies that the production function now exhibits increasing returns to scale. To see this, suppose we increase the labor and capital inputs by some factor . This will increase the average labor supply by the same factor. Hence, the effect on output will be

Because the human capital effect is external, the increasing returns will not affect individual firms’ profit maximizations.

YYKLLhKLLh aaaaaaaa 1111



Growth with human capital investmentTaking logs and differentiating the production function gives us

On a balanced growth path where output and capital grow at the same rate g, we will have

Hence, as in the case of exogenous technical progress, we will have positive growth per capita, but due in this framework to the productivity enhancing effects of economic activities associated with human capital accumulation.

K

Kaanh

K

Ka

L

La

L

Lah

Y

Y

11

1

nhg 1



Income distributionAs in the Solow model, factor shares are given by

Also as in the Solow model, wages grow over time since output grows more rapidly than population. Since capital and output grow at the same rate, interest rates do not grow.

K

Yar

L

Yaw

1



Steady-stateWe can also replicate the dynamic analysis from the Solow model.

Then

akL

Yy

L

Kk

LLhL

1

ˆˆ

ˆ

and Define

nhK

K

L

L

K

K

k

k 1ˆ

ˆ



Since

Steady-state capital stock is then

nhskk

k

kk

y

K

Y

k

Ys

K

K

a

a

1

have we

while

a

nh

sk

1

)1(ˆ

Saving BehaviorSaving Behavior

Last missing ingredient to a fully specified economic modelHandle by positing preferences over consumption over time

A key parameter is consumer’s degree of patience or impatience, measured by how little or much they discount future utility

Consumers face budget constraints which allow them to trade off consumption today for consumption tomorrow

Saving generates returns in excess of the actual amount saved when interest rates are positive

Saving Behavior:Mathematics


Two formulations of consumer model: overlapping generations and dynastic models In both models, we assume time is split up into discrete periods (planning time). In overlapping generations model, consumers live finite lives. We will make the

simplifying assumption that the number of periods of life is 2. Overlapping generations optimization problem is then

subject to

Here, 0<is the consumer’s discount factor, which measures the degree of her impatience

r is the market interest rate (which, recall, will be determined by the production side of the economy), and c2/(1+r) is the present value of second-period consumption determined by the market interest rate

21

,lnln

2

ccc

max1c

*1

21 Y

r

cc



The easiest way to solve this problem is to substitute for second-period consumption from the budget constraint into the utility function, and then take a first-order condition with respect to first-period consumption:

The required first-order condition is

11 *1lnlnmax1

cYrcc

*1

1ˆ

1

*ˆ

0*

11

2

1

111

Yrc

Yc

cYcdc

dU

and



In terms of this model, the per capita rate of growth in consumption is given by

From the growth equations, the per capita growth rate for consumption from the technology side of the economy is give by g-n. Hence, we will have and we can determine the equilibrium interest rate as

So, interaction of growth induced by technology together with degree of patience determines equilibrium interest rates.

Cases: Exogenous population growth only: g=n

Growth with technical progress: g-n=q>0

1111

2 rc

c

11 rng

1

1

ng

r

11

r

11

qr



Dynastic ModelFor this model, we assume consumers act to optimize the discounted utility

stream of a long-lived family, and hence solve

subject to

Analysis of this model is much harder than for the overlapping generations model, so we will simplify by looking not at the market version of the model, but at a social planning version

tclnmax

0t

t c

income of PV*

10

Yr

c

tt

t



Social planner’s problem:

subject to

Substituting from constraints into the objective function, the problem simplifies to

tclnmax

0t

t c

at

at

ttt

KLY

KYc

1

1

0

11lnmax

tt

at

at KKL K



Analyzing the social planner’s problem for this model is not simply a matter of taking first-order conditions and solving. To see why, we normalize the labor supply to 1, since it doesn’t change over time. Then, taking first-order conditions gives us the so-called Euler equations:

In the exercises for this module, you show that by letting =aand making a suitable change of variable, the second-order difference equation generated by the first-order conditions can be converted into the first-order difference equation

t

att

at

at

KKKK

Ka

111

1

11

tt z

z

11



Graphing the difference equation gives us the following picture

. where1

1a

t

tt K

Kz

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Z(t)

z(t+

1)



Properties of the difference equation

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Z(t)

z(t+

1)

Two steady-states, one at z=and the second at z=1If we start below the first steady- state, or anywhere between it and the steady-state at 1 and iterate the difference equation, we will converge to the first steady-state.If we start anywhere to the right of the steady-state at 1, we will diverge.

Only the lower steady-state is optimal. At the upper steady-state, with z=1, we would have which says that all production is being devoted to the accumulation of capital, with none for consumption.

aKK 1



It is possible to show that the planner’s problem can be represented in the simpler form

provided we know the so-called value function V(K). While the math is beyond the scope of what we can do here, it can be shown that this function will exist under suitable assumptions about discounting.

This formulation of the optimal capital accumulation program for the economy is known as the dynamic programming formulation. In the exercises for this module, you showed that for this model, the value function is

11lnmax1

tttK

t KVKKKVt

kkV ln1

ln1

1ln1

1)(

Economic Growth Core hypothesis: Economic growth doesn’t just happen; rather, it is endogenous, and depends on the choices society makes about political.

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