334 neoclassical growth model 2013(1)

8/13/2019 334 Neoclassical Growth Model 2013(1)

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Economics 334:

Approaches to Economic Growth

The Solow Growth Model


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Source: OECD - Maddison (2001) The World Economy: A Millennial Perspective


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Real GDP per capita growth rates, selected countries

1820-1998

Country 1820-70 1870-1913 1913-1950 1950-73 1973-98

United Kingdom 1.26 1.01 0.92 2.44 1.79

Germany 1.09 1.63 0.17 5.02 1.60

France 0.85 1.45 1.12 4.05 1.61

Italy 0.59 1.26 0.85 4.95 2.07

Western Europe 1.00 1.33 0.83 3.93 1.75

United States 1.34 1.82 1.61 2.45 1.99

Canada 1.29 2.27 1.40 2.74 1.60

Japan 0.19 1.48 0.89 8.05 2.34

China -0.25 0.10 -0.062 2.86 5.39

Source: OECD - Maddison (2001) The World Economy: A Millennial Perspective


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Theories of Economic Growth

Classical Economists: David Hume (1711-1776)

Adam Smith (1723-1790)

Thomas Malthus (1766-1834)

David Ricardo (1772-1823)

Karl Marx (1818-1883)

Twentieth Century

David Harrod-Evsey Domar 1940s (Keynesian) Robert Solow 1950s (Neo-classical)

Paul Romer 1980s (Endogenous Growth)

4


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SOLOW, R. M. (1957)Technical Change and the

Aggregate Production Function,Review of Economics

and Statistics, Vol 39, No 3, pp. 312-320

The problem:What explains growth?

A case study of the U.S. economy between 1909 and 1949.

During this period, the economy grew approximately 80percent.

What explains this average annual growth rate? Howmuch is due to:

growth in the labour force?

capital accumulation?

technological change?

The growth accounting framework has developed fromSolows model and is widely used to determine the relativecontribution of these factors in economic growth


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His argument:

That the amount of output produced by a given K(t) and L(t) increases

over time as technology A(t) grows.

How does Solow make his case?

He doesn't have data for A(t) but he argues that growth in technology can

be inferred from the growth in output capital and hours worked. To do this

he brings in a more general economic framework, a production functionwith constant returns to scale. (Recall the definition of constant returns to

scale?)

The most convenient (and widely accepted) way to illustrate the growth

accounting framework is to use a Cobb-Douglas production function:

Y =AKL

where Y=output, A="technical progress", K=capital stock, L=labour supply,

and represent the shares of capital and labour in production.

See Diagram I


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ExtensiveGrowth

Y=Af(K)

sY=sAf(K)

Capital (K)

Output (Y)

Y2

Y1

K1 K2


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But we want to disaggregate the contribution of each

of the factors of production and technology to growth

in output, Y over time. This can be accomplished by a mathematical

manipulation of the Cobb-Douglas production function

(taking the logarithm and differentiating with respect to

time*) to get the equation in linear form. (Why wouldwe want to do this?)

A simple representation of the results of this

manipulation is:Y

Y=

AA

+ KK

+ (1 ) LL

;

Where represents "change", eg. L2 -L1; So,L

L=growth rate of labour and so on.

This equation represents

"extensive growth"

* For those of you who may wish to know, .

The mathematics behind the derivation of the growth accounting equation is not necessary for this

course.ln(xy)

=

lnx+

lny


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The contribution of technical progress can bedetermined by subtracting the contribution of theother factors from output growth.

This is the Solow Residual.

That is,A

A=

YY

KK

(1 ) LL


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But what about "intensive growth?" We are

interested in growth in per capita incomes.

To get to the intensive form of the growth accounting equation we divide (1) byL:

where y=output per capita (or per worker) and k=capital tolabour ratio.

