# Lecture 4

Macroeconomics for Public Policy
Economic growth
Tran Lam Anh Duong
26 Feb 2018

Economic growth theory
• The issue of economic growth theory:

– Average growth rate of GDP per capita in long-run
– Determinants of the growth

⇒ Why does the theory focus on GDP per capita rather than
just GDP?
• The theory deals with the followings contrasts:

– Some countries enjoy high income per capita for a long time,
while some other countries cannot get out of poverty trap.
– Some countries enjoy high growth rate for a long time, while
other countries remain in a "low-growth trap"
– There exist both high and low growth periods within a same

country.
2

Data on economic growth
GDP per capita of Vietnam
1600
1400
1200
1000
800
600
400
200
0

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

Source：WDI, USD, base year 2010
3

Data on economic growth

4

Data on economic growth
Transition of GDP per capita：South Korea vs Liberia
30000
25000
20000
15000
10000
5000

1970
1972
1974
1976

1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010

0

South Korea

Liberia
リベリア

Source：Penn World Table, real, base year: 2005
5

Economic growth vs long-run analysis
• Economic growth theory deals with long-run analysis
⇒ The base theory: long-run analysis theory
• Revisit the most important conclusion of models in long
run:
The equilibrium GDP is only determined by the total supply
conditions (the production technology and production
factors of firms). It is not influenced by the total demand
conditions
⇒ Analysis from now focuses on the production activity of
firms.
6

Production function
• Production function：determines how much output is
produced from given amounts of production factors.
• Production factors: factors that are introduced from outside in
order to produce goods and services. There are two types of
production factors here.
– Capital stock：𝐾𝐾
– Labor：𝐿𝐿

Production function：

point of time

𝑌𝑌 = 𝐹𝐹(𝐾𝐾, 𝐿𝐿)

Gross product 𝑌𝑌 is an increasing function of 𝐾𝐾 and 𝐿𝐿.
Technology level（Total factor productivity）
7

Capital stock and Production
• An assumption: 𝐿𝐿 is constant
⇒ the relation between capital stock 𝐾𝐾
and gross production 𝑌𝑌 is shown in graph
1.
• Properties of graph 1:

The slope is positive（ 𝑌𝑌 is an increasing
function of 𝐾𝐾.）
The increase in the slope slows down
toward the right.

Graph １
𝑌𝑌

𝐿𝐿: constant

2 examples：

– Factory A : 𝐿𝐿 =100, 𝐾𝐾 = 1 computer
– Factory B : 𝐿𝐿 =100, 𝐾𝐾 =200 computer
One day, both factories employ one more
computer. How will the production become?

𝐾𝐾
8

Marginal productivity of capital
• Marginal productivity of capital：the slope in
graph 1.

• Definition of marginal productivity of capital：evaluates
how many units of production will increase if one unit of
capital stock increases（Labor is constant).

• Properties of graph 1:

– The slope is positive ⇒Marginal productivity of
capital is positive.
– The increase in the slope slows down toward the
right ⇒ diminishing marginal productivity of capital
9

Labor and Production
• Graph 2： the relation between
labor 𝐿𝐿 and gross production 𝑌𝑌
under an assumption of that 𝐾𝐾 is
constant
• The slope: Marginal productivity
of labor.
• The increase in labor 𝐿𝐿 lowers
the increase in the slope:
diminishing marginal
productivity of labor

Graph 2
𝑌𝑌

K: constant

𝐿𝐿
10

Technology level and Production
• Increase in technology level of firms：
leads to an increase in production
volume with the same amount of
capital stock 𝐾𝐾 and labor 𝐿𝐿.

• Graph 3： the relation between capital
stock 𝐾𝐾 and gross production 𝑌𝑌 under
the constant labor volume 𝐿𝐿

• Properties of graph 3： Increase in
technology level shifts the entire curve
upward
※ The same analysis is applied to the
case of constant 𝐾𝐾.

