INTERMEDIATE MACROECONOMICS-II

B.A.(H) Economics, Semester-IV

Topic-1: Technological Progress & Elements of Endogenous

Growth
Lecture Notes
(Ref: Jones, Introduction to Economic Growth, 2nd ed. Ch-4 &
5.)

Department of Economics,
Hansraj College, Delhi University.
INTERMEDIATE MACROECONOMICS-II

Economics of Ideas

Ideas → Nonrivalry → Increasing Returns → Imperfect Competition

INTERMEDIATE MACROECONOMICS-II

FIXED COSTS AND INCREASING RETURNS

INTERMEDIATE MACROECONOMICS-II

FIXED COSTS AND INCREASING RETURNS

INTERMEDIATE MACROECONOMICS-II

Engine of Growth
Basic Elements of the Model

Y = K α (ALy )1−α (1)

where, K represents capital stock, Ly labor employed in the
production, Y represents output, A represents the stock, and
α ∈ [0, 1] is a parameter.

K̇ = sK Y − dK . (2)
where, sK rate of savings or rate of forgoing consumption,
depreciates at the exogenous rate d, and n represents the rate of
growth of population.

L̇
=n (3)
L
INTERMEDIATE MACROECONOMICS-II

Number of new ideas produced: Ȧ = δ̄LA . (4)

where, δ̄ represents the rate of discovery of new ideas, and LA
represents the population engaged in research.

Rate of discovery: δ̄ = δAφ (5)

where δ and φ are constants. φ > 0 represents the productivity of
research increases with the stock of ideas that have already been
discovered; φ < 0 corresponds to fishing out.

Ȧ = δLλA Aφ (6)
where, λ ∈ [0, 1] represents the productivity of labor in terms of
producing new ideas. Externality associated with φ is referred to as
standing on shoulders effect and the externality associated with
λ is referred to as stepping on toes effect.
INTERMEDIATE MACROECONOMICS-II

Distribution of total labor force L, into labor engaged in research

LA and labor engaged in production LY .

LY + LA = L (7)
LA
The model assumes that a constant fraction, = sR , of the labor
L
force engages in research to produce new ideas, and the remaining
fraction, 1 − sR , produces output.
INTERMEDIATE MACROECONOMICS-II

Growth in Romer Model

Along a balanced growth path, for rate of growth of per capita

output, gy , rate of growth of capital-labor ration, gk , and of
growth of stock of ideas, gA .

gy = gk = gA (8)
And rate of technological progress,

Ȧ LλA
= δ 1−φ (9)
A A
INTERMEDIATE MACROECONOMICS-II

Along a balanced growth path, as Ȧ/A ≡ gA is constant.

Therefore,

L˙A Ȧ
0=λ − (1 − φ) (10)
LA A
Also, along a balanced growth path, growth rate of number of
researches must be equal to growth rate of the population i.e.
L˙A
LA = n. Thus,

λn
gA = (11)
1−φ
INTERMEDIATE MACROECONOMICS-II

Special Examples

Case 1: Let λ = 1, φ = 0, LA is a constant, and the productivity

of researchers δ is constant. Then

Ȧ = δLA

The economy generates a constant number of new ideas Ȧ = δLA ,

each period. Therefore, the growth rate of stock of ideas Ȧ/A falls
overtime, eventually approaching zero. And, the growth rate of per
capita output also falls overtime and economy doesn’t operate at a
sustainable growth rate even with a constant research effort.
INTERMEDIATE MACROECONOMICS-II

Case 2: Let λ = 1, φ = 1, LA is a constant, and the productivity

of researchers δ is constant. Then

Ȧ = δLA A

Ȧ
=⇒ = δLA
A
This suggests that the growth rate of stock of ideas is constant in
each period. Therefore, the growth rate of per capita ouput is also
constant and even with constant number of researchers, the
economy operate at a sustainable growth rate .
INTERMEDIATE MACROECONOMICS-II

