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Subject ECONOMICS

Paper No and Title 3: Fundamentals of Microeconomic Theory

Module No and Title 14: Other Production Related Concepts, Euler’s Theorem,
Multi-Product Firm, Technical Progress
Module Tag ECO_P3_M14

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
____________________________________________________________________________________________________

TABLE OF CONTENTS
1. Learning Outcomes
2. Euler’s Theorem
3. Technical Progress
4. Multi-Product Firm
5. Summary

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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1. Learning Outcomes
After studying this module, you shall be able to

 Know the concept of product exhaustion theorem and how it was solved by
Wicksteed using Euler’s Theorem
 Why technical progress is essential for growth?
 Technical progress classification and its importance
 Distinction between Hicks and Harrod concept of technical progress
 Multi-product firm and its operation

2. The Euler’s Theorem and Product Exhaustion Theorem

The product exhaustion theorem states that if all the factors of production are paid
equal to their marginal products, they will exhaust the total product. As soon as it
was brought forward that all the factors of production are paid equal to their
marginal products, a difficult problem cropped up over which raised a serious debate
among the economists. The difficult problem that has been put forward was that if all
factors were paid equal to their marginal products, would the total product be just exactly
exhausted? The problem of proving that if all factors are paid rewards equal to their
marginal products, they will exhaust the total product has been called “Adding- up
Problem” or Product Exhaustion Problem. Philip Wicksteed was the first economist who
not only posed this problem but also provided a solution for it.

2.1 Solution of Product Exhaustion Theorem: The three solutions proposed for the
problem of product exhaustion theorem were
a) Philip Wicksteed Solution: Euler’s Theorem
b) Wicksell, Walras and Barone’s Solution
c) J.R. Hicks and R.A. Samuleson: Perfect Competition Model

Let us study these solutions in detail.

a) Philip Wicksteed Solution: Euler’s Theorem: Philip Wicksteed was the first
economist who proposed this problem and provided a solution using Euler’s
Theorem. Euler’s Theorem is a mathematical proposition which states that if a
production function is homogeneous of degree one (i.e. Constant Returns to
Scale) and the factors are paid equal to their marginal products, the total product
is exhausted with no surplus and deficit.

Euler’s Theorem: Euler’s Theorem was developed by Swiss mathematician

Leonhard Euler. According to him, it is a mathematical relationship that applies to
any homogeneous function.
A function f(x) is said to be homogenous of degree t (scalar), if and only if, for
ECONOMICS Paper 3: Fundamentals of Microeconomic Theory
Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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𝑓(λx) = λt 𝑓(𝑥), for all λ > 0 and 𝑥 =

(𝑥1 , 𝑥2 , … , 𝑥𝑛 ).
Here, t is the parameter of returns to scale.
 If the function f(x) is homogeneous of degree α = 1, the function
exhibits constant returns to scale.
 If α > 1, the function exhibits increasing returns to scale.
 If α < 1, the function exhibits decreasing returns to scale.

Let f(x) be a production function with two factors of production of capital and labour.
Then the homogeneous production function of degree t can be mathematically expressed
as:
λt . 𝑌 = 𝑓[λK, λL]
Where Y is output
K is capital
L is labour
tis parameter of returns to scale.
 If t = 1, the function exhibits constant returns to scale.
 If t > 1, the function exhibits increasing returns to scale.
 If t < 1, the function exhibits decreasing returns to scale.

If production function is homogeneous of degree t, then

(i) The Marginal Rate of technical Substitution is constant along rays extending,
from the origin and
(ii) The derived cost function from the corresponding production function is
homogeneous of degree 1/α.

Euler’s Theorem
The theorem says that for a homogeneous function f(x) of degree t, then for all x

𝜕𝑓(𝑥) 𝜕𝑓(𝑥) 𝜕𝑓(𝑥) 𝜕𝑓(𝑥)

𝑥1 . + 𝑥2 . + 𝑥3 . + ⋯ + 𝑥𝑛 . = 𝑡𝑓(𝑥)
𝜕𝑥1 𝜕𝑥2 𝜕𝑥3 𝜕𝑥𝑛

Proof: For a homogeneous function f(x) of degree t,

𝑑𝑓(𝜆𝑥) 𝑑𝑓(𝜆𝑥) 𝑑𝑓(𝜆𝑥) 𝑑𝑓(𝜆𝑥) 𝑑𝑓(𝜆𝑥)
= 𝑥1 . + 𝑥2 . + 𝑥3 . + ⋯ + 𝑥𝑛 .
𝑑𝜆 𝜕𝜆𝑥1 𝜕𝜆𝑥2 𝜕𝜆𝑥3 𝜕𝜆𝑥𝑛

𝑑𝜆𝑡 𝑓(𝑥)
= 𝑡. 𝜆𝑡−1 𝑓(𝑥)
𝑑𝑡
If setting λ=1, the theorem follows.

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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Euler’s Theorem and Production Function: It is based

on some postulations. These are:
i. It assumes a linear standardized production of first degree which implies
invariable returns to scale.
ii. It assumes that the factors are complementary, i.e. if a variable factor increases; it
increases the marginal productivity of the fixed factor.
iii. It assumes that factors of production are perfectly divisible.
iv. The relative shares of the factors are invariable and independent of the level of the
product.
v. There is a stationary, reckless economy where there are no profits.
vi. There is perfect competition.
vii. It is applicable only in the long run.

Let us assume that the production function is homogeneous of degree 1. The

homogeneous production function of the first degree can be written as:

λ𝑌 = 𝑓[λK, λL]
And the Euler’s Theorem can be written as
𝜕𝑌 𝜕𝑌
𝑌 = 𝐾. + 𝐿.
𝜕𝐾 𝜕𝐿
Where Y : Output
K : Capital
L :Labour
𝜕𝑌
: Marginal Product of Capital
𝜕𝐾
𝜕𝑌
: Marginal Product of Labor
𝜕𝐿
Marginal product of capital is the addition to the total output attributable to addition of
one more unit of capital. It is calculated by partially differentiating output with respect to
capital keeping labor constant. Similarly, Marginal product of labor is the addition to the
total output attributable to addition of one more unit of labor. It is calculated by partially
differentiating output with respect to labor keeping capital constant.
Euler theorem states that the marginal product of capital multiplied by the amount of
capital plus the marginal product of labor multiplied by the amount of labor equals to the
total product of the firm.
Example: Let us take the Cobb Douglas Production Function.
𝑌 = 𝑓(𝐾, 𝐿, 𝐴) = 𝐴𝐾 𝛼 𝐿1−𝛼
∂Y 𝐾 𝛼−1
𝑀𝑃𝐾 = = 𝐴𝛼𝐾 𝛼−1 𝐿1−𝛼 = 𝐴𝛼 ( )
∂K 𝐿
𝛼
∂Y 𝐾
𝑀𝑃𝐿 = = 𝐴(1 − 𝛼)𝐾 𝛼 𝐿−𝛼 = 𝐴(1 − 𝛼) ( )
∂L 𝐿
Putting the values in Euler’s theorem, Y=K. MPk + L. MPL , we get
𝑌 = 𝐴𝛼𝐾 𝛼−1 𝐿1−𝛼 𝐾 + 𝐿𝐴(1 − 𝛼)𝐾 𝛼 𝐿−𝛼
𝑌 = 𝐴𝛼𝐾 𝛼 𝐿1−𝛼 + 𝐴(1 − 𝛼)𝐾 𝛼 𝐿1−𝛼
𝑌 = 𝐴𝐾 𝛼 𝐿1−𝛼

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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 If production function is homogeneous of degree

1, then marginal products are homogeneous of degree zero.

Y = K. MPK + L. MPL or K. FK + L. FL
Differentiate with respect to K,
K. FKK + L. FLK + FK = FK
K. FKK + L. FLK = 0. FK = 0
Similarly the same is true for labour.
We have seen that Wicksteed is able to explain the product exhaustion theorem with the
help of Euler’s theorem when production function exhibits constant returns to scale.
Wicksteed proved that if all the factors are paid equal to their marginal products, the total
product will be exhausted.

Wicksteed’s Solution and Criticism

I. First drawback of Wicksteed’s Solution: Wicksteed was able to explain the product
exhaustion theorem with the help of Euler’s theorem. But this solution was criticized by
Walras, Edge worth, Barone and Pareto. According to them, returns to scale are not
constant in real world i.e. production function is not homogeneous of degree one. The
Edgeworth commented on the Wicksteed’s solution that, “there is a magnificence in this
generalization which recalls the youth of philosophy. Justice is a perfect cube, said the
ancient sage, and rational conduct is homogeneous function, adds the modern savant”.

Economist pointed out that production function is such that it yields longrun average cost
curve which is of ‘U’ shaped. LAC curve is also known as “envelope curve” as it
envelopes short-run average cost curves. The long-run average cost curve is U-shaped i.e.
itinitially falls, reaches a minimum and rises thereafter. Initially, the long-run average
cost of production falls as output increases because of increasing returns to scale and then
rises beyonda certain level of output because of decreasing returns to scale. So, if a firm
is working with increasing returns to scale and factors are paid equal to their marginal
products, the total factor reward would exceed the total product. And similarly if a firm is
working with decreasing returns to scale and factors are paid equal to their marginal
products, the total factor reward would not fully exhaust the total product. As the total
factor reward is less than the total product, it would result in surplus. So, the Euler’s
theorem does not apply when firms are working with either increasing returns to scale or
decreasing returns to scale.

II. Second drawback of Wicksteed’s Solution: If production function exhibits constant

returns to scale, then the shape of the long run average cost curve would be horizontal
straight line parallel to x-axis. The horizontal straight line shape of the long run average
cost is not compatible with the perfect competitive market structure as firm would not be
able to determine the equilibrium position.
ECONOMICS Paper 3: Fundamentals of Microeconomic Theory
Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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a. Wicksell, Walras and Barone’s Solution to

Product Exhaustion problem: After Wicksteed, the more satisfactory solution to
the problem of product exhaustion theorem was provided independently by
Wicksell, Barone and Walras. They assumed that
 The production function was not homogeneous of degree one.
 The production function was such that it yields long run average cost curve to be
of ‘U’ shape.

They pointed out the applicability of the product exhaustion theorem in long run in
perfect competition market. In perfect competition market, the industry is in equilibrium
in the long-run when all the firms are in equilibrium and producing a price which is equal
to minimum of long-run average cost. In long-run all the firms are earning zero economic
profit and no firm has an incentive to enter or leave an industry. Thus, the condition
required for the product exhaustion theorem i.e. production function exhibits constant
returns to scale was fulfilled at the minimum point on the long run average cost curve
where returns to scale are constant within the range of small variations of output.
So, under perfectly competitive long run equilibrium if factors are paid rewards equal to
their marginal product, the total product would be exactly exhausted.

b. Hicks and Samuelson Solution to Product Exhaustion Theorem: We have seen that
Wicksell, Barone, Walras pointed out the applicability of the product exhaustion theorem
in case of longrun perfectly competitive equilibrium. And Wicksteed provided a solution
to the product exhaustion theorem with the help of Euler’s theorem and assumed linearly
homogeneous production function. But as all production functions are not linear
homogeneous, the controversy remained unresolved. It does not make any difference
whether we are under perfectly competitive market structure and dealing with usual ‘U’
shaped long run average cost curve, the controversy remained unresolved.
Hicks and Samuelson resolved this controversy and showed that the solution of the
product exhaustion theorem depends not on the property of production function but on
the market conditions of the perfect competition.

In perfect competition market structure, firms are earning zero economic profits. Thus the
solution to product exhaustion problem in case of perfectly competitive factor markets
where factors are paid equal to their marginal products, the existence of perfect
competition in the product market will ensure zero economic profits in the longrun.

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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Figure:1 Long run Equilibrium: In perfect competition market structure, firms are earning
zero economic profits

The zero economic profit condition under perfect competition can also be explained
mathematically. The zero economic profit condition implies that value of total output is
equal to the total cost of production. Let capital (K) and labor (L) be the two factors of
production used by perfectly competitive firm to produce output (Q). Let P be the price of
the product. The value of output is AR multiplied by the output. And total cost is the sum
of the amount spent on each to produce a given level of output.
So, Zero Economic Profit » Value of Output= Total Cost
𝑃. 𝑄 = 𝐿. 𝑤 + 𝐾. 𝑟 … … . (1)
According to marginal productivity theory, each factor is paid equal to the value of their
marginal products. Thus,
𝑤 = 𝑉𝑀𝑃𝐿 = 𝑃𝑥𝑀𝑃𝑃𝐿
𝑟 = 𝑉𝑀𝑃𝐾 = 𝑃𝑥𝑀𝑃𝑃𝐾
where VMP is the value of marginal product. Now substitute these values of r and w in
equation (1) , we get
𝑃. 𝑄 = 𝐿. 𝑃. 𝑀𝑃𝑃𝐿 + 𝐾. 𝑃. 𝑀𝑃𝑃𝐾

It shows that for a given price, if the factors are paid equal to their marginal physical
product, the total payments to factors would be equal to the total product Q and thus total
product would be exactly exhausted.

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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3. Technical Progress
Technical progress is defined as improvement in technology. In other words, technical
progress means
i. More output can be produced from the same amount of factor inputs or same
output can be produced by smaller amount of one or more of the factor inputs. Or
ii. Any qualitative improvement in the existing product Or
iii. Production of entirely new products.

Technical progress is the most important factor in determining the rate of growth of the
economy.

Technical Progress and Production Function: The technical progress can be

represented with the help of production function. Here we are making use of per worker
production function. The assumption of constant returns to scale permits us to write an
aggregate production function into a per worker form.

𝑌 = 𝐹(𝐾, 𝐿).....(1)

If we multiply capital and labor by any constant λ, then output is also multiplied by the
same number i.e. 𝜆𝑌 = 𝐹(𝜆𝐾, 𝜆𝐿).
𝑌 𝐾
Let us put λ=1/L, then 𝐿 = 𝐹( 𝐿 , 1) ....... (2)
where Y/L is output per worker and K/L is capital labor ratio.

Let y=Y/L and k=K/L, then equation (2) can be written as

𝐾
𝑦 = 𝑓(𝑘) = 𝐹( 𝐿 , 1)........(3)
This is the per worker form of the production function where output per worker depends
upon capital per worker.

If there is no usage of inputs, output would be zero i.e. 𝑦 = 𝑓(𝑘) if 𝑘 = 0 𝑡ℎ𝑒𝑛𝑦 = 0. It

ensures that the per worker production function starts from the origin.

Diagrammatic Representation of Per Worker Production Function:In fig.2,The K/L

ratio is measured on x-axis and Y/L ratio on y-axis. The curve starts from the origin as
output is zero when no factor is employed. The Output per worker is increasing but at a
diminishing rate as the ratio K/L ratio rises.

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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Figure: 2 Per worker Production Function: The output per worker depends upon capital
per worker

Technical Progress Shifts the Production Function Upward: The technical progress
shifts the production function upward. It is shown in the figure 3. The production
function without any technical progress is shown by curve 𝑓(𝑘, 𝑡0 ). After the technical
progress the curve shifts upward to 𝑓(𝑘, 𝑡1 ). At any level of K/L ratio on the new
production function except zero, more output per worker is produced.

Figure 3: Shift in Production Function

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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Factor Augmenting Representation of Technical Change: Technical progress shifts

the production function upward such that more output is produced with the same quantity
of factors i.e. without any increase in the usage of capital and labor. If the factors of
production had been augmented then the production function can be written as
𝑌 = 𝐹[𝐴(𝑡)𝐾, 𝐶(𝑡)𝐿]

where Y is output
K is Capital
L is Labor
t is Time
A and C are Factors
A(t) K is Effective Capital
C(t) L is Effective Labor
Here the capital and labor force are multiplied by factors A and C which are functions of
time.

 If Ȧ(t)>0 i.e. the rate of change is positive then effective capital stock increases as
time goes on even though the actual capital stock remains constant.
 If Ċ(t)>0 i.e. the rate of change is positive then effective labor stock increases as
time goes on even though the actual labor stock remains constant.

The technical change can be characterized as

(i) Purely Capital Augmenting- The technical change is said to be purely
capital augmented if Ȧ(t) is positive and C(t) is equal to 1 i.e. Ȧ(t)>0 and
C(t)=1.
(ii) Purely Labor Augmenting- The technical change is said to be purely labor
augmented if Ċ(t) is positive and A(t) is equal to 1 i.e. Ċ(t)>0 and A(t)=1.
(iii) Equally Capital and Labor Augmenting-- The technical change is said to be
equally capital and labor augmented if Ȧ (t) and Ċ (t) both are positive.

Classification of Technical Progress: The classification of technical progress as labor

saving, capital saving or neutral was originally the work of Harrod (1948) & Hicks
(1932) and there exists differences in the criteria of classification. Harrod employs the
concept of capital output ration to explain technical progress classification whereas Hicks
used the concept of Marginal rate of Substitution between factors leaving output
unchanged. Let us study them in detail.
(1) Hicks Classification of Technical Progress
a. Classification in terms of Marginal Product Ratio
𝑀𝑃𝐾 (𝑡) 𝑀𝑃𝐾 (0)
 The technical progress is said to be labor saving if > Or Any
𝑀𝑃𝐿 (𝑡) 𝑀𝑃𝐿 (0)
technical progress resulting in upward shift in the production function is said to be

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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labor saving, if any constant value of the capital-

labor ratio, the ratio of 𝑀𝑃𝐾 to 𝑀𝑃𝐿 has increased.
 Here 𝑀𝑃𝐾 (𝑡)𝑎𝑛𝑑𝑀𝑃𝐿 (𝑡) are the marginal products before the technical progress
and 𝑀𝑃𝐾 (0)𝑎𝑛𝑑𝑀𝑃𝐿 (0) after the technical progress.
𝑀𝑃 (𝑡) 𝑀𝑃 (0)
 The technical progress is said to be capital saving if 𝑀𝑃𝐾(𝑡) < 𝑀𝑃𝐾(0) Or Any
𝐿 𝐿
technical progress resulting in upward shift in the production function is said to be
labor saving, if any constant value of the capital- labor ratio, the ratio of 𝑀𝑃𝐾 to
𝑀𝑃𝐿 has decreased.
𝑀𝑃𝐾 (𝑡) 𝑀𝑃𝐾 (0)
 The technical progress is said to be Hicks Neutral if = Or Any
𝑀𝑃𝐿 (𝑡) 𝑀𝑃𝐿 (0)
technical progress resulting in upward shift in the production function is said to be
labor saving, if any constant value of the capital- labor ratio, the ratio of 𝑀𝑃𝐾 to
𝑀𝑃𝐿 has remains constant.

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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Classification in terms of Ratio of Relative Shares

 The technical progress is said to be labor saving if at any constant value of

𝑟𝐾
capital- labor ratio, the ratio of relative shares i.e. 𝑤𝐿 is increasing.
 The technical progress is said to be capital saving if at any constant value of
𝑟𝐾
capital- labor ratio, the ratio of relative shares i.e. 𝑤𝐿 is decreasing.
 The technical progress is said to be Hicks Neutral if at any constant value of
𝑟𝐾
capital- labor ratio, the ratio of relative shares i.e. 𝑤𝐿 remains constant.
(2) Hicks Classification of Technical Progress
 Any technical progress resulting in upward shift in the production function is said
to be labor saving, if any constant value of the capital- output ratio, the 𝑀𝑃𝐾 is
increasing. In other words, technical progress is said to be labor saving if at any
𝑟𝐾
constant value of capital- output ratio, the ratio of relative shares i.e. is
𝑤𝐿
increasing.
 Any technical progress resulting in upward shift in the production function is said
to be capital saving, if any constant value of the capital- output ratio, the 𝑀𝑃𝐾 is
decreasing. In other words, technical progress is said to be capital saving if at any
𝑟𝐾
constant value of capital- output ratio, the ratio of relative shares i.e. is
𝑤𝐿
decreasing.
 Any technical progress resulting in upward shift in the production function is said
to be Harrod neutral, if any constant value of the capital- output ratio, the 𝑀𝑃𝐾 is
remains unchanged. In other words, technical progress is said to be Harrod
neutral if at any constant value of capital- output ratio, the ratio of relative shares
𝑟𝐾
i.e. 𝑤𝐿 remains constant.

5. Multi-Product Firm
Multi-product firms are firms that are producing more than one good. As multiproduct
firms are dealing with multiple products they have to deal with allocating inputs more
properly in order to obtain higher level of output. This is a greater problem than the one
single-product firms face, the maximization of profit problem, since multiproduct firms
must allocate their factors not only to produce one good, but multiple goods.

The production analysis can be extended to include multiproduct firms. There is a

possibility that two products can be jointly produced in varying proportions by a single
production process. In case of joint products, production of two or more products is
technically dependent on each other. This we can explain with the help of production
possibility curve.

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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Production Possibility Curve: A Production Possibility Curve (PPC) sometimes called

Production Possibility Frontier or Transformation Curve is a curve which shows different
possibilities of two goods that can be produced with the available resources and given
technology. We know that resources are very much limited and have alternative uses.
Suppose a firm decides to produce two goods that are rice and cloth, More of cloth the
firm produces the less of rice it would be able to produce. So by utilizing more resources
in the production of cloth means the lesser resources would be available for the
production of rice and vice versa.

Assumptions:
i. The amount of productive resources is given and remain fixed (i.e. resources can
be shifted from the production of one good to another).
ii. Resources are neither unemployed nor under-employed but utilized efficiently.
iii. Economy is working at full employment level and trying to achieve maximum
possible level of production.
iv. There is no change in technology.

Figure: 4 Production Possibility Curve

Production Possibility Curve is a curve which shows different possibilities of two goods
that can be produced with the available resource and given technology

In the figure 4, rice is measured on X-axis and cloth on Y-axis. On one extreme we are
utilizing all our resources in the production of cloth and on other extreme we are utilizing
all our resources in the production of rice alone. Between these two extreme points, there
aremany possibilities of rice and cloth which a firm can produce. By joining all
production possibilities point, we derive Production Possibility Curve. It is also known as
Production Possibility Boundary.

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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With the given resources and available technology, all the points within and on boundary
of production possibility curve are attainable combinations. All the points on production
possibility boundary represent efficient utilization of resources. It implies that production
of one good can be increased only by decreasing the resources from the production of
other good. All the points within the production possibility curve represent inefficient
utilization of resources. Thus, resources remain under-utilized inside the production
boundary.

Slope of Production Possibility Curve: The Production Possibility Curve is negatively

sloped and “bowed outward”. In short, as larger and larger quantities of resources are
transferred from the production of one output to another, the addition to the production of
second product declines. The slope of production possibility curve is known as marginal
opportunity cost. The marginal opportunity cost refers to the loss of good Y that we need
to sacrifice in order to produce 1 more unit of good X.
∆Y
Marginal Opportunity Cost = Slope of PPC =
∆X

As we are moving from left to right on production possibility curve, its slope increases.
The increasing slope implies that when we are withdrawing resources from the
production of Y to produce more and more of good X, the loss of output of good Y for
each additional unit of good X tends to increase.

Iso Revenue Lines: An iso-revenue is defined as the locus of product combinations that
will earn the same revenue. In other words, all the combinations of rice and cloth lying on
this line give the same revenue when sold in the market. At any given fixed prices, the
iso-revenue line would be a straight line. Higher the iso-revenue line higher is the
revenue earned by selling larger combinations of two goods. For a given fixed price, the
iso- revenue lines are parallel to one another. The slope of the iso-revenue line is equal to
the ratio of price of the product.
𝑃
Slope of the Iso-revenue Line=𝑃𝑥 where 𝑃𝑥 is the price of good X and 𝑃𝑦 is the price of
𝑦
good Y.

Optimum Combination: The aim of the producer is to maximise profit. The revenue
would be maximum when given production possibility curve is tangent to iso-revenue
line. At this point, the marginal rate of transformation i.e. the slope of the production
possibility curve is equal to the ratio of prices of commodities X and commodity Y.

𝑃𝑥
= 𝑀𝑅𝑇𝑥𝑦
𝑃𝑦
And the second condition required for optimum combination of two products is that
production possibility curve must be concave from below.

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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6. Summary

We have seen under product exhaustion theorem that when all the factors are paid equal
to their marginal products, the total product is exactly exhausted. The solution to this
problem was provided by Wicksteed with the help of Euler’s Theorem. Technical
progress is the most important determinant of economic growth. It can be classified into
labor saving, capital saving or neutral. In the last we have studied how the production
analysis can be extended when firms are producing more than one product.

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 14:Other Production Related Concepts, Euler’s Theorem, Multi-
Product Firm, Technical Progress
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Subject ECONOMICS

Paper No and Title 3: Fundamentals of Microeconomic Theory

Module No and Title 15: Special Production Functions- Cobb-Douglas, CES,

VES, Translog and their properties
Module Tag ECO_P3_M15

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 15: Special Production Functions- Cobb-Douglas, CES, VES,
Translog and their properties
____________________________________________________________________________________________________

TABLE OF CONTENTS
1. Learning Outcomes
2. Introduction
3. Cobb Douglas Production Function
4. CES Production Function
5. VES Production Function
6. Translog Production Function
7. Summary

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 15: Special Production Functions- Cobb-Douglas, CES, VES,
Translog and their properties
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1. Learning Outcomes

After studying this module, you shall be able to

 Know the concept of various production functions and their properties

 Analyze the reason for the use of Cobb Douglas production function in economics
 Understand CES production function and its properties
 Understand VES production function and its form when assumption of constant
elasticity of substitution is relaxed
 Evaluate the Trans log production function and its various properties

2. Introduction
Production Function

Nature imposes technological constraints on firms; only certain combination of inputs are
feasible ways to produce a given amount of output, and the firm must limit itself to
technologically feasible production plans. The easiest way to describe feasible production
plans is to list them. The set of all combinations of inputs and outputs that comprise a
technologically feasible way is called a production function. The relation between inputs
and output of a firm is known as production function. Production function is a purely
technical relation which connects factor inputs and output. It shows the maximum
amount of output that can be produced from any specified set of inputs given the existing
technology. It is a flow concept so production refers to units of output over a period of
time.
 It refers to the relation between inputs and outputs of a firm.
 It is a flow concept so production refers to units of output over a period of time.

Mathematically, the production function can be expressed in a functional form as:-

Y  f ( X 1 , X 2 , , X n )
where
Y  Output
X 1 , X 2 ,, X n  Quantity of factor inputsSuch as land , labour, capital or raw material 
f shows the functional relationship between inputs and output.

Concept of Product: There are three important concepts regarding physical production
of factors:-

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 15: Special Production Functions- Cobb-Douglas, CES, VES,
Translog and their properties
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a. Total Product of Labour (TPL)

b. Average Product of Labour (APL)

c. Marginal Product of labour (MPL)

a. Total Product of Labour (TPL): Total product of a variable factor is the

maximum amount of an output that can be produced by a given amount of
variable factors, keeping quantity of other factors such as capital, land fixed.
When the amount of the variable factor increases, the total product increases.

TPL  APL  L
or
TPL   MPL
Where TPL  Total product of labour
APL  Average product of labour
L  Labour or Variable factor
MP  MP of Labour
  SumTotal
Total Product (TPL) initially rises at an increasing rate (so the slope of the TP L curve is
rising in the beginning) but after a point TP L curve starts rising at a diminishing rate,
reaches a maximum and then starts falling as the usage of variable factor increases.

b. Average Product of Labour (APL): Average Product of a variable factor is the

total product divided by the amount of labour (variable factor) employed with a
given quantity of capital (fixed factor) used to produce a commodity. It measures
the average output per unit of the factor.

