Download ECONOMIC DEVELOPMENT PLANNING MODELS

European Scientific Journal
April 2013 edition vol.9, No.10
ISSN: 1857 – 7881 (Print) e - ISSN 1857- 7431
ECONOMIC DEVELOPMENT PLANNING MODELS: A
THEORETICAL AND ANALYTICAL EXPOSITION
Bashir Olayinka Kolawole
Department Of Economics, Lagos State University, Ojo, Lagos State, Nigeria
Abstract
This paper explores some planning models that have, in one period or the other, been
employed by both developed and less developed countries to forge development of their
respective economies. Using theoretical basis of analysis, the paper shows that while some
models are weak in their applicability, certain models like the Leontief Input-Output model
and the Linear programming model, however, are relevant in efficacy to the development of
economies via sectoral and inter-industry interdependence, aggregate demand, and growth in
output.
Keywords: Economic planning, Econometric, Development, Market mechanism, Growth
Introduction
In the literature of economics, any economy whose economic activity is not marketdriven is often described to be government-intervened. Such economy is usually referred to
as a centrally planned economy at least, in the traditional parlance of the economic system.
However, whether an economy is market-driven or state-controlled, there is the rationale for
planning in such country in order to improve and strengthen the market mechanism.
According to Ghatak (1995), since the product and the factor markets in less developed
countries (LDCs) are usually imperfect, market forces fail to attain efficient allocation of
resources. Hence, state intervention in the form of planning is necessary to obtain an efficient
allocation of resources, as prices are wrong signals to the decision makers. Although, markets
have created benefits over the long run, but only through trial and errors, yet they leave
behind many scars of failures with negative externalities. As such, despite its apparent
plausibility, Kooros and McManis (1998) are of the opinion that markets by themselves
cannot provide an accelerated and well-coordinated comprehensive economic plan, and
therefore each country must develop a blueprint for its own future economic well-being. Such
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blueprints, however, would take the form of economic models which are frequently used to
construct economic planning.
Ordinarily, economic models are useful in the setting out of the objective and targets
to be achieved, the constraints which have to be overcome, and the interrelationships among
the different economic variables which would indicate the general structure of the economy.
In the light of such exercise there is, therefore, a need for comprehensive economic planning.
By this, it means determining the country's core competencies, resources, and long-term
comparative advantage, and formulating the country's priorities, and the manner by which its
objectives can be met. Since the outputs of the market are determined by trail-and-error, and
over a long period of time, the development of such a comprehensive blue print is extremely
crucial. As large urban centers or even a comprehensive university cannot be designed expost facto, after the problems have emerged, nor can such problems be mitigated through ad
hoc trial-an-error, or the market system, in which some economists have developed
irrationally infinite confidence, because markets are not coordinated. More so, some markets
are also manipulated by oligopolies (see Kooros and Badeaux, 2007).
Thus, since economic models are frequently used to construct economic planning and
for the fact that such models should have the dual characteristics of clarity and consistency
aside the property of being selective so that only the behavior of the major variables is
analyzed, and quantitative (Streeten, 1966), this paper thus explores some empirically tested
aggregate and multi-sectoral planning models in the developed and developing countries. The
objective of the paper is to examine the theoretical and analytical bases surrounding each
model and also determine the efficacy of such models, as economic models provide
systematic and logical frameworks for economic planning in order to obtain feasible and
optimal solutions in the light of available information.
The rest of the paper is structured with the concepts of economic planning and
development in the second section, and theories and models of economic development plan in
section three. Section four gives empirical discussion, while the conclusion is in the fifth
section.
Concepts of Economic Planning and Development:
Economic Planning
Economic planning, otherwise known as economic development planning, has
become one of the main instruments of achieving a higher growth rate and better standard of
living in many less developed countries (LDCs). Planning in different forms has also been
accepted as an important policy instrument to attain specific targets in most LDCs. It is
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frequently advocated as an alternative to the market mechanism, and the use of market prices,
for the allocation of resources in developing countries. As a holistic approach to development
in developing economies, it promotes the idea and practice of matching development
planning with economic planning as the economy is regarded as the bedrock for a nation’s
development.
Essentially, economic visions and programs cannot be realized without viewing
developmental issues in a holistic way which entails improvement in all human endeavors. In
this sense, development surpasses the economic criteria often measured by economic growth
indices and must be conceived of as a multidimensional process involving changes in social
structures, destructive attitudes, ineffective national institutions and plan for an increase in
par capita output. Thus, development planning presupposes a formally predetermined rather
than a sporadic action towards achieving specific developmental results. In essence,
economic planning entails direction and control towards achieving set objectives. Following
this line of thought, Jhingan, (2005) sees development planning as a deliberate control and
direction of the economy by a central authority for the purpose of achieving definite targets
and objectives within a specified period of time. According to Ghatak (1995), planning can
be defined as a conscious effort on the part of any government to follow a definite pattern of
economic development in order to promote rapid and fundamental change in the economy
and society.
Economic Development
The veritable concept of development is based on the fact that economic, social,
political and physical environment, all combine to characterize the structure of the economy
and the entire social system, as well as the capabilities of the people and their aspirations for
better life. The UNDP Human Development Report 2002 asserted that “politics is as
important to successful development as economics”. But the concept of development goes
even beyond economics and politics. As Todaro and Smith (2003) put it: “Any realistic
analysis of development problems necessitates the supplementation of strictly economic
variables such as incomes, prices, and savings rates, with equally relevant non-economic
institutional factors, including the nature of land tenure arrangements; the influence of social
and class stratifications; the structure of credit, education, and health systems; the
organization and motivation of government bureaucracies; the machinery of public
administration; the nature of popular attitudes toward work, leisure, and self-improvement;
and the values, roles, and attitudes of political and economic elites.”
