CEE Economic Growth & Development

Fall Semester, 2014

Lecture 7. Economic Growth:

The Role of Technological Progress

• The Solow model: Deriving steady state

• The Solow model and technological progress

• Growth accounting

Class Outline

The Vicious Circle of Poverty

GDP

Low savings

Scarce investment

capital

Scarce jobs,

Poor capital

Low output and

income

POVERTY

Breaking the Cycle

GDP

Better

institutions

Investment

capital

Better jobs,

Technology

Higher output

ECONOMIC

GROWTH

Solow-Swan Model of Economic Growth(1956)

Overview

Production function

• Diminishing returns to factor inputs

GDP per capita

( ) ( )( , )Y F K L

,1

( )

t t t

t t

t t

Y K KF F

L L L

y f k

y k

Diminishing Returns to Factor Inputs

( )y f k k

Implication: Countries

with small capital stock

are more productive =>

grow faster

Output per capita

Economic Growth and Capital Accumulation

Increase in capital stock (K)

Investments of firms

Savings of households provide investment funds to firms

s - exogenous savings rate

New additions

to the capital stock

Replacement of the

worn-out capital

(depreciation)

1

1

(1 )

(1 )

t t t

t t t

K I K

K sY K

Net investment = Total savings – Replacement of the depreciated capital

K sY K

Economic Growth and Capital Accumulation (Cont.)

Increase in capital stock (K)

Investments of firms

New additions

to the capital stock

Replacement of the

worn-out capital

(depreciation)

K sY K

• If capital stock is growing

• If capital stock is shrinking

• Break-even investment

sY K

sY K

sY K

0

5

10

15

20

25

0 100 200 300 400 500

y

k

y k

i s k

k

Steady State Level of Capital

0 0I K K k

The economy would grow as long as

*k

*y

( )sf k k

Steady state capital (k*)

and output (y*) per capita

The Solow-Swan Model: Steady State

Steady state: the long-run equilibrium of the economy

•Savings are just sufficient to cover the depreciation of the capital stock

N!B! Savings rate is a fraction of wage, thus is bounded by the interval [0, 1]

In the long run, capital per worker reaches its steady state for an exogenous s

Increase in s leads to higher capital per worker and higher output per capita

Output grows only during the transition to a new steady state (not sustainable)

Economy will remain in the steady state (no further growth)

Economy which is not in the steady state will go there => Convergence

Government policy response?

The Solow-Swan Model: Numerical Example

0.5 0.5( , )Y F K L K L

0.5 0.5 0.5

0.5

;

Y K L K

L L L

K Yk y

L L

Production function

Production function in per capita terms

GDP per capita:

Savings rate:

Depreciation rate:

Initial stock of capital per worker:

y k

30%s

10%

0 4k

Year k y i c δk Δk

1 4

2

…

Consumption: C = (1-s)Y

Consumption per capita C/Y = c

Steady state capital/labor ration:

2*

2

ss k k k

The Solow-Swan Model: Numerical Example (Cont.)

The Solow-Swan Model: Convergence to Steady State

N!B! Regardless of , if two economies have the same s, δ, N, they will

reach the same steady state

0k

• If countries have the same steady state, poorest countries grow faster

•Not much convergence worldwide

Different countries have different institutions and policies

• Conditional convergence: comparison of countries with similar savings rates

0

5

10

15

20

25

0 100 200 300 400 500

y

k

y

i

k

Solow Model: Convergence to Steady State

Convergence to steady state

*k

Solow Model: Increase in Savings Rate

0

5

10

15

20

25

0 100 200 300 400 500

y

k

• Savings rate increases from 30 % to 40 %

• Economy moves to a new steady state => Higher capital and output per capita

y

k

i

newi

*k*

newk

*y

*

newy

What effect would

have increase in δ on

the steady state?

Source: Mankiw (2009)

Log income per capita in 1960 (100=1996)

World Wide Convergence

Changes

in Log

income

per capita

in 1960 -

1990

The Golden Rule Level of Capital

Increasing savings rate means less present consumption

What is the optimal savings rate?

* * * maxc k k

N!B! Optimal savings

rate maximizes

consumption per capita

The Solow-Swan Model: Population Growth

Labor force is growing at a constant rate n =10%

( , )t t tY F K L

( )k sy n k

• Per capita capital stock is affect by investment, depreciation, and population growth

2*

2( )

( )

ss k n k k

n

Steady state:

• Population growth increases Y (level effect)

• Population growth reduces k* and y*

0

5

10

15

20

25

0 100 200 300 400 500

y

k

Solow-Swan Model: Population Growth (Cont.)

y k

( )n k

i s k

*k

*y

( )newn k

*

newk

*

newy

Economies with high rates of population growth will have lower GDP per capita

Government policy response?

