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Chapter 6
A Simple Model
of Growth and
Development
Charles I. Jones
Next Step: Technology Diffusion
A review of the growth models:
Technological progress is always the key.
Solow models take technological progress to be exogenous.
Romer and Schumpeterian models endogenize growth by
providing microeconomic underpinnings to innovation.
The next step is to show how technology gets diffused across
countries, and to explain why some countries’ level of
technological development is still low.
Basic Model
The basic setup is very similar to the Romer model.
Firms produce output Y using labor L and a range of capital
goods x j . The “number” of capital goods is limited by the skill
level h.
h
1

Y

L
x
The production function is
 j dj
0
Low-skilled workers use spades, high-skilled workers use both
spades and excavators.
Key distinction: this time, new designs of capital goods due to
innovation are already available from the more developed part
of the world. Think of this model as discussing a less developed
economy.
Capital Goods Production
As before, one unit of raw capital good produces exactly one unit
of the intermediate capital good.
Total capital stock in the economy must equal to the total stock of
capital goods:

h t 
0
x j t dj  K t 
Intermediate goods are treated symmetrically: x j  x for all j.
The aggregate production function then takes the following form
that treats skill level exactly as a labor-augmenting productivity
factor:
1

Y  K hL 
Capital Accumulation
Capital stock changes with time due to savings/investments and
capital depreciation in a standard way:

K  sK Y  K
In chapter 3, the skill level h was the number of schooling years.
In this model, skill h is defined as the range of the intermediate
goods an individual has learned how to use.
Learning to Use Advanced Goods
Workers progress from using hoes and oxen to using tractors
and fertilizers according to

h  eu A h1 ,   0,0    1
Here u is the amount of time an individual spends accumulating skill
instead of working. A is the best-practice frontier.
Spending additional time (u) accumulating skill will increase the
skill level proportionally (why?).
The change in skill is a geometrically weighted average of the
frontier skill level A and the individual skill level h (why?).
Skill Accumulation
Let us rewrite the skill accumulation equation like this:

h
u  A 
 e  
h
h

The closer the skill level h is to the frontier A (i.e. the lower the
A
value of
, the harder it is to learn a new skill.
h
Computers were difficult to learn when they just appeared
compared to today when they are common, i.e. far from the frontier.
The technological frontier is assumed to evolve because of
investment in research by the advanced countries: 
A
g
A
Balanced Growth Path
We assume that sK and u are constant and exogenous.

We also assume that
exogenously given.
L
 n,
L
where n is also constant and
Since the skill level h is entering the production function like A
enters it in Chapter 3, the growth rate in h will determine the
growth rates of y  Y and k  K

L
L
A
h
The growth rate will be constant if
is constant, which
means A and h must grow at the same rate.
h
h
Along the balanced growth path then,
g y  gk  gh  g A  g
Steady State
The growth rate of the less developed economy is determined by
the growth in the skill level, which in turn is determined by the rate
of growth of the world’s technological frontier.
*
The steady state capital-output ratio is given by
sK
K
  
 Y  n  g 
Substituting it into the production function, we obtain:
 sK
y * t   
 n  g 




1
h* t 
The more time individuals spend learning skills, the closer the
economy is to the best-practice technological frontier:
1
  u  
h
    e 
 A
g

*
Evolution of Output per Worker
Output per worker in our economy is evolving with time and is a
function of exogenous parameters and the evolution of the
world’s best practice technological frontier:

