Intermediate Macroeconomics
Lecture 5 - Endogenous growth models
Zsófia L. Bárány
Sciences Po
2014 February
Recap: Why go beyond the Solow model?
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we looked at the Solow model with technological progress and
found that it matches the Kaldor facts well
we looked at why economists moved beyond the Solow model:
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capital does not move from rich to poor countries
growth accounting → technological improvements contribute
significantly to growth
development accounting → there are large differences in the
level of technology across countries
this week we look at endogenous growth models
1. learning by doing
2. human capital
3. research and development
and look at international technology transfer
Learning by doing
Learning by doing
Based on Romer (1989).
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Main idea: skills or knowledge are accumulated during the
production
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⇒ the skills or knowledge accumulation is free and is a
by-product of production
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the marginal product of capital diminishes at the firm level
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BUT when a firm invests, other firms learn from its experience
too, i.e. investment by a firm generates a positive
externality for the economy
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⇒ no diminishing marginal product of capital at the
aggregate level, i.e. ’AK’ for aggregate capital
Model
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A representative firm i’s production function
Yi = Kiα (BNi )1−α
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α < 1 ⇒ diminishing marginal product of capital
B is the stock of economy-wide knowledge, the firm takes this
as given
The economy wide stock of knowledge is proportional to the
economy-wide stock of capital
B = λK
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λ > 0 represents the idea of a positive externality
the higher aggregate capital, K , and thus aggregate output,
Y , the higher is productivity, B
Since firm i is a representative firm, it represents aggregate output
and capital
K = Ki
and
N = Ni
and
Y = Yi
Use B = λK in the production function:
Y = K α (BN)1−α = K α (λKN)1−α = K (λN)1−α
⇒ this is an ’AK’ production function, with no diminishing
marginal returns to capital at the aggregate level
The capital accumulation equation
K 0 − K = sK (λN)1−α − dK
Let’s assume that the population is constant
k 0 − k = s(λN)1−α k − dk
the growth rate of capital per person
k0 − k
= s(λN)1−α − d = x = constant
k
⇒ if x > 0, then there is long run endogenous growth
this is satisfied if the saving rate, s is sufficiently high
Endogenous growth in the learning-by-doing model
i
s(λN)1−α
dk
k
Assuming s(λN)1−α − d > 0
Implications of the learning-by-doing model
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there is endogenous growth because the stock of knowledge
is determined by the endogenous level of K through
learning-by-doing
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the saving rate affects not only the level of income but also
the growth rate, as x depends on s
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the growth rate is constant in both the short and the long
run, so there is no convergence
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there are ’scale effects’: the growth rate depends on the size
of the population
larger N implies stronger knowledge spillovers and therefore
higher growth rate, x
Implications of the learning-by-doing model
I
there is endogenous growth because the stock of knowledge
is determined by the endogenous level of K through
learning-by-doing
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the saving rate affects not only the level of income but also
the growth rate, as x depends on s
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the growth rate is constant in both the short and the long
run, so there is no convergence
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there are ’scale effects’: the growth rate depends on the size
of the population
larger N implies stronger knowledge spillovers and therefore
higher growth rate, x
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Is this model prediction problematic?
Implications of the learning-by-doing model
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there is endogenous growth because the stock of knowledge
is determined by the endogenous level of K through
learning-by-doing
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the saving rate affects not only the level of income but also
the growth rate, as x depends on s
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the growth rate is constant in both the short and the long
run, so there is no convergence
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there are ’scale effects’: the growth rate depends on the size
of the population
larger N implies stronger knowledge spillovers and therefore
higher growth rate, x
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Is this model prediction problematic?
one way to remove this scale effect is to replace B = λK by
B = λk, i.e. knowledge depends on capital per worker
Human capital
Human capital
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skills are required to put ideas or knowledge into practice
for the OECD countries and in most parts of the world
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average years of schooling are increasing
fraction of college graduates is increasing
as opposed to the process of learning-by-doing, there are
costs and returns to education
Human capital
Based on Lucas (1988).