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World Bank, in its 1991 World Development Report opined that the challenge of
development . . . is to improve the quality of life. Especially in the world’s poor countries, a
better quality of life generally calls for higher incomes- but it involves much more. It
encompasses as ends in themselves better education, higher standard of health and nutrition,
less poverty, a cleaner environment, more equality of opportunity, greater individual
freedom, and a richer cultural life. Development, thus, must be conceived of as a
multidimensional process involving major changes in social structures, popular attitudes and
national institutions, as well as the acceleration of economic growth, the reduction of
inequality, and the eradication of poverty.
In more relevance, development theories of modern days revolve around questions
about what variables or inputs correlate or affect economic growth the most: elementary,
secondary, or higher education, government policy stability, tariffs and subsidies, fair court
systems, available infrastructure, availability of medical care, prenatal care and clean water,
ease of entry and exit into trade, and equality of income distribution (for example, as
indicated by the Gini coefficient), and how to advise governments about macroeconomic
policies, which include all policies that affect the economy. For instance, education enables
countries to adapt the latest technology and creates an environment for new innovations.
According to Todaro and Smith (2003), “Development, in its essence, must represent the
whole gamut of change by which an entire social system, tuned to the diverse basic needs and
desires of individuals and social groups, within that system, moves away from a condition of
life widely perceived as unsatisfactory toward a situation or condition of life regarded as
materially and spiritually better.” In other words, they imply that “. . . . development is the
sustained elevation of an entire society and social system toward a ‘better’ or ‘more humane’
life”.
Irrespective of the specific components of better life, Todaro and Smith (2003) yet
hold the view that development in all societies must have at least the following three
objectives:

To increase the availability and widen the distribution of basic life-sustaining goods
such as food, shelter, health, and protection.

To raise levels of living, in addition to higher incomes, the provision of more jobs,
better education, and greater attention to cultural and human values, which will serve
not only to enhance material well-being but also to generate greater individual and
national self-esteem.
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
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To expand the range of economic and social choices available to individuals and
nations by freeing them from servitude and dependence not only in relation to other
people and nation-states but also to the forces of ignorance and human misery.
Economic development, as distinguished from economic growth, results from an
assessment of the economic development objectives with the available resources, core
competencies, and the infusion of greater productivity, technology and innovation, as well as
improvement in human capital, resources, and access to large markets. Economic
development transforms a traditional dual-system society into a productive framework in
which everyone contributes and from which each one receives benefits accordingly.
Also, economic development occurs when all segments of the society benefit from the
fruits of economic growth through economic efficiency and equity. Economic efficiency will
be present with minimum negative externalities to the society, including agency, transaction,
secondary, and opportunity costs.
Theories and Models of Economic Development Plan:
In development planning, according to Thirlwall (1983), there are four basic types of
models that are typically used. There are macro or aggregate models of the economy which
may either be of the simple Harrod-Domar, or of a more econometric nature, consisting of a
series of n equations in an unknown variable which represents the basic structural relations in
an economy between, say, factor inputs and product output, saving and income, imports and
expenditure. There are also sectoral models which isolate the major sectors of an economy
and give the structural relations within each, and perhaps specify the interrelationships
between sectors. Thirdly, there are inter-industry models which show transactions and
interrelationships between producing sectors of an economy, normally in the form of an
input-output matrix. The fourth comprises of models and techniques for project appraisal and
the allocation of resources between industries.
In similar but with slight categorization, most development plans, according to
Todaro and Smith (2011), have traditionally been based initially on some more or less
formalized macroeconomic model which can be divided into two basic categories as: the one
in which aggregate growth models, involving macroeconomic estimates of planned or
required changes in principal economic variables, and the other of multisector input-output,
social accounting and computable general equilibrium (CGE) models which ascertain, among
other things, the production, resource, employment, and foreign-exchange implications of a
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given set of final demand targets within an internally consistent framework of inter-industry
product flows. These models are presented as follows.
The Harrod-Domar Models
The Harrod-Domar models of economic growth are based on the experiences of
advanced economies. These models are primarily addressed to a developed capitalist
economy and intend to analyze the requirements of steady growth in such economy. Based on
the assumptions of a closed economy, initial full employment equilibrium level of income,
absence of government intervention, among others, Harrod and Domar assign a key role to
invest in the process of economic growth. Though they arrive at similar conclusions, the
different details of each of the models are discussed as follows.
The Domar Model
Domar (1946) builds his model by forging a link between aggregate supply and
aggregate demand through investment. He did this as an answer to the question: “Since
investment generates income on the one hand and creates the productive capacity on the
other, at what rate should investment increase in order to make the increase in income equal
to the increase in productive capacity, so that full employment is maintained?” Beginning the
analysis, Domar connotes the supply side as ‘Increase in Production capacity’, using the
following identities:
I =
annual rate of investment
S =
annual productive capacity per dollar of newly created capital. It represents the
ratio of increase in real income or output to an increase in capital or is the reciprocal of the
accelerator or the marginal capital-output ratio.
Is = the productive capacity of I dollar invested per year.
σ = the net potential social average productivity of investment (=ΔY/I)
Iσ (as Iσ ˂ Is) = the total net potential increase in the output of the economy and is
known as the sigma effects. In Domar’s word this “is the increase in output which the
economy can produce. It is the “supply side of our economy.”