TPL
Thus, APL 
L
Where APL  Average Pr oduct of Labour
TPL  Total Pr oduct of Labour
L  Number of labour employed

Average Product (AP) is an inverted ‘U’ shape curve. The AP curve first rises, reaches a
maximum and then falls thereafter as the usage of the variable factor increases. Average
Product at any point on the total product curve is the slope of the straight line from the
origin to that point on the total product curve.

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 15: Special Production Functions- Cobb-Douglas, CES, VES,
Translog and their properties
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c. Marginal Product of Labour (MPL): Marginal Product of a variable factor is the

addition to the total product by the usage of one more unit of a variable factor. It
measures the rate at which output changes as a variable factor changes. Thus,

TPL Change in Total product

MPL  
L Changein Labour
or
MPL for nth unit  TPn  TPn1
Where n is the number of units of labor

It measures the slope of the total product curve. Marginal Product (MP) is an inverted ‘U’
shape curve. The MP curve first rises, reaches a maximum and then falls thereafter as the
usage of the variable factor increases. When total product starts falling, Marginal product
becomes negative. Marginal Product is the slope of the tangent line to the total product
curve.

Some of the special production functions are:

i. Cobb Douglas Production Function

ii. CES Production Function
iii. VES Production Function
iv. Translog Production Function

Let us now discuss these production functions in detail.

3. Cobb Douglas Production Function

In Economics, the Cobb Douglas production function is widely used to represent the
relationship between inputs and output in an economy. The two most important
neoclassical production functions are the Constant Elasticity of Substitution (CES) and
the Cobb Douglas. The Cobb Douglas production function was created by Charles Cobb
(Mathematician) and Paul Douglas (Economist) in 1927. But its functional form was
proposed by Knut Wicksell (Economist) in the 19th Century. The Cobb Douglas
production function lies between linear and fixed proportion production function with
elasticity of substitution equal to one. It is very popular among economist because of its
flexibility and ease of use.

Mathematical Form: The mathematical form of the Cobb Douglas production function
for a single output with two factors can be written as
𝑌 = 𝑓(𝐾, 𝐿, 𝐴) = 𝐴𝐾 𝛼 𝐿1−𝛼
where Y: Output
K: Capital input
L: Labour input
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Translog and their properties
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A: Level of technology or total factor productivity (A>0)

α: Constant between 0 and 1( 0< α <1)

Constant Returns to Scale: The return to scale is a long run concept when all the
factors of production are variable. In long run output can be increased by increasing all
the factor of production. An increase in scale means that all factors are increased in the
same proportion, output will increase but the increase may be at an increasing rate or at a
constant rate or at a decreasing rate.

Three situations of Return to the Scale

The three situations of Return to the Scale are:-

(i) Increasing Returns to scale: Increasing return to scale occurs when output
increases in a greater proportion than increase in inputs. If all factors are increased
by 20% then output increases by say 30%. So by doubling the factors, output
increases by more than double.

(ii) Constant Returns to scale: Constant Return to Scale occurs when output
increases in the same proportion as increase in input. If all factors are increased
by 20% then output also increases by 20%. So doubling of all factors causes a
doubling of output then returns to scale are constant. The constant return to scale
is also called linearly homogenous production function.

(iii) Decreasing Returns to scale Decreasing return to scale occurs when output
increases in a lesser proportion than increase in inputs. If all the factors are
increased by 20% then output increases by less than 20%.

The Cobb-Douglas production function exhibits constant returns to scale. Constant

returns to scale occurs when output increases in the same proportion as increase in input.
Under constant returns to scale the sum of two exponents for capital and labour is one i.e.
α + (1-α) =1.

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

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Translog and their properties
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Fig 15.1 Constant Returns of Scale

The Constant Return to Scale occurs when output increases in the same proportion as
increase in input .

It is shown in Fig 15.1.The labour and capital is shown on X-axis and Y-axis. Y1, Y2 and
Y3 are the isoquant curves showing different levels of output. . Under constant return to
scale the distance between successive isoquants remain same as we expand output from
100 to 200 to 300 units. On straight line OR starting from origin the distance OA, AB and
BC all are equal.

If the sum of the two exponents for capital and labour is greater than one then the
function exhibits increasing returns to scale. And if the sum of the two exponents for
capital and labour is less than one then the function exhibits decreasing returns to scale.

Isoquant are Convex to the Origin: Under Cobb-Douglas production function, isoquant
are convex to the origin. Fig 15.2 represents an isoquant map. An isoquant map refers to
the family of isoquant curves where higher the isoquant, higher is the level of production.
In the figure, labor is measured on X-axis and capital on Y-axis. Y1, Y2 and Y3 are the
isoquant curves showing various possible combinations of inputs physically capable of
producing a given level of output. If you can operate production activities independently,
then weighted averages of production plans will also be feasible. Hence the isoquants will
have a convex shape.
The isoquant Y1 represents 100 units of output whereas isoquant Y2 represents 200 units
of output and the level of output is higher on isoquant Y2 than Y1. Isoquant Y3 shows 300

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 15: Special Production Functions- Cobb-Douglas, CES, VES,
Translog and their properties
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units of output which is higher than the level of output as shown by isoquant Y1 and Y2
and so on. So, higher the isoquant, higher is the level of output.

Fig 15.2 Isoquants are convex to the origin

The isoquants under Cobb Douglas production function are convex to the origin. This
occurs because of diminishing marginal rate of technical substitution.

Here the isoquant are:-

 Downward sloping.
 Convex to the origin: Diminishing MRTS.
 Higher the isoquant, higher the level of output.

Marginal Products and Average Products:

(i) Under Cobb Douglas production function, the average product and marginal
products of factor depend upon the ratio in which the factors are combined to
produce output.
∂Y 𝛼−1 1−𝛼
𝐾 𝛼−1
𝑀𝑃𝐾 = = 𝐴𝛼𝐾 𝐿 = 𝐴𝛼 ( )
∂K 𝐿

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 15: Special Production Functions- Cobb-Douglas, CES, VES,
Translog and their properties
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Y 𝐴𝐾 𝛼 𝐿1−𝛼 𝐾 𝛼−1
𝐴𝑃𝐾 = = = 𝐴( )
K 𝐾 𝐿

The average product of capital depends on the ratio of capital and labor (K/L)
and does not depend upon the absolute quantities of the factors used. The
same is true for labor.

(ii) The marginal product is proportional to the output per unit of its factor.

∂Y 𝐾 𝛼−1 𝑌
𝑀𝑃𝐾 = = 𝐴𝛼𝐾 𝛼−1 𝐿1−𝛼 = 𝐴𝛼 ( ) = 𝛼( )
∂K 𝐿 𝐾

∂Y 𝛼 −𝛼
𝐾 𝛼 𝑌
𝑀𝑃𝐿 = = 𝐴(1 − 𝛼)𝐾 𝐿 = 𝐴(1 − 𝛼) ( ) = (1 − 𝛼) ( )
∂L 𝐿 𝐿

Linear in Logarithm: The Cobb-Douglas production function is linear in logarithm.

𝑌 = 𝑓(𝐾, 𝐿, 𝐴) = 𝐴𝐾 𝛼 𝐿1−𝛼
Now applying log both sides, we get
ln 𝑌 = ln 𝐴 + 𝛼 𝑙𝑛𝐾 + (1 − 𝛼) 𝑙𝑛𝐿
where,
 α is the partial elasticity of output w.r.t capital. It measures the percentage change
in output for, say, one percentage change in the capital input holding the labor
input constant.
 (1-α) is the partial elasticity of output w.r.t labour. It measures the percentage
change in output for, say, one percentage change in the labour input holding the
capital input constant.

The Cobb Douglas production function is linear in parameter. It can be estimated using
least squares method.

Properties of Cobb Douglas Production Function

i. Constant Returns to Scale: The Cobb Douglas production function exhibits

constant returns to scale. If the inputs capital and labor are increased by a
positive constant, λ, then output also increases by the same proportion i.e.
𝑓(λK, λL, A) = λ f(K, L, A) for all λ>0.

.
𝑌 = 𝑓(𝐾, 𝐿, 𝐴) = 𝐴𝐾 𝛼 𝐿1−𝛼
𝑓(λK, λL, A) = 𝐴(𝜆𝐾)𝛼 (𝜆𝐿)1−𝛼
= 𝐴𝜆𝛼 𝐾 𝛼 𝜆1−𝛼 𝐿1−𝛼

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Module 15: Special Production Functions- Cobb-Douglas, CES, VES,
Translog and their properties
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= 𝜆𝐴𝐾 𝛼 𝐿1−𝛼
= 𝜆𝑌
 If the function exhibits decreasing returns to scale then 𝑓(𝜆𝐾, 𝜆𝐿, 𝐴) <
𝜆𝑌 𝑓𝑜𝑟 𝑎𝑛𝑦 𝜆 > 1.
 If the function exhibits increasing returns to scale then 𝑓(𝜆𝐾, 𝜆𝐿, 𝐴) >
𝜆𝑌 𝑓𝑜𝑟 𝑎𝑛𝑦 𝜆 > 1.

ii. Positive and Diminishing Returns to Inputs: The Cobb Douglas production
function is increasing in labor and capital i.e. positive marginal products.

∂Y ∂Y
(i) > 0 𝑎𝑛𝑑 >0
∂K ∂L
𝑌 = 𝑓(𝐾, 𝐿, 𝐴) = 𝐴𝐾 𝛼 𝐿1−𝛼
∂Y
𝑀𝑃𝐾 = = 𝐴𝛼𝐾 𝛼−1 𝐿1−𝛼
∂K
∂Y
𝑀𝑃𝐿 = = 𝐴(1 − 𝛼)𝐾 𝛼 𝐿−𝛼
∂L
Assuming A, L and K are all positive and 0 < 𝛼 < 1, the marginal products are positive.
2
(ii) Diminishing Marginal Products with respect to each Input: ∂ Y2 <
∂K
2
0 𝑎𝑛𝑑 ∂ Y <0
2
∂L

2
∂ Y = 𝐴𝛼(𝛼 − 1)𝐾 𝛼−2 𝐿1−𝛼 <0 if α<1
2
∂K

Here, any small increase in capital will lead to a decrease in the marginal product of
capital. Any small increase in capital cause output to rise but at a diminishing
rate. The same is true for labor.

iii. Inada Conditions

(i) The marginal product of capital (labor) approaches infinity as capital
(labor) goes to zero.
∂Y ∂Y
lim = lim =∞
𝐾→0 ∂K 𝐿→0 ∂L

(ii) The marginal product of capital (labor) approaches zero as capital (labor)
goes to infinity.
∂Y ∂Y
lim = lim =0
𝐾→∞ ∂K 𝐿→∞ ∂L

iv. The Cobb Douglas production function has elasticity of substitution equal
to unity.

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Translog and their properties
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𝑑 𝐾/𝐿 𝑀𝑅𝑇𝑆
𝜎= 𝑥
𝑑 𝑀𝑅𝑇𝑆 𝐾/𝐿
where 𝑀𝑅𝑇𝑆 = 𝑀𝑃𝐿 /𝑀𝑃𝐾

𝑀𝑃 𝐴(1−𝛼)𝐾𝛼 𝐿−𝛼 (1−𝛼) 𝐾

𝑀𝑅𝑇𝑆 = 𝑀𝑃 𝐿 = =
𝐾 𝐴𝛼𝐾𝛼−1 𝐿1−𝛼 𝛼 𝐿

𝐾 𝛼
» 𝐿 = 1−𝛼 ∗ 𝑀𝑅𝑇𝑆,
𝑑(𝐾/𝐿) 𝛼
=
𝑑𝑀𝑅𝑇𝑆 1 − 𝛼
𝑑 𝐾/𝐿 𝑀𝑅𝑇𝑆
𝜎= 𝑥 =1
𝑑 𝑀𝑅𝑇𝑆 𝐾/𝐿

v. Constant Income Shares of Output: The exponent of capital (labor),α (1- α),
represents the contribution of capital (labor) to output. This is the same as the
portion of output distributed to capital (labor) i.e. capital (labor) income share.

𝑌 = 𝑓(𝐾, 𝐿, 𝐴) = 𝐴𝐾 𝛼 𝐿1−𝛼

The real wage of labour (w) is calculated by partially differentiating Y w.r.t. L, which is
nothing but marginal product of labor (𝑀𝑃𝐿 ).

∂Y 𝛼 −𝛼
𝐾 𝛼 𝑌
𝑤 = 𝑀𝑃𝐿 = = 𝐴(1 − 𝛼)𝐾 𝐿 = 𝐴(1 − 𝛼) ( ) = (1 − 𝛼) ( )
∂L 𝐿 𝐿
Total wage bill=𝑤. 𝐿 = 𝑀𝑃𝐿 . 𝐿 = 𝐴(1 − 𝛼)𝐾 𝛼 𝐿1−𝛼

𝑇𝑜𝑡𝑎𝑙 𝑊𝑎𝑔𝑒 𝐵𝑖𝑙𝑙

The labor share in real National Product is 𝑅𝑒𝑎𝑙 𝑁𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑃𝑟𝑜𝑑𝑢𝑐𝑡

𝑇𝑜𝑡𝑎𝑙 𝑊𝑎𝑔𝑒 𝐵𝑖𝑙𝑙 𝑤𝐿 𝐴(1−𝛼)𝐾𝛼 𝐿1−𝛼

=𝑅𝑒𝑎𝑙 𝑁𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 = = =1−𝛼
𝑌 𝐴𝐾𝛼 𝐿1−𝛼

In the same way, the capital income share remains constant at 𝛼.

4. CES Production Function

Another most popular neo classical production function is constant elasticity of
substitution (CES) production function. The CES production function was developed by
Arrow, Chenery, Minhas and Solow as a generalisation of the Cobb Douglas production
function that allows for non-negative and constant elasticity of substitution.

Functional Form: The standard CES production function can be written as:

ECONOMICS Paper 3: Fundamentals of Microeconomic Theory

Module 15: Special Production Functions- Cobb-Douglas, CES, VES,
Translog and their properties
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Y = A [δ𝐾 −𝜌 + (1 − δ) 𝐿−𝜌 ]−1/𝜌

where Y is output, K and L are capital and labor inputs, and A,δ,ρ are the parameters.
 A, technology determines the productivity and A𝜖[0, ∞).
 δ determines the optimal distribution of inputs and δϵ[0,1].
1
 Ρ determines the elasticity of substitution (σ) where ρϵ[-1,0)U(0,∞) and 𝜎 = 1+𝜌

Special Cases under CES Production Function

(i) For ρ→0, σ approaches 1» Cobb Douglas Production Function.
(ii) For ρ→+∞, σ approaches 0» Leontief Production Function.
(iii) For ρ→-1, σ approaches ∞» Perfect Substitutes.

The CES production function is linearly homogeneous and therefore exhibits constant
returns to scale. It is non linear in parameters, so cannot be estimated using least squares
method.

Properties of CES Production Function

i. Constant Returns to Scale: The Cobb Douglas production function exhibits

constant returns to scale.

Y = A [δ𝐾 −𝜌 + (1 − δ) 𝐿−𝜌 ]−1/𝜌

𝑓(𝜆𝐾, 𝜆𝐿, 𝐴)= A [δ(𝜆𝐾)−𝜌 + (1 − δ) (𝜆𝐿)−𝜌 ]−1/𝜌

= A λ[δ𝐾 −𝜌 + (1 − δ) 𝐿−𝜌 ]−1/𝜌

=λY.

ii. Positive and Diminishing Returns to Inputs: The marginal products of the
input are

∂Y 𝛿
𝑀𝑃𝐾 = = 𝜌 (𝑌/𝐾)𝜌+1
∂K 𝐴
∂Y 1 − 𝛿
𝑀𝑃𝐿 = = (𝑌/𝐿)𝜌+1
∂L 𝐴𝜌

They both are positive for K,L>0. With any small increase in capital or labor increases
the output but at a diminishing rate.

iii. Inada Conditions

(i) The marginal product of capital (labor) approaches infinity as
capital (labor) goes to zero.
∂Y ∂Y
lim = lim =∞
𝐾→0 ∂K 𝐿→0 ∂L

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(ii) The marginal product of capital (labor) approaches zero as capital

(labor) goes to infinity.
∂Y ∂Y
lim = lim =0
𝐾→∞ ∂K 𝐿→∞ ∂L

𝟏
iv. The Elasticity of Substitution is 𝝈 = 𝟏+𝝆.
The elasticity of substitution is calculated using formula:

𝑑 𝐾/𝐿 𝑀𝑅𝑇𝑆
𝜎= 𝑥
𝑑 𝑀𝑅𝑇𝑆 𝐾/𝐿
where 𝑀𝑅𝑇𝑆 = 𝑀𝑃𝐿 /𝑀𝑃𝐾

𝑀𝑃𝐿 1 − 𝛿
𝑀𝑅𝑇𝑆 = = (𝐾/𝐿)𝜌+1
𝑀𝑃𝐾 𝛿

𝑑 𝑀𝑅𝑇𝑆 1 − 𝛿
= (𝐾/𝐿)𝜌 (1 + 𝜌)
𝑑 𝐾/𝐿 𝛿
𝑑 𝐾/𝐿 𝑀𝑅𝑇𝑆 1
𝜎= 𝑥 =
𝑑 𝑀𝑅𝑇𝑆 𝐾/𝐿 1+𝜌

5. VES Production Function

In the last section we have read about CES production function. Once we relax the
assumption of constant elasticity of substitution we arrive at variable elasticity of
substitution (VES). The VES production function can be expressed as:
𝑌 = 𝛾𝐾 𝛼(1−𝛿𝜌) [𝐿 + (𝜌 − 1)𝐾]𝛼𝛿𝜌
Where
Y: Output
K: Capital
L: Labor
α,ρ,δ andγ: Parameters
 𝛾 > 0, α > 0
 0< δ <1
 0≤ ρδ ≤1
𝐿 1−𝜌
 > 1−ρδ
𝐾

Here, α is the parameter of returns of scale. If the value of α is 1, the production function
exhibits constant returns to scale.

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Elasticity of Substitution: The elasticity of substitution σ for the for the VES production
function is
𝜌−1 𝐾
σ = σ(K, L) = 1 +
1 − 𝛿𝜌 𝐿
 The elasticity of substitution σ varies with the capital labor ratio around the
intercept term of unity.
 The elasticity of substitution is greater than zero over the relevant range of K/L.
𝐿 1−𝜌
 For σ>0 requires that 𝐾 > 1−𝛿𝜌 .

Properties of VES Production Function

i. Positive and Diminishing Returns to Inputs: The VES production function

is increasing in labor and capital i.e. positive marginal products.

∂Y ∂Y
(iii) > 0 𝑎𝑛𝑑 >0
∂K ∂L

𝑌 = 𝛾𝐾 𝛼(1−𝛿𝜌) [𝐿 + (𝜌 − 1)𝐾]𝛼𝛿𝜌
∂Y 𝑌 𝐿 1−𝜌
𝑀𝑃𝐿 = = 𝛼𝛿𝜌 > 0 𝑓𝑜𝑟 0 ≤ 𝛿𝜌 ≤ 1 𝑎𝑛𝑑 >
∂L 𝐿 + (𝜌 − 1)𝐾 𝐾 1 − 𝛿𝜌
∂Y 𝑌 𝑌
𝑀𝑃𝐾 = = 𝛼(1 − 𝛿𝜌) + 𝛼𝛿𝜌(𝜌 − 1) >0
∂K 𝐾 𝐿 + (𝜌 − 1)𝐾
Here any small increase in capital will lead to a decrease in the marginal product of
capital. Any small increase in capital cause output to rise but at a
diminishing rate. The same is true for labor.

ii. VES Production Function reduced to Cobb Douglas and Linear

production Function
 For 𝜌 = 0 » Harrod Domar Fixed Coefficient Model
 For 𝜌 = 1 » Cobb Douglas Production Function
 For 𝜌 = 1/𝛿(> 1) »Linear Production Function

iii. The Elasticity of Substitution can vary along an Isoquant: The VES
requires that the elasticity of substitution be the same only along a ray through
the origin.

6. Translog Production Function

The translog production function can be expressed as:
𝑌 = 𝑓(𝑥1 , 𝑥2 … . , 𝑥𝑛 )

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𝑛 𝑛
1/2[∑𝑛
𝑗=1 𝛽𝑖𝑗 ln 𝑥𝑗 ]
= 𝛼0 ∏ 𝑥𝑖𝛼𝑖 ∏ 𝑥𝑖
𝑖=1 𝑖=1
Where
Y: Output
𝛼0 : Efficiency Parameter
𝛼𝑖 𝑎𝑛𝑑 𝛽𝑖𝑗 : 𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 (𝑈𝑛𝑘𝑛𝑜𝑤𝑛)
𝑥𝑗∶ 𝐼𝑛𝑝𝑢𝑡 𝑗
Logarithm Form: Taking log both sides, we obtain
𝑛 𝑛 𝑛

ln 𝑌 = 𝑙𝑛𝛼0 + ∑ 𝛼𝑖 ln xi + 1/2 ∑ ∑ 𝛽𝑖𝑗 ln xi 𝑙𝑛𝑥𝑗

𝑖=1 𝑖=1 𝑗=1
The translog production function is a generalisation of the Cobb Douglas production
function. For 𝛽𝑖𝑗 = 0, the log form of translog function reduces to a Cobb Douglas
production function. It is linear in parameters, so can be estimated using least squares
method.

Monotonicity and Translog Production Function: The marginal product i.e addition to
total product due to addition of one more factor is
∂Y ∂lnY 𝑌
𝑀𝑃𝑖 = = .
∂xi ∂lnxi 𝑥𝑖
∂lnY
Where ∂lnx is the production elasticity which can be calculated from log form.
i
𝑛
∂lnY
= 𝛼𝑖 + ∑ 𝛽𝑖𝑗 ln xj (i = 1,2, … . , n)
∂lnxi
𝑗=1
𝑛
Y
So, 𝑀𝑃𝑖 = [𝛼𝑖 + ∑ 𝛽𝑖𝑗 ln xj ]
𝑗=1 xi

 MP of xi can be positive for a range in values of xj but can be negative if 𝛽𝑖𝑗 > 0
(all i,j) and xj → 0.
 If there exist at least one 𝛽𝑖𝑗 < 0 then 𝑀𝑃𝑖 < 0 𝑎𝑠 xj → ∞.
Thus, the translog function is not monotonic.

Is Isoquants under Translog Function Convex: The isoquants are strictly quasi-convex
if the Bordered Hessian matrix is negative definite. In order to construct Bordered
Hessian matrix we need to derive second direct and cross partial derivatives using chain
rule.

𝑛 𝑛
2Y 𝑌
𝑓𝑖𝑖 = ∂ 2 = 2 [𝛽𝑖𝑖 + (𝛼𝑖 + ∑ 𝛽𝑖𝑗 ln xj − 1) (𝛼𝑖 + ∑ 𝛽𝑖𝑗 ln xj )
∂K 𝑥𝑖
𝑗=1 𝑗=1

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2Y 𝑌
𝑓𝑖𝑗 = ∂ = [𝛽𝑖𝑗
∂xi ∂xj 𝑥𝑖 𝑥𝑗
𝑛

+ (𝛼𝑖 + ∑ 𝛽𝑖𝑗 ln xj ) (𝛼𝑗

𝑗=1
𝑛

+ ∑ 𝛽𝑖𝑗 ln xi )
𝑖=1
The Bordered Hessian matrix is
0 𝑓1 ⋯ 𝑓𝑛
[𝑓1 𝑓11 ⋱ ⋮ ]
𝑓𝑛 ⋯ 𝑓𝑛𝑛
Here the values of the first and second partial derivatives vary with input levels, there
is no guarantee that the isoquants are globally convex.

7. Summary

Nature imposes technological constraints on firms; only certain combination of inputs

are feasible ways to produce a given amount of output, and the firm must limit itself
to technologically feasible production plans. The easiest way to describe feasible
production plans is to list them. The set of all combinations of inputs and outputs that
comprise a technologically feasible way is called a production function. The relation
between inputs and output of a firm is known as production function. Production
function is a purely technical relation which connects factor inputs and output. The
two popular neoclassical production functions that are widely used in economics are
Cobb Douglas production function and CES production function. The Cobb Douglas
production function lies between linear and fixed proportion production function with
elasticity of substitution equal to one. Cobb Douglas production function is very
popular among economist because of its flexibility and ease of use. CES production
function is a generalization of the Cobb Douglas production function that allows for
non-negative and constant elasticity of substitution. The VES production function
requires that the elasticity of substitution be the same only along a ray through the
origin. The translog production function is also a generalization of the Cobb Douglas
production function. The log form of translog function reduces to the Cobb Douglas
production function.

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Economics 207, 2019 Allin Cottrell

The Cobb–Douglas Production Function

1 Introduction

In general, a production function is a specification of how the quantity of output behaves as a func-
tion of the inputs used in production. This concept can be applied at the level of individual firms,
industries, or entire economies. Since we’re doing macroeconomics we will be considering an ag-
gregate production function, applying at the economy-wide level.
Various specific mathematical forms have been put forward for the production function, but the most
commonly used is that developed by Charles Cobb and Paul Douglas in the second quarter of the
20th century. Here’s their specification:

Y = AK α N 1−α 0<α<1 (1)

Here Y represents aggregate output, K the capital input, and N the labor input (capital and labor
being the two “factors of production” in this function). The A term represents Total Factor Produc-
tivity (TFP for short); you can think of this as a “quality” factor—as opposed to K and N which are
just quantitative. The value of A reflects the state of technology as well as the skill and education
level of the workforce. All being well, we’d expect A to be gradually increasing over time.

2 Marginal product, diminishing returns

A particularly important aspect of a production function is the marginal product of the factors. Take
first the marginal product of labor (or MPN for short)—that is, the change in output that results
when the labor input is varied, holding the capital input and TFP constant. We find this by taking
the first derivative of equation (1) with respect to N:
dY
MPN = = (1 − α)AK α N 1−α−1 (2a)
dN
= (1 − α) AK α N 1−α N −1


(2b)
Y
= (1 − α) > 0 (2c)
N
Given that Y and N must be positive and α is a positive fraction, we see that the marginal product
of labor must be positive: a greater labor input leads to the production of more output. No suprise
there.
The familiar economic concept of “diminishing returns” leads us to expect that the MPN, while
positive, should be declining: as the labor input is increased, holding K and TFP constant, output
should increase but at a diminishing rate. Does the Cobb–Douglas function satisfy this condition?
To find out we need to take the derivative of the MPN with respect to N, or in other words the

1
second derivative of Y with respect to N.

dMPN d 2Y
= = (−α)(1 − α)AK α N 1−α−2
dN d N2
= (−α)(1 − α) AK α N 1−α N −2

Y
= (−α)(1 − α) 2 < 0
N
We can tell that the second derivative is negative—hence satisfying diminishing returns—because
all terms in the multiplicative expression are positive apart from the negative −α.
Strictly analogous math tells us that the Cobb–Douglas function also exhibits a positive but dimin-
ishing marginal product of capital, MPK. (In this case the thought-experiment is, what happens to
output when K is increased while N and TFP are held constant?)
Positive MPK:
dY
= α AK α−1 N 1−α = α AK α N 1−α K −1


MPK = (3a)
dK
Y
=α >0 (3b)
K

Diminishing returns to capital:

dMPK d 2Y
= = (α − 1) α AK α−2 N 1−α
dK dK2
Y
= (α − 1) α 2 < 0
K

3 Cross partials

A further point relevant for macroeconomic analysis: what (if anything) happens to the marginal
product of labor when the capital input is increased? And conversely, what happens to the MPK
when N increases? In mathematical terms, we’re talking about the so-called “cross-partial” deriva-
tives, dMPN/d K and dMPK/d N.
dMPN
= α(1 − α)AK α−1 N 1−α−1
dK
= α(1 − α) AK α N 1−α K −1 N −1

Y
= α(1 − α) >0
KN
So an increase in capital raises the marginal product of labor. And
dMPK
= (1 − α) α AK α−1 N 1−α−1
dN
Y
= (1 − α) α >0
KN
So raising N also raises the MPK. (And it turns out that the two cross partials are identical.)
Also note: from equations (2a) and (3a) it should be clear that an increase in Total Factor Produc-
tivity, A, will raise the marginal products of both factors.

