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XET 504
LONG-RUN ECONOMIC GROWTH
HARROD – DOMAR MODEL
Japheth
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Awiti, Ph
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February 27, 2020
Japheth
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Awiti, Ph
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XET 504
February 27, 2020
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Outlne
1
2
3
4
5
Introduction
Assumptions
The Model
Applications
Limitations
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Introduction
The model was independently developed by Roy
Harrod (Harrod 1939, 1948) and Evsey Domar
(Domar 1946, 1947)
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Assumptions
The assumptions underlying the model include
the following:
a
...
b
...
That is,
K
the capital–labour ratio, , is constant
...
O
...
D
...
)
c
...
e
...
Y
Total new investment spending is
determined by total savings
...
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The Model
If we assume that the economy consists of only
two sectors, households and firms, we can
express the national income equation as
Yt = Ct + St
(1)
where Yt = GDP, Ct = consumption, and St =
saving
...
O
...
D
...
)
For the economy to be in equilibrium, investment
spending must equal savings
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Model (Contd
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(3)
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Model (Contd
...
If we assume capital depreciates
over time at the rate δ, we can express the
evolution of the capital stock over time as
follows:
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Model (Contd
...
(4)
= (1 − δ) Kt + It
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Model (Contd
...
O
...
D
...
)
then it follows that
K = vY
and
∆K
=v
∆Y
∆K
where
is the incremental capital–output
∆Y
ratio (ICOR)
...
O
...
D
...
)
Since total saving is some proportion of output,
the savings function can be written as
St = sYt
(5)
where s is the savings rate (also referred to as
the savings ratio)
...
O
...
D
...
)
Using the results that Kt = vYt and
It = St = sYt (since total new investment is
determined by total savings), we can rewrite
Equation (4) as
vYt+1 = (1 − δ) vYt + sYt
...
O
...
D
...
)
If we divide through by v , simplify, and then
subtract Yt from both sides of Equation (6), we
get Equation (7):
i
hs
− δ Yt
...
O
...
D
...
)
If we divide Equation (7) through by Yt , we
obtain Equation (8):
s
Yt+1 − Yt
= −δ
Yt
v
where
(8)
Yt+1 − Yt
is the growth rate of GDP
...
O
...
D
...
)
Yt+1 − Yt
, then the
Yt
Harrod–Domar growth equation can be written
as
s
g = − δ
...
O
...
D
...
)
According to this equation, the growth rate of
GDP is jointly determined by the savings rate, s,
the capital–output ratio, v , and the rate of
depreciation, δ
...
O
...
D
...
)
The higher the savings rate and the lower the
capital–output ratio (or the lower the amount of
capital required to produce a single unit of
output in the economy) and the lower the
depreciation rate, the faster the economy will
grow
...
O
...
D
...
)
Assuming the depreciation rate is low and
ignoring it, we can conclude that the key driver
of growth is the savings ratio, since the
capital–output ratio is assumed fixed
...
O
...
D
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Per Capita GDP (Contd
...
This
allows us to rewrite Equation (6) in per capita
variables as:
We let yt =
vyt+1Lt+1 = (1 − δ) vYt + sYt
...
O
...
D
...
)
Dividing both sides of Equation (10) by Lt gives
vyt+1
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Lt+1
= (1 − δ) vyt + syt
...
)
Further, dividing both sides of Equation (11) by
vyt gives
s
yt+1 Lt+1
= (1 − δ) +
...
O
...
D
...
)
If we let g ∗ be the growth rate in per capita
yt+1
GDP, y , it can be shown that
= 1 + g ∗
...
We can,
Lt
therefore, rewrite Equation (12) as
s
(1 + g ∗) (1 + n) = (1 − δ) +
...
O
...
D
...
)
Upon simplification, we have
g ∗ + g ∗n =
Japheth
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s
− δ − n
...
)
Further simplification shows that
1 s
∗
−δ−n
...
O
...
D
...
)
For small values of g ∗ and n, however, g ∗n u 0
...
O
...
D
...
v
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Applications of the Model
For a given capital–output ratio, v , which is
assumed fixed, we can use the model to
determine the savings ratio required to achieve a
particular targeted growth rate
...
O
...
D
...
)
If the actual savings ratio is less than the
required savings ratio, policies can be pursued to
raise the savings ratio (from both private sources
and public sources)
...