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introduce a production function for human capital
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the production of new human capital is proportional to
existing human capital
⇒ ’AK’ for the production of human capital
no diminishing marginal product in the production of human
capital
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how good someone is in accumulating human capital (A)
might depend on his years of education
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prediction: positive growth in the long run
Model
The consumer consumes all his wage income:
C = wuH s
w - real wage per efficiency unit of labor
u - fraction of time devoted to working (exogenous)
I H s - stock of human capital supplied
I ⇒ uH s total units of efficiency labor supplied
Future human capital
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H s 0 = b(1 − u)H s
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depends on current human capital, H s
on the time devoted to training and education, 1 − u
b - efficiency of human capital accumulation
Model
The representative firm’s production function:
Y = zuH d
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z TFP
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uH d amount of efficiency units of labor in production
the profits are:
π = Y − wuH d = zuH d − wuH d = (z − w )uH d
The competitive equilibrium:
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the labor market has to clear: H d = H s = H
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the goods market has to clear: C = Y = zuH
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human capital accumulation: H 0 = b(1 − u)H
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the equilibrium growth rate of human capital is:
H0 − H
= b(1 − u) − 1
H
⇒ if b(1 − u) > 1 ⇒ human capital increases forever
⇒ there is endogenous growth
Endogenous growth in the human capital model
H0
b(1 − u)H
45◦
H
Assuming b(1 − u) > 1
The role of u
remember, u is the fraction of time devoted to working, while
1 − u is the fraction of time devoted to studying, and
C = zuH
what is the effect of a decrease in u on Y and C ?
immediate effect
fewer hours worked (at constant H) ⇒ a decline in Y ⇒ a decline
in the level of C
long run effect
after the immediate change u is again constant, thus
C0 − C
H0 − H
=
= b(1 − u) − 1
C
H
⇒ lower u ⇒ higher 1 − u, more time devoted to studying ⇒
higher growth rate of H and C
The role of u
Is a lower u necessarily better?
Is there convergence?
imagine country A and country B have the same characteristics
uA = uB , zA = zB , bA = bB
but country A has a higher initial level of human capital
HA (0) > HB (0) → will they converge?
Is there convergence?
imagine country A and country B have the same characteristics
uA = uB , zA = zB , bA = bB
but country A has a higher initial level of human capital
HA (0) > HB (0) → will they converge?
Is there convergence?
imagine country A and country B have the same characteristics
uA = uB , zA = zB , bA = bB
but country A has a higher initial level of human capital
HA (0) > HB (0) → will they converge?
No convergence – countries grow at the same rate
Convergence in the Solow model
in the Solow model what happens to two countries, A and B,
which have the same characteristics
sA = sB , nA = nB , dA = dB , same technology zF (K , N)
but country A has higher initial capital per person than country B:
kA (0) > kB (0)?
Convergence in the Solow model
in the Solow model what happens to two countries, A and B,
which have the same characteristics
sA = sB , nA = nB , dA = dB , same technology zF (K , N)
but country A has higher initial capital per person than country B:
kA (0) > kB (0)?
Convergence in the Solow model
in the Solow model what happens to two countries, A and B,
which have the same characteristics
sA = sB , nA = nB , dA = dB , same technology zF (K , N)
but country A has higher initial capital per person than country B:
kA (0) > kB (0)?
Conditional
convergence –
poor country
grows faster
Research and development
Research and development
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new ideas and knowledge are developed in the market through
devoting resources to research and development
for the OECD countries
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R&D spending as a fraction of GDP is increasing over time
number of researchers as a fraction of the total employment is
Research
increasing
over timeand Development
What is the key difference between ideas and physical
capital?
Like capital, ideas are economic goods
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there is a cost to producing them
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they can be used in production
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there is a price for them (the value for a patent)
There are some differences along the following attributes
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rivalrous vs non-rivalrous
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degree of excludability
Can you come up with examples for each?
Examples
rivalrous
excludable
non-excludable
non-rivalrous
One country R&D model
Based on Romer (1990).
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introduce a production function for ideas
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production of new ideas is proportional to the existing stock
of ideas, i.e. ’AK’ for the production of ideas
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the coefficient depends on, for example, the number of
researchers
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there is no diminishing marginal product in the production of
ideas
Model
The production function is
Y = ALY
where LY is the number of workers engaged in producing output
and A is the level of knowledge (or technology)
The output per worker is then
Y
ALY
LY
LA
y=
=
=A
=A 1−
= A(1 − γA )
L
L
LY + LA
LA + LY
where γA is the fraction of workers engaged in R&D
Model
We assume that the production of new knowledge leads to the
following growth rate of knowledge
0
b ≡ A − A = γa L
A
A
µ
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proportional to the number of workers engaged in R&D, γA L
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µ captures the cost of new inventions
If γA is constant ⇒ y is proportional to A since y = A(1 − γA )
⇒ the growth rate of y is the same as the growth rate of A
0
b = A − A = γa L
yb = A
A
µ
The role of γA
γA is the fraction of labor which works in R&D
what is the effect of an increase in γA ?