2
4 Returns to scale

We’ve shown that the Cobb–Douglas function gives diminishing returns to both labor and capital
when each factor is varied in isolation. But what happens if we change both K and N in the same
proportion?
Suppose an economy in an initial state has inputs K 0 and N0 and produces output Y0 :
Y0 = AK 0α N01−α
Now suppose we scale the inputs by some common factor λ. (For example, λ = 2 would mean that
we double each input.) We’ll then have inputs K 1 = λK 0 and N1 = λN0 and will produce output
Y1 . The question is, how does Y1 relate to Y0 ? Let’s see:
Y1 = AK 1α N11−α
= A (λK 0 )α (λN0 )1−α
= A λα K 0α λ1−α N01−α
= λα+1−α AK 0α N01−α
= λY0
So if we scale both inputs by a common factor, the effect is to scale the output by that same factor.
This is the defining characteristic of constant returns to scale. From the math above we can see that
this occurs in the Cobb–Douglas function because the exponents on capital and labor, α and 1 − α,
add up to 1.
We could imagine a generalization of Cobb–Douglas in which the exponents on capital and labor
are (say) α and β respectively, preserving the requirement that each exponent be a positive fraction
(this is needed to give positive but diminishing marginal products) but dropping the requirement
that they sum to 1. In that case we’d get increasing returns to scale if α + β > 1 and decreasing
returns to scale if α + β < 1.

5 Factor shares

You may be familiar with this point from microeconomics: in a “perfectly competitive” economy,
profit-maximizing behavior on the part of firms tends to ensure that the factors of production are
paid a return equal to their respective marginal products. Now we saw above—in equations (2c) and
(3b)—that the marginal products of labor and capital according to the Cobb–Douglas production
function are
Y
MPN = (1 − α)
N
Y
MPK = α
K
These are the earnings “per unit” of the factors, under the perfect competition assumption. To get
the total earnings of the factors we have to multiply by their respective quantities, N and K . Then
we get
Y
Labor earnings = N × (1 − α) = (1 − α)Y
N
Y
Capital earnings = K × α = αY
K

3
So we see that (1 − α) is labor’s share in total output, Y, and α is capital’s share. (We also see that
the factor shares add up to 100 percent of output only if the Cobb–Douglas exponents sum to 1.)
It would be a serious stretch to suppose that the US economy conforms to the textbook model of
perfect competition. Nonetheless, if we’re willing to fudge a bit we may take the factor shares in US
National Income (a measure which is closely related to GDP) as indicative of “ballpark-realistic”
values of the Cobb–Douglas exponents. Figure 1 shows the share of “Compensation of Employees”
from 1960 to 2017; it varies between about 0.61 and 0.68. Very roughly, we may think of α being
about 1/3 and (1 − α) about 2/3—though note that the recent data show labor receiving appreciably
less than 2/3 of income.

0.69

0.68

0.67

0.66

0.65

0.64

0.63

0.62

0.61

0.6
1960 1970 1980 1990 2000 2010

Figure 1: Labor share in National Income, USA 1960–2017

4
____________________________________________________________________________________________________

Subject ECONOMICS

Paper No and Title 12: Economics of Growth and Development - I

Module No and Title 28: Classical theory of development- Contributions of Adam

Smith
Module Tag ECO_P12_M28

ECONOMICS Paper 12: Economics of Growth and Development - I

Module 28: Classical theory of development- Contributions of Adam
Smith
____________________________________________________________________________________________________

TABLE OF CONTENTS

1. Learning Outcomes
2. Introduction
3. Determinant factors of economic development
4. Balanced development policy
5. Human resource and economic development
6. Legend of development
7. Development process
8. Static state
9. Merits of the Adam Smith’s theory
10. Demerits of the theory
11. Significance of Adam smith model in less developed countries
12. Summary

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1. Learning Outcomes
After studying this module, you shall be able to

 Understand the Determinant factors of economic development

 Understand human resource and economic development
 Analyse the merits and demerits of Adam Smith theory

2. Introduction

The father of Economics, Adam Smith, defined the concept of economics in his famous
book, “ An Enquiry into the Nature and Causes of the Wealth of Nations” in the year as
the study of wealth. He analysed the basic economic factors that can help a nation to build
its wealth; thereby giving the concept of development economics. He has, however, did
not propound the systematic theory of development and this was left for the economist of
later years.
Over the years, Development economics has emerged as a separate branch of economics
which specifically deals with the developmental issues of middle and low-income
countries. Its subject matter has changed over the years, for instance, in the decade of 1960-
70s the developmental economists were preoccupied with deciding the superiority of state
controlled or market owned economies which shifted the debate over traditional industries
or the modern sector encompassing globalization in 1980s and 90s. In the recent years it
has emerged as a more holistic concept to define objectives and economic policies for the
emerging countries.
In order to build the understanding of Development Economics, we will begin with the
theory of Adam Smith. In his theory, he has explicitly focussed on the six broad areas
which are defined as follows: -
(A) Determinant factors of economic development
 Nature law or laissez faire policy
 Division of labour and market perfections
 Capital accumulation is the engine of growth.
(B) Policy of balanced growth

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(C) Human resource /capital and economic development

(D) Legends of development
(E) Process of growth
(F) Static state
The above mentioned six major areas are described at length in the following section

3. Determinant factors of Economic Development

Economic development is based on several factors; however, Adam smith has classified
and defined the following three major factors as determinants of economic development
for low and middle-income countries:
(I) Natural law: - Smith believed that growth and economic development can only be
possible in an independent and free environment. Any restriction or interference with
economic activities checks (stops) the process of economic development. He supports the
obvious and simple system of natural liberty.
According to Smith every individual is the best caretaker of his interest and any kind of
intervention that prevents the individuals from carrying out the self interest thing may not
lead to good market outcomes.
Adam smith strongly supported the policy of free trade and laissez faire (minimum
intervention) in economic activities. He asserted the existence of invisible hand or a system
that brings about the equilibrium in perfectly competitive market and hence, maximises the
national output.
(2) Division of labour and market perfection: - Division of labour is the beginning point of
the theory of economic growth. According to Smith, the development of a nation depends
upon its level of production and quantity of output being produced that further depends
upon the division of labour.
The division of labour ensures the right placement of the workforce which enhances the
total productivity. The benefits that an economy can derive from the division of its labour
force can be listed out as follows:
a) Increased skill of each labour.
b) Saves time in production of the product.
c) Introduction of new innovations.
d) Increment in productivity.
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According to Smith, an economic model for development of the economies should be
competitive in nature; existence of any form of monopolistic tendencies or monopoly and
mal practices can act as an inhibitor for the process of development.

(3) Capital accumulation is the engine of growth :- According to Smith, the growth of any
society depends upon its capital accumulation which acts as an engine of growth. Capital
accumulation reflects the productive capacity of any economic system and it further
depends upon the real savings of the society. Also, according to him judicial expenses
increase capital accumulation.
In an economic system, the factor payments are made to four factors of production i.e. land,
labour, capital and entrepreneur in the form of rent, wages, interest and profits. It would be
unjust to expect savings from the labourers as the amount they receive in the form of wages
is barely enough for their subsistence and is largely spent on their consumption. Therefore,
savings should be channelized from the remaining factors of production as given below:

a) Wages: According to Smith, it is the capitalists and landlords who can do savings
and not the workers as the wages earned by them are just enough to meet their basic
requirements.

b) Profit: - Profit is the reward for the capitalist/ entrepreneur for bearing the risk. The
level of profits can be enhanced by the expansion of economic activity that depends
upon the investment of savings.

c) Interest: - Interest is the reward for the capital invested in the economic activity.
According to Smith due to increment in prosperity, development, progress and
population, rate of interest decline so as to increase in the supply of capital.

d) Rent: -Rent is the reward for the use of land for any economic activity. Smith
thought that with the process of development, it is also possible that there will be
increment of monetary and real rent. Rent as a proportion of national income can
also increase national income.

4. Balanced development policy

Adam smith was indeed a strong supporter of industrialisation; he however, did not
undermine the importance of agricultural and small scale industries in the initial stages of
development. In order to achieve the sustained growth in long run, it is crucial to adopt the

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policy of balanced development in the starting phase of development which focus on all
the sectors of economy i.e. agriculture, industries and trade and service sector rather than
focusing only on the industrial sector.
In the process of development, Smith favored the growth and development of agricultural
sector in the first stage, followed by industrial sector in the second stage and in the last the
trade and service which can be presented as follows:

Further, Smith also understood the role of externalities in the process of development.

5. Human resource and economic development

According to Adam Smith, human resource (labour) is an active factor of development. He
clearly indicates that “labour is that treasure which provides all the facilities and necessities
for annual consumption to nation”. Keeping in view the deficiency of capital, capital output
ratio must be maintained so that the rate of development may not get slow down. Low
standard population, however, must necessarily be controlled.

6. Legend of development
According to Adam Smith following are the legend for economic development of any
society:
(1) Farmers
(2) Trader
(3) Producer

It is impossible to attain the economic growth without the presence of these three economic
legends as their presence is the pre-requisite for the process of capital accumulation and
economic development. Also, as per the understanding of Smith, the working of these three
is closely inter-related to each other and none of them can work in isolation.

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7. Development process
In the opinion of Smith, the process of economic development is a continuous, regular and
cumulative process. The growth and development of agriculture, industries and commerce
brings about the changes in various macroeconomic and developmental variables. A few
of the macroeconomic and developmental variables are listed out below:
(I) Capital accumulation
(ii) Technological progress
(iii) Expansion of market
(iv) Population growth
(v) Division of labour and
(vi) Increase in profits

A word of caution is that the changes continue to occur in the above mentioned variables
with the process of economic development; however, this process of change does not
continue at the same level.

8. Static state
The above said process of development will continue till the time economy has not
developed its resources fully. Once the economy has developed its resources fully, then
following changes will take place in the economic variables:

(1) Competition between labours is likely to bring the wage rate down at the subsistence
level
(2) Competition between the traders can reduce their profit level.

Once the reduction in the level of profit begins, it will continue further and will lead to the
following:
I) Reduction in increment
(II) Capital accumulation stops
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(III) Population become static
(IV) Profits reached to minimum level
(V) Wage rate reduced at minimum levels
(VI) Per capital income become static
(VII) Economy reached at a static condition and Smith called this condition as static state

Explanation of Adam smith model by diagram

𝑑𝑘
On X axis time period and on Y axis rate of capital accumulation 𝑑𝑡

Economy grows till ‘T’ time from K to S. Once obtaining the position T, economy reaches
to its static state. This is because rent became so high that profit became zero and hence
capital accumulation stops.

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9. Merits of the Adam Smith’s theory

(I) Adam Smith model has indicated the following features of development which are
necessary for the development of any economy:
a) Saving
b) Capital formation
c) Market expansion
d) Free trade policy
e) Balance growth
In the opinion of Adam Smith improved technology has special importance for economic
development. The use of technology enhances division of the labour force and expansion
of market which further induces the production. The above mentioned points can be
explained as follows:

(1) Saving: -Saving is very essential for the process of economic development as savings
leads to capital accumulation which enhances the productive capacity of resources.

(2) Capital formation: - Adam Smith assumed capital accumulation as main factor of
economics of development which can be elaborated with the help of the following diagram:

The high capital accumulation leads to increasing capital formation which enhances the
rate of economic development.

(3) Expansion of market: The increasing rate of economic development will enhance the
employment opportunities; thereby providing income to individuals which in turn will
increase the purchasing capacity of individuals and increase the aggregate demand; thus
leading to expansion of the market.

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(4) Laissez faire policy: - This approach of Adam Smith is relevant in many areas, for
instance, international trade and production in private sector. This refers to minimum
intervention of the government in economic activities.

(5) Balanced growth: - According to Adam Smith there exist an inter dependency among
farmers, traders and producers which is crucial for the process of economic development.
The inter-relationship between the three can be understood as the expansion in the
agricultural produce will provide increased amount of raw materials to industries; the
expansion of industries will further encourage the trade and commerce. Therefore,
balanced growth is required for the development of commerce and manufacturing
industries; therefore, for the entire process of development.
Smith principally viewed economic development: “as a process embedded in, and limited
by a particular physical, institutional, and social environment. More specifically, Smith
conceives of economic development as the filling-up with people and physical capital
(‘stock’) of a spatial container (‘country’) that encompasses a given endowment of natural
resources and is shaped internally and bounded externally by laws and institutions”.

10. Demerits of the theory

The theory propounded by Adam Smith is criticized on the following grounds:

(1) Rigid division of society: This theory of Smith divides society in two parts which are
as follows:
(i) Capitalist or landlord class
(ii) Labour class or land less class
It is, however, wrong to assume the existence of only two classes as in the real world, there
also exists a third class known as Middle class and this class cannot be neglected.
(2) One side saving base: - According to Smith only capitalist and landlords (money lender)
saves money but this was not true always because even working class also saves money
(3) Perfect completion is unrealistic assumption

(4) Negligence of public sector due to his policy of laissez faire: Smith fails to explain the
role of public sector in capital accumulation and economic development

In these days there is growth of public sector in every country which is helpful in capital
formation
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(5) Negligence of industry and entrepreneurship: - smith neglected the important factors
such as industry and entrepreneurship. but in reality entrepreneur organised all the factors
and they brought new innovations which formed capital and development
(6) Unrealistic assumption of static state

(7) Static model: - according to Hicks, Smith model is a static model because this model
due not give any sequence of development for less developed countries

11. Significance of Adam smith model in less developed countries:

The significance of Smith’s model is quite limited in less developed countries because of
various reasons. A few of them are listed below:

(i) Market size is small (limited): The size of the market in less developed countries is small
which does not provide the economies of scale and therefore, the production is no longer
profitable.

(ii) Saving capacity is low: Savings are very essential for capital formation which is needed
for the process of economic development. In less developed countries, however, the
capacity of individuals to save is low which restricts the economic development.

(iii) Lack of induced investment (because the level of profits is low in this economies):
There is lack of induced investment in less developed countries owing to the lower levels
of profit and this act as a barrier in the process of growth and economic development.

(iv) Size of market depends upon level of income and output produced: The size of market
in an economic system depends on the level of income and production of the society. Since
in less developed countries the level of income and output is less and therefore the size of
market also tends to be small.

(v) Productivity is low: Due to low level of savings there is lack of capital formation which
lowers down the productivity of the economic system.

(vi) Level of income is low due to low productivity: Owing to the low level of savings,
capital formation and productivity, the size of income also tends to be on lower side in less
developed countries.

(vii) Vicious circles of poverty applied form both demand side and supply side: In less
developed countries, there exists vicious circle of poverty from both demand and supply
ECONOMICS Paper 12: Economics of Growth and Development - I
Module 28: Classical theory of development- Contributions of Adam
Smith
____________________________________________________________________________________________________
side. Owing to various factors such as low level of savings, investment etc, there tends to
be low level of productivity and therefore low level of employment. This again leads to the
problem of poverty in less developed countries.

Due to the above mentioned reasons Smiths’ theory based on division of labour and size
of the market does not apply on less developed nations. He further argued on the basis of
following grounds:
(B) (i) Perfect competition does not exist in these nations
(ii) The market structures are imperfect in these nations.

(C) In the opinion of Smith, without the interference of government development in free
economic activities is impossible and therefore, it is essential for the government to
interfere in the economic activities of the state.

Besides all this, the model of Smith told us about some factors which can enhance the
process of development in less developed countries i.e.
(i) Saving
(ii) Balance growth

(iii) Three legants stated by Smith for under developed countries are farmers, producers
and traders; these three legends increase production in their own sector and are helpful in
capital accumulation and development of the nation.

(iv) In the absence of free trade market system government can also help in increasing the
production by giving the economic assistance.

ECONOMICS Paper 12: Economics of Growth and Development - I

Module 28: Classical theory of development- Contributions of Adam
Smith
____________________________________________________________________________________________________

12. Summary
Now let us summarise what we have learnt in this module:
 The father of Economics, Adam Smith, defined the concept of economics in his
famous book, “An Enquiry into the Nature and Causes of the Wealth of Nations” in the
year 1776 as the study of wealth.
 Economic development is based on several factors; however, Adam smith has
classified and defined the three major factors as determinants of economic development
for low and middle-income countries
 Adam smith was indeed a strong supporter of industrialisation; he however, did not
undermine the importance of agricultural and small scale industries in the initial stages
of development.
 In order to achieve the sustained growth in long run, it is crucial to adopt the policy
of balanced development in the starting phase of development which focus on all the
sectors of economy i.e. agriculture, industries and trade and service sector rather than
focusing only on the industrial sector.

 The significance of Smith’s model is quite limited in less developed countries

because of various reasons.

ECONOMICS Paper 12: Economics of Growth and Development - I

Module 28: Classical theory of development- Contributions of Adam
Smith
 Subject ECONOMICS

Paper No and Title Paper-5 Advance Microeconomics

Module No and Title Module-10: Macro Theories of Distribution—Kalecki and

Kaldor’s
Module Tag ECO_P5_M10

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 10: Macro Theories of distribution – Kalecki and Kaldor’s
 Table of Contents

1. Learning Outcomes
2. Introduction
3. Kalecki’s Degree of Monopoly Theory of Income Distribution
4. Kaldor’s Theory of Income Distribution
5. Summary

TABLE OF

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 10: Macro Theories of distribution – Kalecki and Kaldor’s
 1. Learning Outcome

After studying this module you will learn:

 How different factors like monopoly power, material cost that wage cost, savings
behavior of capitalist class and wage earners etc. affect the share of entrepreneurs’
income and share of workers’ income.
 How degree of monopoly power is redefined.
 The concept of stability of economy in the context of income distribution of
factors.

2. Introduction

The distribution of factor’s income depends on the degree of monopoly power as shown by
Kalecki. He suggests that there is an inverse relation between relative share of wages in gross
national product and both the degree of monopoly power and the share of raw material cost
relative to labour costs. On the other hand Kaldor provides an alternative theory of
distribution that incorporates the multiplier concepts developed by Keynes to determine the
relative shares of factors’ in the income. Based on the assumption of full employment Kaldor
shows that given the constant marginal propensity to save of workers and capitalists the share
of profit depends on the ratio of investment to income. In this module we derive and explain
both Kalecki and Kaldor’s theories of distribution.

3. Kalecki’s Degree of Monopoly theory of Distribution

Kalecki developed the degree of monopoly theory of distribution. According to him the
relative share of labourer’s income or wages in gross national product is negatively related to
both the degree of market power and the ratio of raw material costs relative to labour costs.

In order to show this Kalecki uses the Lerner measure of the degree of monopoly power.
𝑝−𝑀𝐶
It is defined as 𝛽 = where 𝑝 is the price per unit of product, MC denotes the marginal
𝑝

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 10: Macro Theories of distribution – Kalecki and Kaldor’s
 cost and 𝛾 is inverse of elasticity of demand. Marginal cost is
assumed here to consider cost of labour, cost of raw materials and overhead cost. He also
assumes that there exists idle capacity. This implies that firm’s short run marginal cost curve
is constant and parallel to horizontal axis and it also coincides with the short run average
variable cost curve. The amounts of profit, interest, depreciation and salaries depend on by
how much the price of product exceeds the marginal cost. The formal derivation of Kalecki’s
relation of factors’ income distribution is given below.

Let us consider

𝜋𝑎 = Average profit per unit of product,

𝑜𝑎 = Average overhead of interest, depreciation and salaries.

𝑤𝑎 = Average wages.

𝑟𝑎 = Average raw material costs.

Further we assume that 𝑀𝐶 = 𝑜𝑚 + 𝑤𝑚 + 𝑟𝑚 , Where 𝑜𝑚 , 𝑤𝑚 , and 𝑟𝑚 represent marginal

overhead cost, marginal wage cost and marginal raw material cost respectively. Then price can
be determined as:

𝑃 = 𝜋𝑎 + 𝑜𝑎 + 𝑤𝑎 + 𝑟𝑎

Then 𝑃 − 𝑀𝐶 = 𝜋𝑎 + 𝑜𝑎 − 𝑜𝑚 + 𝑤𝑎 − 𝑤𝑚 + 𝑟𝑎 − 𝑟𝑚 ,

Since we assume that marginal wage costs and marginal raw material costs are constant, then
𝑤𝑎 − 𝑤𝑚 = 0 and 𝑟𝑎 − 𝑟𝑚 = 0 and also assume that marginal overhead cost is negligible.
Hence, we obtain

𝑃 − 𝑀𝐶 = 𝜋𝑎 + 𝑜𝑎

Multiplying both side by output of the firm ( 𝑦) we have

𝑦(𝑃 − 𝑀𝐶) = 𝜋𝑎 . 𝑦 + 𝑜𝑎 . 𝑦
𝑃−𝑀𝐶
Since from definition we know 𝛽 = 𝑃

Then 𝑃 − 𝑀𝐶 = 𝛽𝑃

Or 𝛽𝑃𝑦 = 𝜋𝑎 . 𝑦 + 𝑜𝑎 . 𝑦

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 10: Macro Theories of distribution – Kalecki and Kaldor’s
 Then summing over the economy we obtain

∑ 𝛽𝑃𝑦 = ∑ 𝑦(𝜋𝑎 + 𝑜𝑎 )

Now denote aggregate entrepreneurial income and aggregate overhead cost in the above relation
as 𝐶 and 𝑂 respectively. We have

∑ 𝛽𝑃𝑦 = 𝐶 + 𝑂

The aggregate turnover or total revenue in Kalecki model is defined as 𝑇𝑅 = ∑ 𝛽𝑃𝑦

𝐶+𝑂
Then 𝛽̅ = 𝑇𝑅 is referred as the weighted average of the degree of monopoly in the economy.
That is, 𝛽̅ measures the degree of monopoly power of the whole economy. The domestic
aggregate output or national income of the economy can be written as 𝑌 = 𝐶 + 𝑂 + 𝑊 where

𝑌 represents domestic national income or output and 𝑊 is aggregate wage bill. Then the
following relation can be written as

𝐶+𝑂 =𝑌−𝑊

Then

𝑌−𝑊
𝛽̅ =
𝑇𝑅
or
𝑇𝑅 𝑌−𝑊
𝛽̅ 𝑊 = 𝑊

𝑇𝑅 𝑌
or 𝛽̅ 𝑊 = 𝑊 − 1

𝑌 𝑇𝑅
or = 1 + 𝛽̅ 𝑊
𝑊

𝑊 1
or = ̅ 𝑇𝑅
𝑌 1+𝛽
𝑊

𝑊
The factor is the share of wages in domestic national income. The above relation reveals that
𝑌
the share of wages inversely depends on the degree of monopoly power and the ratio of total
turnover to aggregate wage bill.

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 10: Macro Theories of distribution – Kalecki and Kaldor’s
 Kalecki again reformulated his theory of distribution by dropping the
Lerner measure of degree of monopoly power. Kalecki in his new model defined degree of
monopoly power of a single firm as the ratio of average price to average cost of production and
for the economy as a whole it is the ratio of aggregate proceeds to aggregate prime cost (i.e. sum
of wage cost and material cost). Kalecki’s reformulated theory of distribution can be stated as

𝐴=𝑊+𝑀+𝑂+𝑃

Where 𝐴 = Aggregate proceeds or sales revenue

𝑊 = Aggregate wage bill

𝑀 =Aggregate raw material cost

𝑂 =Aggregate overhead cost

𝑃 =Aggregate profit

Kalecki redefines the degree of monopoly power by taking ratio of aggregate proceeds to
𝐴
aggregate prime costs. This is denoted as 𝑘 = 𝑊+𝑀

Since
𝐴=𝑊+𝑀+𝑂+𝑃

We can write

𝐴
(𝑊 + 𝑀) = 𝑊 + 𝑀 + 𝑂 + 𝑃
𝑊+𝑀
or

𝑘(𝑊 + 𝑀) − (𝑊 + 𝑀) = 𝑂 + 𝑃

or

(𝑘 − 1)(𝑊 + 𝑀) = 𝑂 + 𝑃

Now adding total wage bill on both sides we obtain

𝑂 + 𝑃 + 𝑊 = (𝑘 − 1)(𝑊 + 𝑀) + 𝑊

Then

𝑊
𝑤=
𝑊 + (𝑘 − 1)(𝑊 + 𝑀)

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 10: Macro Theories of distribution – Kalecki and Kaldor’s
 Where 𝑤 denotes the share of wages in the value added of the
industry.

1
𝑤=
𝑀
1 + (𝑘 − 1)(1 + 𝑊 )

𝑀
By denoting 𝑊 as 𝑗 Kalecki reaches at

1
𝑤=
1 + (𝑘 − 1)(1 + 𝑗)

The above relation clearly reveals that the relative share of wages in the value added depends on
the degree of monopoly power (𝑘) and the ratio of the material cost to total wage bill (𝑗). For a
given value of𝑗, the higher the value of 𝑘, the lower will be the relative share of wages.

Kaldor’s theory of Income Distribution

By using the concept of Keynesian multiplier concept Kaldor determines the relative factor
shares in his alternative theory of income distribution. Kaldor suggests that given marginal
propensities to save of capitalists and workers, the share of profit depends on the ratio of
investment to income.

In Kaldor’s theory the economy is assumed to have a state of full employment for a given level
of output. The income in the economy (𝑌) can be classified into two broad categories: wages
(𝑊) and Profit (Π). The wages consists of income generated from manual labour as well as
salaries, while profits are comprised of not only the income from entrepreneurs but also the
income of property owners. The difference between these two income group lies in their attitude
towards consumption and savings i.e the marginal propensity to save of wage earners is
relatively far lower than that of the profit earners or capitalists. Let 𝑆𝑊 and 𝑆Π denote aggregate
savings of wage and profit earners respectively. Now the income identities can be written as:

𝑌 =𝑊+Π

𝐼=𝑆

𝑆 = 𝑆𝑊 + 𝑆Π

Now for a given level of investment, the savings of both income groups are assumed to be
proportionally related to their corresponding incomes. That is, the saving functions can be
written as:

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 10: Macro Theories of distribution – Kalecki and Kaldor’s
 𝑆𝑊 = 𝑠𝑤 . 𝑊 and 𝑆Π = 𝑠Π . Π

Hence from income identities, we obtain

𝐼 = 𝑠𝑤 . 𝑊 + 𝑠Π . Π

I = 𝑠𝑤 . (𝑌 − Π) + 𝑠Π . Π

Or
𝐼 = 𝑠𝑤 . 𝑌 + (𝑠Π − 𝑠𝑤 ). Π

Or
𝐼 Π
= 𝑠𝑤 + (𝑠Π − 𝑠𝑤 ). (1)
𝑌 𝑌
And
Π 1 𝐼 𝑠𝑤
= . − (2)
𝑌 (𝑠Π − 𝑠𝑤 ) 𝑌 (𝑠Π − 𝑠𝑤 )

The equations (1) & (2) imply that for given marginal propensity to saves 𝑠𝑤 and 𝑠Π the share of
capitalist’s income, i.e. the ratio of profit to total income depends on the ratio of investment to
total income. In this model the share of profit in income is invariant with the changes in the two
saving propensities 𝑠𝑤 and 𝑠Π . This invariant relation along with the assumption of full
employment also implies that aggregate demand determines the price level and money wage and
with the increase in investment causes to rise in demand thereby leading to increase in price
levels. The rise in price level will in turn raise the profit margins. On the other hand the fall in
investment and thus in aggregate demand will depress the price level and thereby cause a
compensatory rise in real consumption. As the price level is assumed to be flexible, the
economic system in this model attains stability at the full employment level.

