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Transcript of Economics Lecture 2
7/21/2019 Economics Lecture 2
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Lecture 2. Growth Facts, Production Function
and Growth Accounting
ECON30009 Macroeconomics
Shuyun May Li
Department of Economics
The University of Melbourne
Semester 2, 2014
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Outline
1. Introduction
2. Modeling Production
3. Accounting for Growth
Required reading: Chap. 1 of AK
Further reading: “On productivity: the influence of natural
resource inputs”, Productivity Commission report (2013)
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1. Introduction
• Advanced economies produce a multitude of goods and
services. Despite the varied means to produce them, certain
features are common to all production processes.
• All production processes use basic inputs to generate final
products–the outputs, using a technology for combining
inputs to make outputs.
• Macroeconomists are concerned with a country’s total
production of final goods and services–its gross domestic
product, or GDP.
•
Economists refer to increases over time in real GDP aseconomic growth.
• Economic growth leads to improvement in a country’s living
standard which is measured by per capita GDP, or output
per person.
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US GDP per capita since 1960
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Australian GDP and GDP per capita since 1901
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Levels and growth rates of per capita GDP
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Output per person in 6 rich countries
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Convergence in per capita GDP
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• Some central questions of growth theory:
– How economic growth happens over time?– Why standards of living and growth rates vary substantially
across countries?
– Can poor countries catch up with rich countries?
• Intuitively, economic growth depends on various factors:– the availability and quality of inputs (capital accumulation,
population growth, human capital accumulation)
– the efficiency with which inputs are combined to produce
output (learning-by-doing)
– scientific and engineering advances that permits producing
more output from given inputs (technological progress)
– the state of the economy (recessions or expansions).
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• Correspondingly, there are many different channels through
which output grows.
• Various growth models differ in what growth channels they
strive to explain.
• In the Solow-Swan growth model, both labour and technology
are exogenous, while capital is endogenous, so the model
explains how growth happens through capital accumulation(to a certain degree).
• The life-cycle model we’ll introduce represents another well
known growth model–the overlapping generations growth
model, which further complete the story of growth through
capital accumulation.
• Before we get into the model, we first briefly review some
concepts and measurement issues around the production
function.
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2. Modeling Production
• Economists conceptualise the production process of final goods
by a mathematical relationship known as the production
function:
Y t = F (At, K t, Lt).
– Y t refers to the amount of output produced during a
particular time period.– Labour, Lt, refers to the total efforts supplied by workers,
managers and owners of business enterprises.
– Capital, K t, refers to the amount of nonhuman inputs
employed in the production (buildings, machinery, etc).
– Productivity, At, refers to the efficiency with which capital
and labour are used.
– F refers to a mathematical function that describes the
dependence of Y t on K t, Lt and At.
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• A widely-used production function is the Cobb-Douglas (C-D)
production function:
Y t = AtK βt L1−βt ,
where At denotes multifactor productivity (MFP) or total
factor productivity (TFP), and β ∈ (0, 1) is a parameter.
– In a competitive economy operating with C-D production
function, the income share paid to capital and labour are β
and 1 − β , respectively.
– Constant returns to scale
– Diminishing returns to each of the inputs
• Historically, capital’s share of US GDP has been remarkablyclose to 1/3, so β often takes the value of 1/3 or 0.3.
• C-D production function is viewed as a good representation of
aggregate production for advanced economies.
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• Measuring capital
– In any modern economy, a wide variety of capital goods are
used to help produce output: equipment, business
structures, inventories, residential structure, land, etc.
– One approach to measure capital is to convert all types of
capital goods to constant dollar terms, then add them up.
– Capital depreciates (wears out) over time, and capitalstock increases through investment–the acquisition of new
capital goods.
– The change in capital stock (productive capital available for
production) between two time periods:
∆K t ≡ K t+1 − K t = I t − Dt, i.e. (1)
K t+1 = K t − Dt + I t. (2)
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– K t: the capital stock at the beginning of period t
I t: new purchases of capital in period t, gross investment
Dt: depreciation of the capital stock that occurs in period t
I t − Dt: Net investment
– Eq. (2) gives another approach to measure capital, known
as Perpetual Inventory Method .
·
The basic idea is to construct the series of capital stockrecursively using initial capital stock and data series on
depreciation and investment.
– Dt is typically defined as a fraction of K t,
Dt = δK t,
where δ varies for different types of capital goods, then (2)
becomes
K t+1 = (1 − δ )K t + I t. (3)
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• Measuring labour: Economists typically measure labour input
by the number of hours worked.
(Total hours=working age population × employment ratio ×
number of hours per employed.)
• Labour productivity
– Labour productivity is defined as the amount of output per
unit of labour input, Y t/Lt.– Rewriting the C-D production function in its intensive
form:
Y tLt
= AtK βt L
1−βt
Lt
= AtK βt L−βt = At
K tLt
β
,
or equivalently
yt = Atkβt , (4)
where yt and kt are labour productivity and capital-labour
ratio, respectively.
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• MFP
– MFP plays a crucial role in increasing the economy’s
capacity to produce. Improvements in MFP make it
possible to produce more output without additional input.