The Demand side is ‘Required Increase in Aggregate Demand’
ΔY = the annual increase in income
ΔI = the increase in investment
α(= ΔS/ΔY) = marginal propensity to save
Then the increase in income will be equal to the multiplier (1/α) times the increase in
investment. That is
ΔY = ΔI 1/α
since (1/α ≡ 1/1-mpc)
(1)
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thus,
ΔS = ΔI
(2)
At equilibrium, that is
AD = AS
(3)
ΔI = αIσ
(4)
and
implying that
ΔI 1/α = Iσ = ΔY
(5)
The above equation shows that in order to maintain full employment, the growth rate
of net autonomous investment (ΔI/I) must be equal to ασ (the MPS times the productivity of
capital). This is the stage at which investment must grow to assure the use of potential
capacity in order to maintain a steady growth rate of the economy at full employment.
The Harrod Model
Harrod (1939) holds the view that once the steady or equilibrium growth rate is
interrupted and the economy falls into disequilibrium, cumulative forces tend to perpetuate
this divergence thereby leading to either secular deflation or secular inflation. He, therefore,
tries to show in his model how steady growth rate may occur in the economy.
Harrod (1939) based his model upon three different rates of growth: the actual growth
rate (G); the warranted growth rate (Gw); and the natural growth rate (Gn). The first
fundamental equation of the model takes root in the actual growth rate as
GC = s
(6)
where G is the rate of growth of output in a given period of time and can be expressed
as ΔY/Y; C is the net addition to capital and is defined as the ratio of investment to the
increase in income. That is I/ΔY. S is the average propensity to save, S/Y.
Substituting the ratios into (6) gives
I=S
(7)
The equation for the warranted growth, Gw is given by Harrod to be
GwCr = s
(8)
where Gw is the “warranted rate of growth” or the full capacity rate of growth of
income which will fully utilize a growing stock of capital that will satisfy entrepreneurs with
the amount of investment actually made. It is, thus, the value of ΔY/Y. Cr is the “capital
requirements.” It denotes the amount of capital needed to maintain the warranted rate of
growth. That is, the required capital-output ratio. (It is the value of I/ΔY, or C). S is the
average propensity to save, S/Y.
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Essentially, equation (8) states that if the economy is to advance at the steady rate of
Gw that will fully utilize its capacity, income must grow at the rate of S/Cr per year. That is,
Gw = s/Cr. If income grows at the warranted rate, the capital stock of the economy will be
fully utilized and entrepreneurs will be willing to invest the amount of saving generated at
full potential income. According to Harrod, Gw is, therefore a self-sustaining rate of growth
and if the economy continues to grow at this rate it will follow the equilibrium path. The
economy will be in disequilibrium when Gw is not equal to G.
Incorporating the natural growth rate, Gn Harrod specifies that
Gn.Cr = or ≠ s
(9)
where Gn is the natural rate of advancement the increase of population and
technological improvements allow. This rate depends on the macro variables like population,
technology, natural resources and capital equipment. In other words, it is the rate of increase
in output at full employment as determined by a growing population and the rate of technical
progress.
As such in this situation, for full employment equilibrium growth, the below must
hold as
Gn = Gw = G
(10)
As a caveat, Harrod stresses the fact that the relation in equation (10) above is but a
knife-edge balance. He maintained that for once there is any divergence between natural,
warranted and actual rates of growth, conditions of secular stagnation or secular inflation will
be generated in the economy.
In comparison, however, the Harrod and Domar models are similar to an extent.
Given the capital-output ratio, as long as the average propensity to save is equal to the
marginal propensity to save, the quality of saving and investment fulfills the conditions of
equilibrium rate of growth. Also, putting the models side by side, Harrod’s s is Domar’s α.
Harrod’s warranted rate of growth, Gw is Domar’s full employment rate of growth, ασ. Thus,
Harrod’s Gw = s/Cr ≡ Domar’s ασ.
By implication, therefore, in an economy, s has to be moved up or down as the
situation demands. The model's assumption that labor and capital are used in fixed
proportions is untenable. Generally, labor can be substituted for capital and the economy can
move more smoothly towards a path of steady growth. Also, the restrictive assumption of a
constant saving-income ratio cannot hold considering Kaldor (1960) model that is based on
the classical saving function which implies that saving equals the ratio of profits to national
income. That is, S = P/Y.
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The Fel‘dman Model
Fel’dman (1928) presents his model on a theoretical basis which is concerned with
long term planning. The was built on the assumptions that there is no government
expenditure except on consumption and investment, production is independent of
consumption, there is no lags in the growth process, capital is the only limiting factor, among
others.
Given the assumptions, Fel’dman (1928) follows the Marxian division of the total
output of an economy (W) into category 1 and category 2. The former relates to capital goods
that are meant for both producer goods and consumer goods, while the latter category relates
to all consumer goods including raw materials for them. The production of each category is
expressed as the sum of constant capital (C), variable capital (wages), V, and surplus value S.
It can be represented as
W1 = C1 + V1 + S1
+ W2 = C2 + V2 + S2
W=C+V+S
(11)
The fraction of total investment allocated to category 1 is the key variable to the
model as the rate of investment is rigidly determined by the capital coefficient and the stock
of capital in the first category. Fel’dman (1928) employed the following notations to
demonstrate the two-sector model:
𝛾 = the fraction of total investment allocated to category 1;
I = the annual rate of net investment allocated to the respective categories, so that
I = I1 + I2;
t = the time, as measured in years;
V = the marginal capital coefficient for the whole economy, as V1 and V2
represent marginal capital coefficients of the respective category;
C = the annual rate of output of consumer goods;
Y = the annual net rate of output/income of the whole economy;
α = the average propensity to consume;
𝜶′ = the marginal propensity to save;
t0, C0, and Y0 = the respective initial magnitudes of t, C, and Y; and
I1 = 𝜸I = the annual rate of net investment allocated to category 1.