O
...
D
...
)
If it is not possible to raise the savings ratio,
then the savings gap can be bridged through
foreign borrowing or foreign aid
...
O
...
D
...
)
For example, if a particular country discovers
that to achieve its targeted growth rate of GDP
the savings rate should be s ∗ but the actual
savings rate is s < s ∗, then the amount of
resources to be obtained from foreign sources
(either in the form of foreign aid or foreign
borrowing) would (s ∗ − s) of GDP
...
O
...
D
...
The model assumes that the capital–output
ratio, v , is fixed
...
v
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Limitations (Contd
...
This could
change over time
...
O
...
D
...
)
b
...
Implicit
here is the assumption that foreign aid will
go to investment one to one
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Limitations (Contd
...
The model assumes zero substitutability
between capital and labour, in particular the
capital–output ratio and the capital–labour
ratio are assumed to be constant
...
O
...
D
...
)
The implication here is that for the economy to
continue being in equilibrium (that is, a situation
of full employment of both labour and capital),
the economy must grow at the same rate as the
growth of the capital stock and the growth of
the labour force
...
O
...
D
...
)
If we assume the rate of depreciation is zero, this
implies that the equilibrium growth path is given
s
by g = n =
...
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Limitations (Contd
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Example 1
Suppose that for a particular country, the
savings rate is 20%, the capital–output ratio is
4, the depreciation rate is 1%, and the rate of
growth of the population is 2% per year
...
O
...
D
...
)
a
...
c
...
What is the rate of per capita GDP growth?
What should the savings rate be to get the
growth rate of overall GDP to 8%?
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Example (Contd
...
What should the savings rate be to get the
growth rate of per capita GDP to 8%?
e
...
O
...
D
...
In this case
s = 0
...
01
...
O
...
D
...
)
Therefore,
g =
=
s
−δ
v
0
...
01
4
= 0
...
01
= 0
...
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Solution (Contd
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Solution (Contd
...
Here,
s = 0
...
01, n = 0
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Solution (Contd
...
2
=
− 0
...
02
1 + 0
...
02
1
...
01961
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Solution (Contd
...
2
− 0
...
02
0
...
02
...
O
...
D
...
)
Per capita GDP, therefore, grows at
approximately 2% per year
...
O
...
D
...
)
c
...
In this particular case,
g = 0
...
01
...
O
...
D
...
)
Therefore,
s = v (g + δ)
= 4 (0
...
01)
= 0
...
The savings rate should be 36%
...
O
...
D
...
)
d
...
In our case,
g ∗ = 0
...
01, n = 0
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Solution (Contd
...
08 (1 + 0
...
01 + 0
...
0816 + 0
...
02]
= 0
...
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Solution (Contd
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Solution (Contd
...
It can be shown that
v=
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s
...
)
Since
g ∗ = 0
...
2, δ = 0
...
02,
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Solution (Contd
...
2
0
...
02) + 0
...
02
=
0
...
1116
= 1
...
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Solution (Contd
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Practice Exercise 1
Suppose that the country of Xanadu saves 20%
of its income and has a capital–output ratio of 4
...
O
...
D
...
)
a
...
Using the Harrod–Domar model, calculate
the rate of growth of total GDP in Xanadu
...
O
...
D
...
)
c
...
Imagine, now, that the economy of
Xanadu suffers violent labour strikes every
year, so that whatever the capital stock is in
any given year, a quarter of it goes unused
because of these labour disputes
...
O
...
D
...
)
If population growth is 2% per year, calculate
the rate of per capita income growth in Xanadu
under this new scenario
...
O
...
D
...
2K , 0
...
2Y
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Example (Contd
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Example (Contd
...
b
...
O
...
D
...
In equilibrium,
Y = min (0
...
5L)
= min (40, 100)
= 40
...
O
...
D
...
)
b
...
e
...
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Example (Contd
...
2K = 0
...
O
...
D
...
5
...
)
K
...
5
200
For K = 200, L =
= 80
...
5
This means that L =
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Practice Exercise 2
An economy is described by the following
equations:
Y = min (0
...
5L)
K = 800
L = 800
S = 0
...
O
...
D
...
)
where Y is output, K is capital input, L is
labour input, and S are savings
...
O
...
D
...
)
a
...
What is the equilibrium output level?
How much capital and labour are employed
in equilibrium?
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End
End
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