When the two marginal propensities to save differ to each other, i.e. 𝑠𝑤 ≠ 𝑠Π and 𝑠𝑤 < 𝑠Π ,the
model will operate. If 𝑠Π < 𝑠𝑤 , a fall in price would cause a decline in demand and thus lead to
a further fall in prices. Similarly increase in price also have cumulative effect and will drive the
system away from the stability. When 𝑠Π > 𝑠𝑤 , the stability will be achieved. The intuition
behind the mechanism of stability is that at the full employment situation when 𝑠Π > 𝑠𝑤 and
investment exceeds exante saving, the aggregate demand will increase and thereby leading to
Π
increase in prices and profit margins and also share of profit to income ( 𝑌 ). This redistribution of
income in favour of capitalists will also raise aggregate real saving. Opposite situation will arise
when investment falls leading to the aggregate demand. This will consequently decrease the
price level as well as profit margin and finally the share of profit to income. The degree of
1
stability depends on the factor (𝑠 . This is known as the “coefficient of sensitivity of income
Π −𝑠𝑤 )
distribution” as it measures the change in share of profit in income due to change in share of
ECONOMICS PAPER No. 5: Advanced Microeconomics
MODULE No. 10: Macro Theories of distribution – Kalecki and Kaldor’s
 investment in output. When 𝑠𝑤 = 0, the amount of profit equals to the
1
amount of investment and the consumption of capitalist is then Π = 𝐼. According to Kalecki,
𝑠Π
this situation implies that the earnings of capitalists depend on what capitalists spend.

The share of wages on the other hand can be derived as:

𝑊 Π 𝑊 1 𝐼 𝑠𝑤
Since𝑌 = 𝑊 + 𝑃, therefore = 1 − 𝑌 or = 1 − [(𝑠 . − (𝑠 ]
𝑌 𝑌 Π −𝑠𝑤 ) 𝑌 Π −𝑠𝑤 )

Now for stability of the system we must consider the case in which𝑠Π > 𝑠𝑤 . If this condition is
Π
𝑑
not satisfied, one of the factor shares will become negative. It can be shown that 𝑌
<0
𝑑𝑠Π
W Π
𝑑 𝑑
and𝑑𝑠 > 0. When
𝑌 𝑌
< 0 holds, it can said that the greater the capitalists spend, the higher
w 𝑑𝑠Π
W
𝑑
𝑌
will be their earnings. The opposite will happen for workers. The positive sign of implies
𝑑𝑠w
that the share of workers’ income will increase if they save more.

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 10: Macro Theories of distribution – Kalecki and Kaldor’s
 Summary

 Kalecki’s degree of monopoly theory of distribution suggests that share of wages to

domestic national income inversely depends on degree of monopoly power and ratio of
total revenue to wage costs, while the degree of monopoly power is measure by Lerner’s
index.
 Kalecki redefined the degree of monopoly power as the ratio of price to average unit cost
and used this concept instead of Lerner’s measure of monopoly power in order to
reformulate his theory of income distribution.
 This reformulated theory shows that the share of wages relative to value added inversely
depends on the monopoly power and the ratio of aggregate material costs to total wage
bills.
 Kaldor provides an alternative theory by adopting the Keynesian theory of multiplier.
 Kaldor’s model suggests that at the full employment situation and given marginal
propensities to save of capitalists and wage earners the share of profit to income is a
function of the share of investment relative to income.
 In Kaldor’s model the stability of the system can be achieved when capitalists’ marginal
propensity to save is greater than the workers’ marginal propensity to save.
 The intuition behind this theory of distribution is that if capitalists spend more and save
less, their share of profit relative to national income will increase, while for workers the
opposite will happen.

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 10: Macro Theories of distribution – Kalecki and Kaldor’s
____________________________________________________________________________________________________

Subject ECONOMICS

Paper No and Title 12: Economics of Growth and Development I

Module No and Title 11: Kaldor and Pasinetti Growth Model

Module Tag ECO_P12_M11

ECONOMICS Paper 12: Economics of Growth and Development I

Module 11: Kaldor and Pasinetti Growth Model
____________________________________________________________________________________________________

TABLE OF CONTENTS
1. Learning Outcomes
2. Introduction
3. Kaldor’s Model of Economic Growth
4. The Pasinetti model
5. Summary

ECONOMICS Paper 12: Economics of Growth and Development I

Module 11: Kaldor and Pasinetti Growth Model
____________________________________________________________________________________________________

1. Learning Outcomes

After studying this module, you shall be able to

 Know the concept of Kaldor’s model of economic growth and its distinctive
attributes
 Learn that the Pasinetti’s model is an improvement over the Kaldor’s theory of
distribution
 Identify the superiority of Kaldor’s model to the earlier neo-classical growth
models.
 Evaluate which is a better model
 Analyse the intricate details

2. Introduction

Unlike other neo-classical growth models such as the Harrod-Domar model and Solow
model, Kaldor’s model of economic growth (Kaldor, 1957) considers the causation of
technical progress as endogenous and provides a framework that relates the genesis of
technical progress to capital accumulation. The model is based on Keynesian techniques
of analysis and the well-known dynamic approach of Harrod in regarding the rates of
changes in income and capital as the dependent variables of the system. Pasinetti’s model
has made a correction to the Kaldor’s theory of distribution and points out that in any
type of society, when any individual saves a part of his income, he must also be allowed
to own it, otherwise he would not save at all.

3. Kaldor’s Model of Economic Growth

Kaldor’s growth model is based on certain assumptions. The basic assumptions of the
model are the following:

ECONOMICS Paper 12: Economics of Growth and Development I

Module 11: Kaldor and Pasinetti Growth Model
____________________________________________________________________________________________________

Assumptions
1. The model assumes that in a growing economy the general level of output at any
point of time is limited by the availability of resources and not by effective
demand. In other words, the model assumes full employment in the strictly
Keynesian sense – a state of affairs in which the short-run supply of aggregate
goods and services is inelastic and irresponsive to further increase in monetary
demand.
2. Technical progress depends on the rate of capital accumulation and technical
invention.
3. The variables of the model such as income, capital, profits, wages, savings and
investment are expressed in real terms i.e. values are expressed at constant prices.
4. The model assumes an investment function which makes investment of any period
partly a function of the change in output and partly of the change in the rate of
profit on capital in the previous period.
5. Monetary policy is assumed to play a passive role – which means that interest
rates follow the standard set by the rate of profit on investment in the long-run.
The model is consistent with continued price-inflation (with money wages rising
faster than productivity) or with a constant price level. It is also consistent with
constant money wages.
6. It is assumed that there are no effects of a change in the share of profits and
wages, and of a change in rate of profit on capital (or of interest rates) on the
choice of techniques adopted.

Framework of the model

Kaldor postulates the following three relationships:

(i) Given savings propensities for profit-earners and wage-earners.

ECONOMICS Paper 12: Economics of Growth and Development I

Module 11: Kaldor and Pasinetti Growth Model
____________________________________________________________________________________________________

(ii) Investment decisions in any period are governed by the desire to maintain the
capital stock in a given relationship to turnover, modified by any change in the
rate of profit on capital.
(iii)A given technical relationship between the rate of growth in productivity per
person and the rate of growth in capital per person.

Let , , , and denote respectively real income, capital, profits, savings and
investment at time . Income is divided into two categories, wages and profits, where
wages comprise salaries as well as the earnings of manual labour, and profits comprise
entrepreneurial incomes and also incomes accruing to property owners. The familiar
saving-investment identity is represented in the following equation

The three relationships mentioned above can be represented through linear equations as
follows:

(1) Savings Function

(1)
where

(2) Investment Function

where and

(3) Technical Progress Function

ECONOMICS Paper 12: Economics of Growth and Development I

Module 11: Kaldor and Pasinetti Growth Model
____________________________________________________________________________________________________

where and

Equation (1) shows total savings as consisting of a proportion, of total profits and a
proportion of total wages . Equation (2) shows the stock of capital at time
(which is assumed to be equal to the desired stock of capital at time ). Capital stock
at time is equal to a constant fraction, of the output of the preceding period plus a
constant proportion, of the rate of profit of previous period, multiplied by the output of
the preceding period. Equation (3) gives the investment demand function which is
derived from (2) by difference equation, and shows that investment in period ( )
assumed to match to the difference between desired and actual capital at time , and is
equal to the increase in output over the previous period multiplied by the
relationship between desired capital and output in the previous period plus a
proportion, of the change in the rate of profit over that period, multiplied by the
output of current period, since it is implicit in equation (2) that
.

Equation (3) implies that investment of period (expressed as a proportion of the

existing stock of capital) is equal to the expected rate of growth of turnover (which is
assumed to be equal to the actual rate of growth in turnover for the previous period) if the
rate of profit on capital is constant, and it is greater (or smaller) than this if the rate of
profit on capital is rising (or falling). Equation (4) shows the rate of growth of income
(and labour productivity) as an increasing function of the rate of net investment expressed
as a proportion of the stock of capita.

Equations (1), (2) and (3) taken together provide the mechanism underlying savings and
investment. In simple words, the savings function indicates that savings are determined
by the propensity to save out of profits and wages respectively. The investment function
shows that investment is determined by the rate of profit and the changes associated with

ECONOMICS Paper 12: Economics of Growth and Development I

Module 11: Kaldor and Pasinetti Growth Model
____________________________________________________________________________________________________

it. The technical progress function shows that the rate of

growth of income is a function of the rate of investment, capital per person and the
technical progress.

Taking into account the assumptions and the implications of the three functions, the
operation of the model can be examined under two conditions: (a) constant working
population, and (b) Expanding population. In the former case the proportionate rate of
growth in total real income, , will be the same as the proportionate rate of growth in
output per head, , and in the latter case the proportionate change in total real income
will be the sum of the proportionate change in productivity, , and the proportionate
change in the working population, . These two versions of the model are discussed
below:

(a) Constant Working Population

Starting from an arbitrary point of time, , the existing stock of capital, can be
considered as a datum which is inherited from the past. can be taken as given which the
fully employed labour force produces with the capital stock . Similarly, the income
and capital in the previous period, and respectively are given.
Capital stock satisfies the following condition

Since , and are given, equation (3) can be rewritten as

Equation (6) shows that the rate of investment in period 1, as proportion of income of that
period equals the rate of growth of income over the previous period multiplied by the
capital-output ratio of the current period, plus a proportion of the change of the rate of
profit over the previous period. Equation (6) can be written as
ECONOMICS Paper 12: Economics of Growth and Development I
Module 11: Kaldor and Pasinetti Growth Model
____________________________________________________________________________________________________

and equation (1) can be written as

These two equations (7) and (8) jointly determine both the distribution of income
between profits and wages, and the proportion of income saved and invested at .
Given a particular income distribution, the level of profits acts as the equilibrating
mechanism between savings and investment. The level of profits has to be such as to
induce a rate of investment that is just equal to the rate of savings at that particular
distribution of income.

[Figure 1]

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This mechanism is illustrated in Figure-1. Profits as a

ratio of income are measured on X-axis, and savings and investment as a ratio of

income and respectively are measured on Y-axis. The line represents the

saving function given by equation (7) and

represents the investment function given by equation (8). The point of intersection
indicates the short-run equilibrium level of profits and investment as a proportion of
income. If the level of profits is lower than the equilibrium value, investment will tend to
exceed savings until the difference is eliminated through the consequential rise in profits.
Similarly, if the level of profits is higher than the equilibrium value, savings will exceed
investment. The equilibrium will be stable if the slope of the curve exceeds the slope
of the curve, which implies that the following condition is satisfied

The implication of this condition is that the stability of the equilibrium depends upon the
changes in the acceleration coefficient which in turn depends on the changes in the rate of
profit and ultimately on the income distribution. According to Kaldor this is only a
necessary condition of the equilibrium growth path. The stability of the model depends
on two further restrictions (known as the sufficient condition for the stable equilibrium of
the system). These two restrictions are:

(10)

where stands for minimum wages and represents minimum margin of profit.

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The first restriction implies that profits should not be

greater than the surplus available after the labour force has been paid a subsistence wage.
If this condition were not satisfied, investment would be less than that indicated by
equation (3). The second constraint implies that profits should be higher than the
minimum margin of profits ( that the entrepreneurs must obtain in order to continue
further investment. In other words,

the prevailing rate of profit should be adequate so as to induce the entrepreneurs to

continue further investment. According to Kaldor, this minimum margin of profit is the
degree of monopoly. If this condition were not satisfied, full-employment savings
indicated by equation (1) would exceed investment, so that income and employment
would be reduced below the full-employment level to the point where savings generated
by that income are no more than sufficient to finance investment. Figure-2 elucidates this
point.

[Figure-2]

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In the above diagram, the dotted line represents the minimum level of profits ( . If this
minimum level were to fall to the right of E, the equilibrium would not be at point E, but
at point Q as shown in Figure-2 and income will fall to the point where the savings-
income ratio is reduced to the level indicated by Q.

Assuming these conditions are satisfied, the technical progress function ensures the
growth of income and capital from onwards, and the gradual shift of the economy

from a short-run equilibrium to a long-run equilibrium of steady growth. This is

illustrated in Figure-3.

[Figure-3]

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The proportionate growth of capital is measured

on X-axis and the proportionate growth of income is measured on Y-axis.

represents the technical progress function. Point denotes the long-run equilibrium
where the proportionate growth of income is equal to the proportionate growth of capital.
Suppose that the initial rate of investment at , is less than the equilibrium rate of

growth of capital as shown in Figure-3. This implies that the growth of output in

successive

units of time will be greater than the growth of capital, the rate of investment will
increase in the subsequent period so as to make equal to ( , , etc. denote the

rates of growth of income corresponding to the points , , etc. in the diagram.). This
in turn will raise the growth of income in the second period to . By similar reasoning,
the growth of output in the third period will increase to and so on until is reached at
which the rates of growth of income and capital are equal. The indirect effects through
changes in the rates of profit on capital will reinforce this process and any associated
change in the rate of profit on capital will make the rate of increase in investment even

greater.

The long-run equilibrium rate of growth of income and capital is independent of the
savings and investment functions. It depends only on the technical progress function, and
is given by

which is the equilibrium rate of growth in productivity, i.e. that particular rate of growth
of productivity which makes the growth rates of income and capital equal, and which
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(under the hypothesis of a constant population) is itself

equal to both of them. Therefore, it can be concluded that the technical progress function
is essential for establishing the dynamic equilibrium between savings and investment, and
moving the economy towards the steady growth path.

(b) Expanding Population

The model discussed earlier was modified by Kaldor to account for the expansion of the
population. This version of the model assumes the Malthusian theory that population
growth is a function of the rate of increase of the means of subsistence which is assumed
to be equal to the rate of increase in total production.

Kaldor takes into account two limitations of the Malthusian theory. These two limitations
are as follows:
1. For a given fertility rate, the rate of growth of population cannot exceed a certain
maximum, regardless of how fast is the growth of real income.
2. The rate of population growth will rise only moderately as a function of the rate
of growth in income over some interval of the latter before that maximum is
reached.

Given these limitations, and denoting , for the rates of growth of population and
income at time , and for the maximum rate of growth of population, Kaldor express
the relationship between population growth and income growth algebraically as follows.

Assuming to start with that the rate of population growth is (i.e ), in equation

(3) is replaced by and by

Hence, the long-run equilibrium rate of growth of both income and capital becomes
(13)
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where

If (and hence ), the rates of growth of income and population will

continually accelerate until the latter approaches λ. Therefore, in the long-run equilibrium
population must grow at its maximum rate which is indicated by the horizontal section of
the curve in Figure-4.

[Figure-4]

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[Figure-5]

In Figure-4, the rate of growth of income is measured horizontally and the rate of growth
of population is measured vertically. OM curve shows the income growth path. The rates
of growth of income and population will increase continually till the rate of growth of
population approaches λ. This assumes that the shape and position of the technical
progress function as given by the coefficients and in equation (4), and hence
remain unaffected by changes in population. This implies that there are constant returns
to scale. In other words, an increase in numbers, given the amount of capital per head,
leaves output per head unaffected. This assumption may be valid enough in the case of a
relatively under-populated country, but in the case of over-populated countries, the
scarcity of land will cause diminishing returns. With given techniques and capital per
head, an increase in population will cause a fall in productivity. Given the rate of the flow
of new ideas, the curve denoting technical progress function will be lowered by an extent
depending on the rate of increase in population. In this situation, the technical progress
function will cut the capital-axis positively as shown in Figure-5. This implies that in
order to maintain output per head at a constant level, a certain percentage growth in

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capital per head will be required. In this case we

have two points of intersection of the curve with the diagonal line, and , of which
only the latter is stable. If the economy is in a position which is to the left of , the rates
of growth of income and capital will steadily diminish until the growth of income and
capital come to a complete standstill. In this situation it is even possible that the position
of the technical progress function curve is below the diagonal line as shown by the
dotted line in Figure-5.
In this case no long-run equilibrium is attainable, and this may give rise to a situation of
stagnation in the economy.

Whether an expanding population will be consistent with an equilibrium of growth or not

will primarily depend on the relative magnitude of two factors: (1) the maximum rate of
population increase λ, and (2) the rate of technical progress which causes a certain
percentage increase in productivity, in equation (4) when both population and capital
per head are held constant.

4. THE PASINETTI MODEL

The Pasinetti model (1962) postulates a simple relation between the rate of profit, the
income distribution and the rate of economic growth. The model has made a correction to
the Kaldor’s theory of distribution and points out that in any type of society, when any
individual saves a part of his income, he must also be allowed to own it, otherwise he
would not save at all. This implies that the stock of capital which exists in the system is
owned by those (capitalists and workers) who in the past made the corresponding
savings. In other words, even if workers save and own a part of the capital stock (either
directly or indirectly through loans to the capitalists) they will also receive a share of the
total profits. Thus, total profits must be divided into two categories: profits accrue to
capitalists and profits received by workers (Kaldor’s theory of distribution did not
consider this distinction). Pasinetti reformulated the model to eliminate the confusion
regarding the two different concepts of distribution of income: distribution of income
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between profits and wages, and distribution of income

between capitalists and workers. These two concepts of income distribution only coincide
in the particular case in which workers do not save (i.e. there is no saving out wages).

Formulation of the model

Total income ( ) is divided into two broad categories, wages ( ) and Profits ( ).
Aggregate savings ( ) is the sum of worker’s savings ( ) and capitalist’s savings ( ),
and total profits is the sum of profits accruing to the capitalists and profits
accruing to the workers . Therefore,

(1)
(2)
(3)
Substituting identity (3) into identity (1), the latter can be written as

(4)

The savings functions of the workers and the capitalists are defined as given below

(5)
(6)

The condition under which the system will remain in a dynamic equilibrium is the
saving-investment equality ( ). Substituting the savings functions into the saving-
investment identity, the equilibrium condition becomes

(7)

From equation (7) we obtain

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Equation (8) expresses the distribution of income between

capitalists and workers. But, the distribution of income between profits and wages is
something different, and to obtain it, we need to add the share of workers’ profit to

both sides of equation (8). Equation (9) represents the ratio of a part of profits ( ) to total
capital. In order to obtain the ratio of total profits to total capital (the rate of profit), we
must add the ratio to both sides of equation (9). We need to find suitable

expressions for

and

We already know ( ) from equation (9). Writing for the amount of capital that
workers own indirectly – through loans to the capitalists – and for the rate of interest on
these loans, we obtain

We require an expression for ( ) which is given below. In dynamic equilibrium:

Substitution of equation (11) into equation (10) finally gives us:

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By exactly following the same procedure, equation (10)

can be written as

Equation (8) shows the distribution of income between capitalists and workers, equation
(14) expresses the rate of profit, and equation (15) represents the distribution of income
between wages and profits.

Rate and Share of Profits in Relation to the Rate of Growth

In a long-run equilibrium model, it is obvious to assume that the rate of interest is equal
to the rate of profit. If we formulate such a hypothesis and substitute for in equation

(14), we get

Provided that (otherwise the ratio would be indeterminate) the whole

expression becomes

Similarly, equation (15) becomes

Equations (16) and (17) represent the most striking result of the Pasinetti model. It shows
that in the long-run workers’ propensity to save influences the distribution of income

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between capitalists and workers, but, not the distribution

of income between profits and wages. Further, it has no impact on the rate of profit.

The model is based on the institutional principle that profits are distributed in proportion
to the ownership of capital. This implies that in the long-run, profits are distributed in
proportion to the amount of savings contributed. In other words, profits are proportional
to savings, and the ratio of profits to savings are same for the workers and capitalists.
Thus,

In order to determine the actual value of the ratio of profits to savings for the whole
system, substitute the savings functions into equation (18). So, we obtain

which can also be written as

Equation (19) states that in the long-run, when workers save they receive an amount of
profits ( ) such as to make their total savings exactly equal to the amount that the
capitalists would have saved out of worker’s profits ( ) if these profits remained to
them. In other words, the workers will always receive, in the long-run, an amount of
profits proportional to their savings, whatever the rate of profit may be. Hence, the rate of
profit is indeterminate on the part of the workers.

However, there is a direct relation between savings and profits in the case of capitalists
since all their savings come out of profits. Therefore, for any given , there is only one
proportionality relation between profits and savings which makes the ratio equal
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to . According to Pasinetti, “This proportionality

relation can be nothing but , which will therefore determine the ratio of profits to
savings for all the saving groups, and consequently also the income distribution between
profits and wages and the rate of interest for the whole system”.

The Pasinetti model illustrates that there exists a distribution of income between profits
and wages which keeps the system in long-run equilibrium. In order to maintain full
employment over time, a specific amount of investment has to be undertaken which is
uniquely and exogenously (from outside the economic system) determined by technology
and population growth. In this case, the equilibrium rate of profit is determined by the
natural rate of growth ( ) divided by the capitalists’ propensity to save ( ). This is given
by the following equation (the complete derivation is given in the Appendix).

It is important to note that the equilibrium rate of profit is independent of other variables
of the model. This rate of profit as determined by equation (20) keeps the system on the
dynamic path of full employment. In a system where full employment investments are

actually carried out, and prices are flexible with respect to wages, the only condition for
stability is .

Implications of the Model

There are two implications of the model. First, the rate of profit and the income
distribution between profits and wages are determined independently of the workers’
propensity to save ( ). Second, the proportion that profits must bear to savings in the
whole system is given by the capitalists’ saving propensity ( ). The workers’ decisions to
save are irrelevant in this respect. The share of total profits accruing to workers ( ) is
predetermined, and the workers cannot influence it at all.
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Appendix
Suppose there exists an infinite number of possible techniques which is expressed by the
following production function
(21)
The production function shows that output ( ) is a function of capital ( and labour ( ).
It is assumed to be homogeneous of first degree and invariant to time. Further, it is
assumed that labour ( ) is increasing at an externally given rate of growth , so that
. Using these notations, equation (16) can be written as
(22)
By defining , so that , we may write

Substituting into equation (22), we obtain

But on the steady growth path , so that the equilibrium relation is

whence

The equilibrium rate of profit is determined by the natural rate of growth divided by the
capitalists’ propensity to save.

5. Summary

• Kaldor assumes that the saving rate remains fixed. But assuming so he ignores the
effects of 'Life-Cycle' on savings and work.
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• Kaldor model fails to describe that behavioral

mechanism which could tell that distribution of income will be such that the steady
growth is automatically attained.

• There are two implications of the model. First, the rate of profit and the income
distribution between profits and wages are determined independently of the workers’
propensity to save ( ). Second, the proportion that profits must bear to savings in the
whole system is given by the capitalists’ saving propensity ( ).

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 Subject ECONOMICS

Paper No and Title Paper-5 Advance Microeconomics

Module No and Title Module-9: Macro Theories of Distribution—Ricardian and

Marxian
Module Tag ECO_P5_M9

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 9: Macro Theories of Distribution – Ricardian, Marxian
 Table of Contents

1. Learning Outcomes
2. Introduction
3. Ricardian Theory of Distribution
4. Marxian Theory of Income Distribution
5. Summary

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 9: Macro Theories of Distribution – Ricardian, Marxian
 1. Learning Outcome
After studying this module you will learn:

 The theoretical formulation of Ricardian concept of factor income distribution.

 How Marx extended the concept of factor income distribution which is based on
Ricardian surplus theory.

2. Introduction

The principal problem in the political economy by Ricardo is the discovery of those regulations
or principles which govern the shares of factor income distribution. Since Ricardo several
theoretical models were developed to solve this principal problem of income distribution. In this
module first the Classical theory or Ricardian theory of distribution is illustrated this is followed
by how Marx developed his own theory of distribution based on Ricardo’s surplus theory. The
analytical differences between these two theories are explained in the second part.

3. Ricardian Theory of Distribution

Ricardo’s theory emphasizes the fact that the principal problem in the political economy is the
formulation of laws that govern the distributive shares of national income to the factors used in
production. At the outset of his theory of distribution he mentioned the historical fact that “in
different stages of society the proportions of the whole produce of the earth which will be
allotted to each of these (three) classes under the names of rent, profit and wages will be
essentially different”. Although his major concern in the problem of distribution is not only due
to problems related to distributive shares, but to the belief that the whole mechanism of
economic system can be understood by the theory of distribution.

Ricardian theory of distribution was based on the marginal principle and surplus principle. The
marginal principle is used to explain the share of rent and the surplus principle focus on the
distribution of residual part of the value of production between wages and profit. In order to
explain Ricardian model, the following assumptions need to be highlighted:

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MODULE No. 9: Macro Theories of Distribution – Ricardian, Marxian
 (1) The economy is broadly divided into two sector, agriculture and
industry.
(2) In agricultural sector a giant farm produces only one output, say corn. In order to produce corn
three factors land, labour and capital are used. Total land used in production is fixed by the
nature. The other two factors labour and capital are also used in fixed proportion to produce corn.
But the production technology between land and labour are assumed to be a variable coefficient.
(3) The production technology is subject to diminishing returns. This implies that as more and more
of the composite of capital and labour are employed on a given amount of land, the marginal
productivity of variable factor diminishes.
(4) The Malthusian law of population is assumed to operate. This implies that the population will
increase (or decrease) when wages are above (or below) the subsistence level and in the long run
the population will remain stationary at natural rate of wage (i.e., wage at subsistence level).
(5) Capital employed in production is classified in to fixed capital and circulating capital. As the
turnover period is relatively long a small part of annual wage is used as capital, while the
circulating capital is equal to the proper ‘wage fund’.

Based on the above assumptions we can explain how the forces operating in agricultural sector
help determine the distribution of factors’ income in industry.

In the agricultural sector the distribution of income can be explained with the help of figure-2.1.
In the diagram the vertical axis measures quantities of corn and the horizontal axis measures the
amount of labour employed in the production of corn. At given technology, the 𝐴𝑃 curve
represents average product of labour and 𝑀𝑃 curve shows the marginal product of labour.
Because of the assumption of diminishing returns these two separate curves exist. For a given
amount of labour the corn output is uniquely determined. That is, at 𝑂𝐿1 unit of labour total
output is measured by the rectangle 𝑂𝑅𝐺𝐿1. The rent is equal to the difference between product
of labour on marginal land and product on average land. That is the difference between average
and marginal productivity depends on the elasticity of 𝐴𝑃 curve. However, the marginal
productivity of labour (or produce minus rent) is the sum of both wage and profit rather than
simply equal to wage. The wage rate in Ricardian model is determined by the constant supply
price in terms of corn and is independent of marginal productivity of labour. According to
modern economic theory the Ricardian hypothesis implies that at a given wage 𝑂𝑊 there is
infinitely elastic supply of labour. This assumption of infinitely elastic supply of labour is based
on the Malthusian theory of population. It states that population will increase indefinitely when
wages are above the subsistence level and decline when wages are below the subsistence level.
The demand for labour in the Ricardian model is determined by the accumulation of capital
which determines as to how many workers are employed at the wage rate 𝑂𝑊. The equilibrium
is obtained not by the intersection of MP curve and the supply curve of labour, but by the
aggregate demand for labour in terms of corn i.e., wages fund.