Graph 3
𝑌𝑌

𝐿𝐿: constant

Increase in
technology level

𝐾𝐾
11

An example of production function
• Cobb-Douglas production function：
𝑌𝑌 = 𝐴𝐴𝐾𝐾 𝛼𝛼 𝐿𝐿1−𝛼𝛼

– This production function contains of all properties
discussed above ⇒ good example and often applied.
– 𝐴𝐴：technology level（TFP ）, positive constant.
– 𝛼𝛼： a constant lies between 0 and 1.
• 𝛼𝛼 = 0.5 ⇒ 𝑌𝑌 = 𝐴𝐴 𝐾𝐾 𝐿𝐿
• 𝛼𝛼 = 1⁄3 ⇒ 𝑌𝑌 = 𝐴𝐴𝐾𝐾

1⁄ 2⁄
3
3

𝐿𝐿

12

Slow – Swan model
• A model which analyzes the mechanism of
economic growth caused by capital
accumulation.
• In 1956, Robert Solow and Trevor Swan
developed this model independently.
• In 1987, Robert Solow got his Novel prize for
this model.
13

Slow – Swan model
• In a country, labor (𝐿𝐿) and technology level (𝐴𝐴)
are constant.
• Production function：diminishing marginal
productivity of capital
⇒ The only way to increase gross production
(𝑌𝑌) is to increase capital stock (𝐾𝐾)
⇒ How to increase capital stock (𝐾𝐾)?

14

Process of capital accumulation
• Revisit investment (𝐼𝐼)：Goods are bought not only for
present use, but also for future use.
⇒Invest this period, capital stock of next period will increase.
• Capital stock depreciation(𝐷𝐷)：A part of capital stocks that
becomes unusable or destroyed after a time of usage
⇒If no new capital formation is made, the capital stock of
the whole economy will decrease.
• Change in capital stock from this period to the next period
(∆𝐾𝐾)
∆𝐾𝐾 = 𝐼𝐼 − 𝐷𝐷
15

Determinants of capital depreciation
and investment
• Capital depreciation(𝐷𝐷)：assumed to be proportional to the
capital stock 𝐾𝐾 for this period
𝐷𝐷 = 𝑑𝑑𝑑𝑑

– Capital depreciation rate(𝑑𝑑)：Percentage of how much capital stock
become unusable in a year. This rate is assumed to be constant (0 <
𝑑𝑑 < 1)

• Determinants of capital （ 𝐼𝐼 ）：

investment

– Assume that there is no agent from abroad, the equilibrium of goods
market becomes：investment = savings from the whole economy
𝐼𝐼 = 𝑆𝑆
– A part of gross production is transferred to gross savings： 𝑆𝑆 = 𝑠𝑠𝑠𝑠 =
𝑠𝑠𝑠𝑠 𝐾𝐾, 𝐿𝐿
– Savings rate（𝑠𝑠）is assumed to be constant (0 < 𝑠𝑠 < 1)
16

Basic equation of economic growth
• Rewrite the equation that shows changes in capital
stock from this period to the next period（ ∆𝐾𝐾 = 𝐼𝐼 − 𝐷𝐷）
∆𝐾𝐾 = 𝑠𝑠𝑠𝑠 𝐾𝐾, 𝐿𝐿� − 𝑑𝑑𝑑𝑑

Basic equation of economic growth
– The equation of only variable 𝐾𝐾
– If 𝐾𝐾 of this period is given, we can calculated 𝐾𝐾 of all other
periods.
– It becomes the most basic formula for economic growth
model

17

Analysis from graphs
Gross production・
savings

Gross production
（𝐹𝐹(𝐾𝐾, 𝐿𝐿)）
Gross saving
（𝑠𝑠𝑠𝑠(𝐾𝐾, 𝐿𝐿)）

Capital depreciation・
Gross savings

𝐴𝐴

Capital depreciation
（𝐷𝐷 = 𝑑𝑑𝑑𝑑）
𝐸𝐸

Gross savings
（𝑠𝑠𝑠𝑠(𝐾𝐾, 𝐿𝐿)）

𝐵𝐵
Capital stock （𝐾𝐾）

𝐾𝐾0

𝐾𝐾 ∗

Capital stock （𝐾𝐾）

K1= K0 + dentaK0
K2= K1 + dentaK1
18

• Steady state is at point 𝐸𝐸

– “Increasing power” and “Decreasing power" are just
balanced.
– Capital stock does not increase or decrease over time.