Immediate effect
fewer people working in production ⇒ a decline in Y ⇒ a decline
in C
long run effect
the growth rate of A increases ⇒ the growth rate of Y increases
The role of γA
One-Country Model R&D – One-Country Mode
Shifting
ShiftingLabor
Laborinto
intoR&D
R&D
nto
ntoR&D
R&D
Ln(A)
Ln(y)
e of drawing figure in log from handout 1
Slide #33
Summary of the one-country model
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there are scale effects
the growth rate depends on the size of the population, L
a larger L implies more workers engaged in R&D (for given γA )
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this implies that countries with larger populations have higher
growth rates, higher levels of technology and are richer
→ are these good predictions?
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one interpretation of the model is that a country’s level of
technology depends on R&D done around the world
this is a reasonable assumption if there is international
technology transfer
→ two-country model
Two country R&D model
There are two countries, labelled 1 and 2.
The production function for each country j = 1, 2 is
Yj = Aj (1 − γAj )Lj
Countries acquire new technologies either by invention or by
imitation.
The option of imitation is open only to the less developed country,
the ’technology follower’.
Assume L1 = L2 = L and γA1 > γA2 .
⇒ country 1 will be the technology leader and country 2 the
follower in the steady state
The cost of imitation
Assume that the cost of imitation is
A1
µc = c
A2
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c(·) is downward sloping
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c(·) tends to zero as A1 /A2 tends to infinity
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c(·) tends to the cost of invention, µi as A1 /A2 tends to one
⇒ the cost of imitating a given technology is less than the cost of
reinventing the technology
µc < µi
The
costofofImitation
imitation
Cost
Cost of Imitation
Slide #38
The steady state
What can the growth rates of technology (and hence output) be in
the two economies?
bj =
theR&D
growth –
rate
has the
same form asModel
before: A
Two
-Country
country 1 is the leader ⇒ µ1 = µi
Steady
State
country
2 is the follower ⇒ µ2 = µc if A2 < A1
Steady
State
n the steady
tate, the growth
ates of A1 and
A,1
L
A2 are equal .
i
There is a steady
tate level of
A, 2
L
A1/A2 .
c
Growth rate of technology
γAj
µj
L
The steady state
In the steady state, the growth rates are the same
γA1
γA
L = 2L
µi
µc
⇒
µc =
γA2
µi
γA 1
The steady state level of relative technologies, A1 /A2 can be found
using c(·)
A1
γA
c
= µc = 2 µi
A2
γA1
The role of γA2
what happens if the follower increases R&D effort, i.e. γA2
increases?
steady state effect
more resources in R&D ⇒ growth curve shifts up ⇒ lower level of
steady state A1 /A2
The role of γA2
what happens if the follower increases R&D effort, i.e. γA2
increases?
R&D
Two-Country Model
steady
state –
effect
more resources in R&D ⇒ growth curve shifts up ⇒ lower level of
An
increase
in R&D in
Ansteady
increase
inthe
thefollower
follower
stateinAR&D
1 /A2
An increase in
γA2 shifts up the
growth rate of A2.
The new steady
state level of
A1/A2 is lower.
Slide #41
The role of γA2
immediate effect
fewer people in production ⇒ drop in output, Y
then
temporary increase in the growth rate of the follower
The role of γA2
Slide #41
immediate effect
fewer people in production ⇒ drop in output, Y
then
Two
-Country Model
temporary increase in the growth R&D
rate of –the
follower
R&D – Two-Country Model
An
Anincrease
increasein
inR&D
R&Din
inthe
thefollower
follower
ase in
ease
inR&D
R&Din
inthe
thefollower
follower
Ln(y)
se in
oa
y
n the
e for
er
w does this compare to the one-country model?
S
Slide #42
Recap of economic growth models
1. Malthusian Model
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population growth increasing in per capita consumption
stagnation in the long-run
2. Solow Model
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capital accumulation
no (endogenous) growth in the long-run
conditional convergence
model with labor-augmenting technological progress matches
the Kaldor facts well
3. Endogenous Growth Models
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role of technology and its origin
no convergence