sK
y t   
 n  g 
*




1
1
  u   *
 e  A t 
g

In this model, economies grow because they learn how to use new ideas developed
throughout the world.
Investing more increases growth, while rapidly growing population produces a downward
pressure on growth.
Economies that spend more time accumulating skills will be closer to the technological
frontier.
Differences in the technological level are explained by differences in u, the amount of time
individuals invest in accumulating skills.
The TFP Problem
In our model, Total Factor Productivity is the same in all
countries.
However, one of the stylized facts we examined says that the TFP
levels vary considerably across countries.
The possible explanations for TFP differences are
•Lack of economic institutions that allocate labor and capital to
their most productive uses
•State intervention may result in productive inefficiencies
•Firms may be reluctant to invest in skill accumulation
Technology Transfer
The key assumption so far was that the intermediate capital goods
are readily and freely available for use from the more advanced
countries. However:
•Steering wheel may need to be placed on the other side
•Electric appliances might need to conform to a different standard
•Movies may need to be modified even for the same language (e.g.
Australian vs American English)
Patents
•Property rights protection is necessary to ensure innovation
•Do patents issued in one country get enforced in another?
• If yes, innovation is stimulated even more
• However, in this case one has to pay for the intermediate
goods
Effects of International
Property Rights Protection
Consider a model of two countries:
•The North, at the technological frontier
•The South, far below the frontier, imitating the designs invented in
the North
If international property rights are respected in the South, imitation
is more difficult, so it reduces  in equation for skills growth:

h
 A
 eu  
h
h

Along the balanced growth path, the reduction of growth rate of
skills leads to a reduction in the ratio of skills to technological level:
1
  u  
h
    e 
 A
g

*
Effects of International
Property Rights Protection
However, better protection of the international property rights will
induce the North to innovate since every new idea results in higher
profits, so more labor in the North is engaged in research: the
Northern share of researchers s R will go up.
This increases the level of A.
Since in the South, the balanced-growth path output per capita is



determined by y t    s
, the decrease in  and an

 e  A t 
*
K
 n  g  



1
1

g

u


*
increase in A will have mutually offsetting effects.
Helpman (1993) finds that the net effect is negative: more
innovation in the North is not enough to offset slower imitation in
the South
The Case of Appropriate Technology
In some countries, imitation only makes sense once an appropriate
level of technological development is reached.
•Computers are useless without electricity network
•Bullet trains are not needed in a country with no railroads
In case the South is not sufficiently developed to adopt the Northern
technologies, the A in the South will be lower.
However, if the North modifies its technologies for use in the South
due to the international property rights protection in the South, the
level of A will be higher due to more incentives to innovate in the
North, so that the net effect will be positive.
Recent Developments
TRIPS: Agreement on Trade-Related Aspects of Intellectual Property
Rights
As off 1995, ratifying TRIPS became compulsory for anyone willing
to join the WTO (World Trade Organization)
The frontier countries (US, Europe) pushed for TRIPS
Developing countries negotiated a delay in the adoption of this
agreement
In 2005, the window to implement TRIPS ended for the less developed
countries, but it was extended until 2013 for the least developed
countries.
Globalization and Trade
One of our stylized facts was that openness to the international trade
is positively associated with growth.
Let us modify the production function to account for this link:
1
Y L
h
x
0

j
dj
becomes
Y  L1
h m

x
 j dj
0
The number of intermediate goods varieties is equal to h, those
goods produced at home, PLUS m varieties imported from other
countries.
The Capital Stock Identity
Let z be the number of units of each of the h types of intermediate
goods that a country has learned how to make.
The value of domestically produced capital will be ht zt   K t 
Since the domestic final-goods sector uses x  x j units of each of the h
intermediate goods, the country keeps ht xt  for domestic production.
As a result, the country can pay K t   ht xt  for the intermediate
goods produced abroad.
Since there are m different types imported, and x(t) units of each
imported variety use, the following identity hods:
K t   ht xt   mt xt 
Interpreting the Capital Stock Identity
K t   ht xt   mt xt 
Since ht zt   K t  , it follows that ht zt   xt   mt xt 
Net intermediate goods produced domestically are shipped as
exports to foreign countries in exchange for imports of mx in
foreign intermediate goods.
Alternatively, we can invoke the FDI (foreign direct investment)
reasoning: the home country owns K units of capital, but only hx
of those are located inside the country, while the other mx units are
located abroad.
•Intel’s chip plant in Costa Rica
An equivalent amount of foreign capital is invested in home
country:
•Toyota’s assembly plant in Tennessee
Production Function with Trade
The capital stock identity can be rearranged to obtain
K t   xt ht   mt 
Combining with the modified production function, we can write:
Y K