Thus, since only I1 increases the capacity of category 1, then it follows that
𝑑𝐼
𝐼
𝛾𝐼
= 𝑉1 = 𝑉
𝑑𝑡
1
1
[since I1 = 𝛾I]
(12)
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In time t, total investment will grow at an exponential rate
I = 𝑒 𝛾𝑡/𝑉1
(13)
In other words, total investment will grow at a constant exponential rate of 𝛾/V1.
Similarly, the annual rate of net investment allocated to category 2 is given by I2 =
(1−𝛾)I. And I2 being the source of increased capacity in category 2,
𝑑𝐶
𝑑𝑡
𝐼
= 𝑉2 =
(1−𝛾)
𝑒 𝛾/𝑉1 𝑡
𝑉2
2
[since I = 𝑒 𝛾/𝑉1 𝑡 ]
(14)
The annual rate of output of consumer goods is given by
1−𝛾
C = C0 + (
𝛾
𝑉
) 𝑉1 (𝑒 𝛾/𝑉1 𝑡 − 1)
(15)
2
The elements which determine the national income and the growth rate of the
economy are given by
Y=I+C
(16)
By substituting the values of I and C in the above equation, it gives
1−𝛾
Y = 𝑒 𝛾/𝑉1 𝑡 + C0 + (
𝛾
𝑉
) 𝑉1 (𝑒 𝛾/𝑉1 𝑡 − 1)
1−𝛾
Y = 𝑒 𝛾/𝑉1 𝑡 – 1 + 1 + C0 + (
𝛾
1−𝛾
𝛾
)
𝑉1
𝑉2
𝑉
) 𝑉1 (𝑒 𝛾/𝑉1 𝑡 − 1)
(18)
2
1−𝛾
Y = (𝑒 𝛾/𝑉1 𝑡 – 1) + 1 + C0 + (
Y = [1 + C0 + (
(17)
2
𝛾
𝑉
) 𝑉1 (𝑒 𝛾/𝑉1 𝑡 − 1)
(19)
2
+ 1] (𝑒 𝛾/𝑉1 𝑡 − 1)
(20)
Assuming that I0 = 1, the equation becomes
Y = I0 + C0 + [(
1−𝛾
Y = Y0 + [(
𝛾
1−𝛾
𝛾
𝑉
𝑉
) 𝑉1 + 1] (𝑒 𝛾/𝑉1 𝑡 − 1)
) 𝑉1 + 1] (𝑒 𝛾/𝑉1 𝑡 − 1)
2
(21)
2
[Since Y0 = I0 + C0]
(22)
The fundamental equation shows that C and Y each represent a sum of a constant and
an exponential in t. Their rates of growth will differ from 𝛾/V1. The values of C and Y will be
greater than the value of I. With the passage of time, the exponential 𝒆𝜸/𝑽𝟏 𝒕 will dominate the
scene and the rates of growth of C and Y will gradually approach 𝛾/V1. But this may take
quite a long time, unless of course it so happens that
C0 =
(1−𝛾) 𝑉1
𝛾
𝑉2
(23)
in which case the constants will vanish, and C and Y will grow at the rate of 𝛾/V1 from
the very beginning.
By implication, if the purpose of economic development is the maximization of
investment or national income at a point of time, or of their respective rates of growth, or of
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integrals overtime, 𝛾 should be set as high as possible. This is always true for investment and
nearly always for income, the only exception being when V1 greatly exceeds V2 and even then
for a short period of time. A high 𝛾 does not imply, however, any reduction in consumption.
With capital assets assumed to be permanent, even 𝛾 = 1 would merely freeze consumption
as its original level. If assets were subject to wear and tear, consumption would be slowly
reduced by failure to replace them.
The Mahalanobis Model
Mahalanobis, (1953 & 1955) developed a single-sector, two-sector, and a four-sector
model that fit into development planning of the Indian economy. Initially making national
income and investment the variables in his single model, Mahalanobis (1953) further
developed a two-sector model where the entire net output of the economy was to be produced
in the investment goods sector and the consumer goods sector. The model assumes an
economy that is related to a closed economy; non-shiftable capital equipment once installed
in any of the sector; a full capacity production in both the consumer and capital goods
sectors; determination of investment by the supply of capital goods; and no changes in prices.
