In growth and development policy investment ratio is an important policy instrument. However, there is no well defined framework to determine what should be the investment ratio for a given growth target. This paper explains the potential of Solow (1956) and Solow (1957) to explain the relationship between the target growth rate and investment ratio. Hypothetical data are used for illustration.Investment ratio, Growth targets, Growth accounting, Total Factor Productivity, Neo classical growth model.

B Bhaskara Rao

English

Research Papers in Economics

Growth & Investment 1
Investment Ratio and Growth
B. Bhaskara Rao
rao b@usp.ac.fj
University of the South Paciﬁc, Suva (Fiji)
Abstract
In growth and development policy investment ratio is an important
policy instrument. However, there is no well deﬁned framework to
determine what should be the investment ratio for a given growth
target. This paper explains the potential of Solow (1956) and Solow
(1957) to explain the relationship between the target growth rate and
investment ratio. Hypothetical data are used for illustration.
JEL: E15.000, E15.001, E15.003, E15.004;
KEYWORDS: Investment ratio, Growth targets, Growth accounting,
Total Factor Productivity, Neo classical growth model.2 B. B. Rao
1. INTRODUCTION
A legacy of the Harrod-Domar growth models for growth and develop-
ment policy is that to increase the growth rate, the ratio of investment
to output (investment ratio) should be increased. In contrast, the neo
classical growth model of Solow (1956) and his growth accounting
framework (GAF), based on Solow (1957), imply that the key to per-
manently raise the growth rate is to raise total factor productivity
(TFP). An increase in the investment ratio only increases the growth
rate during the transition of the economy from one steady to another
steady state. However, it is well known that the East Asian countries,
especially Singapore, have achieved high growth rates for several years
in spite of modest increases in TFP and mainly through high invest-
ment ratios. Therefore, it would be interesting to understand the re-
lationship growth and investment ratio. We ﬁrst use the simpler GAF
and then the more informative growth model framework (GMF) of
Solow (1956). We show that the simple GAF gives good approximate
empirical results for the medium term of 5 to 10 periods/years.
2. THE TWO FRAMEWORKS
GAF is essentially used as an identity to estimate TFP and TFP
based on GAF is known as the Solow residual. Therefore, GAF can
be used to estimate the required rate of growth of capital, as a resid-
ual, provided reliable estimates are available for other parameters and
variables in this identity.
We assume a Cobb-Douglas production function with constant re-
turns and Hicks neutral technical progress:
Yt = At Kα L(1−α) (1)
where Y is output, A is stock of knowledge, K is capital and L is
labour. It implies that
˙ Yt = g + α ˙ Kt +( 1− α) ˙ Lt (2)
where g the rate of growth of TFP and dot on the other variables are
proportionate changes. Equation (2) can be solved for ˙ Kt to get:
˙ Kt =
˙ Yt − (1 − α)˙ Lt − g
α
=
θ − (1 − α)n − g
α
(3)
where θ is the rate of growth of output (our target), n is rate of
growth of employment and g is growth of TFP. Thus, the requiredGrowth & Investment 3
rate of growth of (net) capital, for a given target growth rate (θ) can
be computed as a residual if the values of all other parameters and
variables are known.
Using the deﬁnition of gross investment, the investment ratio can
be expressed as:
It
Yt
=
Inet
t
Yt
+
δKt
Yt
=
Inet
t
Yt
+ δΠt (4)
where δ is depreciation rate and Π is the capital-output ratio. Substi-
tuting for Inet, from the discrete version of (3) into (4), gives:
It
Yt
= s =
[θ − (1 − α)n − g]
α
Kt−1
Yt
+ δΠt
=
￿
(θ − (1 − α)n − g)
α (1 + θ)
+ δ(1 + γ)
￿
Πt−1 (5)
where s is investment ratio and γ is the rate of change in the capital
output ratio and equals zero if this ratio is constant. That capital
output ratio is fairly constant is a stylized fact.
Given the target rate of growth and the values for other variables,
equation (5) can determine the required investment rate. A main
limitation of GAF is that, since it is an identity, it ignores the dynamic
adjustments in the growth model. Therefore, it implies that the target
rate of growth can be maintained indeﬁnitely. However, if in reality
such dynamic adjustments are small and time consuming, GAF is a
good predictor of the growth rate in the medium term of 5 to 10
periods. Given its simplicity, it is an attractive alternative approach
to determine the investment rate for a given growth target.
The main driving force in the economy, moving it towards its steady
state, is the fall in the marginal productivity of capital as capital stock
increases. Such eﬀects are taken into account in the neo classical
growth model of Solow (1956). However, it is not obvious, from its
familiar textbook versions, how long the relationship between growth
rate and investment ratio will be signiﬁcantly above the new steady
state growth implied by TFP or TFP and factor accumulation when
the behavior of the level of output is of interest.
This relationship can be derived, for example, by asking what
should be the investment ratio to double output, say over 14 peri-
ods, implying that the average growth rate during these 14 periods is
5%. For this purpose, we use the following closed form solution for4 B. B. Rao
output for the neo classical growth model:1
Yt = A0egt L0ent
￿
s
n + g + σ
￿
1 − e(1−λt)
￿
+
￿Y0
A0
￿[(1−α)/α]
e−λt
￿ α
1−α
(6)
where s is investment ratio and λ =( 1− α)(n + g + δ), t =0···t is
time. To simplify our observation that GAF is satisfactory for medium
term policy to determine the investment ratio s to achieve a given
target rate of growth (on the average) we shall use a simple simulation
exercise with plausible values for the parameters and variables.
3. SIMULATION RESULTS
We start with closed form solution in equation (6), say for a develop-
ing country; and assume that growth of employment n =0 .01, TFP
g =0 .0125, share of proﬁts α =0 .425 and depreciation rate σ =0 .095.
Needless to say these values can be changed with more reliable esti-