• Economic growth caused by capital accumulation

– How about country whose capital stock 𝐾𝐾 is smaller than
capital stock 𝐾𝐾 ∗ at steady state?
⇒ 𝐾𝐾 will increase across time, converging to the steady
state, then capital accumulation stops there.
– Why does this happen?
⇒ Because of the assumption of diminishing marginal
productivity of capital.
19

• 3 factors can affect the steady state

2. Technology level

1. Saving rate

3. Capital depreciation
rate

The higher the saving rate is, The higher the technology
level is, the higher the steady The lower the capital
depreciation rate is, the higher
state becomes.
becomes.
Capital
Gross production・
savings

depreciation

𝐸𝐸𝐴𝐴
𝐸𝐸𝐵𝐵

𝐾𝐾𝐵𝐵∗

level A

𝐸𝐸𝐴𝐴

Savings A
Savings B

𝐾𝐾𝐴𝐴∗

Gross production・
savings

𝐾𝐾

𝐸𝐸𝐵𝐵

𝐾𝐾𝐵𝐵∗

Technology
level B

𝐾𝐾𝐴𝐴∗

𝐾𝐾

Capital
depreciation B
Capital
depreciation A

Gross production・
savings

𝐸𝐸𝐵𝐵

𝐾𝐾𝐵𝐵∗

𝐸𝐸𝐴𝐴

Savings

𝐾𝐾𝐴𝐴∗

20

𝐾𝐾

• With this model, every economy has its own steady state.
⇒ Capital accumulation stops there.
⇒ No more growth at last.
⇒ It is not enough to understand the reality.
• An important message from this model：
With only capital accumulation, economic growth
does not last forever.

Behind this conclusion： the assumption of diminishing
marginal productivity of capital
21

Human capital
• Common capital stock：Physical production equipment such
as machine, factory, office building
⇒ “Physical capital”
• There is another type of capital that is very important to
economic growth：”human capital”
• Human capital：Knowledge, skills, health which are
accumulated in the worker's brain and body.
• There are some common features between “physical
capital” and “human capital”：
― Techniques and skills are not archived by just a day, but it must
be accumulated over time
― To accumulate human capital is trying to gain higher future
income at the expense of current income

22

Population growth and Economic growth
• An assumption：Consider labor and population growth
similar, that labor (𝐿𝐿) increases over time is also called
population growth
• Labor（𝐿𝐿）：continue to increase at a certain rate every
period ⇒ Constant population growth rate
• Method of analysis：a similar analysis used for capital
stock can be applied on capital stock per capita (𝐾𝐾⁄𝐿𝐿)
• The result： At the steady state, the capital stock per
capita 𝐾𝐾⁄𝐿𝐿 converges to a constant.
⇒ 𝑌𝑌⁄𝐿𝐿 will be constant
⇒ The growth of 𝐾𝐾 will be equal to population
growth rate

23

Technology progress and Economic growth
• The problem of the basic model：Technical level is constant over
time （unreality）
⇒ Introduce “Technology progress” into Slow-Swan model
• Revisit production function： 𝑌𝑌 = 𝐴𝐴𝐾𝐾 𝛼𝛼 𝐿𝐿1−𝛼𝛼
⇒ An assumption on “Technology progress” ： 𝐴𝐴 continues to grow at
a certain rate
• The result：
-

Capital stock per capita and income per capita will continue to grow
without stopping
The higher the rate of advancement of technology progress is, the
higher growth rate is

• The most important message of Slow-Swan model：
The ultimate source to keep income per capita growing is technology
progress
24

Technology progress
• How to make technology progress?
１．Research & Development (R&D), technology
revolution
２．Technology transfer：Transfer technology
developed in developed countries to developing
countries

25

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