On the basis of the above assumptions, the economy is divided into λk, that is, the
proportion of net investment used in the capital goods sector; and λc, the proportion of net
investment used in the consumer goods sector. Thus,
λc + λk = 1
(24)
Further, at any point of time (t), net investment (I) is divided into λkIk, the part that
increases the productive capacity of the capital goods sector, and λcIc the part that increases
the productive capacity of the consumer goods sector. In the form that
It = λcIt + λkIt
(25)
If taking β as the total productivity coefficient when βk and βc are the capital-output
ratio of the capital goods sector and consumer goods sector, then it can be shown that
𝛽=
β k Ik + β c Ic
𝜆𝑘 + 𝜆𝑐
(26)
The income identity equation for the entire economy is
Yt = It + Ct
(27)
As national income changes, investment and consumption also change. The change in
investment depends upon previous year’s investment (𝐼𝑡−1 ) and so does consumption depends
on previous year’s consumption (𝐶𝑡−1 ). Hence, the increase in investment in period t, is
ΔIt = It – It-1
(28)
and increase in consumption is
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ΔCt = Ct – Ct-1
(29)
Essentially, the increase in the two sectors is related to the linking up of productive
capacity of investment and the output-capital ratio. Initially, the investment growth path is
determined by the productive capacity of investment in the capital goods sector (λk Ik) and its
output-capital ratio (βk), such that
It – It-1 = λkβkIt-1
(30)
It = It-1 + λkβkIt-1
(31)
It = (1 + λkβk) It-1
(32)
Inserting different value for t (t= 1, 2, 3, . . .,) the solutions to equation (32) become
I1 = (1 + λkβk) I0
(33)
I2 = (1 + λkβk) I1
(34)
I2 = (1 + λkβk) (1 + λkβk) I0
(35)
I2 = (1 + λkβk)2 I0
(36)
Similarly, by putting the value of t in equation (36), it gives
It = I0 (1 + λkβk)t
(37)
It – I0 = I0 (1 + λkβk)t – I0
(38)
It – I0 = I0 (1 + λkβk)t – 1
(39)
Also, by inserting the value of t (t= 1, 2, 3, . . .,) in the consumption growth path, as
Ct – C0 = λcβcI0
(40)
C2 – C1 = λcβcI1
(41)
Ct – C0 = λcβc (I0 + I1 + I2 + . . . + It)
(42)
By substituting the values of I1, I2, . . ., It in equation (39) and its related equations, it
can be solved as below
Ct – C0 = λcβc [I0 + (1 + λkβk)I0 + (1 + λkβk)2I0 + . . . + (1 + λkβk)t I0]
(43)
Ct – C0 = λcβcI0 [1+ (1 + λkβk) + (1 + λkβk)2 + . . . + (1 + λkβk)t]
(44)
(1 + 𝜆 𝛽 ) 𝑡 −1
Ct – C0 = λcβcI0 [ (1 + 𝜆𝑘 𝛽𝑘 ) – 1 ]
𝑘 𝑘
Ct – C0 = λcβcI0 [
(1 + 𝜆𝑘 𝛽𝑘 ) t −1
𝜆𝑘 𝛽𝑘
]
(45)
(46)
As such, the growth path of income for the whole economy, given equation (46), is
or
ΔYt = ΔIt + ΔCt
(47)
Yt −Y0 = (It− I0) + (Ct− C0)
(48)
By substituting the values of equations (39) and (46) in equation (48), it gives
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Yt −Y0 = [I0 (1 + λkβk)t – 1] + λcβcI0 [
Yt −Y0 = I0[(1 + λkβk)t – 1] [1 +
Yt −Y0 = I0 [(1 + λkβk)t – 1] [
𝜆𝑐 𝛽𝑐
𝜆𝑘 𝛽𝑘
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(1 + 𝜆𝑘 𝛽𝑘 ) 𝑡 −1
𝜆𝑘 𝛽𝑘
]
]
𝜆𝑘 𝛽𝑘 + 𝜆𝑐 𝛽𝑐
𝜆𝑘 𝛽𝑘
(49)
(50)
]
(51)
Supposing that I0 = α0Y0 and substituting it in equation (51) above, it gives
Yt −Y0 = α0Y0 [(1 + λkβk)t – 1] [
Yt = α0Y0 [(1 + λkβk)t – 1] [
Yt = Y0 [1 + 𝛼0
𝜆𝑘 𝛽𝑘 + 𝜆𝑐 𝛽𝑐
𝜆𝑘 𝛽𝑘 + 𝜆𝑐 𝛽𝑐
𝜆𝑘 𝛽𝑘
𝜆𝑘 𝛽𝑘 + 𝜆𝑐 𝛽𝑐
𝜆𝑘 𝛽𝑘
]
(52)
] + Y0
(53)
𝜆𝑘 𝛽𝑘
] [(1 + λkβk)t – 1)
(54)
where α0 is the rate of investment in the base year, Y0 and Yt are the gross national income in
the base year and year t, respectively.
Intuitively, the ratio
𝝀𝒌 𝜷𝒌 + 𝝀𝒄 𝜷𝒄
𝝀𝒌 𝜷𝒌
of the above equation is the overall capital coefficient.
If, on assumption that βk and βc are given, the growth rate of income will depend upon α0 and
λk. Assuming further that α0 to be constant, the growth rate of income depends upon the
policy instrument, λk.
In the economy, if βc > βk, it implies that the larger the percentage investment in
consumer goods industries, the larger will be the income generated. However, the expression
(1 + λkβk)t in equation (54), shows that after a critical range of time, the larger the investment
in capital goods industries, the larger will be the income generated. Thus, initially a high
value of λk increases the magnitude (1 + λkβk)t., and lower the overall capital coefficient
𝝀𝒌 𝜷𝒌 + 𝝀𝒄 𝜷𝒄
𝝀𝒌 𝜷𝒌
. But as time passes, a higher value of λk would lead to higher growth rate of
income in the long run.
On the other hand, if βc = βk, then the reciprocal of the overall capital coefficient, that
is,
𝝀𝒌 𝜷𝒌
𝝀𝒌 𝜷𝒌 + 𝝀𝒄 𝜷𝒄
= λk equals marginal rate of saving. By extension, the important policy
implication of the model is that for a higher rate of investment (λk), the marginal rate of
saving must also be higher. Thus, a higher rate of investment on capital goods in the short run
would make available a smaller volume of output for consumption. But in the long run, it
would lead to a higher growth rate of consumption. See Jones (1975).
The Leontief Input-Output Model
The input-output model or technique is used to analyze inter-industry relationship in
order to understand the interdependencies and complexities of the economy and thus the
conditions for maintaining equilibrium between supply and demand. According to Ghatak
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(1995), the technique usually delineates the general equilibrium analysis and the empirical
side of the economic system of production of any country. It is also known as “inter-industry
analysis.”
As a finest variant of general equilibrium analysis, Jhingan (2004) enumerates three
main features of input-output analysis to be: concentration on an economy which is in
equilibrium as it is not applicable to partial equilibrium analysis; it does not concern itself
with the demand analysis as it deals exclusively with technical problems of production; and it
is based on empirical investigation. The assumptions upon which the technique operates,
according to Ghatak (1995), are that no substitution takes place between the inputs to produce
a given unit of output and the input coefficient are constant – the linear input functions imply
that the marginal input coefficients are equal to the average; joint products are ruled out, that
is, each industry produces only on commodity and each commodity is produced by only one
industry; and external economies are ruled out and production is subject to the operation of
constant returns to scale.
