Long run economic growth, part 3.

The Solow growth model, with technological progress

Plan of the lecture

• Convergence

• Solow growth model – full version, with technological progress

Conditional convergence

• Solow model predicts that, other things equal, “poor” countries (with lower Y/L and K/L) should grow faster than “rich” ones.

• "The convergence hypothesis or the catch-up effect - poorer countries’ income per capita (measured by GDP / person) will grow faster than GDP per capita of the rich countries. According to this hypothesis, the relative backwardness can facilitate the economic development of the country; hence – can be an advantage .

https://www.nbportal.pl/slownik/pozycje-slownika/konwergencja

Conditional convergence

• Note however, that the Solow model predicts CONDITIONAL convergence- poorer countries’ income per capita (measured by GDP / person) will grow faster than GDP per capita of the rich countries, provided that both the rich and poor countries have the same steady-state.

• What the Solow model really predicts is conditional convergence - countries converge to their own steady states, which are determined by saving, population growth, and education.

• There is no reason the expect ABSOLUTE convergence (i.e. that poor countries will grow faster than the rich countries); but – if the Solow model is correct – we should observe conditional convergence

Conditional convergence

• Convergence stems from the assumption of declining marginal revenue of capital: the same amount of capital invested in a poor country, where capital is scarce, it will bring much greater benefit than in rich country (neoclassical models).

Conditional convergence

• To analyse the process of convergence, let’s look at the growth of capital per worker :

• Since y=f(k) is neoclassical, the share of average product of capital y/k is declining with k

• A Cobb-Douglas production function example

)()(

ndk

ys

k

kndsy

k

k

)()( 1 ndsk

k

kndsk

k

k

Condictional convregence 6

(n+d)

k*

0

k

skα-1

k

k

k

k

Convergence

• Econometric research almost undoubtedly shows that conditional convergence exists

• The econometric research shows that there is a statistically significant relationship between GDP per capita and it’s growth rate, when one controls for the investment rate, the stock of human capital, international trade, financial depth, government spending, etc

Convergence – states in the USA

8

Source: Sala-i-Martin, 1996

Convergence – states in the USA

Source: Sala-i-Martin, 1996

Convergence – European regions

10

Source: Sala-i-Martin, 1996

Convergence in the EU

11

Źródło: Bower & Turrini (2009); http://ec.europa.eu/economy_finance/publications/publication16470_en.pdf

Convergence in the EU 12

In general - lack of covergence across the world 13

But according to IMF (and other researchers)

• Convergence has entered into a new phase…

• Since the 90’s the developing countries are growing much quicker that the rich countries

• This is explained by globalization forces, favorable demographic trends (slowdown in population growth plus the still young society), better quality of institutions (governance), the improvement of macro-economic stability, sectoral transformation, etc The big question: will this trend continue?

• See IMF Data Mapper for the most recent trends

Convergence in recent times

Źródło: Dervis, 2012

Convergence in recent times

• Rodrik, 2011

Adding technological progress

In the „simple” Solow model we had studied so far,

– the production technology is held constant (A=1).

– income per capita is constant in the steady state.

Neither point is true in the real world:

– 1904-2004: U.S. real GDP per person grew by a

factor of 7.6, or 2% per year.

– examples of technological progress abound

Technological progress in the Solow model

• Assume: Technological progress is labor-augmenting:

• where N A = the number of effective workers.

• It increases labor efficiency at the exogenous rate g:

A

Ag

),( ANKFY

Intuition – a one-time increase in technology

kAy 0

y

k

ksAsy 0

kdn

*k

ksAsy 1

kAy 1

Technological progress in the Solow model

Notation:

= output per effective worker

= capital per effective worker

can be expressed as:

AN

Kk

AN

Yy

ˆ

ˆ

)ˆ(ˆ

),(

kfy

ANKFY

• Like in case of the „simple” Solow model, we want to understand the dynamics of capital – but capital per effective worker

Technological progress in the Solow model

kngdysngkkdys

ngkAN

dKsY

N

N

A

Ak

AN

K

AN

NANAKANK

AN

Kk

ˆ)(ˆ)(ˆˆˆ

)(ˆ)(

)(ˆ)(

)(

)()(ˆ2

Fundamental equation of the Solow growth model

kgndysk ˆ)(ˆˆ

(d + n + g)k = effective depreciation or break-even investment: the amount of investment necessary to keep k constant.

Consists of:

– d k to replace depreciating capital

– n k to provide capital for new workers

– g k to provide capital for the new “effective” workers created by technological progress

Steady-state

The steady-state

• In the steady-state:

ngng

y

y

N

N

A

A

yAN

yNAyNA

yAN

yAN

Y

Y

ggy

y

A

A

yA

yAyA

yA

yA

y

y

k

k

y

y

0ˆ

ˆ

ˆ

)ˆ)((ˆ)(

ˆ

ˆ

0ˆ

ˆ

ˆ

)ˆ(ˆ)(

ˆ

ˆ

0ˆ

ˆ

ˆ

ˆ

Steady-state growth rates in the Solow model with tech. progress

n + g Y Total output

g (Y/ N) Output per worker

0 Y/(NA ) Output per

effective worker

0 K/(NA ) Capital per

effective worker

Steady-state

growth rate Variable Variable

The Golden Rule once again

• Find the steady-state, for which consumption is maximized

• In the steady-state:

• Find steady-state capital per effective worker, for which consumption is maximized:

*ˆ)(*ˆ*ˆ kgndyc

gndkf

gndkd

yd

kd

cd

*)ˆ('

0)(*ˆ

*ˆ

*ˆ

*ˆ

The Golden Rule once again

• Just like in the case of the simple Solow growth model & a Cobb-Douglass production function…

• ..the steady-state consumption per „AN” is maximized, when s=α

• What happens with steady-state consumption, when the saving rate „s” goes up?

• What happens with steady-state consumption when the population growth rate „n” goes up?

Summary

• Adding technological progress to Solow model allows to explain long run economic growth of GDP per capita.

• The only (!) engine of long run growth of GDP per capita is technological progress.

• However, the Solow model does not explain, where technological progress comes from!

• Modern growth models try to fill this gap.