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MODULE No. 9: Macro Theories of Distribution – Ricardian, Marxian
 When there is accumulation of capital, the labour force will grow. Hence the ‘agricultural wages
fund’ is determined by the area 𝑂𝑊𝐸𝐿1 . For any given labour 𝐿1 , the profits are determined by
the residue generated from the difference between marginal product of labour and the rate of
𝑃𝑟𝑜𝑓𝑖𝑡𝑠
wages. Hence, the ratio of profits to wages ( 𝑊𝑎𝑔𝑒𝑠 ) determines the rate of profit per cent on how
much capital is employed. This rate of profit per cent of capital employed is indeed equal to the
ratio of profit to wages when capital is assumed to be turned over once a year in such a way that
capital employed is equal to the annual wage bill. In equilibrium situation, both agricultural
sector and industrial sector must generate the same money rate of percentage of profit earned on
capital; otherwise the capital will shift from one form of employment to other.

However in agricultural sector the money rate of profit cannot deviate from the rate of profit
measured in terms of its own product, that is, ‘corn rate of profit’. This happens as in agriculture
both input (in terms of wages) and output is comprised of same product, i.e., corn, while in
industrial sector, the input and output do not consist of same commodities. In agriculture the cost
per worker is fixed in corn, but in industry for a given state of technology the product per worker
is fixed in terms of manufacturing goods. Therefore, the equality in the money rate profit in both
the sectors may occur only through the price adjustment between industrial and agricultural
commodities. The rate of profit in terms of money in industry depends on the rate of profit in
terms of corn produced in agriculture. The rate of profit in terms of corn is subject to margin of
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MODULE No. 9: Macro Theories of Distribution – Ricardian, Marxian
 cultivation. For a given state of technology this margin of cultivation
reflects the extent of capital accumulation. Thus according to James Mill the profit can fall due
to diminishing fertility of soil.

In order to make the whole structure of the Ricardian economy more logically consistent it is
important to assume that wages are not only fixed in terms of corn, but the entire wage income
should be spent on corn. If we relax this assumption, any change in relation between industrial
and agricultural prices will change real wages so that it will no longer be possible to derive the
size of surplus and the rate of profit on capital from the corn rate of profit. The corn rate of profit
can be defined as the relationship between the product of labour and the cost of labour working
on the marginal land. Let us assume that the agricultural products are considered as wage goods
and industrial products are considered as nonwage goods. Now the annual wages fund can be
determined by the total corn output (shown by the area 𝑂𝑅𝐺𝐿1in figure-2.1). Of this total corn
output 𝑂𝑊𝐸𝐿1 is used in agriculture and 𝑊𝑅𝐺𝐸 is employed in rest of the economy. Now
suppose due to protection to agriculture, any increase in 𝑂𝑊𝐸𝐿1 will depress the rate of profit
and reduce the rate of growth. In the same manner all the taxes other than those levied on land
are imposed on profit and these will also reduce the rate of accumulation and growth.

4. Marxian Theory of Distribution

The Marxian theory is mainly based on the Ricardo’s ‘surplus theory’. But it differs analytically
from Ricardo’s concept in many respects. Unlike Ricardo, Marx ignores the concept of
diminishing returns and hence according to him there is no distinction between rent and profit.
Marx considers only two factors of production- labour and capital. The capitalist class owns the
stock of capital and hires labour to produce commodities. The supply price of labour was
considered fixed in terms of general commodities and not in terms of ‘corn’, while share of
profits in total output is determined by the surplus value of output per unit of labour over the
supply price of labour.

In the process of capitalism, capitalist first enters into market with money to
purchase labour power and other means of production. After completion of production he returns
to the market for selling his products and earn money. This entire process is expressed as
𝑀 – 𝐶 − 𝑀/ where 𝑀/ is larger than 𝑀. Marx refers to the difference between 𝑀/ and 𝑀 as
surplus value. According to him this surplus value arises due to higher productivity of labour
relative to value of labour power. The value of any product according to Marx is determined by
the amount of labour required to produce that product. Similarly, the value of labour power is the
labour hour necessary for production and also the reproduction of this labour. This means that

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MODULE No. 9: Macro Theories of Distribution – Ricardian, Marxian
 the labour power is the value of means of subsistence required for
maintaining the livelihood of the labourer. The wage rate received by the worker is equal to the
value of labour power. In the capitalist system the amount of labour time engaged in production
is divided into necessary labour and surplus labour. The portion of total product which is
generated from the necessary labour is given to the labourer in the form of wage and the rest of
the product in which surplus labour is used, is appropriated by the capitalist in the form of
surplus value.

Marx gives the three concepts of capital generated in production under capitalism. These are
value of constant capital, value of variable capital and the surplus value. The value of output is
distributed among these three different components of capital. This relation can be written as

𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑜𝑢𝑡𝑝𝑢𝑡 = 𝐶 + 𝑉 + 𝑆

Where 𝐶 = Constant capital (Fixed capital and raw materials).

𝑉 = Variable Capital (wages & salaries).

𝑆 = Surplus value (interest, dividends and the amount for reinvestment).

According to Marx the value of net product has only two components 𝑉 (wage share) and
𝑆(profit share).

The distribution of surplus value among the various classes namely the capitalist, landlords and
others has been less emphasized in the Marxian theory.

Marx provides the concepts of three important ratios to explain his theory of capitalism. First he
defines the rate of surplus value or rate of exploitation by the ratio of surplus value to variable
capital i.e., 𝑆/𝑉. The magnitude of this rate depends on three factors; the length of labour time,
the amount of product necessary for real wage and finally the productivity of labour. The rate of
surplus can be raised either by increasing labour time or improving labour productivity or
lowering the real wage rate or by some combination of these three.
𝐶
Secondly, the ratio of constant capital to the sum of constant and variable capital, i.e. 𝐶+𝑉 defines
the concept of organic composition of capital in Marxian system. The organic composition of
capital is determined by real wage rate, the productivity of labour, the production technique and
the amount of capital accumulation.
𝑆
Finally the ratio of surplus value to the sum of constant and variable capitals i.e. measures
𝐶+𝑉
the rate of profit. According to Marx there is no share of surplus value obtained by landlord in
the form of rent and surplus value is considered as profit. The rate of profit only depends on the
amount of actual capital employed in production and not on the total investment. Thus the rate of

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 profit varies with the changes in rate of surplus value and the organic
composition of capital. In notation this can be written as

𝑆 𝑆 𝑉 𝑆 𝑆
𝑝= = = [1 − ]
𝐶+𝑉 𝑉 𝐶+𝑉 𝑉 𝐶+𝑉

Or 𝑝 = 𝑠 / (1 − 𝑞)

Where 𝑝 = rate of profit

𝑠 / = Rate of surplus value

and 𝑞 = organic composition of capital.

Marx explains the nature of capital accumulation by using the above relation. The accumulation
of capital happens when the surplus value of production is converted into capital. Thus in the
capitalist system the activities of production, accumulation and reproduction are carried on
continuously.

The demand for labour power increases with the progress of capital accumulation. For any
commodity when demand for a commodity rises, there will be a resulting increase in price of that
commodity and this will continue until the equilibrium is obtained. At equilibrium the price will
be equal to its value. According to Marxian theory this principle of demand theory is not
followed in case of labour power. The equilibrium obtained through demand supply mechanism
is missing in case of labour power under the process of capitalism. Although capital
accumulation increases the labour power, but the equality between wages and labour power is no
longer assumed to happen.

Interestingly, this problem does not exist in the Ricardian model as he adopted the Malthusian
theory of population in order to determinate the wage rate. When capital accumulation takes
place the market wage rate will be higher than the natural wage rate and this will cause a rise in
population. The supply of labour will increase and consequently the market wage rate will fall to
its subsistence level. Marx did not follow the Malthusian theory of population. He solved this
problem by suggesting that there exist a pool of unemployed labour called the reserve army of
labour which causes a continuous downward pressure on the wage level through their active
competition in the labour market. The reserve army constitutes of that part of labour force which
has been displaced by machinery and additional working population. Now under capitalism with
the installation of new machinery, the individual capitalist always attempts to minimize his wage
cost. The overall effect of this wage cost minimizing behavior of all capitalists will result in a
rise of unemployment. This in turn has a negative impact on wage level. This implies that when
there is a tendency for a rise in demand for labour power, there is the counteracting pressure of
the reserve army developed to retard the process of wage increase and vice versa.

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 9: Macro Theories of Distribution – Ricardian, Marxian
 In Marxian theory, there is progressive mechanization of production
process with the growth of capital accumulation. In this context the concept of mechanization
means an increase in the organic composition of capital. From this concept Marx derived his
famous law of the falling tendency of the rate of profit. It has been explained earlier that the rate
of profit is related with the rate of surplus value and the organic composition of capital. That is in
notation 𝑝 = 𝑠 / (1 − 𝑞).

Now if we assume that the rate of surplus value (𝑠 / ) remains constant, the rate of profit (𝑝)
declines with the increase in organic composition of capital. In the process of development 𝑞
shows an increasing trend and hence it can be said that there will be a tendency of 𝑝 to fall. At
the same time the change in 𝑠 / may exactly or more than offset the effects of change in 𝑞.

According to Marx, productivity of labour and relative share of profit increase with technical
progress, while the share of wage declines. This happens due to the act that labourers fail to raise
their real wages with the increase in labour productivity. Thus capitalists appropriate all
additional outputs generated due to technical progress. Marx conformed with Ricardian view that
there exists an inverse relation between share of profit and share of wage. However, there is a
contradiction between them in regard to the direction of the movement of these shares. Marx
suggested that there will be upward movement of profit share, while Ricardo’s prediction was
that share of wage will rise.

Summary

 Ricardian theory of distribution was based on the marginal principle and surplus principle. The
distribution of share of rent is explained by marginal principle concept, while the surplus
principle is used to explain how the residual part of the value of production is distributed
between wages and profits.
 In the Ricardian theory the economy is mainly divided into two sectors-- agriculture and
industry.
 According to Ricardo the rent to the landlord is determined by the difference between product of
labour on marginal land and product on average land.
 In the Ricardian model the marginal productivity of labour (or produce minus rent) is the sum of
both wage and profit rather than simply equal to wage. Wage rate here is determined by the
constant supply price in terms of corn and is independent of marginal productivity of labour.
 The demand for labour in the Ricardian model is determined by the accumulation of capital.
 The profits in this model are determined by the residue generated from the difference between
marginal product of labour and the rate of wages.

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 9: Macro Theories of Distribution – Ricardian, Marxian
  In equilibrium situation of Ricardian model, both agricultural sector
and industrial sector must generate the same money rate of percentage of profit earned on
capital; otherwise the capital will shift from one form of employment to other.
 In Ricardian theory the equality in the money rate of profit in both the sectors may occur only
through the price adjustment between industrial and agricultural commodities. The rate of profit
in terms of money in industry depends on the rate of profit in terms of corn produced in
agriculture.
 The Marxian theory of distribution followed the concept of Ricardo’s ‘surplus principle’.
 According to Marx there is no distinction between rent and profit as he ignores the concept of
diminishing returns.
 Marx considered the supply price of labour determined in terms of general commodities and not
in terms of ‘corn’, while share of profits in total output is determined by the surplus value of
output per unit of labour over the supply price of labour.
 Marx gives the three concepts of capital generated in production under capitalism. These are
value of constant capital(C), value of variable capital (V) and the surplus value(S).
 Marx provides the concepts of three important ratios to explain his theory of capitalism. First he
defines the rate of surplus value or rate of exploitation by the ratio of surplus value to variable
capital i.e. 𝑆/𝑉. Secondly, the ratio of constant capital to the sum of constant and variable
𝐶
capital, i.e. defines the concept of organic composition of capital in Marxian system. And
𝐶+𝑉
𝑆
finally the ratio of surplus value to the sum of constant and variable capitals i.e. 𝐶+𝑉 measures the
rate of profit.
 According to Marx the accumulation of capital takes place when the surplus value of production
is converted into capital and thus in the capitalist system the production, accumulation and
reproduction are carried on continuously.
 In Marxian theory, there is progressive mechanization of production process with the growth of
capital accumulation (increase in organic composition of capital).
 According to Marx, the productivity of labour and relative share of profit increase with technical
progress, while the share of wage declines.
 Marx argued that there will be upward movement of profit share, while Ricardo’s prediction was
that share of wage will rise.

ECONOMICS PAPER No. 5: Advanced Microeconomics

MODULE No. 9: Macro Theories of Distribution – Ricardian, Marxian
____________________________________________________________________________________________________

Subject ECONOMICS

Paper No and Title 12: Economics of Growth and Development- I

Module No and Title 29: Ricardian Theory of Development

Module Tag ECO_P12_M29

ECONOMICS Paper 12: Economics of Growth and Development- I

Module 29: Ricardian Theory of Development
____________________________________________________________________________________________________

TABLE OF CONTENTS

1. Learning Outcomes
2. Introduction
3. Inter-relationship between factors of production
4. Access of capital accumulation
5. Critical appraisal
6. Summary

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1. Learning Outcomes

After studying this module, you shall be able to –

 Explain the relative share of inputs in national output.

 Describe the interdependence of agricultural and industrial sector
 Express the role of saving/capital accumulation as expounded by David Ricardo
and implication for growth

2. Introduction

This chapter deals with the theory given by David Ricardo. He was one of the greatest
pessimist classical economists. He presented his views on economic development in
the famous book titled as, “The principles of Political Economy and Taxation”,
written by him in the year 1817. He predicted that capitalist economies would end up
in a stationery state where there would be no growth and diminishing returns would
occur in the agricultural sector of these economies.
However, in the opinion of Schumpeter, Ricardo never propounded any theory of
development; he rather, simply discussed the theory of distribution. Therefore, the
theory given by him is to be considered as a detour.
The opinion expressed by Schumpeter is, however, one of the opinions but it does not
undermine the importance of Ricardian theory in development economics. Hence, the
present chapter makes an attempt to elaborate this theory of development.
In order to understand the theory given by Ricardo, we will first begin by the
assumptions given by him. He basically assumed a simple economy with two factors
of production producing only one commodity (i.e. corn) with a perfectly inelastic
demand.
Assumptions of the Ricardian theory
The theory of development is based on the following assumptions:
i. All land is used for the production of one commodity i.e. corn.
ii. The law of diminishing returns operates on land.
iii. The supply of land is fixed.

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iv. There are two inputs i.e. labor and capital in the production process and the inputs
are used in fixed proportion.
v. All laborers are working at subsistence wage.
vi. Technology remains constant.
vii. Cost of labor i.e. wage remains fixed.
viii. Demand for labor depends upon the accumulation of capital.
ix. Productivity of labor does not affect the labor demand.
x. Capital accumulation results from profit.
xi. Perfect competition exists in the market.
xii. Demand for corn is perfectly – inelastic.

3. Inter- relationship between factors of production

Ricardo developed his theory based on the inter-relationships among three factors of
the production i.e. (i) landlords (ii) capitalists and (iii) laborers. He elaborated the
inter-relationships among these factors based on the above mentioned assumptions
which can be explained as follows:
The entire production of the economy (i.e. total product of corn) is distributed among
these three factors of production in the form of factor payments i.e. rent for landlords,
profit for capitalists and wages for laborers. Given the total output of corn, the share
of each group can be determined by the following methods:-

Rent:
Rent is defined as the reward for the use of land. It is determined by the difference
between the average product (AP) and marginal product (MP) of labor working on
land. Mathematically, it can be expressed as follows:
Per unit rent = (AP – MP)
Total Rent = (AP – MP) x L
where, AP = Average product of labor
MP = Marginal product of labor
L = No. of laborers working on land

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Profit:
Profit is defined as the reward for the risk undertaken by the entrepreneur and is
defined as the difference between marginal product and wage rate. Mathematically, it
can be expressed as follows:
Per unit profit = (MP – W)
Total profit = Total Output (Bill + Rent)
OQRM = (OQRT + OWLM)
Or Total profit = WLTP
Diagrammatically, estimation of profit can be presented as given below in the figure
1.
Figure 1: Estimation of profit

On the horizontal axis, we have measured amount of labor employed in agriculture

and on the vertical axis, we have measured production of corn.
Further, with the economic development, total output increases and as a result of
which the wage fund also rises. This situation leads to a proportionate increase in the
quantity of labor employed. As there is an increase in the population; the demand for
corn also rises. With the increase in the demand for corn, the prices of corn further
rises. Moreover, with the application of the diminishing returns on land, rent
ECONOMICS Paper 12: Economics of Growth and Development- I
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continues to rise and profits have a tendency to fall. Eventually, increasing population
gradually eliminates profits.
This situation can be explained with the help of the diagram given above (Figure 1).
If the units of labor employed increases from OM to ON, then
Total output = OABN
Total wage bill = OWSN
Rent = WABS
Profit = (MP – W) x L
= (NS – OW) x ON
= (NS – NS) x ON
= (O) ON
= O = NIL
i.e. the profits would turn out to be zero

Rate of profit
Rate of profit can be defined as the ratio of profits to the wages. Mathematically, it
can be expressed as:
Profit
Rate of profit =
Wages
This is to say that the rate of profit depends on wages and there exists inverse
relationship between wages and profit.
This inverse relationship can be understood with the help of the following procedure.
The improvement in agriculture would imply that either the productive capacity of
land has increased or there is employment of better machineries with lesser workers
and producing more output. Now, the increase in the output of corn would cause a fall
in its prices. As a result, the subsistence wage would also fall owing to the
employment of lesser workers but there would be an increase in the level of profits
which would result into capital accumulation. This would further increase the demand
for labor and as a result of this; the wage rate would also rise. This would lead to
increase in population; along with this, the demand for corn would further rise and
this would result into rise in price of corn. Therefore, the economy will witness an
increase in wage rate and profit level will decline.
In this context, David Ricardo gave the concept of stationery state which can be
explained as follows:

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Stationery State:
According to David Ricardo, there is a natural tendency for the profit rate to fall in
the economy so that the country eventually, reaches at the stationery state. The
stationery state can be explained with the help of the following diagram given below
(figure 2).
Figure 2: Total product, rent and total population

On the horizontal axis, we have measured level of population and on vertical axis we
have measured the difference between total product and rent i.e. Total product – Rent.
The nonlinear OP curve is the function of population which shows that as population
increases, then population function will increase at the decreasing rate due to the
operation of law of diminishing returns.
A straight line OW, which passes through the origin measures constant real wage.
The vertical distance between the horizontal axis and the wage rate line OW measures
total wage bill at the different level of population and can be expressed as follows.

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Level of population wage bill

At ON1 W1 N1
ON2 W2 N2
ON3 W3 N3
and it continues further

Foreign trade:
Ricardo has always argued in favor of free trade and he considered it as an important
element for the economic development of any nation.

Taxes:
According to Ricardo, taxes adversely affect the level of investment. The imposition
of taxes also curtails the level of income, profit and thus capital accumulation.
Therefore, he was never in favor of imposition of taxes.

Dynamic Theory:
This theory is dynamic in nature. It studied the changes among various variables
which follow a proper sequence; this reflects the dynamic nature of the theory.

4. Access of capital accumulation

According to Ricardo, capitalists are responsible for capital accumulation which

depends on the profits that are reinvested. A capitalist earns profit and when these
profits are reinvested, it leads to capital accumulation. Therefore, capital
accumulation can be considered as the outcome of the profits
Basically, capital accumulation (CA) depends upon two factors which are given
below:
(i) Willingness to save (WS)
(ii) Capacity to save (AS)
The above discussion can be elaborated with the help of the following functional
form:
CA =F () ---------------------------------------------- (i)
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CA = F (AS, WS) ---------------------------------------- (ii)

Equation (i) depicts capital accumulation as the function of profits and equation (ii)
depicts it as a function of capacity to save and willingness to save.
Further, capacity to save is the main instrument of capital accumulation and it
depends upon the net income of the society which can be expressed as follows:

AS = F (Y)
where Y = Net income of the society

Net income can be defined as the difference between total production (TP) and cost of
laborers working at subsistence wage rate (CW). This can be expressed as follows:
Y = TP- CW
or, Net income = Surplus

Note: the larger is the surplus; the large will be capacity to save (AS).

This surplus is invested by landlords and capitalists and the size of surplus depends
on the rate or profit.

This entire procedure can be expressed with the help of the following flow chart:
Profit  capital accumulation  Total production  wage fund  population
 Demand for corn  price for corn 

The Ricardian theory of development can be summarized in the following points:

* As population increases, interior grade land is also brought under cultivation to
fulfill the increasing demand for corn.
* Rent on the superior grade land further rises and hence, absorb greater share of
the output produced on this type of land. This reduces the share of capital and
labor.
* Reduction in the share of capital and labor causes decline in profit and wages tend
to fall to subsistence level.

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* This process of rising rents and declining profits continues till the output from
marginal land just covers subsistence wages of labor employed. Then profit tends
to zero and hence, the stationery state arrives.

In this situation of stationery state the variables will behave in the following fashion:
i. Capital accumulation tends to stop
ii. Population does not grow
iii. The wage rate continues to be at the subsistence levels.
iv. Technical progress ceases to improve

Therefore, Profit defined as the difference between total product and rent whole
divided by total wage bill (mathematically, Total Product – Rent / Total wage bill) at
different level of population can be expressed as given below:
P N
a) Profit at ON1 Population level = W1 N1 = P1 W1
1 1
P2 N2
b) Profit at ON2 Population level = W = P2 W2
2 N2
P N
c) Profit at ON3 Population level = W2 N2 = P3 W3
1 1
SN
d) Profit at ON2 Population level = SN = UNITY
i.e. the process will continue till profit disappears altogether at point S, and this point
is referred to as stationery state point.

5. Critical Appraisal

The theory of development as propounded by David Ricardo is criticized on the

following grounds:

i. Emphasis on agricultural development: According to Ricardo, agriculture

development plays an important role in economic development of any nation
because industrial development to a great extent depends on the development
of agriculture.

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ii. Profit rate: The rate of profit is very important for economic development as it
determines the capital accumulation which is vital for the development of a
nation.
iii. Importance of saving: Ricardo lays emphasis on capital accumulation through
saving, which is again important for economic development.

6. Summary

The theory of development as propounded by David Ricardo though suffered from some
limitations; is still an important theory in the development economics. He elaborated how
an economy would gradually reach to a state of stationary where capital accumulation
would become stagnant or stop; population would not grow further; labor would be paid
at subsistence level of wage rate and there would not be any technical progress. He
assumed a simple economy producing only one commodity with two factor inputs (labor
and capital); among them the total production of the economy is distributed. Further,
encouraged international trade and did not support the taxation regime. According to him
with the increase in the level of population the inferior quality of land would also be
brought under cultivation.
David Ricardo, however, gave too much importance to agricultural sector while in an
economic system, giving too much importance to any one sector would not result in
sustained growth. Both the sectors of the economy i.e. traditional agricultural sector and
modern industrial sector are inter-dependent as the former provides raw material to the
latter and latter provides market for the former sector.
Overall, the theory of development given by David Ricardo is a benchmark in the history
of development economics.

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Subject ECONOMICS

Paper No and Title 12: Economics of Growth and Development- I

Module No and Title 31: Karl Marx and development of Capitalistic Economy,
Immutable laws of capitalist development; Crisis in capitalism
Module Tag ECO_P12_M31

ECONOMICS Paper 12: Economics of Growth and Development- I

Module 31: Karl Marx and development of Capitalistic Economy, Immutable
laws of capitalist development; Crisis in capitalism
____________________________________________________________________________________________________

TABLE OF CONTENTS

1. Learning Outcomes
2. Introduction
3. Logics of the Marxian theory
4. Theory of capital crisis
5. Criticism of Marxian theory
6. Summary

ECONOMICS Paper 12: Economics of Growth and Development- I

Module 31: Karl Marx and development of Capitalistic Economy, Immutable
laws of capitalist development; Crisis in capitalism
____________________________________________________________________________________________________

1. Learning Outcomes

After studying this module, you shall be able to –

 Explain a very significant departure from the Ricardian-Theory

 Express the theory of exploitation of labour
 Describe the nature of capital

2. Introduction

The entire analytical structure of Marxian theory of distribution is based upon ‘labor
theory of value” which clearly shows that the value of commodity that includes both
goods and services is essentially determined by the labor time which is necessary for
carrying out its production.
The main feature of the Marx analysis is not the labor theory as given by Adam Smith
and David Ricardo in their classical theory rather this theory is further applied to
determine the value of labor.
The theory of Marx assumed that the supply price of labor i.e. real wage level or it can be
termed as subsistence wage will be just enough to permit the labor force to survive and
reproduce.

3. Logics of the Marxian Theory

Karl Marx has presented certain arguments in order to build his theory of capital crisis.
He has explained the following six logics to justify the theory:
1. Labor is absolute and ultimate source of all economic values and it is the result of
pre-labor. In the opinion of Marx, while considering factors of production labor
force shall be considered before taking into account capital.
2. Minimum wages: According to Marx, real wage or subsistence wage should be
determined by the quantity of labor force required to turn out the goods and
services that constitute minimum level of subsistence necessary to maintain the
labor force intact rather than focusing on the welfare of the labor force.

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laws of capitalist development; Crisis in capitalism
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3. To earn surplus: Every capitalist tries to receive surplus from his labor force. And
the surplus, also known as profit is defined as the difference between output
produced per worker and minimum wages paid per worker. It is denoted by S.
4. Every capitalist wants to increase the surplus: The main objective of capitalists is
to increase the surplus value. Therefore, the capitalist always tries to maximize
the level of his profits. He can increase his profit by opting any of the following
ways:
i. By prolonged working hours: If the capitalists prolong the working hours
then the amount of surplus will certainly increase.
ii. By increasing the productivity of labors.
iii. By decreasing the number of hours required to produce the output
equivalent to labors subsistence.
In the opinion of Marx, second method i.e. increase in the productivity of labor is much
more effective than the first and third method as these methods suffers from their own
limitations.
Therefore, the capitalists increase his surplus value by bringing the improvement in
productivity of labor. In order to receive large stocks of capital, capitalists reinvest the
surplus value earned by them. The productivity of labor is directly affected by the
increased amount of capital.
Profit or surplus is determined by the amount of capital invested in the process of
production. Further, Marx splits the capital into two parts which are given below:
(i) Constant capital which is denoted by (c), and
(ii) Variable Capital which is denoted by (v)
The two types of capital can be defined as following:
1. Constant capital (c): It includes plants, machinery and raw material used in the
process of production.
2. Variable Capital (v): Capital devoted to the purchase of labor powers in the form
of wages is known as variable capital (v). It is also known as wage bill.
Now, the total output consists of the following three elements:
i. Variable capital (v)
ii. Constant capital (c) and
iii. Surplus Value (s)
Therefore, the value of total output denoted by o can be expressed in the following form:

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Module 31: Karl Marx and development of Capitalistic Economy, Immutable
laws of capitalist development; Crisis in capitalism
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o = (c + v) + s --------------------------------1
where, (c + v) is the total capital invested in the process of production.
Organic composition of capital: It is defined as the ratio of the constant capital to the
variable capital. Mathematically, organic composition of capital can be expressed in the
form of c/v.
The ratio of surplus: It is defined as the ratio of surplus value to the variable capital.
Mathematically, the ratio of surplus can be expressed in the form of s/v. It is also known
as the rate of exploitation.
Rate of profit: It is denoted by r and is defined as the ratio of total profit to total capital.
Mathematically, the rate of profit can be expressed in the following form:
Rate of profit = Total profit / Total capital
𝑠
Rate of profit = 𝑐+𝑣
Now, dividing numerator and denominator by v, we get
𝑠/𝑣
Rate of profit = r = 𝑐 ------------------------------------------ 2
+1
𝑣

The formula depicted in equation 2 reflects the following conclusions:

i. If s/v = constant then there is a negative relationship between organic composition
of capital (c/v) and rate of profit (r).
ii. If s/v increase but less than the increase in c/v, then the rate of profit, r, will go
down.
iii. If s/v increase faster than c/v, then it is possible to know that in the long run the
rate of profit, r, will fall in any case.
𝜕𝑟
iv. 𝜕 > 0 i.e., there is positive relation between r and s/v
𝜕( )
𝑣

𝜕𝑟
v. 𝜕 < 0 i.e., there is positive relation between r and c/v.
𝜕( )
𝑣

5. The capitalist uses more of constant capital rather than variable capital to increase
their surplus. Further, according to Marx
 Constant capital is the production of pre-labor, and
 Variable capital is the production of current labor.
6. The constant capital is used by capitalist and as a result of this the following will
take place:
i. There will be increase in the total production.