Taking the logarithm and derivative with respect to time once again, we have:

So growth in per capita income is a function of technological change andcapital accumulation. But if capital is subject to diminishing returns, then in thelong run growth in output is determined by technological change.

Solow's findings: 7/8ths of output growth in the period 1909-1949 was due to"technological progress."

y

y=

AA

+ kk

Y

L=A

K

L

L

L

This gives us y =Ak


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Another way to represent the model:

Formal refinements of this model take into accountdepreciation, dand population growth, n.

Firms must invest sufficient capital to replace wornout machinery and equipment

And to keep growing, an economy must provide thegrowing labour force with capital

- That is, to maintain the capital to labour ratio thatallows the growth of per capita income to remainat its desired rate

Taking into account depreciation only we canrepresent this mathematically as : K

(Investment)=sF(K,L)-dK Taking into account both depreciation and population

growth in extensive form:

K=sF(K,L)-(d+n)K or K=sY-(d+n)K


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Again, we are interested in intensive growth:

That is, we are interested in determining the savingsnecessary to maintain the desired growth in per capita

income after taking in to account the depreciation of

capital and population (work force) growth.

- the STEADY STATE level of capital accumulation

is such that....

sy=(d+n)k

THAT IS, WHERE SAVINGS PER CAPITA IS JUST

SUFFICIENT TO COVER DEPRECIATION AND

POPULATION GROWTH


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We can represent these relationships

graphically:

y*

THE ECONOMY WILL

TEND TOWARD THESTEADY STATE RATE

OF GROWTH.

k*k1* k2*

sy=sf(k)


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What if the population growth rate falls?

9

y*

y2*

k* k2*

sy=sf(k)


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What if the savings rate rises?

So increasing theproportion ofincome allocatedto savings, raises

per capita income.

....But whatdetermineswhether themembers of a

society raise theshare of savingsfrom income(individually or inaggregate)?

k2*

sy2=s2f(k)y2

*

sy=sf(k)


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How would we represent the Solow model with

technological change graphically?


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two issues to think about here:

Firstly, that it is usually the case that when technologicalchange occurs, further investment in capital is required to

implement the change

That links to a second assumption, that technical change

tends to be labour augmenting(that it makes each worker

more productive)

Y=f(K,AL)

We can now think of capital per worker in terms of

capital per effective worker, since now each worker

is made more productive by technological change.

Note: for a good mathematical representation of this model, see jones, c. (2001) Introduction to Economic

Growth. For a nice tutorial on the neoclassical growth model with some interactive pages see

www.Fgn.Unisg.Ch/eurmacro/tutor/neoclassicalgrowth-index.Html .
http://www.fgn.unisg.ch/eurmacro/tutor/neoclassicalgrowth-index.htmlhttp://www.fgn.unisg.ch/eurmacro/tutor/neoclassicalgrowth-index.htmlhttp://www.fgn.unisg.ch/eurmacro/tutor/neoclassicalgrowth-index.htmlhttp://www.fgn.unisg.ch/eurmacro/tutor/neoclassicalgrowth-index.html


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Secondly, if population is growing at the rate of n, and

technology is growing at the rate of a, then the required

investment to maintain the steady state rate growth of output,

output per worker (now effective worker), and consumption percapita is (n+d+a)k.


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This is why technology is seen to be crucial to increasing

the standard of living. Without technological change, thesteady state growth rate of output is just sufficient to keep

up with population growth (n) and depreciation (d). With

technological change, output per worker will rise and

potentially incomes as well. What determines whether

workers incomes will rise?

Question: If technological change is labour augmenting

and the rate of population growth is falling, what

implications does this have for economic growth? What

are the options available to an economy characterized bythis situation (Europe and other industrialized countries)?


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Question: If the nature of existing technological

change is labour augmenting and the birth ratesare falling, what implications does this have for

economic growth?

What are the options available to an economy

characterized by this situation (Europe and

some other industrialized countries)?


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334 neoclassical growth model 2013(1)

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