Growth Effect vs Level Effect

Let the share of population engaged in research sector increases
permanently. Also, let λ = 1 and φ = 0. Let sR increases
Ȧ sR L
permanently to sR0 . Also, = δ
A A
INTERMEDIATE MACROECONOMICS-II
INTERMEDIATE MACROECONOMICS-II

 ∗  α/1−α
y sK
= (1 − sR )
A n + gA + d
 α/1−α
∗ sK δsR
y (t) = (1 − sR ) L(t)
n + gA + d gA
INTERMEDIATE MACROECONOMICS-II

Economics of the Model

The Final-Goods sector

A
X
Y = L1−α
Y xjα (12)
j=1

where, Output Y , is produced using labor, LY , a number of

different capital goods, Xj , also referred to as intermediate goods,
and A measures the number of capital goods.

Y = L1−α α 1−α α 1−α α

Y x1 + LY x2 + · · · + LY xA

Z A
≈ L1−α
Y xjα dj
0
INTERMEDIATE MACROECONOMICS-II

At wage rate w and rental price of each capital good pj , firms

decide the level of labor and each capital good, in order to
maximize the profit function.
Z A Z A
max L1−α
Y xjα dj − wLY − pj xj dj
LY ,xj 0 0

From the first-order conditions, we get

Y
w = (1 − α) (13)
LY

pj = αL1−α
Y xj
α−1
(14)
INTERMEDIATE MACROECONOMICS-II

The Intermediate-Goods sector

Consists of monopolists, each produces a particular capital good.
Firms gain monopoly power by purchasing the design for a specific
capital good from research sector and produce each unit of capital
at a fixed cost of raw material r .
Profit maximization problem for a firm in this sector is
max πj = pj (xj )xj − rxj
xj

where pj (x) is the demand function for the capital good x.

From the first-order conditions, we get
p 0 (x)x + p(x) − r = 0

x r
p 0 (x) +1=
P P
1
p= p 0 (x)x
r
1+ p
INTERMEDIATE MACROECONOMICS-II

The elasticity p 0 (x)x/p can be calculated from the demand

function of each capital good, i.e. pj = αL1−α
Y xj
α−1
. And
0
p (x)x/p = (α − 1). Such that,
1 1
p= p 0 (x)x
r= r
1+ α
p
This suggests that each capital good is sold at the same price.
Therefore, by keeping the quantity of each good same i.e. xj = x,
we get profit as

Y
π = α(1 − α)
A
And the total demand for capital goods will be equal to the total
capital stock in the economy, K :
Z A A
X
xj dj = xj = xA = K
0 j=1
INTERMEDIATE MACROECONOMICS-II

The final goods production function can be written, using xj = x,

as
Y = AL1−αY x
α

−α α
Y = AL1−α
Y A K

= K α (ALY )1−α
INTERMEDIATE MACROECONOMICS-II

Research Sector
Let PA be the price of the new design, P˙A be the capital gain by
reselling the design, r be the prevailing rate of interest in the
market, and π be the profit earned.
Then from the arbitrage equation, we get

rPA = π + P˙A

π P˙A
r= +
PA PA
Along a balanced growth path, r is constant. Therefore, π/PA
must also be constant, which requires both π and PA to grow at
the same rate, which turns out to be population growth rate n.
Therefore, we get
π
PA =
r −n
INTERMEDIATE MACROECONOMICS-II

Labor working in the final-good sector earns a wage,

Y
wY = (1 − α)
LY

Labor working in the research sector, based on their productivity δ̄

and price of new design PA , earns a wage,

wR = δ̄PA

In the context of labor, being indifferent to engage in either

production or research sector, wY = wR , which gives,
1
sR = −n
1 + rαg A
INTERMEDIATE MACROECONOMICS-II

The interest rate in this three sector economy is, r = α2 Y /K ,

which is less than the marginal product of capital i.e. αY /K .
So, as oppose to the solow model, under the romer model, as
the production in the economy is characterized by the
increasing returns and all factors cannot be paid their
marginal products.
INTERMEDIATE MACROECONOMICS-II

Optimal R&D