The requirement of these assumptions is that if the total output of say Xi of the ith
industry be divided into various number of industries, 1, 2, 3, n, then it gives, according to
Leontief (1951 & 1986), the balance equation as
Xi = xi1 + xi2 + xi3 + . . . + xin + Di
(55)
and if the amount say Yi absorbed by the outside sector is also taken into
consideration, then the balance equation of the ith industry becomes
or
Xi = xi1 + xi2 + xi3 + . . . + xin + Di + Yi
(56)
∑𝑛𝑗=1 𝑥𝑖𝑗 + 𝑌i = Xi
(56^)
where Yi is the sum of the flows of the products of the ith industry, to consumption,
investment and exports, net of imports.
Equation (56^) shows the conditions of equilibrium between demand and supply. It
illustrates the flows of outputs and inputs to and from one industry to other industries and
vice versa. In the analysis of input-output, the system of equations (55) and (56) presents the
conditions of internal consistency of the plan. The plan would not be feasible without them
because if these equations are not satisfied, there might be excess of some goods and
deficiency of others.
As xi2 represents the amount absorbed by industry 2 of the ith industry it then follows
that xij stands for the amount absorbed by the jth industry of ith industry. Thus, the technical
coefficient or input coefficient of the ith industry is denoted by
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aij =
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𝑥𝑖𝑗
(57)
𝑋𝑗
where xij is the flow from industry i to industry j, Xj is the total output of industry j
and aij is a constant which is called technical coefficient or flow coefficient in the ith industry
and it shows the number of units of one industry’s output that are required to produce one
unit of another industry’s output.
Cross-multiplying the terms in equation (57) gives
𝑥ij = aij . Xj
(58)
By substituting the value of 𝑥ij into equation (56^) and transposing the terms gives
the basic input-output system of equations in the form
Xi− ∑𝑛𝑗=1 𝑎𝑖𝑗 𝑥𝑗 = Yi
(59)
where n represents the number of sectors in the economy. If on assumption that n = 2,
that is, two-sector economy, there would be two linear equations that could be stated
symbolically in the form
x1 – a11x1 – a12x2 = Y1
(60)
x2 – a21x1 – a22x2 = Y2
(61)
which can be represented in matrix notation as
or
X − [𝐴]X = Y
(62)
X[𝐼 − 𝐴] = Y
(63)
where matrix (I−A) is known as the Leontief Matrix and is further extended as
(I−A)-1 (I−A) X = (I−A)-1 Y
(64)
such that
X = (I−A)-1 Y
(65)
and I, is the identity matrix of the form,
1
I=[
0
0
]
1
(66)
Hence,
[
𝑋1
1
] = {[
𝑋2
0
−1 𝑌
0
] − [𝐴]} [ 1 ]
𝑌2
1
(67)
For analytical illustration purpose, from Ghatak (1995), that there are only two
sectors, agriculture (X1) and textiles (X2) in the economy. The Input-Output coefficient table
depicts economic activity as
Agriculture
Textiles
Agriculture
0.6
0.4
Textiles
0.2
0.3
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then
A=(
0.6 0.2
)
0.4 0.3
(68)
and
1 0
0.6
)−(
0 1
0.4
0.7⁄0.2
[𝐼 − 𝐴 ]−1 = (
0.4⁄0.2
I−A = (
0.2
0.4 −0.2
)=(
)
0.3
−0.4 0.7
0.2⁄0.2
3.5 1
)=(
)
0.4⁄0.2
2 2
(69)
(70)
If the final demand is given by
10
)
5
(71)
𝑥 = [𝐼 − 𝐴]−1 D
(72)
D=(
such that,
thus given that
40
3.5 1 10
) ( )=( )
(73)
5
30
2 2
Implying that the agricultural sector (X1) would produce 40 units and the textiles
𝑥=(
sector (X2) would produce 30 units. The analysis can be extended to include many other
sectors, like health, education, communication, transportation, manufacturing, banking,
foreign trade and balance of payments, and so on, in the economy.
The above presentation, however, is an open static model of Input-Output analysis.
But in reality, most economic variables are dynamic in nature as cause and effect, and action
and reaction do not occur immediately after one and other: it takes some time for certain
economic activities to happen as a result of some causes or actions. Thus, according to
Sandee (1988), the analysis becomes dynamic when it is closed by the linking of the
investment part of the final bill of goods to output. The dynamic input-output model extends
the concept of inter-sectoral balancing at a given point of time to that of inter-sectoral
balancing over time.
In a Leontief dynamic input-output model, the output of a given period is supposed to
go into stocks (or capital goods), which in turn are distributed among industries. The dynamic
balance equation is of the form
Xi(t) = xi1(t) + xi2(t) + xi3(t) + . . . + xin(t) + (Si1 + Si2 + Si3 + . . . + Sin)
+ Di(t) +
Yi(t) (74)
The Linear Programming (Optimizing) Model
The main task of development strategy is to ensure that resources will be forthcoming
to meet the goals of a development program, and that the resources are allocated efficiently
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subject to certain constraints. The programming model can provide a simultaneous solution
for the three basic purposes of development planning, which are the optimum allocation of
resources, efficiency in the use of resources through the proper valuation of the resources,
and the avoidance of social waste, and thirdly, the balance between different branches of the
national economy. Linear programming can be considered as providing an operational
method for dealing with economic relationships, which involve discontinuities. It is a specific
approach within the general framework of economic theory (see Koutsoyiannis, 1989).