h  m L
1
The term h+m enters as a labor-augmenting technology.
More foreign inputs increase output.
Rearranging slightly, we obtain: Y  K

hL 
1
1
m

1  
h

This is a familiar aggregate production function augmented by the ratio
of foreign to domestic inputs.
Balanced Growth Path with Trade
Since mathematically the aggregate production function is similar to
the one analyzed at the beginning of the chapter, all familiar
conclusions apply. In particular, along the balanced growth path,
 sK 

y t   
 n  g  
*

1
  u 
 e 
g 
1

 m *
1   A t 
 h
In case the economy is closed, i.e. m=0, we are back to the solution
we obtained earlier in this chapter.
Trade Openness and Growth
Ratio of imports to GDP can act as a crude measure of the country’s
openness:
Im mx
m K


GDP Y m  h Y
m
h
As the ratio
increases (e.g. think of increasing m with fixed h),
the ratio of imports to GDP goes up, too.
 sK 

From the production function y t   
 n  g  
*

1
  u 
 e 
g 
1

 m *
1   A t  ,
 h
m
h
an increase in
acts in the same way as an increase in education u or
savings rate sK, by increasing the level of technology A.
After opening up, growth will be higher, and extent of openness will be
higher, too. That is explaining the stylized fact on trade and growth.
Trade Openness and Growth Miracles
Korea: imports accounted for 13% of GDP in 1960, rising to 54%
by 2008.
China: imports are 3% of GDP in 1960, but they are 27% in 2008.
Both countries have substantially increased the amount of foreignproduced intermediate parts in their production factor mix.
South Korea
•Cell phone parts
•Automobile parts
Foreign-Produced Inputs and Growth
Along the balanced-growth path, h is growing at the rate g.
m
Unless m is growing at the same rate or higher, the ratio
will be
h
falling.
Broda, Greenfield and Weinstein (2010) document the following:
•A median country increased the number of imported intermediate
inputs from 30000 in 1993 to 41000 by 2003.
•That translates into a growth rate of about 3.5% per year
•92% of the increase in the value of imports is accounted for by the
increase in the variety of imported goods rather than the volume of
the already imported goods
•The empirical model implies that an increase in the variety of
imported inputs increases median growth from 2% to 2.6% with the
increased growth effects in place for 75 years!
Same Long-Run Growth Rate?
Technology diffusion model implies that in the long run each
country’s growth rate will be the same and equal to the rate of
expansion of the best-practice technological frontier A.
Belgium and Singapore are good examples of countries that grow by
actively borrowing the best-practice ideas from the rest of the world
(both have small populations).
Our stylized facts, though, say that average growth rates vary
enormously across countries.
Large variation in growth rates are explained by transition dynamics.
Different growth rates are possible while countries are changing their
position within the world’s income distribution.
Different Growth Rates and the Steady State
We know that countries that are below their steady state will grow
faster than average, while being above the steady state slows down
growth.
What makes countries be away from their steady states?
•Shock to capital stock (e.g. war)
•Policy reform affecting investment
•Policy reform affecting educational system
United States and United Kingdom
1870-1950: US grows at
1.7% per year > 0.9%
for UK
After 1950: US and UK
grow at 2.03% and
2.17%, respectively
We have to be careful interpreting average growth figures over
long periods of time!
Transition Dynamics and Technology Diffusion
Transition dynamics in the Solow models is a feature of capital
accumulation equation.
In the model of technology diffusion, transition dynamics involves
not only capital accumulation, but also technology diffusion in

h  eu A h1
Suppose a country reduces import tariffs, which increases the rate
of growth in the skill level h, and the level of steady-state level of

1
income
 sK 1   u    m  *
*
  e  1   A t 
y t   
 n  g    g   h 
The principle of transition dynamics says that, since the economy is
now below its steady-state level of per capita income, it will start
growing faster.