mates based on actual data. If the investment ratio s =0 .15, this
economy takes about 24.25 periods to double its income with an im-
plied growth rate θ =0 .0286. On the other hand if the investment
rate s =0 .25, it takes about 14 periods to double income with an im-
plied average growth rate θ =0 .05 for the 14 periods. In other words
a 0.01 increase in the investment ratio, on the average, increases the
growth rate by 0.275 and the implied elasticity of growth with respect
to investment ratio is 3.1. Such increases in the growth rate, however,
are not perpetual and converge, in the limit, to the rate of 0.0225,
implied by the assumed TFP and factor accumulation.
It is important to note from the simulation the average growth rate
during the ﬁrst 14 periods. With s =0 .25 the average growth rate is
0.050 and with s =0 .15 0.026. The growth proﬁles of the economy
with s =0 .15 and s =0 .25 are given below in Figure 1. It is assumed
that initially the economy is in the steady state, with a growth rate
of 0.0225.
A noteworthy feature is that when s increases by a substantial
amount, high growth rates continue for several periods and the econ-
omy takes several decades to converge to its steady state. When
s =0 .25 growth after 20 periods is still above 3%. In about 50
periods it reaches 0.95% of its steady state value. Consequently, it
may be said that the need to increase the investment ratio to raise
the growth rate, during the intermediate periods of up to 10 to 15
1 This solution is given in Jones (2002). Jones acknowledges that originally this
was due to Sato (1963).Growth & Investment 5
periods, is also an important policy option in addition to the current
emphasis on policies to increase TFP. This is of importance because
considerable time and political will are necessary to raise TFP and
therefore increasing the investment ratio is a pragmatic option for the
short and intermediate policies.
FIGURE 1
Time in horizontal axis & Growth on vertical axis
Suppose, we use GAF to capture the eﬀects of a rise in the invest-
ment ratio. How good are the predictions of the growth rates, say
over 5 to 10 years? To examine this, it is reasonable to emphasize on
the predictions of the average growth rates from these GAF and GMF
because GAF predicts a constant growth rate for all the periods.
To generate predictions of growth rates with GAF, we assume con-
stant capital output ratio i.e., γ = 0. Therefore, we select a value
for the capital output ratio (Π) that satisﬁes the assumption that the
growth rates of output and net capital are the same. This yielded
Π=1 .7529 and we shall use the approximate value of Π = 0.75. Fur-
thermore, the rate of growth of TFP used earlier (g =0 .0125) needs
adjustment because it is equal to (1 − α) ∗ g‘ where g‘ is the measure
based on Hicks neutral TFP. Therefore, g‘=[ g/(1 − α)] = 0.0217 in
GAF. With these adjustments GAF predicts that 5% growth rate can
be achieved with s =0 .255 and this is close to s =0 .25 from GMF.
The only weakness of GAF is that it implies that this growth rate of
5% is constant, since it does not incorporate the dynamic adjustments
that drive the economy towards a new steady state. Therefore, if6 B. B. Rao
such adjustments are fast, GAF is unlikely to accurately predict the
dynamics of the growth rate. The average growth rate in GMF for
the ﬁrst 5, 10 and 14 periods, respectively, are 0.064, 0.055 and 0.050.
Therefore, while GAF somewhat under predicted the average growth
rate for the ﬁrst 5 periods, its prediction for the ﬁrst 10 and 14 periods
is very good. Furthermore, the possibility of the economy converging
to the new steady state during even 10 periods is remote. Therefore, it
may be said that GAF is a simple and useful framework in the medium
term for determining investment ratios for given growth targets. The
diﬀerences between the predictions from these two frameworks will be
further reduced if in fact the implied rate of decline in the marginal
productivity of capital is not rapid enough to take the economy to its
new steady state in 10 or 15 periods. In fact such slow adjustments
might have enabled the East Asian economies to achieve high growth
rates for substantial periods with high rates of factor accumulation
and low rates of TFP.
4. CONCLUSIONS
In this paper we developed two frameworks, using Solow (1956) and
Solow (1957), to examine the relationship between the growth rate
and investment ratio. This helps to determine investment ratios re-
quired to achieve given growth rate targets. Investment ratio seems to
be still important in policies for growth and development. We found
that both frameworks imply that substantial improvement in the ac-
tual growth rates can be achieved for several periods by increasing
the investment ratio. Although the framework based on Solow (1956)
is generally preferred, GAF, based on Solow (1957), is also satisfac-
tory for 5 to 14 year period predictions. The widely accepted view
that only improvements to TFP can permanently raise growth rates
is valid for the long run of 50 periods or more. Although such long
run implications should not be ignored, it is also important to note
that improvements to the investment ratio are also important for the
medium term policies.Growth & Investment 7
References
Jones, C.I., (2002) Introduction to Economic Growth, (W.W. Norton
& Co, New York) 2nd Edition.
Sato, R., (1963) “Fiscal Policy in a Neo-Classical Growth Model: An
Analysis of Time Required for Equilibrium Adjustment,” Review of
Economic Studies, pp.16-23.
Solow, R.M., (1956) “A Contribution to the Theory of Economic
Growth”, Quarterly Journal of Economics, pp.65-94.
———– (1957) “ Technical Change and Aggregate Production Func-
tion”, Review of Economics and Statistics, pp.312-320.

A Contribution to the Theory of Economic Growth”,

Fiscal Policy in a Neo-Classical Growth Model: An Analysis of Time Required for Equilibrium Adjustment,” Review of Economic Studies,

Introduction to Economic Growth,

Technical Change and Aggregate Production Function”,

Investment Ratio and Growth

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