– A variety of factors can cause MFP to change:
· improvements embodied in capital and labour inputsthat increase the quality of capital and labour
· disembodied changes that boost productivity in a more
general way
– Disembodied MFP changes refer to technological change,
but also reflect the presence of any productive factors not
measured as capital or labour inputs
– So MFP or TFP is hard to measure directly.
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3. Accounting for Growth
• An important application of any production function is to use
it to conduct growth accounting.
• It aims to answer the question: how much of output growth
between any two time periods is due to growth in inputs
(capital and labour) or improvements in productivity?
• Growth accounting was pioneered by Abramovitz (1956) and
Solow (1957).
• Using the C-D function, we can derive the growth accounting
formula (see Appendix) that decompose output growth
between any two time periods into 3 sources:– growth in TFP
– growth in capital weighted by capital’s factor share β
– growth in labour weighted by labour’s factor share.
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∆Y tY t
= ∆At
At
+ β ∆K t
K t+ (1 − β )
∆Lt
Lt
, (5)
where ∆Xt
Xt
≡ Xt+1−Xt
Xt
stands for the growth rate in X from
period t to t + 1.
Note that this formula also exhibit constant returns to scale
and diminishing returns to growth in each factor.
• We have data on output, hours worked and capital, and 1 − β can be estimated by the income share of labour.
• So the growth rate of A (TFP) is usually computed as a
residual, known as the Solow residual method.
∆At
At =
∆Y tY t − β
∆K tK t − (1 − β )
∆Lt
Lt .
– The Solow residual reflects the amount of output growth
that cannot be explained by (or that’s left over after)
growth in the quantities of capital and labour.
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• The growth accounting formula can be rewritten as
∆Y tY t
−∆Lt
Lt
= ∆At
At
+ β
∆K t
K t−
∆Lt
Lt
, (6)
where ( ∆Y tY t
− ∆Lt
Lt
) is the growth rate of Y L
, and ( ∆K tK t
− ∆Lt
Lt
) is
the growth rate of K L
.
– This formula decomposes the growth in labour productivity
into 2 sources: MFP growth, and growth in capital-labour
ratio (capital-deepening).
– It is more widely used in empirical growth accounting.
• There have been many studies on growth accounting that build
on the basic framework or extends the basic framework (e.g.,
extract improvements in input quality from growth in MFP).
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Growth accounting for six rich countries
TFP growth accounts for most growth in the rich countries
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Asia Miracles? Growth rates: 1960-1990
Input growth accounts for most growth in many fast-growing
countries.
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Growth accounting for the US (BLS)
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Growth accounting for Australia
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Contribution of ICT capital
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Annual average labour productivity growth
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Review Questions
• What is a common feature of all production processes?
• What are the central questions of growth theory? Name a few
channels through which output grows.
• What is a production function?
• Write down the Cobb-Douglas production function. What does each
term refer to, and what properties the function exhibit?
• Write down an equation that describes how the stock of capital
accumulates over time. Understand the concepts of gross
investment, depreciation and net investment.
• Have a general understanding on how to measure capital and labour.
• Understand the concepts of embodied and disembodied changes in
MFP.
• What is labour productivity? How does it relates to MFP and
capital-labour ratio, under the Cobb-Douglas production function?
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• Write down the two growth accounting formulae. How do the
growth accounting formulae decompose output growth, and growth
in labour productivity?
• How to measure the growth rate of MFP or TFP? Does the Solow
residual purely reflect technological progress?
• Have a general idea on how to conduct growth accounting in
practice (The Productivity Commission report gives you a growth
accounting exercise).
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Appendix: Deriving the growth accounting formula
Taking the log of both sides of the C-D production function yields
lnY t = lnAt + β lnK t + (1− β) lnLt.
For period t + 1, the equation above is written as
lnY t+1 = ln At+1 + β lnK t+1 + (1− β) lnLt+1.
Taking the difference of the two equations above, we have
lnY t+1 − lnY t = lnAt+1 − lnAt + β(ln K t+1 − lnK t) + (1− β)(lnLt+1 − lnLt),
or equivalently,
ln
„Y t+1
Y t
«= ln
„At+1
At
«+ β ln
„K t+1
K t
«+ (1 − β) ln
„Lt+1
Lt
«. (7)
Now we show that in the equation above, the ln terms are approximately the
corresponding growth rates. Recall that the growth rate of a variable Xt is definedas
∆Xt
Xt
≡
Xt+1 −Xt
Xt
= Xt+1
Xt
− 1,
soXt+1
Xt
= 1 + ∆Xt
Xt
.
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Then we have
ln
„Xt+1
Xt
«= ln
„1 +
∆Xt
Xt
«≈
∆Xt
Xt
, (8)
where the “≈” follows a mathematical approximation result:
ln(1 + x) ≈ x
for x that is close to zero. Eq. (8) implies that the growth rate of a variable can be
approximately calculated as the log difference of the variable:
∆Xt
Xt
≈ ln„Xt+1
Xt
«= ln(Xt+1)− ln(Xt).
This is exactly what statisticians do in practice. However, be aware that the
approximation is good only when the growth rate is small (the closer it is to zero,
the better is the approximation).
Applying this approximation result to Eq. (7), we obtain the growth accounting
formula: ∆Y t
Y t≈
∆At
At
+ β∆K t
K t+ (1− β)
∆Lt
Lt
.
Note that the equality only approximately holds (you’ll see this in one of your
tutorial questions for week 3).
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