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.

3



US GDP per capita since 1960

4



Australian GDP and GDP per capita since 1901

5



Levels and growth rates of per capita GDP

6



Output per person in 6 rich countries

7



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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Economics Lecture 2