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laws of capitalist development; Crisis in capitalism
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ii. There will be increase in the level of unemployment.

iii. There will be decrease in the wages of labor.

The theory of Marx is based on the above mentioned six logics. Further, Marx believed
that the technological progress tends to increase the organic composition of capital (c/v).
Since the rate of profit is inversely related to c/v, therefore it tends to decline with the
capital accumulation. Marx explained this tendency of rate of profit as the “law of rate of
profit to fall”. Further, he explained this tendency with the help of the following formula
and graph 1.
Formula for rate of profit:
𝑠/𝑣
𝑟=𝐶
+1
𝑉

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laws of capitalist development; Crisis in capitalism
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Explanation of the diagram: The amount of capital is measured on horizontal axis and
total output is measured on vertical axis. Total wage is OT, which is constant and parallel
to horizontal axis. This horizontal line shows that labor supply is constant.
With the initial amount of capital equals to OK, then the total production will be equal to
AK and total profit of entrepreneurs will be equal to AS. Therefore, the rate of profit will
be equal to:
𝐴𝑆1
Profit rate = tan 𝛼 = 𝐴𝐾1

If an entrepreneur increases the amount of capital from OK1 to OK2, then in this situation
the total output will be equal to BK2 and profit will be equal to BS2. Therefore, the rate of
profit will be equal to:
𝐵𝑆2
Profit rate = tan β= ,
𝐵𝐾2

where tan β = tan α1

Now, if we compare both the profit rates, we get:
tan> tanβ
or
𝐴𝑆1 𝐵𝑆2
>
𝐴𝐾1 𝐵𝐾2

Therefore, now we can say that as more amount of capital is engaged in the process of
production, then positively the rate of profit will decline.

4. Theory of capital crisis

It is the capitalist who increases the degree of exploitation either by reducing wages or
prolonged working hours or increase in the productivity to stop the diminishing tendency
of profit. Because each capitalist is busy in using labor saving and cost minimizing
techniques, so ratio of labor (value of surplus) to total production reduces. And as a result
of these factors, the rate of profit also gets reduced. Therefore, with the decline in the rate
of profit, production will no longer remain profitable. Under these circumstances capital
crisis starts to instigate in the economy.
According to Marx, the cause of economic crisis is the poverty and low purchasing power
of general public in the economy of the society. Economic crisis appears in the form of
over production of goods as these goods face difficulties in finding new markets for
themselves leading to fall in prices; thereby forcing the entrepreneurs to curtail down the
production of goods on a large scale. The economy hence witnesses a rise in the level of

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laws of capitalist development; Crisis in capitalism
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unemployment, recession, cut in the wages of laborers, end of credit facilities and thereby
destroying the small capitalists as a result of the economic crises.
However, in accordance with Marx it is not compulsory that the use of labor saving
technology will always follow the above mentioned course of action. It is very much
possible that the revival of economy begins quickly with the change in the technology as
with the fall in prices, reduction in the wage rate and end of speculation crisis there are
huge possibilities that the rate of profit might witness an increase. The increasing level of
profit would then result into new investments and expansion of the economic activities.
As pointed out by Karl Marx, crisis is always a starting point of new and large
investment. Therefore considering the societal point of view these crisis form the base of
next turn over cycle.
The cycle of crisis leads to a period of recession and as recession is always followed by
recovery and therefore it ultimately leads to a situation of economic boom. The period of
economic boom will soon be followed by the situation of crisis and again the same cycle
will be initiated. This leads to the witnessing of Development of capitalist production.
This sequence of action can be expressed with the help of the following figure 1:

Figure 1: Cycle of crisis

Crisis Recession Recovery Boom

In each period of the crisis, bigger capitalist tends to ruins the smaller ones. Also there
increases the exploitation of labor class and thereby towards the end, the “Dooms day” of
capitalist finally arrives.

5. Criticism of the Marxian Theory

The theory of capital crisis as given by Karl Marx is an interesting theory in itself; it
however suffers from certain limitations. For instance a few of its assumptions such as
existence of surplus values, increasing unemployment due to technological innovations
etc have attracted the criticism from various other researchers. The criticisms of the
Marxian theory of capital crisis are given below:
1. The existence of surplus value being unrealistic: The entire analysis of the
Marxian theory is based on the existence of surplus value. The critics, however,
argued that in the real world the relationship between variables are based on real
prices rather than being determined by the values.
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2. Technological progress helpful in increasing employment: According to Marx, the

use of labor saving technology reduces the ratio of labor to total production and
eventually it would lead to increase in the level of unemployment. Therefore he
believed that the technological progress increases unemployment in the society. It
is, however, nothing but the exaggeration of a short term phenomenon.
Technological progress may replace a few of the laborers with the capital
intensive technology but in the long run it will have exactly the opposite effect on
the economy. In the long run, because of the technical progress there would be
more generation of employment opportunities; thereby increasing the level of
income as well as total demand. Hence it would be wrong to assume that owing to
technological progress the level of unemployment would increase in the economy
rather it reflects the pace of economic development of any society.
3. The law of the tendency of the rate of profit to fall is not correct: In the opinion of
Marx, with the process of economic development or technological progress there
is an increase in the organic composition of capital i.e. c/v. Since the rate of profit
is inversely related to organic composition of capital therefore the rate of profit
tends to decline. In the entire context, Marx, however, failed to understand that it
is not compulsory for technological innovation to be a labor saver technique.
There is no denying from the possibility of the technological progress being
capital savers. This would further lead to a decline in the capital output ratio as
due to the technological innovation the productivity of capital increases and
therefore lesser amount of capital would be required to produce the same level of
output. The improvement in the productivity is mostly likely to increases the level
of profits as well as an improvement in the wages being paid to the laborers.
4. The cyclical theory is not appropriate: Karl Marx was the first person who
assumed “cyclic” nature of the crisis as an integral part of his analytical structure.
He stressed on the fact that with the increase in the capital accumulation, the
demand for the consumption goods decreases and therefore, this would lead to
decline in the level of profits. He, however, could not explain the fact that while
the process of economic development is going on, it is not necessary that the
proportion of wages in total income would decline and also the demand for
consumption goods. There is no reason to believe that this decline in the wages
and consumption goods would accompany the process of economic development.
5. Static analysis: According to Schumpeter, the theory of capital crisis developed
by Marx is static in its nature. The economy is, however, dynamic in its nature
rather than being static and therefore this theory is criticized on the ground of its
nature of analysis.

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laws of capitalist development; Crisis in capitalism
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6. Summary

The theory propounded by Karl Marx is based on his six major logics. His theory
basically revolves around the capital crisis which arises due to the use of labor saving
technology. In the situation of crisis the exploitation of workers increases as there would
be a decline in their wage rate. He has been, however, criticized on certain grounds and
Marx failed to provide any justifications to the arguments forwarded against him. There
is no reason to believe that the technological progress would always be labor saving and
it cannot be capital saving. Similarly the law of tendency of rate of profit to fall is also
not a compulsory phenomenon in the long run. With the technological innovations, it is
expected that the rate of profit would certainly increase. Despite of all the criticisms
given against this theory of capital crisis, even in the present day world the capitalists are
governed by the profit motives rather than social motives. This class of people is most
likely to make working class people more unstable and vulnerable compared to any other
class. Therefore, the capitalists might increase in the level of their profits; the conditions
of the laborers might not show the sign of improvement; it might get more miserable.

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laws of capitalist development; Crisis in capitalism
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Subject ECONOMICS

Paper No and Title 12: Economics of Growth and Development I

Module No and Title 4: Solow Growth Model

Module Tag ECO_P12_M4

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TABLE OF CONTENTS
1. Learning Outcomes
2. Introduction
3. Assumptions of the Solow Model
4. Structure and working of the Solow’s model
5. Determinants of the steady state
6. Golden-rule steady state
7. Dynamic adjustment in the Solow’s model
8. Stability of the equilibrium in the Solow’s model
9. Technological progress under Solow’s model
10. Solow’s criticism of Harrod’s model
11. Summary

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1. Learning Outcomes

After studying this module, you shall be able to

 Know the concept of Solow growth model

 Learn the importance of growth model
 Identify the effects of the different factors
 Evaluate the important features according to Solow growth model
 Analyse the Solow growth model

2. Introduction

Economic growth refers to an increase in the level of aggregate output or income an

economy is able to produce in a unit of time. A commonly used measure to compare the
growth across countries during any time period or to compare the growth within a
country across different time periods is the proportional rate of growth of output. It is
denoted by g Y where Y stands for the aggregate output in an economy during a given
time period. If time is measured in discrete units then g Y is defined in percentage form as
 Y 
   100% where Y  Yt  Yt 1 . If time is instead measured continuously
 Y 
1  dY 
then g Y     100% . Here dY stands for a very small change in Y during an
Y  dt 
infinitesimally small period of time dt .

Mathematical and theoretical foundations of neoclassical growth models

Production function of any product is the technological relation between the amounts of
different inputs and the amount of output. Similarly we may define an aggregate
production function for the economy’s aggregate output as composite good. Although
different goods and services have different production functions but the simplification of
writing an aggregate production function allows us to study growth in aggregate output as
caused by the growth in aggregate inputs employed in the economy. Aggregate
production function can be described in mathematical form as

If we shorten the notation by dropping the variable time then aggregate production
function is written as

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Here Y is the amount of aggregate output during any given time period. K and L are the
amounts of labour and physical capital employed during that time period. In the above
production, the parameter A is the technological parameter which denotes the level of
efficiency or productivity of the inputs of labour and capital or equivalently the level of
technology at any point in time. The production function implies that the level of
aggregate output that can be produced from any given amount of physical resources
depends not just on the amount of resources employed but also upon the productivity of
these resources. Productivity of a given kind of machine for example depends upon how
much skills the worker operating on it has. Similarly productivity of a given skilled
worker depends upon the kind of machine she operates on. The functional relation F can
take many forms. For example it might be a Leontief Production of the form
as used by Harrod and Domar. It might also be a Cobb-Douglas
production function of the form where α is a constant and 0≤α≤1. Cobb-
Douglas production function has very nice properties which made it useful to Solow in
his analysis. The main difference between the Leontief function and the Cobb-Douglas
function is in terms of the possibility of substitution between the labour and capital.
While Leontief function does not allow any substitution, Cobb-Douglas on the other hand
allows various substitution possibilities between the two factors. The lack of substitution
between the two factors in the Leontief function implies that both factors will always be
used in a fixed proportion . This ratio is also known as the capital-labour ratio. On the
other hand the possibility of substitution between the labour and capital in Cobb-Douglas
function implies a variable proportion of capital and labour i.e. a variable capital-labour
ratio.

We have studied in microeconomic theory a law called the ‘law of variable proportions’,
which states that if we increase the use of any factor relative to others factors then the
marginal productivity of the relatively more used factor eventually starts to decline and
may even become negative after some level of relative use. The production function
which allows for substitution between factors therefore must be of the form that allows
for diminishing marginal productivity of the both labour and capital given the level of
technology.

Growth in the level of aggregate output can occur either due to growth in resources
employed or growth in their productivities. In other words, growth would occur if either
of K, L or A increases over time as no factor would be underutilized in the long run.

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3. Assumptions of the Solow Model

1. Solow assumes that there is a composite good whose production level is given
by .
2. Capital and labour are the only factors of production.
3. Output in any period is net of depreciation.
4. Aggregate production function is linearly homogeneous i.e. has
constant returns to scale.
5. There is perfect substitutability between capital and labour.
6. Both factors have positive but diminishing marginal products, i.e.
while . Solow made this assumption as a necessary condition
for existence of a unique and stable equilibrium.
7. Population is growing exponentially at a constant rate of growth . Therefore the
labour supply curve is given by . The constant growth rate of labour is
given exogenously to the Solow’s model.
8. Stock of capital is in the form of accumulation of the composite good.
9. Economy in every period saves a constant proportion of that period’s output.
10. Investment planned is always equal to savings planned. Thus Solow does away
with the short run Keynesian framework since according to him the neoclassical
model is best to analyse growth which is inevitably a long run phenomenon.
11. There is always full employment of factors of production. It requires smooth
clearing of goods, labour and capital markets. Markets clear to always bring about
equality between demand and supply.
12. There are no adjustment lags in any market in the economy.

4. Structure and working of the Solow’s model

Structure and working of the Solow’s model

Let us first look at the production function used by Solow. The aggregate production
function of the composite good is given by . Since Solow assumes constant
returns to scale

where

If we choose then we can write the above production function as .

Now if we define as and as then the aggregate production can be rewritten as

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where . We can draw this production function in a two-dimensional

diagram by placing output per capita on the axis and capital per capita on the
axis as shown in the figure below

Figure 1

Notice that the production function is increasing but shaped concave to the origin evident
from falling slope of tangent to the curve as increases. In other words
but . The particular shape of this one variable production function owes itself to
positive but diminishing marginal productivity of capital in the aggregate production
function . We can show this by using some basic calculus.

&

Since Solow assumes that and therefore and .

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Solow assumed such fine properties of the production function in order to ensure the
existence of unique stable equilibrium in his model. The following conditions are known
as Inada conditions:

1. for all
2. for all
3.
All these properties are satisfied by Cobb-Douglas production function.

Further the savings per capita in any time period are given by and since investment
per capita is assumed to be equal to savings per capita always therefore flow of gross
investment per capita during any time period is given by . Investment leads to
addition to the stock of capital per capita. Due to investment therefore, capital per capita
gets increased at the rate of in each period. But during any given time period
capital per capita also gets reduced due to increase in the size of the labour force which
grows at the constant rate and the depreciation of capital equipment. Let the rate of
depreciation be denoted as . The capital per capita therefore gets reduced at the rate
of .

The rate of change of capital stock per capita is therefore given by

This is the basic differential equation of the Solow’s model.

According to this equation,

1.
2.
3.
In words we can say that capital per capita i.e. capital-labour ratio grows so long as gross
investment per capita is more than the reduction in capital per capita and it stops growing
(or changing) when net investment per capita is zero. The capital per capita that will
persist in the very long run in the Solow’s model is known as the Steady-state capital
stock per capita. It is denoted as . Here the steady state is defined as the state of the
economy which when reached will continue. In such a state, also known as the balanced
growth path, the ratio of capital to output also remains constant. This can be seen from
the production function . Since remains constant in steady state and so will the

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per capita output . Therefore the ratio also remains constant. In fact, the
steady state to be precise is defined as the state of an economy in which output capital
ratio remains constant over time.

The steady state level of capital per capita can be obtained graphically as shown in the
figure 2 below. In the figure, the investment per capita function is drawn as the red
coloured concave curve . Given the production function, position of gross
investment curve is dependent on the exogenously given savings ratio . A higher savings
ratio raises the curve upwards and as , the investment per capita curve approaches
the output per capita curve shown in blue. The green coloured straight line is the
reduction in capital curve due to growth of population and depreciation.

Figure 2

At any given capital-output ratio (say ), the difference in the height of the output per
capita curve and investment per capita curve tells us the consumption per capita whereas
the difference in the height of the investment per capita and the reduction in capital per
capita line tells us the net investment per capita in the economy. If net investment per
capita is positive, which happens for all positive capital-labour ratios less than , capital
stock per capita in the economy grows. If net investment per capita is negative then
capital stock per capita in the economy falls. This occurs at capital-labour ratios more
than . Only at , addition and reduction to capital per capita offset each other and
therefore eventually remains steady at .

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5. Determinants of the steady state

We determine the steady state capital stock per capita by solving the following equation

The left hand side of this equation is the slope of the line joining the origin to point on the
output per capita curve . Since this function is concave, so the left side of the
above equation falls as increases and rises as decreases.

This implies that the steady state which satisfies the above equation would fall (rise)
according to as the right hand side increase (decrease).

Given the production technology, steady state capital stock per capita therefore

1. Rises as increases. It means that economies which save more will have higher
capital-stock per capita and therefore higher output per capita in the steady state.
2. Falls as increases. It means that economies whose population are growing faster
will have smaller capital-stock per capita and therefore smaller output per capita
in the steady state.
3. Falls as increases. It means that economies with higher depreciation rates of
capital equipment will have smaller capital-stock per capita and therefore smaller
output per capita in the steady state.
We can thus write as a function of , i.e. .

6. Golden-rule steady state

Note that the above analysis tells us that given the rate of population growth and rate of
depreciation, all possible capital stock per capital can be maintained steadily by choosing
an appropriate savings ratio in the economy. In a centrally planned economy where
government can control the rate of savings, it will choose that rate of savings and in turn
that level of steady state capital stock per capita which maximizes consumption per capita
in the steady state. Such a rate of savings is called the golden rule level of savings and
denoted as . that maximizes consumption per capita is known as golden rule steady
state capital stock per capita and denoted as . To derive , let us maximize steady state
consumption per capita with respect to .

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Note that in steady-state, net investment per capita is zero and thus savings per capita
must be . The necessary first order condition for maximization is that

Figure 3

We have shown the golden rule steady state in the figure 3 above. Notice that here we
have drawn here the investment per capita and output per capita in different steady states.
The gap between the two curves measures the consumption per capita in steady state. The
gap is maximized when the slopes of the two curves are equal as given in the condition
derived just above. A welfare maximizing planner would choose s_g in order to
maximize the consumption per capita of the future generations.

7. Dynamic adjustment in the Solow’s model

Dynamic adjustment in the Solow’s model

The process of eventual adjustment towards a steady state is not instantaneous. It may
take lots of decades depending upon the present state of an economy and level of steady
state capital stock per capita. Let us analyse the adjustment over time of an economy
towards steady state under the following two situations:
1.
If initially the economy happens to be at a lower level of capital labour ratio
compared to steady state then the economy will converge towards the steady state
level of as shown in figure 4 below.
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Figure 4
We may also show the dynamic path of k as a time path as shown in figure 5 below.

Figure 5

2
If on the other hand the economy happens to be at a capital labour ratio level
higher than that in steady state, then the adjustment of will be downwards
towards steady state as shown in figure 6 below.

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Figure 6

8. Stability of the equilibrium in the Solow’s model

There are two equilibriums in the above Solow’s model. One is at and . The
former equilibrium point is unstable as a slight deviation from it will push the economy’s
capital labour ratio towards . The only stable equilibrium in this model is since
whenever the economy is not at , and then the economy will converge towards it as
seen above.
The existence of such a stable equilibrium in the Solow’s model is guaranteed by the
Inada conditions assumed about the production function. Without such assumptions, there
may be no stable equilibrium with positive capital labour ratio in the Solow’s model as
shown in the figure 7 below

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Figure 7
In the above figure, at all positive levels of , net investment is negative and thus the
economy will converge to zero capital.

9. Technological progress under Solow’s model

Solow treated technological progress as exogenously determined from his model. He

assumed that let the rate of growth of technology be given by . He further assumed that
this increase in productivity is labour-augmenting and therefore the aggregate
production function can be written as .Such a technical progress is known as
Harrod-neutral technical progress. Now the output can be thought of a function of two
factors, capital and effective labour . We can denote the output and capital per
effective labour as and respectively. The rate of growth of effective labour
is given by rate of growth of technology and rate of growth of population, i.e. . The
rest of the analysis for determining the steady state is same as earlier, except that now the
reduction in capital is not just due to growth of labour but due to growth of effective
labour. The figure 8 below shows a similar analysis to what we have done earlier.

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Figure 8

1. In the steady state under technological progress, the capital per effective labour
now remains constant at and output per effective labour remains constant
at .
2. The output per unit labour however grows at the rate of exogenous technological
progress . This implies that according to Solow, the only source of sustainable
growth in per capital output or income is the technological progress.
3. The difference in rates of technological progress will explain the persistent
difference in the rate of growth of per capita output across economies.

10. Solow’s criticism of Harrod’s model

According to Solow the peculiar result of knife-edge instability in the long run
equilibrium growth of an economy in the Harrod’s model arose due to a rigid assumption
of inflexibility in the substitution between labor and capital in the Aggregate production
function. Using the assumption of a Leontief production function within the framework
of his own model, Solow demonstrated that anomalous result of Harrod is just a special
case of a more general Solow’s model under very restrictive assumption regarding
production technology i.e. the assumption of fixed capital-labor ratio.
To show Harrod’s result as a special case, Solow assumes that let the aggregate
production function be given

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The above production function represents a fixed coefficient production function with
every unit of output requiring units of capital and units of labor. The basic differential
equation in the Solow’s model therefore becomes

Since this is piece-wise defined function, we can rewrite it as

If we now draw the per capita savings as a function of , then we get the kinked red line
as shown in figure 9 below

Figure 9

The per capita savings function is a ray from origin with slope till and becomes

horizontal beyond that. Here is nothing but the warranted rate of growth in the Harrod’s
model. Here Solow considers following three possibilities regarding the natural rate of
growth:
1. i.e. natural rate of growth is more than the warranted rate of growth
Notice that the line lies above the kinked per capital savings curve for all
levels of . Therefore and would steadily fall to zero no matter what it
was initially. At any point of time output is given by since capital is the

scarce factor as becomes less than . Note that since capital to labor ratio is

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fixed at in production, a fall in towards 0 implies growing unemployment of

labor.

2. i.e. natural rate of growth is equal to the warranted rate of growth

In equilibrium remains constant but its level is determined by where it was to
begin with. If then labor is in short supply and so will fall to and

stay there over time. It is also visible in the figure above that beyond , the line
lies above the kinked portion of the per capital savings curve and therefore
capital stock per capita must fall. If on the other hand, then remains
at the level since reduction in capital per capita is equal to gross investment per
capita. It means that unemployment rate is preserved at its initial level.

3. i.e. natural rate of growth is less than the warranted rate of growth
In this case, we have a stable equilibrium at . Capital is growing at
the same rate as labor but the capital employed is growing at the same rate as
labor due to fixed proportions. This would lead a growing excess capacity but
would not result in fall in price of capital relative to labor owing to fixed
proportions in production.

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11. Summary

1. In the Solow’s model, the economy will converge to its very long run steady-state
equilibrium which depends upon the savings rate and the growth rate of
population, both given exogenously in the model.
2. In the steady-state, both capital labour ratio and output per capita remain constant.
Therefore in an economy with given production technology, there is no
sustainable source of growth in per capita output.
3. In the steady-state, the output, labour and capital all grow at the same rate .
4. If the savings rate and population growth rate are identical for the two economies
(say one developed and one developing) and they have access to the same
technology, then both the economies would converge to same level of output per
capital over time. Thus given everything else equal, a capital poor low income
economy would eventually catch up with the richer economy.

5. A balance growth path exists only when and not always.

6. Even if a balanced growth path exists, it involves growing unemployment of labor
or capital.
7. The reason for such result lies in the lack of substitutability between factors.
8. A market clearing model in which demand for factors in the form of their
marginal product curves gets equated to supply of factors through smooth adjustment
of their relative prices would guarantee that we can get a stable balanced growth path
for any natural rate of growth of population.

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Subject ECONOMICS

Paper No and Title 12: Economics of Growth and Development-I

Module No and Title 20: Endogenous Growth; Intellectual capital: Role of

learning, education and research
Module Tag ECO_P12_M20

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Module 20: Endogenous Growth; Intellectual capital: Role of learning,
education and research
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TABLE OF CONTENTS
1. Learning Outcomes
2. Introduction
3. Endogenous Growth Models: An Overview
4. Romer’s Model
5. Lucas Model
6. Summary

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1. Learning Outcomes

After studying this module, you shall be able to

 Know the meaning of endogenous growth

 Learn the weaknesses of neo-classical growth models
 Identify the impact of technological spillovers on the aggregate production
function of the economy.
 Evaluate the role of human capital on economic growth of any economy
 Analyse the limitations of the basic models of endogenous growth

2. Introduction

In majority of the neo-classical growth models, technical change is considered as an

exogenous factor, though it is an important driver to growth. However, Kaldor and Arrow
proposed the idea of technology having some endogenous component as well. Further,
the Solow Model which is largely based on the idea of residual effect of technical change
and that in the long run, the economies move on the stagnant growth path was challenged
on several grounds e.g. its unrealistic result for exogenous technical progress accounting
for more than 70 per cent of output growth; prediction of convergence among poor and
rich economies in the long run; failure to explain widened growth disparities and long run
imbalances in factor productivities etc. These shortcomings lead to the birth of
endogenous growth theory, which has been formally developed since mid-1980s
following the pivotal works by Romer (1986, 1990) and Lucas (1988) and which has
soon become the dominant framework for studying economic growth and development.
One of the main reasons why economists have grown interested in endogenous growth is
because of an empirical puzzle. The neo-classical model predicts that countries with low
per-capita incomes grow faster than those with high income, so that over time per-capita
incomes converge. But a broader set of data indicated that the rich countries may follow
an ever increasing growth path while the poor economies move on at much slower pace
leading to divergence among these economies. It is also found that even the same type of
technology and rate of investment may not lead to similar results across the economies.
This has also been found that even the same level of foreign direct investment may lead
to different results in different economies with different human and physical capital.
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Thus, it is important to find some endogenous factors that determine the rate of growth of
the economies.

3. Endogenous Growth Models: An Overview

The endogenous growth models in contrast to the exogenous growth models believe that
the growth of the economy is more influenced by the internal processes and policies,
structure etc. than merely the external factors. Endogenous growth is long-run economic
growth at a rate determined by forces that are internal to the economic system, that
govern the opportunities and incentives to create technological knowledge. In the long
run the rate of economic growth, as measured by the growth rate of output per person,
depends on the growth rate of total factor productivity (TFP), which is determined in turn
by the rate of technological progress. The neoclassical growth theory of Solow (1956)
and Swan (1956) assumed the rate of technological progress to be determined by a
scientific process that is separate from, and independent of, economic forces.
Neoclassical theory thus implies that economists can take the long-run growth rate as
given exogenously from outside the economic system. Endogenous growth theory
challenges this neoclassical view by proposing channels through which the rate of
technological progress, and hence the long-run rate of economic growth, can be
influenced by economic factors. It starts from the observation that technological progress
takes place through innovations, in the form of new products, processes and markets,
many of which are the result of economic activities. The firms may learn from experience
how to produce more efficiently, a higher pace of economic activity can raise the pace of
process innovation by giving firms more production experience. The spread of this
knowledge and consequently, the macroeconomic effects of any technical progress
depend upon the spillover effects that again depend upon the level of human capital in the
society. Here lies the important difference between the endogenous growth theories and
the neo-classical growth theories which take technical progress as exogenous factors. In
this context, we can discuss here, the basic models of endogenous growth given by
Romer and Lucas.