N!B! Sustainable

economic growth

still remains

unexplained

The Role of Technological Progress

• Technological change, increase in factor productivity

Larger output with given quantities of capital and labor

( , )Y F K A L Effective labor

( , , )Y F K L A

How does technological progress translates into larger output?

• State of technology (A)

Labor-augmenting technological progress

• A as labor efficiency

• TP reduced number of workers needed to produce the same output

• TP increases output using the same number of workers

The Solow-Swan Model with Technological Progress

• Technology is improving every year at the exogenous rate (g)

Production function: GDP per effective labor

( ) ( ) ( )( , , )Y F K L A

1t t

t

A Ag

A

( , )

t t

t t t t

Y F K A L

Y KF

A L A L

The Solow-Swan Model with Technological Progress (Cont.)

• From GDP per effective labor to the GDP per capita?

' '

( , )

( )

t t

t t t t

t t

Y F K A L

Y KF

A L A L

y f k

'( )t tt t t

t t t

Y Ky A F A f k

L A L

GDP per

effective

labor

Capital per

effective labor

• We are interested in GDP per capita

0

5

10

15

20

25

0 100 200 300 400 500

y’

k’

' 'y k

( )n g k

' 'i s k

'*k

'*y

Steady state: Constant levels of capital and output per effective worker

The Solow-Swan Model with Technological Progress (Cont.)

The Solow-Swan Model: Technological Progress (Cont.)

• Capital and output per effective worker are constant in steady state

• What about per capita variables?

GDP per capita grows at the rate of technological progress (sustainable growth)

Balanced growth path: growth of variables at the same rate

•Per capita variables (capital, output and consumption) grow at a constant rate g

• Per effective labor variables are not growing in the steady state

* '

*( )ty A f k

N!B! Solow model explains 60 % of cross-country variation of the GDP per capita

by differences in savings rate and population growth

Growth Accounting

Real GDP per capita growth rate for Czech Republic in 2011 was 1.7 %

Real GDP per capita growth rate for the USA in 2012 was 2.2 %

How much of this growth is due to the factors’ accumulation and/or technology?

( , , )Y F A K L

Y A K L

Growth accounting: breakdown of observed growth of GDP into changes in

inputs and technology

A Y K L

Contribution of technology as a residual

Growth Accounting (Cont.)

• Capital (K) increases by 1 unit

What is the effect on output Y?

Marginal product of capita (MPK)

TE Capital stock increased by 10 units and MPK =0.1. What is the impact on

GDP?

unit

( , 1, ) ( , , )F A K L F A K L

( , , )Y F A K L

0.1 10 1Y

Growth Accounting (Cont.)

• Labor (L) increases by 1 unit

What is the effect on output Y?

Marginal product of labor (MPL)

TE Labor force increases by 10 units and MPK =0.3.

units

( , , 1) ( , , )F A K L F A K L

( , , )Y F A K L

0.3 10 3Y

Accounting for the increase in all components

How to account for the technological change?

Solow Residual: the left-over growth of output when growth attributed to the

changes in labor and capital is subtracted

( , , )

A K L

Y F A K L

Y MP A MP K MP L

Solow Residual

A K LMP A Y MP K MP L

Calculate it as a residual

Solow Residual (Cont.)

Where do we get marginal products of capital and labor?

L

L

Y

LF

K

K

Y

KFg

Y

Y Lk

A K LY MP A MP K MP L

Mathematical manipulations

• Transforming changes to growth rates

A K L

Y A K LMP MP MP

Y Y Y Y

GDP

growth

rate

Unobservable

technological

change (g)

Solow Residual (Cont.)

L

L

Y

LMP

K

K

Y

KMPg

Y

Y LK

Share of

capital in

output

Share of

labor in

output

K L

Y K Lg MP MP

Y Y Y

N!B! Key assumption: Factors of production are paid marginal product

• Wages and rental rate of capital reflect productivity of factors

Y K Lg

Y K L

Historical Factor Shares

Labor (2/3)

Capital (1/3)

Source: Acemoglu, 2009

Source: Mankiw, 2009