Essentially, neither economic theory nor linear programming says anything about the
implementation of the optimal plan or solution. They simply derive the optimal solution in
any particular situation. As such, both approaches are ex ante methods aiming at helping the
economic units to find the solution that attains their goal of whether utility maximization,
profit maximization, or cost minimization given their income or factor inputs at any particular
time. Linear programming, however, basically solves economic problems through graphical
or simplex approach. Where the variables whose values must be determined is more than two,
the graphical solution is difficult or impossible because of the need for multidimensional
diagrams. A linear programming can be stated formally in the form below as
Maximise
Y = 𝛼1 𝑋1 + 𝛼2 𝑋2 + 𝛼3 𝑋3 + 𝛼4 𝑋4 + 𝛼5 𝑋5
Subject to
𝑙1 𝑋1 + 𝑙2 𝑋2 + 𝑙3 𝑋3 + 𝑙4 𝑋4 + 𝑙5 𝑋5 ≤ L
𝑘1 𝑋1 + 𝑘2 𝑋2 + 𝑘3 𝑋3 + 𝑘4 𝑋4 + 𝑘5 𝑋5 ≤ K
𝑠1 𝑋1 + 𝑠2 𝑋2 + 𝑠3 𝑋3 + 𝑠4 𝑋4 + 𝑠5 𝑋5 ≤ S
𝑋1 ≥ 0, 𝑋2 ≥ 0, 𝑋3 ≥ 0, 𝑋4 ≥ 0, 𝑋5 ≥ 0
where Y represents aggregate output or firm’s profit, 𝛼𝑖 is contribution from each
sector or production unit to Y, 𝑋𝑖 represents a specific sector or production unit, 𝑙𝑖 , 𝑘𝑖 , and 𝑠𝑖
represents the amount or part of resource employed out of the total quantity of a particular
resource, L, K, and S available in the economy or firm, respectively. 𝑋𝑖 ≥ 0 implies that each
sectoral or production unit output is either positive or zero, and it represents the nonnegativity constraint of the programming model.
The equation which contains Y is the objective function, or total output or profit
function as it expresses the objective of a particular country or firm as the case may be. The
equations which contain L, K, and S are the technical or functional constraints. The technical
constraints are set by the state of technology and the availability of factors of production.
They express the fact that the quantities of factors which will be absorbed in the production
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of total output or a given commodity cannot exceed the available quantities of these resources
(factors of production). See Baumol (1977), Hadley (1962), and Panne (1976).
Macroeconometric Model
The planning exercise and plan formulation in developing countries in recent time has
found basis in macroeconometric models. The demonstration of the application of such
models takes the form of a simple Keynesian framework of analysis such that 𝐶𝑡 represents
consumer expenditure, 𝐼𝑡 is capital formation, 𝑌𝑡 is national income, 𝑟𝑡 is interest rate, 𝑀𝑡 is
the exogenously supplied money, t is time, and 𝑢𝑡 , 𝑣𝑡 , 𝑧𝑡 are error terms as in the equation
below.
𝐶𝑡 = 𝑎0 + 𝑎1 𝑌𝑡 + 𝑎2 𝑟𝑡 + 𝑢𝑡
(75)
𝐼𝑡 = 𝑏0 + 𝑏1 𝑌𝑡 + 𝑏2 𝑟𝑡 + 𝑣𝑡
(76)
𝑌𝑡 = 𝐶𝑡 + 𝐼𝑡
(77)
𝑀𝑡 = 𝑐0 + 𝑐1 𝑌𝑡 + 𝑐2 𝑟𝑡 + 𝑧𝑡
(78)
Such a model, however, is not dynamic as it does not determine prices and it ignores
foreign trade. Also, a change in government taxes and spending (public policy) is not
assigned any role. In essence, Klein (1965) set out a more sophisticated version of the model
which is presented as follow:
𝑌𝑡 −𝑇𝑡
𝐶𝑡 = 𝑎0 + 𝑎1
𝑃𝑡
+ 𝑎2 𝐶𝑡−1 + 𝑢1𝑡
(79)
𝑌
𝐼𝑡 = 𝑏0 + 𝑏1 𝑃𝑡−1 + 𝑏2 𝐾𝑡−1 + 𝑏3 𝑟𝑡−1 + 𝑢2𝑡
𝑡−1
𝐹𝑡 = 𝑐0 + 𝑐1
𝑌𝑡 −𝑌𝑡−1
𝑃𝑡
+ 𝑐2 𝐹𝑡−1 + 𝑐3
𝐸𝑡 = 𝑑0 + 𝑑1 𝑇𝑤𝑡 + 𝑑2
𝑌𝑡
𝑃𝑒𝑡
𝑃𝑡
𝑃𝑓𝑡
𝑃𝑡
+ 𝑢3𝑡
+ 𝑢4𝑡
(81)
(82)
= 𝐶𝑡 + 𝐼𝑡 − 𝐹𝑡 + 𝐸𝑡 + 𝐺𝑡
𝑃𝑡
(80)
(83)
𝑇𝑡 = 𝑒0 + 𝑒1 𝑌𝑡 + 𝑢5𝑡
(84)
𝐼𝑡 = 𝐾𝑡 − 𝐾𝑡−1
(85)
𝑌𝑡
= 𝑔0 + 𝑔1 𝐿𝑡 + 𝑔2 𝐾𝑡 + 𝑢6𝑡
𝑃𝑡
𝑤𝑡 𝐿𝑡
𝑃𝑡 = ℎ0 + ℎ1
𝑤𝑡 − 𝑤𝑡−1
𝑤𝑡−1
𝑌𝑡 𝑃𝑡
= 𝑗0 + 𝑗1
+ ℎ2
𝑃𝑓𝑡
𝑃𝑡
𝑁𝑡 − 𝐿𝑡
𝑁𝑡
(86)
+ 𝑢7𝑡
+ 𝑗2
𝑃𝑡 − 𝑃𝑡−1
𝑃𝑡−1
(87)
+ 𝑢8𝑡
𝑁𝑡 = 𝑘0 + 𝑘1 (𝑁𝑡 − 𝐿𝑡 ) + 𝑘2 𝑤𝑡 ⁄𝑃𝑡 + 𝑢9𝑡
𝑀𝑡
𝑃𝑡
𝑌
= 𝑙0 + 𝑙1 𝑃𝑡 + 𝑙2 𝑟𝑡 + 𝑢10𝑡
𝑡
𝑃𝑒 = 𝑚0 + 𝑚1 P + 𝑢11𝑡
(88)
(89)
(90)
(91)
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In the model as described above, the endogenous variables are C, the real consumer
expenditures, Y, national income in current prices, T represent taxes less transfer payments, p
is the index of general price level, I is net real investment, K, real capital stock, r is interest
rate, F is real imports, E is real export, 𝑃𝑒 , L is employment, w is wage rate, and N is labour
supply. The exogenous variables are p, import prices, 𝑇𝑤 is volume of world trade, G is real
government expenditures, and M is money supply.