4. Romer’s Model

This model challenges the basic assumption of neo-classical model of growth which
refers to applicability of diminishing returns in the long run. This model states that many
economies have been able to exhibit increasing returns to scale even in the long run. This
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long run increase in per capita income has been possible not due to the external factors
but due to the endogenous technical changes. Any change in technology leads to positive
effects on productivity of the factors whose combined effect may lead to increasing
returns. Romer has discussed about the spillover effects of technical change. He states
that a technical change, any innovation, new method of doing things do not remain with
one individual only. He stresses that the physical part of any new technology may be
privately owned but the knowledge part soon becomes a public good. Since, production is
a social phenomenon, any new method used in a production unit soon spreads to another
units as under perfect markets, the factors are perfectly mobile and they carry their
knowledge along with them. A highly productive factor in a production unit not only
himself/herself higher level of productivity but also raises the level of productivity of his
fellow beings in the production unit. Thus any investment in new knowledge, research
and development or human capital may have greater social returns as compared to its
private returns. As a result the national output would be increasing at an increasing rate
and new unit of human capital investment will yield increasing rate of return. In the
simple framework of Romer’s model, we take a single sector Cobb-Douglas type
production function which has inherent assumptions of homogenous sectors and
applicability of constant returns to scale. But in order to include the spillover effects of
technology on aggregate production of the economy, it includes a separate variable of
human capital. Therefore, the production function in Romer’s model can be read as
follows:
Y=A.Kα.L1-α .Kβ
Here, ‘A’ is the efficiency parameter of given level of technology, ‘L’ stands for units of
labour, ‘K’ for units of capital which is presented here both as the physical as well as

human capital. ‘Y’ is the level of output and α and β are the output elasticies of the
respective variables. Above equation cn also be written as
Y=A.Kα+β.L1-α ..................(1)
At any point of time, the change in output would be possible only due to change in
physical as well as human capital and units of labour along with the changes in factor
productivity, therefore,
dy Y K Y L
 .  . ……….(2)
dt K t L t

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The value of different components of this equation can be found by differentiating

dy K
equation (1) partially with respect to ‘K’ and ‘L’ and if we further denote as ΔY,
dt t
L
as ΔK and as ΔL, equation (2) takes the following form:
t
Y K L
 (   )  (1   ) ……(3)
Y K L
If both output and capital stock grow at the same and constant rate, then
Y K
  g and since the growth of labour force depends upon the growth of
Y K
L
population which grows at a natural rate determined exogenously,  n , where, ‘n’ is
L
natural growth rate of population, equation (3) can be rewritten as

g  (   ).g  (1   ).n
 .n
or, g n 
1  (   )
here, g-n shows the growth of per capita income. In absence of any spillover effects of
technological change, the constant returns to scale will be applicable and under such
conditions since β=0, this will mean that g-n will also be equal to zero which indicates
that in absence of spillover effects, the economy will not experience any growth in its per
capital income and hence, the constant returns to scale will be applicable. However, in
Romer’s model all the factors viz. capital stock, labour and technology are assumed to be
working together whose productivities mutually influence each other, the value of β will
always be positive. Hence, if β > 0, the growth of per capita income will also be positive,
therefore, g-n > 0. This is possible only due to the spillover effects of the technology
within as well as across production units.
Although, this model has provided an important breakthrough, yet its applicability
to the developing economies is questionable because many of the assumptions of this
model find little validity in the developing economies e.g. the assumption of single sector
economy may be unrealistic for the dual economies of the developing countries.
Moreover, the developing countries largely face the problem of structural rigidities which
are hardly mentioned by Romer. The analysis of these rigidities is very important in the
context of the spillover effects of technology, research and development or any any type
of knowledge across all production sectors of the economy. Due to these structural
bottlenecks many a times the developing economies are found not to be using the full
capacity of their available capital even though they fight with the problem of shortage of
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capital. Romer Model is silent about all such factors in the context of developing
economies while these very factors actually lead to slower growth of the per capita
income in these economies even though they are using the same technology as has been
used by the developed economies. Romer’s model is silent about the causes and effects of
all such problems of the developing countries. Actually, the developing economies lack
sufficient incentives to invest in physical as well as human capital. This has a great effect
on supply of savings, capital formation and hence on growth of income. Besides, during
the transition phase developing countries also undergo the process of reallocation of
resources which are not generally efficient ones, particularly during the initial phase of
the transition. This inefficient reallocation of resources at any point of time has medium
as well as long term effects upon the growth of income of the ecomnomy. But all these
factors have been ignored by Romer’s model. Hence, the developing economies find little
guidance from this model.

5. Lucas Model

Lucas’ model of growth emphasises the importance of human capital in the growth the
economy. He states that it is difference in attainments of the human capital that has led to
worldwide economic disparities. Lucas states that the developed countries went through
the process of industrial revolution long time back in the history of these economies. The
incentive to earn more profits has led to invest in creation of knowledge so that the
conditions of normal profits can be converted in to the long run capacity to earn
supernormal profits. However, this process was hardly understood by the developing
economies. This is their misconceptions or overindulgence in the idea of capital stock
being the sole and most important determinant of economic growth that has led to wrong
strategies and hence they were not able to experience the same level of growth as
experienced by the economies which have experienced the epochs of industrial
revolution. These misconceptions or little understanding of the importance of human
capital has led to lower investments in human capital for a long time in these economies.
Consequently, these economies have lagged much behind the developed economies who
are growing at faster rate leading to divergence across the poor and rich economies of the
world. Actually, the physical capital and human capital are not substitute to each other,
they are rather essential complements of each other. Therefore, a greater investment in
human capital also leads to greater productivity of the physical capital. In this framework
Lucas model emphasises that the skilled workers and the new technology are inseparable
from each other. In order to measure the effect of human capital accumulation on income,
we can identify two separate components of total savings in an economy – these savings
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can be used for increase in physical capital stock and/or these savings can also be used
for enhancing the level of human capital in the economy which will lead to an increase in
productivity of labour as well as capital in future time period. To further elaborate the
growth model given by Lucas, let us first look at the basic equation of this model. in this
equation output is considered to be the function of physical as well as the human capital
stock.

y  k  h1 ……….…(1)
Here, ‘h’ stands for human capital and ‘k’ for physical capital. As discussed above, part
of the savings are spent in accumulation of physical capital and its part is spent on
accumulation of human capital. These two components of savings can be expressed in the
following manner. Firstly, taking ‘s’ as the part of savings being used for accumulation of
physical capital i.e.
k (t  1)  k (t )  sy (t ) ……………..(2)
Secondly, the proportion of savings spent on accumulation of human capital can be
expressed as
h(t  1)  h(t )  qy (t ) …………….(3)
Thus, sy(t) and qy(t), respectively show the total amount of resources spent on
accumulation of physical and human capital. For self-sustained growth of the economy,
‘y’, ‘k’ and ‘h’ should be growing at the same rate. The rate of economic growth actually
depends upon the growth of investment in physical as well as human capital. Therefore, it
is important to find the ratio of investment in human capital to that of physical capital.
For this, we would have to find the growth of these two types of capital in an economy.
a). Growth of Physical Capital: The growth of physical capital can be derived from
equation (2) by putting the value of y(t) and also dividing both sides by k(t). The resultant
equation can be written as
1−𝛼
𝑘(𝑡 + 1) − 𝑘(𝑡) 𝑠. [𝑘(𝑡)]𝛼 [ℎ(𝑡)]1−𝛼 ℎ(𝑡)
= = 𝑠. [ ]
𝑘(𝑡) 𝑘(𝑡) 𝑘(𝑡)
If we take h(t)/k(t)=r, this equation can be written as
k (t  1)  k (t )
 sr1
k (t )
Similarly, we can also derive the equation for growth of human capital
b). Growth of Human Capital:
h(t  1)  h(t )
 qr 
h(t )

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Since, in the long period the growth of human capital as well as the physical capital are
equal, therefore,
q
sr1  qr  or r 
s
This ‘r’ can be used as long term growth rate and since in the long period, the growth of
income, physical capital and human capital are the same, therefore,
y (t  1)  y (t )
 sr1  qr 
y (t )
 s q1
Thus, the long term growth of the economy depends upon the rate of physical capital
formation as well as human capital formation. It is the human capital which compensates
the fall in growth of output due to applicability of diminishing returns on physical capital.
The human capital investment, rather ensures increasing returns by its internal as well as
external positive and output stimulating effects. The internal and external effects of
human capital formation in any economy can be discussed as below.
(i) Internal Effects of Human Capital: According to Lucas, the total time of a
human being, particularly a worker, can be divided in to two components – the time
spent in production and the time spent in accumulation of human capital. If we
denote the proportion of time spent in production as µ(h), then the time spent in
accumulation of human capital will be 1- µ(h). In any economy, the size of the
labour force as well as its productivity per hour significantly influences the level of
output. Therefore, instead of having merely the size of the work force, Lucas has put
forth the idea of ‘effective labour force’ which is shown as the product of size of the
labour force and the time spent on producing goods and services for a given level of
human capital.

Ne    (h).N (h).dh
0

Here, N(h) is the size of labour force and Ne is the effective labour force. Thus, we
can express production as function of physical capital stock and effective labour
force.
Y=f(K, Ne)
The level of human capital would not only have a macro economic impact upon the
aggregate output of the economy but will also have accrue certain private benefits to
the holders of the human capital as in a competitive market economy, wages are paid
according to the marginal product of workers. Since, the workers with higher level of
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human capital are more productive, they would have higher earnings. Total wages in
an economy for a given level of human capital can be calculated as follows:
Total wages = f ' ( K , Ne ).h. (h)
Where, f ' ( K , Ne ) shows the marginal productivity of labour.
(ii) External Effects of Human Capital: Increase in level of human capital formation
in any economy, undoubtedly increase the level of productivity of a single worker
but also have an overall effect upon the average productivity of the economy as a
whole. Even a single worker with higher human capital in a production unit has
huge ripple effects in the production unit. Same is true for the economy as a whole.
But in a perfectly competitive economy, it is generally assumed that human capital
of an individual would not affect the average level of human capital, yet its
opposite is not true as average level of human capital in any economy determines
the minimum target to be achieved by average workers to ensure their
employability. Hence human capital investments and its attainments by the private
individuals are largely determined by the average level of human capital for the
country as a whole. This is termed as external effect of human capital. In order to
know about the external effect of human capital, it is important to know about the
average level of human capital which can be calculated as below.

 h.N (h).dh
ha  0


 N (h).dh
0

Here, ha is average level of human capital in a country. Now, we can easily adjust
our production function by incorporating internal as well as external effect of
human capital. First of all let us have production as function of capital and effective
labour force.
Q  A.K  (t ) Ne1 (t )
Putting the value of Ne, we get
Q  A.K  (t )[  (t ).h(t ).N (t )]1 
Incorporating the external effects of human capital or the average level of human
capital for the society as a whole
Q  A.K  (t )[ (t ).h(t ).N (t )]1 [ha (t )]
Any increase in time for accumulation of human capital i.e. 1-  (t ) will raise the
individual as well as the average level of human capital of the society which will

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have positive effects upon the level of output at an increasing rate. Thus, the
economy would grow at a faster rate due to applicability of increasing returns to
scale in the production sector. The change in human capital, which is the main force
behind the faster growth of the economy, can be measured as
hˆ(t )  [h (t )] .G[1   (t )] ,
a

here, G is the growth of human capital and it is always positive i.e. G > 0 but the
existing level of human capital or say the knowledge, which the society has attained
so far, will have diminishing returns to output, therefore,  < 0. In order to simplify
the analysis if we simply assume  =1, then the equation showing the change in
human capital at any point of time can be shown as
hˆ(t )  [h (t )].G[1   (t )]
a

We can discuss here two extreme cases, one is when whole of the time is spent in
accumulation of human capital i.e.  (t ) = 0, and the second when whole of the time
is spent in production only i.e. 1-  (t ) = 0 . In the first case, the change in human
capital will be
hˆ(t )
G
[h(t )]
i.e. the economy can achieve the highest growth of human capital equal to G while
in second case there will be no change in growth of human capital and it would be
zero. In absence of any change in human capital, since diminishing returns to scale
are applicable to the existing level of human capital, the economy will also grow at
diminishing rate for any change in its inputs. But in real life, the value of  (t ) or
that of 1-  (t ) varies between 0 and 1 which shows that the economy moves on a
continuous growth path and the rate of growth of the economy will be higher for
higher growth of human capital. This fact points towards the fact that the
applicability of diminishing returns can be postponed by increasing investment in
human capital. Thus, the economies with higher rate of growth of human capital
experience a higher growth of income and the economies with lower investment in
human capital will experience the lower growth of income, leading to divergence
between the two types of the countries. Through this fact, Lucas pointed out that the
the gap between growth of rich and poor economies can be explained by the gap in
investments in human capital in these economies.
Finally, Lucas also differentiated between the optimum growth path and the
equilibrium growth path. By optimal path he means that the society wants to
maximise its utility function by achieving the optimal level of per capita income
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with an optimal combination of K(t), N(t), c(t), h(t) and ha(t). On the other hand,
equilibrium path means that there is simultaneous equilibrium in all firms and
households as well the economy as a whole. Assuming that ha(t) is exogenously
determined and it is expected that each individual will follow the same path so that
the actual behaviour coincides with the expected behaviour and there is no gap
between demand and supply (i.e. AD = AS), for given physical and human capital
stock. The solution will be achieved in both the cases if h(t) = ha(t). Any divergence
between the two will mean the divergence from the equilibrium as well as the
optimal growth path.

Like Romer’s Model, Lucas’ Model also gave little attention to the structural rigidities of
the developing economies. There is no doubt that these economies have low level of
productivity due to low level of human capital but at the same time these economies are
also suffering from the problem of misallocation and misutilisation of the available
resources. These economies not only face the problem of skilled workers but also
underutilisation of existing human capital. Due to lack of opportunities and low returns to
human capital, there is little incentive to invest in human capital by private individuals.
On the other hand, the skilled workers have a greater tendency to migrate to other
countries in search of better opportunities. These are the workers which the economy
needs the most and their emigrations means the drain of most essential and productive
resources. This loss of intellectual capital has a huge and long run adverse effect upon the
economic growth of the poor countries. These aspects are ignored by Lucas’ model of
endogenous growth, yet there is no doubt that the developing economies can learn a lot
from this model that it is the higher level of human capital that can ensure higher
productivity of other resources. So, it must be attained as well as retained.

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6. Summary

The neo-classical growth theories, though recognised the importance of technology but
they took it as exogenous factor and therefore they failed to explain why even in the long
run the richer economies are growing at a rate higher than the poor economies and why
same level of investment and technology do not produce similar results across the
economies. Moreover, there is little evidence of convergence between the two types of
the economies as suggested by Solow’s framework. The answers to many of these
questions can be found in the endogenous growth models which state that the returns to
capital and a given technology mainly depend upon the inherent characteristics of the
economy and therefore, the factors to growth are endogenous rather than being
exogenous. This is the spillover effect of the technology and the spread of human capital
that leads to increasing returns to scale even in the long run. Higher is the speed of these
spillovers, higher will be the rate of growth of the economy. Similarly, the economies
which invest more in human capital are also able to grow at a higher rate than the
economies which spend less on the same. Yet, these models have proposed one sector
economy which finds little applicability in the developing economies which are dual in
character and face the problem of structural rigidities. In the developing economies,
misallocation of the resources is as big an issue as the availability of the same. Though
the endogenous growth models are the important breakthrough in the existing knowledge
of growth economics but they hardly deal with the important issues of the developing
economies which hinder the process of growth even with increase in investment in
human capital as well as creation of technology.

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Subject ECONOMICS

Paper No and Title 12: Economics of Growth and Development - I

Module No and Title 21: AK model — Explanations of cross country differentials

in economic growth
Module Tag ECO_P12_M21

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economic growth
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TABLE OF CONTENTS

1. Learning Outcomes
2. Introduction
3. The Cross-Country Difference in Growth
4. Simple AK Model
5. AK Model with Human Factor
6. Summary

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1. Learning Outcomes

After studying this module, you shall be able to

 To discuss about the basic flaws of the neo-classical growth models.

 To examine the cross-country differentials in economic growth.
 To understand the basic framework of the AK model.
 To examine the long run growth of the economy in simple AK model
 To understand the long run growth of economy by incorporating human capital in
the AK model.
 To review the limitations of the AK model

2. Introduction

The neo-classical approaches to economic growth were largely considered to be

unsatisfactory due to several inherent flaws. These models view improvements in total
factor productivity (technological progress) to be the ultimate source of growth in output
per worker, but they do not provide an explanation as to where these improvements come
from. In the language of economists, long-run growth is determined by something that is
exogenous in the model. Diminishing returns to the accumulation of capital, which plays a
crucial role in limiting growth in the neoclassical model, is an inevitable feature of an
economy in which the other determinants of aggregate output, namely technology and the
employment of labour, are both given. However, there is a class of model in which one of
these other determinants is assumed to grow automatically in proportion to capital, and in
which the growth of this other determinant counteracts the effects of diminishing returns,
thus allowing output to grow in proportion to capital. These models are generally referred
to as AK models, because they result in a production function of the form Y = AK, with
‘A’ constant. The AK model is actually considered the first version of endogenous growth
theory. However, the earlier version of this model go back to Harrod (1939) and Domar
(1946) who assumed an aggregate production function with fixed coefficients. Frankel
(1962) developed the first AK model with substitutable factors and knowledge
externalities, with the purpose of reconciling the positive long-run growth result of Harrod-
Domar with the factor substitutability and market clearing features of the neoclassical
model. The Frankel model showed a constant savings rate, whereas Romer (1986)
developed an AK model with intertemporal consumer maximization. The idea that
productivity could increase as the result of learning-by-doing externalities, was put forth
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by Arrow (1962). Then Lucas (1988) developed an AK model where the creation and
transmission of knowledge occurs through human capital accumulation. Similarly, we can
cite a number of other models which have followed the AK framework. Hence, it is
important here to examine this approach and its contribution to economic theory.

3. The Cross-Country Difference in Growth

There are large differences in per capita income across countries. Of total world
income, 42 per cent goes to those who make up the richest 10 per cent of the world’s
population, while just 1 per cent goes to those who make up the poorest 10 per cent (World
Bank, World Development Indicators, 2014). This points towards not only unequal
distribution of world income across different countries but also differences in their growth
rates. The key sources of these differences can be numerous depending upon the national
policies and institutions. Hence, it is very important to understand how some countries can
be so rich while some others are so poor as the income differences have major welfare
consequences. The differences in growth rate across economies have actually widened the
income inequalities. Acemoglu (2007) has indicated that even in the historically brief post-
war era, the world has witnessed tremendous differences in growth rates across countries
and these have ranged from negative growth rates to average rates as high as 10 per cent a
year. During this period some of the countries have grown at a faster pace, some at a slower
rate and some stagnated after growing for a short period. It is being believed that much of
these differences in economic growth cannot be wholly attributed to the post-war era alone
as during this period, the “world income distribution” has been more or less stable, with a
slight tendency towards becoming more unequal. Further, the Maddison data has suggested
that much of the divergence took place during the 19th century and early 20th century. It
is important to observe that the process of rapid economic growth started in the 19th, or
perhaps in the late 18th century and then takes off in Western Europe, while many other
parts of the world do not experience the same sustained economic growth. The high levels
of income today in some parts of the world are owed to this process of sustained economic
growth, and this process of differences in economic growth has also caused the divergence
among nations. This divergence took place at the same time as a number of countries in
the world started the process of modern and sustained economic growth. Therefore
understanding modern economic growth is not only interesting and important in its own
right, but it also holds the key to understanding the causes of cross-country differences in
income per capita today. The endogenous growth theories largely owe these differences or
divergences in economic growth to the institutions, policies, technologies along with the

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levels of investments and other transitional dynamics. These theories also pointed out the
importance of investment in human capita along with that of physical capital to explain
these divergences. These theories also point out that the technology differences across
countries include both genuine differences in the techniques and in the quality of machines
used in production, but also differences in productive efficiency resulting from differences
in the organization of production, from differences in the way that markets are organized
and from potential market failures and how the human factors handle these technologies
with effects on productive efficiency. Hence, a detailed study of “technology”, physical
capital and human capital as correlates of economic growth is necessary to understand both
the world-wide process of economic growth and cross-country differences. It is important
to examine the sources of income differences among countries that have (free) access to
the same set of technologies, but do not generate sustained long-run growth. A full analysis
of both cross-country income differences and the process of world economic growth
requires models in which technology choices and technological progress are endogenized.
Hence, we can start with simple AK model.

4. Simple AK Model

As we have already discussed that the first version of endogenous growth theory was AK
theory, which did not make an explicit distinction between capital accumulation and
technological progress. In effect it lumped together the physical and human capital whose
accumulation is studied by neoclassical theory with the intellectual capital that is
accumulated when innovations occur. An early version of AK theory was produced by
Frankel (1962), who argued that the aggregate production function can exhibit a constant
or even increasing marginal product of capital. This is because, when firms accumulate
more capital, some of that increased capital will be the intellectual capital that creates
technological progress, and this technological progress will offset the tendency for the
marginal product of capital to diminish. In the special case where the marginal product of
capital is exactly constant, aggregate output Y is proportional to the aggregate stock of
capital K:
Y = AK
where A is a positive constant that reflects the level of technology and ‘K’ here is taken in
a broader sense as it includes physical as well as human capital. This model shows constant
marginal product to capital (as MPk = dY/dK=A) indicating that long run growth is
possible. Thus, AK model is a simple way of illustrating endogenous growth. Assuming a
closed economy, the savings are equal to investment under conditions of full employment.

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Since savings are the function f income and capital depreciates at a constant rate i.e. ‘δ’ the
change in capital stock can be traced through following equations.
I = S = s.Y = s.AK
and, since capital depreciates at a constant rate, the change in capital stock i.e. 𝐾̇ can be
expressed as 𝐾̇ = 𝑠. 𝑌 − 𝛿. 𝐾. This change in capital stock can also be represented by a
diagram given below.

Figure 1: The AK Model

In this figure Y-axis show output per worker while the X-axis show the capital stock. The
line Y=AK having a constant slope shows the constant marginal productivity of capital; the
line S=s.Y is the gross investment line while the line δK shows the depreciation line or the
total replacement investment. The difference between the gross investment line and the
replacement line i.e. area between S=s.Y line and δK line shows net investment in the
economy which is positive and increasing.
The growth of capital stock can be found by dividing both sides of the equation
showing change in capital stock with ‘K’, we get
𝐾̇ 𝑌
= 𝑠. − 𝛿
𝐾 𝐾
Since, Y=AK, i.e. Y/K =A, therefore, above equation can be rewritten as
𝐾̇
= 𝑠. 𝐴 − 𝛿
𝐾
As, growth of output is equal to the growth of capital stock,
𝑌̇ 𝐾̇
= = 𝑠. 𝐴 − 𝛿
𝑌 𝐾

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Further, assuming that s.A > δ, the growth of capital stock as well as growth of output i.e.
𝑌̇
> 0, showing that the economy will be ever increasing as compared to the Solow model
𝑌
which shows that the income increases at a declining rate before stagnating for a given
technology. We can easily compare how the paths of per capita income would differ in
case of Solow model of exogenous growth and the AK model of endogenous growth. This
comparison has been shown through figure 2.

Figure 2 The AK Model and the Solow Model Compared for Rising Saving Rate

Source: Miguel Lebre de Freitas, Introduction to Economic Growth.

Figure 2 compares the impact of rate of change in savings upon the growth of income. The
top part of the diagram shows the levels of income and the bottom part shows the growth
rate of the same. In the upper part, we can see that a once for all increase in saving rate in
t0 time period leads to an ever growing income curve (shown as ln y) in case of AK model
while in case of Solow model, the income increases initially but ultimately reaches at the
same level after t1. This can be observed through the angle ‘γ’. In case of Solow type growth
path, as savings increase or say, due to exogenous change in technology in t0 time period,
the income curve immediately and its slope rises as we can see that the size of angle ‘γ’
increases from γ0 to γ1 but after t1 time period, it again comes back to the previous level

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i.e. γ0. However, in case of AK type growth the increase in income is forever, shown by an
ever increasing curve and once for all increase in the size of angle γ0 to γ1. The growth path
can better be elaborated through growth rate of income in the lower segment of the diagram.
We can see that in t0 time period, the growth rate of income increases immediately from γ0
to γ1 but the Solow type growth path shows that it comes back to γ0 level while the AK
growth path shows that growth of income is constant at stays at γ1 level even in the long
run.