Practically, however, the planner will have to determine different types of the
mentioned variables to render such a model applicable to the special problems of LDCs. The
actual econometric techniques to be used will now depend upon initial specifications of the
equations and the subjective judgement of the planner in the light of the actual state of
information. See Agarwala (1970), Chenery et al (1971), Ghosh (1968), and Ghosh et al
(1974).
Empirical Discussion
The Harrod-Domar model has been applied as a basis to develop more comprehensive
plans for some less developed countries. In its First Five Year Plan (1950-1 to 1955-6), India
employed the Harrod-Domar model to formulate her national plan. The strategy of the plan
was to rehabilitate the Indian economy which had been hit hard by the Second World War
and the Partition. Thus, the emphasis was to create the necessary economic and social
overheads like power, transport, public health, education and to develop agriculture in order
to build a solid foundation for industrialisation in the subsequent plans. The Harrod-Domar
model, however, failed as the implementation informed the private sector control of the
development of local industries and minerals resulting in about six percent public expenditure
on them.
The Mahalanobis model which was adopted in India’s Second Five Year Plan (1955-6
to 1960-1) as four sectors strategy has shown that the value that was chosen for 𝜆𝑘 yielded
inefficient resource allocation as it lay within the feasibility locus between an increase in
employment and a rise in output. In other words, reallocation of investment among the three
sectors apart from the capital goods sector would have resulted in higher output and
employment. Also, the model’s supposition that the supply of agricultural produce is
infinitely elastic is untenable as supply of agricultural produce has failed to meet the
increased demand for food and raw materials ever since the beginning of the planning period.
The problem of capital accumulation that Fel’dman and Mahalanobis models
encountered has neither been solved satisfactorily even by the Brahmananda and Vakil
(1956) model as the abundant labor alone is not enough to achieve a higher level of capital
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accumulation. Thus, it is true that the problem of employment creation in labor-surplus
countries can hardly be exaggerated, and such a problem has not really been solved in the
Mahalanobis model. Komiya (1959) also shows that the value which Mahalanobis chose for
𝜆𝐾 yielded inefficient resource allocation as it lay within the feasibility locus between an
increase in employment and a rise in output.
The input-output model of Leontief was adopted in the Netherlands in 1948 through
1960, as well as in some other developed and less developed countries. The theory basically
centers on the ratios between inputs and output, otherwise called input coefficients. The
model’s analysis of the Netherlands case involves 35 sectors constituting the industrial
sectors, 7 primary sectors, and 6 final sectors. Using the central input-output prediction
experiment, the prediction of the intermediate demand given final demand for the predicted
year and applying the input coefficients for the base year as they are expressed in current
value: The analysis reveals that only 27 out of the 35 industries are considered covering, on
the average, 95 per cent of aggregate intermediate demand. The prediction, according to
Tilanus (1989) is seen to defeat final demand blowup. However, the superiority of inputoutput vanishes if the national accounts data used for the blowup procedure are two or more
years more recent than the input-output table.
Conclusion
The use of an aggregate approach like the Harrod-Domar model provides simplicity
and clarity in its application. Also, the model is complete in the sense that it covers the entire
economy as it is selective and fairly realistic. Moreover, the model does not suffer from any
internal inconsistencies. However, such models is highly aggregated and does not provide
any idea about the internal relationships and interdependencies between different sectors in
the economy. More importantly, the adoption of capital-output ratio is constrained by the
difficulty in estimating capital in LDCs with the difference in regional capital-output ratios
among regions or states in one hand, and difference in the capital-output ratios between the
regions and the central government on the other hand. Thus, it fails to provide any idea about
the consistency between different sectors. Even yet, the disaggregation of the Harrod-Domar
model into the Two-sector model is not sufficient for development planning for LDCs as
experienced by Kenya.
In the linear programming model, prices are regarded as the indicators of the marginal
worth to the society. However, in the LDCs, where the market is mostly imperfect, price will
usually be higher than the marginal cost. Also, the relationships in the optimizing model are
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assumed to be linear while many constraints in the LDCs are nonlinear functions of the
structural variables.
Availability of reliable data is always the bain of the econometric model and such
model also suffers from the difficulties involved in misplaced aggregation and illegitimate
isolation. Thus, given the need for analyzing many important sectors of the economy and
their interrelationships, with sectoral interdependence, to provide greater consistency between
aggregate supply and aggregate demand, development planners have increasingly turned their
attention to the application of the input-output model.
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