5. AK Model with Human Factor

In its more realistic form, we can also add labour as an input along with capital. In this
context, first of all, we can discuss Arrow’s model with knowledge spillovers. In this
model, the production function for final output can be written as
𝑌 = 𝐵. 𝐾 𝛼 𝐿1−𝛼 (1)
which is a Cobb-Douglas type production function showing constant returns to scale with
inputs K and L. In a model with technology and population growth as exogenous factors,
the population, equal to labour input L, can be normalized to one and the individual firm
takes total factor productivity B as given. However, we suppose that B is in fact
endogenously determined. Specifically, the accumulation of capital generates new
knowledge about production in the economy as a whole. In particular, we assume that
𝐵 = 𝐴𝐾 1−𝛼 (2)
where, A is constant and is greater than zero i.e. A > 0
That is, an incidental by-product of capital accumulation by firms in the economy is the
improvement of the technology that firms use to produce. Technological progress,
modelled as a by-product of capital accumulation, is external to the firm. Combining the
two preceding equations gives
𝑌 = 𝐴. 𝐾. 𝐿1−𝛼 (3)
This is exactly the AK model above, noting that L = 1. However, in further formulation of
the AK model, we can include human capital as a separate variable having a positive effect
upon the level of output. Thus, more skilled labour force will be assumed to produce more
output than an unskilled individual, and the total stock of such “skills” is called human
capital. Crucially, human capital can be accumulated through education. Thus, both types
of capital can be accumulated—this turns out to imply that the model has similar properties
to the AK model. In this perspective, we can have a production function of the following
type:
𝑌𝑡 = 𝐴𝑡 . 𝐾𝑡 𝛼 𝐻𝑡 1−𝛼 (4)

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where Kt is physical capital and Ht is human capital. So, growth is determined by
𝑌𝑡̇ 𝐴̇ 𝐾̇ 𝐻̇
= 𝐴𝑡 + 𝛼 𝐾𝑡 + (1 − 𝛼) 𝐻𝑡 (5)
𝑌𝑡 𝑡 𝑡 𝑡
Assuming that like physical capital, human capital also depreciates for given attainments.
This can be understood in this way that if a person does not updates its knowledge, the
knowledge accumulated so far depreciates in a dynamic economy. For simplifying the
analysis, let us assume that both the physical as well as human capital depreciates at the
same rate, then we can easily derive the equations for accumulation of each type of capital
stock. This simplification is expected not to disturb the relevant conclusions of the model.
Hence, the change in physical capital stock is defined as
𝐾̇𝑡 = 𝑠𝑘 𝑌 − 𝛿𝐾 (6)
and the change in human capital stock can be defined as
𝐻̇𝑡 = 𝑠𝐻 𝑌 − 𝛿𝐻 (7)
Similarly, we can also define the capital output ratios for both the physical and human
capital.
The capital output ratio for physical capital is
𝐾
𝑥𝐾 =
𝑌
while for human capital, it is
𝐻
𝑥𝐻 =
𝑌
For finding the growth rate of physical as well as human capital, we can divide the
equations (6) and (7), respectively with ‘K’ and ‘H’ and substitute the above values of
respective capital output ratios, we have
𝐾̇ 𝑠
= 𝑥𝐾 − 𝛿 (8)
𝐾 𝐾
and
𝐻̇ 𝑠
𝐻
= 𝑥𝐻 − 𝛿 (9)
𝐻
In AK model, for a steady growth path, the growth for physical and human capital stock
should be the same i.e.
𝐾̇ 𝑠 𝑠 𝐻̇
= 𝑥𝐾 − 𝛿 = 𝑥𝐻 − 𝛿 = 𝐻 (10)
𝐾 𝐾 𝐻
This implies that
𝑠𝐾 𝑠𝐻 𝑥𝐾 𝑠𝐾 𝐾 𝑠𝐾
= ⇒ = ⇒ = (11)
𝑥𝐾 𝑥𝐻 𝑥𝐻 𝑠𝐻 𝐻 𝑠𝐻
So, along the steady growth path, the stock of human capital can be written as
𝑠
𝐻 = 𝑠𝐻 𝐾 (12)
𝐾

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This value of ‘H’ can be put in equation (4), we get

𝛼
𝑠𝐻 1−𝛼
𝑌 = 𝐴𝐾 ( 𝐾)
𝑠𝐾
Or,
𝑠 1−𝛼
𝑌 = 𝐴. 𝐾 (𝑠𝐻 ) (13)
𝐾
Again, under the steady growth path, growth of income is equal to the growth of capital
stock
𝑌̇ 𝐾̇
=
𝑌 𝐾
𝐾
from equation (8) and putting the value of 𝑥𝐾 = 𝑌 , we get,
𝑌̇ 𝑠𝐾 𝑌 𝑠𝐻 1−𝛼
= − 𝛿 = 𝐴. 𝑠𝐾 ( ) −𝛿
𝑌 𝐾 𝑠𝐾

Hence,
𝑌̇
= 𝐴. (𝑠𝐾 )𝛼 (𝑠𝐻 )1−𝛼 − 𝛿 (14)
𝑌
Thus, allowing for both type of inputs- physical as well as human capital, which are
continuously accumulated produce same results as that of the AK model as equation (14)
is another form of growth of income in AK model with capital input only. In this case, the
steady growth rate is 𝐴. (𝑠𝐾 )𝛼 (𝑠𝐻 )1−𝛼 − 𝛿 instead of AS – δ as in simple AK model. Here,
the simple saving rate has been replaced with a geometric average of the two saving rates
in the two factor model while leaving the broader implications unchanged. Lucas on the
other hand tried to incorporate the role of human capital in terms of effective labour force
and an attempt has also been made to measure the human capital accumulation in terms of
allocation of time between production time and time spent in human capital accumulation.
Thus the effect of human capital accumulation can be observed in the standard AK model.
Although, we have already discussed the Lucas model in the lesson on endogenous growth
models, yet it is important here to look at it briefly in AK framework. For this purpose, we
now consider a simple endogenous growth model with human capital accumulation in
which Y is a function in physical capital K and "effective labour" h. L, where ‘h’ is level of
human capital per person and ‘L’ is the size of the labour force.
𝑌 = 𝐾 𝛼 (ℎ. 𝐿)1−𝛼
Lucas assumes that human capital per person evolves according to
ℎ̇ = (1 − 𝑢)ℎ

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where u is time spend working and (1 – u) is time spend accumulating skills. The growth
rate of h is given by
ℎ̇
= 1−𝑢
ℎ
Since h enters the production function like labour-augmenting technological progress in
the Solow model, we can conclude that the long-run growth rate of output is equal to the
growth of human capital stock which is equal to the growth of physical capital stock on
𝑌̇ ℎ̇ 𝑌̇
steady state equilibrium path. Hence, 𝑌 = ℎ or 𝑌 = 1 − 𝑢
This implies that every policy measure which affects u has an impact on the long-run
growth rate.
Thus, we have observed that AK model gives a new framework for the long run
growth of the economies. However, there are still some reasons to doubt the predictions
about long-run growth generated by this class of models. The first line of criticism is related
with the non-accumulable factors. In the real world, there are factors of production that are
in fixed supply, such as land, or that cannot simply be accumulated indefinitely such as
energy. Remember that the AK model results are of a knife-edge variety: Any move away
from all factors being accumulable, and we move back to the Solow model results.
Moreover, similar treatment to all type of human capital is also criticised by many as they
say that the strict parallel between human capital and physical capital in the model just
described is probably not completely accurate. For instance, not all expenditures on
education will produce the same effect on output. The marginal boost to aggregate output
of primary teaching is altogether different to that of higher education; training the non-
skilled informally and the formal training of the professional also differ in their marginal
returns. By clubbing all these different types of human capital together hardly proposes an
effective policy suggestion for countries with a varied structure of human capital. Another
source of the difficulties faced by the AK model is that it does not make an explicit
distinction between capital accumulation and technological progress. In effect it just lumps
together the physical and human capital. Lastly, it also criticised for not giving any
explanation for any possibility of convergence among the economies.

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6. Summary

Now let us summarise what we have learnt in this module:

 AK model is considered the simplest version of endogenous growth. Although, we

can observe many neo-classical approaches in the AK framework such as that of
Harrod and Domar who assume an aggregate production function with fixed
coefficients but the important implication of modern AK type production function
is that it can explain long-run growth using the same basic assumptions as the
neoclassical model but adding knowledge externalities among firms that
accumulate physical capital.
 The idea that productivity could increase as the result of learning-by-doing
externalities, was most forcefully pushed forward by Arrow and then Lucas
developed an AK model where the creation and transmission of knowledge occurs
through human capital accumulation.
 All these models clearly state that the economies can experience long term growth
or the increasing returns to scale through knowledge spillovers and since the
developed countries have a higher stock of human capital as compared to the
developing ones, the former grow at a faster pace than the latter one and the
economies of two types of economies go on diverging from each other.
 However, like all the endogenous growth models, the AK model also does not
address the structural rigidities of the developing countries which hinder the process
of creation as well as accumulation of human capital which is an important factor
for ensuring sustained long run growth of the economies.

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Subject ECONOMICS

Paper No and Title 12: Economics of Growth and Development - I

Module No and Title 2: Harrod-Domar Growth Model - I

Module Tag ECO_P12_M2

ECONOMICS Paper 12: Economics of Growth and Development - I

Module 2: Harrod-Domar Growth Model - I
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TABLE OF CONTENTS

1. Learning outcomes
2. Introduction
3. Harrod’s Model
4. Structure and working of Harrod’s Model
5. Summary

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1. Learning Outcomes

After studying this module, you shall be able to:

 Know about the Harrod’s model.

 Understand the structure and working of Harrod’s model.
 Learn the types of rate of growth for an economy

2. Introduction

Roy Harrod and Evsey Domar worked separately to develop their highly similar models of
economic growth and business cycles. The two economists expanded the short-run
Keynesian framework to analyze the growth process in the developed economies. Both of
them criticized the basic Keynesian framework of income determination in the short run
for ignoring the role of investment to create more capacity for the production of output.
The investment in physical capital, according to these economists, has a dual role. Dual
role of investment here means that investment spending generates income on one hand and
also increases the productive capacity of the economy on the other hand. Increase in the
income as a result of increase in investment is called the demand side effect while the
increase in the productive capacity of economy due to investment is called the supply side
effect. Both the economists were interested in finding out an equilibrium growth path
which would guarantee a full employment in some sense. Although the two models of
Harrod and Domar are similar in many respects but they have some crucial differences as
well. Let us investigate the two models below in turns

3. Harrod’s Model

The model was first given by Harrod in his 1939 paper in the ‘Economic Journal’. His first
concern was to find that does there exist an equilibrium growth rate of output which if the
economy grows at then it will continue to grow at the same rate moving over time?His
second concern was to investigate that whether such an equilibrium growth path is stable
in the sense that if ever the economy grew at some different rate then would it automatically
move towards equilibrium growth rate in due time?

Basic assumptions of the Harrod’s Model

1. Savings and investment refer to income of the same period. Both Saving and
Investment are net, i.e. over and above the depreciation.
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2. The economy saves a constant proportion of its income, which implies that the
𝑑𝑆
marginal propensity to save (MPS) i.e. is equal to average propensity to save
𝑑𝑌
𝑆
(APS) i.e. 𝑌. Since if MPS ≠ APS, then the latter could not stay constant.
3. Income is determined by investment through the multiplier process while
investment is determined through the process of accelerator.
4. If plans of investment are realized then the firms don’t change the rate of desired
investment whereas if plans are under realized i.e. actual investment is less than
planned investment or over realized i.e. actual investment is more than planned then
firms increase or decrease respectively the rate of desired investment.
5. The economy is assumed to begin with full employment of capital.
6. There are no lags in the adjustment between demand and supply, especially between
investment and creation of productive capacity.
7. Aggregate output in the economy can be written as a function of aggregate physical
capital and aggregate labor measured in suitable units respectively i.e. 𝑌 = 𝐹(𝐾, 𝐿).
It is further assumed that there are constant returns to scale in the aggregate
production function which means that if both the factors are changed by some equal
proportion then output also gets by same proportion.
8. General Price level and the rate of interest remains fixed. It means that the relative
prices of capital and labor remain constant as the economy grows. This assumption
has a very crucial implication for the Harrod model. A constant relative factor price
𝐾
implies a constant capital-labor ratio i.e. 𝐿 in the economy over time. Some authors
𝐾
like Branson formulate the requirement of fixed in the Harrod model using an L-
𝐿
shaped fixed proportion production function of the form 𝑌 = Min(𝑎𝐾, 𝑏𝐿)as
shown below.

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If the production function is conceived to be of a perfectly complementary form as

above then a change in relative factor prices from say (w/r) to (w/r)’will not change
𝑏
the ratio 𝑎 in which the factors are used. Although there is no need to assume such
a functional form of aggregate production provided we assume constant relative
price of factors.
9. The implication of constant returns to scale along with fixed proportion of factors
implies constant capital-output ratio as well as constant labor-output ratio overtime.
The constancy of capital-output ratio is a very important assumption of the Harrod’s
model.
10. There is no role of government in influencing aggregate demand and supply.

4. Structure and working of the Harrod’s Model

4 Harrod wanted to find out that rate of growth of investment or output which will sustain
itself overtime. In order to find that out Harrod does the marriage of multiplier and
accelerator to arrive at his most fundamental growth equation. Keynesian multiplier can be
written says that
∆𝐼
∆𝑌 = (1)
𝑠
Here Y stands for the aggregate output, I for net investment and s for the savings ratio in
the economy.
The accelerator theory of investment tells us that the net investment planned or desired in
any period in an economy is a fixed multiple of the expected change in output during that
period i.e.
𝐼 = 𝐶𝑟 ∆𝑌 (2)
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∆𝐾𝑟
Here 𝐶𝑟 = stands for the desired change in capital stock per unit increment in output.
∆𝑌
𝐶𝑟 is also known as desired incremental capital-output ratio. If we combine equations 1 and
2 above by eliminating ∆Y then we get that rate of growth of planned investment
∆𝐼 𝑠
=𝐶 (3)
𝐼 𝑟
We know that in any short run equilibrium of the economy, planned net savings must equal
planned net investment i.e. 𝐼 = 𝑆 which implies that in a short run equilibrium 𝐼 = 𝑆 = 𝑠𝑌
or 𝐼/𝑌 = 𝑠. In other words, to keep their ratio fixed, planned investment and output must
∆𝐼 ∆𝑌
grow at the same rate i.e. = if the economy has to always remain in short run
𝐼 𝑌
∆𝑌 ∆𝐼
equilibrium at all points in time while growing. Substituting in place of in equation 3
𝑌 𝐼
above, we get
∆𝑌 𝑠
=𝐶 (4)
𝑌 𝑟
Equation 4 is the most fundamental equation in Harrod’s model since it says that if
economy is growing at the rate given in the right hand side of equation 4, then planned
investment is always equal to planned savings along the moving growth path. To see why
this happens we can rewrite the equation 4 as𝐶𝑟 ∆𝑌 = 𝑠𝑌 where the right hand side is
nothing but the planned savings and the left hand side is nothing but the planned
investment. Since we generally assume in Keynesian framework that savings are always
realized and by truism we know that actual savings must always equal actual investment
therefore equation 4 can be understood to give us that growth path where planned
investment must equal actual investment. The assumption number 7 above tells us that if
the plans of firms are met, then there is no reason for them to change their rate of investment
which through multiplier would ensure an unchanged rate of growth. Therefore if the
𝑠
economy grows at the rate 𝐶 then there is no force to deviate it from this constant growth
𝑟
path. Such an equilibrium growth rate is defined as Warranted (Required) rate of growth
by Harrod and denoted as 𝑔𝑤 . In his own words, it is “that rate of growth which, if occurs,
will leave all the parties satisfied that they have produced neither more nor less than the
right amount. … It will put them into a frame of mind which will cause them to give such
orders as will maintain the same rate of growth”. Therefore we may write
𝑠
𝑔𝑤 = 𝐶 (5)
𝑟

In fact he defines the following three different types of rate of growth for an economy.
1. Warranted (Required) rate of growth denoted as 𝑔𝑤 .
2. Actual rate of growth denoted as 𝑔𝑎 .
3. Natural rate of growth denoted as 𝑔𝑛 .

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Using a similar terminology as in case of warranted rate of growth we may write that

∆𝑌 ∆𝑌 𝐼 ∆𝑌 𝐼 𝑠
𝑔𝑎 = = ( 𝐼 × 𝑌) = (∆𝐾 × 𝑌) = 𝐶 (6)
𝑌

Here S is the saving ratio and C is the actual (not desired) ratio of actual change in capital
∆𝐾 𝐼
i.e. actual net investment to the change in output i.e. 𝐶 = = ∆𝑌 . Thus C is the actual
∆𝑌
incremental capital-output ratio.

Natural rate of growth is defined by Domar as that rate of growth of output which is
required to fully employ the entire growing labor force. Since labor-output ratio is assumed
to be constant, therefore the natural rate of growth of output must be equal to the rate of
growth of labor. Suppose if the rate of growth of population is given by 𝑛 then the natural
rate of growth must be equal to 𝑛.
𝑔𝑛 = 𝑛 (7)

You may think of growth of labor force as not just an increase in number of labor but an
increase in number of effective labor to incorporate the increases in labor productivity.

We know that in the short run disequilibrium occurs whenever actual investment is not
equal to planned investment i.e. there is either an unplanned positive or an unplanned
negative addition to stock of inventories. This equilibrium is restored by the firms in
subsequent periods through increase or decrease in output in case of unplanned removal or
unplanned addition to inventories respectively. Does a similar response mechanism also
bring the economy back to required equilibrium growth path in case of any deviation from
it? After deriving the warranted or equilibrium rate of growth, Harrod’s next concern was
to check if the equilibrium is stable i.e. is any deviation from the equilibrium path self-
correcting over time?

Amartya Sen in his introduction to Harrod in the book ‘Growth Economics’ has formulated
the following adaptive expectation response model of the firms wherein
𝐼𝑡 = 𝐶𝑟 ( 𝑌𝑡𝑒 − 𝑌𝑡−1 )
𝑒 𝑒
𝑔𝑡𝑒 − 𝑔𝑡−1 = 𝜆(𝑔𝑡−1 − 𝑔𝑡−1 )

The first equation says the investment in any period is a function of the increase in output
expected in this period over the previous period’s output. The second equation says that
the expected rate of growth of output increases by a constant positive multiple of the
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difference between actual and expected rate of growth of output in the previous observed
period. For example, if actual output in previous period was more than expected output i.e.
actual growth was more than expected growth in previous period then the firms will revise
their expectations of growth of output upwards in the current period implying a larger
expected output in the current period and therefore a larger investment given by the first
equation of Sen. The reverse will hold true whenever actual will be less than expected.

Let us consider the following two scenarios:

1. 𝑔𝑎 > 𝑔𝑤
2. 𝑔𝑎 < 𝑔𝑤

Using equations 5 and 6 above, we may write that if

𝑔𝑎 > 𝑔𝑤 ⟹ 𝐶 < 𝐶𝑟 ⟹ ∆𝐾 < ∆𝐾𝑟 ⟹ 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑝𝑎𝑙𝑛𝑛𝑒𝑑 𝐼 𝑖𝑠 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑
⟹ 𝑔𝑎 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠

𝑔𝑎 < 𝑔𝑤 ⟹ 𝐶 > 𝐶𝑟 ⟹ ∆𝐾 > ∆𝐾𝑟 ⟹ 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑝𝑙𝑎𝑛𝑛𝑒𝑑 𝐼 𝑖𝑠 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒𝑑

⟹ 𝑔𝑎 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒𝑠
The first implication above considers the situation when for some reason the actual rate of
growth in the economy happens to be more than the warranted rate of growth. Under such
a situation, the actual incremental capital per unit of incremental output will be less than
the desired incremental capital per unit of incremental output. A shortfall in actual capital
vis-à-vis planned capital will take either the form of unplanned shortfall in the stock of
inventories or an excess demand for the equipment. Both the situations will be inflationary
and responded by the firms through increase in the rate of planned investment according
to assumption 4. Such an increase will cause a further increase in the actual rate of growth
of output through the multiplier mechanism. It means that whenever the actual growth rate
of output is more than the warranted rate of growth, the response by the firms will make
the actual growth rate even larger than the warranted growth rate. Therefore the response
mechanism which worked to create equilibrium in the short run will create further
disequilibrium in the long run and take the economy further away from the equilibrium
growth path towards a boom.

The second implication above considers the situation when the actual rate of growth in the
economy happens to be less than the warranted rate of growth. Under such a situation, the
actual incremental capital per unit of incremental output will be more than the desired
incremental capital per unit of incremental output. A shortfall in actual capital vis-à-vis
planned capital will again take either the form of unplanned addition to the stock of

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inventories or an excess capacity of the equipment. Both the situations will be deflationary
and responded by the firms through decrease in the rate of planned investment according
to assumption 4. Such a decrease will cause a further decrease in the actual rate of growth
of output through the multiplier. It means that whenever the actual growth rate of output is
less than the warranted rate of growth, the response by the firms will make the actual
growth rate even smaller than the warranted growth rate. Therefore the response
mechanism will create further disequilibrium in the long run and take the economy further
away from the equilibrium growth path towards a depression. The above analysis
concludes that the equilibrium growth path of Harrod is completely unstable and any
deviation is further aggravating rather than self-correcting. For this reason, the equilibrium
growth is said to have a Knife-Edge Stability. If you make a slip on either side, you keep
falling.

The instability of equilibrium in the Harrod model was used by him to explain the business
cycles above and below the trend path of warranted growth. But we know that business
cycles always have peaks and troughs. Harrod explained these through the use of natural
rate of growth of output. Remember from equation 7 that 𝑔𝑛 = 𝑛. Harrod argues that in an
economy with constant capital-labor ratio, the actual growth rate of output can never be
more than the natural rate of growth of output i.e. 𝑔𝑎 ≤ 𝑔𝑛 = 𝑛. In other words, growth
is constrained by labor.

Here again we consider two possible scenarios:

1. 𝑔𝑎 ≤ 𝑔𝑛 < 𝑔𝑤
2. 𝑔𝑛 > 𝑔𝑤

In the first situation, natural rate is less than the warranted rate which implies that the actual
rate which has to be less than or equal to the former will also be less than the warranted
rate. Such a situation as we have seen above will push the actual rate further below and
bring a deflationary gap in the economy. During this gap, neither the labor nor the capital
will be fully employed. The less than full employment of capital is obvious since at all rates
of growth below the warranted, actual capital stock must be more than planned capital
stock. Correction will only be possible if the warranted rate of growth falls below to
become equal to the actual. A fall in warranted rate requires a fall in savings or an increase
in𝐶𝑟 . But an increase in latter is not profitable and thus possible under conditions of
deflation. In fact, 𝐶𝑟 is bound to fall. But Harrod argues that the savings rate in the economy
will fall during the deflation due to redistribution in income against the capitalists and will
fall more in proportion to fall in 𝐶𝑟 thereby decreasing the warranted rate. When it has

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fallen enough, then actual will once again become greater than warranted and begin to
recover thereafter.

In the second situation, natural rate is more than the warranted rate. In this situation, if
actual is equal to warranted then there the economy will move along a constant growth path
where there will full employment but persistent growing unemployment of labor. However,
if actual is more than warranted, then actual rate will keep growing till it reaches its bounds
of natural rate of growth. Although the unemployment will keep o reducing but the capital
will be in excess demand putting inflationary pressures. But this rate high rate of growth
will not be sustained forever since due to inflation there will be a redistribution of income
in favor of capitalists implying an increase in savings rate faster than the increase in desired
incremental capital output ratio. On the other hand, if actual is less than warranted, then
the economy will move further in depression with high labor unemployment levels and
also unused capacity. This will keep happening unless the warranted rate falls below
through the mechanism discussed in the above paragraph. Note that although warranted
growth rate changes, but the warranted growth path does not which requires that actual rate
of growth must be equal to warranted rate of growth.

The only situation in the Harrod’s model where there is full employment of both capital
and labor under equilibrium is when
𝑔𝑎 = 𝑔𝑤 = 𝑔𝑛

This is so improbable a situation that Mrs. Joan Robinson has termed it ‘Golden Age’4.5
Other factors

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5. Summary

1. The equilibrium rate of growth which maintain itself is called the warranted rate of
𝑠
growth and given by 𝑔𝑤 = 𝐶
𝑟
2. Along this warranted growth path, there is full employment of capital and firms
have all the satisfaction to continue investing at a rate that the same growth rate is
achieved.
3. The equilibrium is unstable and any diversion from this growth path will result in
further aggravation of the gap between actual and warranted growth rate. The figure
below shows the time path of the growth rate of an economy under three possible
situations.

In this figure we assume that natural rate is more than the warranted rate.
4. Only if 𝑔𝑎 = 𝑔𝑤 = 𝑔𝑛 , there is full employment of both labor and capital under
equilibrium growth path.

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Subject ECONOMICS

Paper No and Title 12: Economics of Growth and Development I

Module No and Title 3: Harrod – Domar Growth Model – II

Module Tag ECO_P12_M3

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TABLE OF CONTENTS
1. Learning outcomes
2. Introduction
3. Domar’s model of Economic growth
4. Structure of Domar’s model
5. Working of Domar’s model
6. Comparison between Harrod’s and Domar’s model
7. Summary

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1. Learning Outcomes

After studying this module, you shall be able to

 Know about the basis of Domar’s model of economic growth

 Learn about the structure of Domar’s model
 Understand the working of Domar’s model
 Analysis the comparison between the Harrod and Domar’s model

2. Introduction

In this module, we will discuss in detail about the Domar’s model of economic Growth.
Evsey Domar published his work on “Capital Expansion and Growth” in 1946, seven years
later than a very similar work by Roy Harrod. Although there are many similarities in
conclusions of the two models by Domar and Harrod but there are some very important
differences in the two models.

3. Domar’s Model of Economic growth

Domar’s starting point was to criticize Keynesian framework for its incompleteness in
addressing two important issues related to the long-run
1. Keynesian analysis considers investment only to be an income generating (i.e.
demand side) instrument through the multiplier while ignoring the essential
productive capacity changing (i.e. supply side) role of investment. It is precisely
because of this reason that Keynes assumed that employment of labor is a function
of national income. But Domar points out that this is true only in the short run. He
assumes that overtime employment of labor is a function of ‘ratio of national
income to productive capacity’.
2. Keynesian analysis, while giving over importance to the need of full employment
of labor, almost ignores the issue of unemployment of capital, which in turn
becomes the source of labor unemployment. According to Domar, it is the
premature obsolescence of capital equipment that discourages investment and
growth, thereby causing the labor unemployment.

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Main Assumptions of Domar’s model

1. Savings and investment refer to income of the same period.

2. Both Saving and Investment are net, i.e. over and above the depreciation.
3. Depreciation is measured not in respect to historical costs of the depreciated
equipment but in respect to the cost of replacement of the depreciated asset by
another one of same productive capacity. The latter cost may be different from the
former owing to technological progress.
4. There is a constant general level of prices.
5. No adjustment lags are present in any market.
6. Productive capacity of an asset or of the entire economy can be measured. This
assumption is very strong since the productive capacity of a given physical stock
of capital does not depend just upon the technical factors but economic and
institutional factors as well, for example distribution of income, preferences of
consumers’, wage rates, relative prices, and structure of industry, etc.
7. Average and marginal propensities to save are equal for the economy.
8. The ratio of productive capacity of capital to the size of capital for the entire
economy is equal to that of the new investment projects. It means nothing but that
average potential output to capital ratio is equal to marginal potential output capital
ratio.
9. Employment rate depends upon the ratio of actual output and productive capacity,
also known as the utilization ratio.

4. Structure of Domar’s Model

1. National Income (output) is determined by investment through the multiplier as per

the following relation
𝐼𝑡
𝑌𝑡 = … (1)
𝑠
Where 𝑠 (assumed to be constant) is the marginal propensity to save.

2. Productive capacity of an economy (or capital stock) is defined by Domar as its

output when the entire labor force is fully employed in some conventional sense.

3. For the new investment projects, the ratio of potential productive capacity created
by these new investment projects to the size of capital invested in them (i.e.𝐼), is
denoted by 𝜂. Domar in his original work had used a different symbol though.

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𝑑𝑃
𝜂= … (2)
𝑑𝐾
𝑃 𝑑𝑃
Using assumption 8, we can write 𝜂 = 𝐾 = 𝑑𝐾 .

4. The addition in potential productive capacity of the entire economy’s capital stock
may be less than the potential productive capacity of just the new investment
projects. This is because operation of new projects may involve transfer of scarce
labor from older projects making the latter less productive and thereby increasing
the overall productive capacity of the economy due to new investment less than the
potential. The ratio of change in productive capacity of the economy (due to new
investment) to the amount of investment is termed by Domar as the “potential social
average investment productivity” and is denoted by 𝜎. Therefore

𝑑𝑃⁄𝑑𝑡
𝜎= … (3)
𝐼

where 𝑃 stands for the productive capacity of the economy.

5. Domar assigns the following characteristics to 𝜎:

a. Its magnitude depends on the technological progress (as embodied in new
investment projects). Thus we can say that 𝜎 refers to increase in capacity
which accompanies rather than caused by investment.
b. 𝜎 refers to increase in potential capacity to supply. Its realization depends
upon the presence of adequate demand for it.
c. The maximum value which 𝜎 can attain is equal to 𝜂. The shortfall would
depend upon the magnitude of the rate of investment, growth of labor force
and natural resources, technological progress, misdirection of investment
etc. Therefore
𝜎≤𝜂
d. It is assumed by Domar that 𝜎 & 𝜂 remain constant.

6. When 𝜎 < 𝜂 then following an investment 𝐼 new projects with productive capacity
of 𝐼𝜂 are built. The productive capacity of entire economy however increases only
by 𝐼𝜎(< 𝐼𝜂). This implies that somewhere in the economy (not excluding new
projects because of misdirection of investment), productive capacity must be
reduced by 𝐼𝜂 − 𝐼𝜎. But since Domar has assumed in assumption number 8 above
that the ratio of productive capacity of capital to the size of capital for the entire
economy is constant at 𝜂, therefore every year an amount of capital (or capital
𝐼(𝜂−𝜎)
value) equal to becomes useless. Such an untimely (unintended) demise of
𝜂

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capital equipment (over and above depreciation) is termed by Domar as “Junking”.
Junking of capital is a result of unprofitability of old type of capital assets which in
turn may be caused by either shortages in supply of labor or loss of demand of
produce because of lower costs and better quality of newer products.

7. The incentives for investment are provided by the rate of growth of output and hurt
by the amount of junking. We may write that
+ −
Δ𝐼 ⏞ ⏞
Δ𝑌 𝜂−𝜎
= 𝑓( , )
I Y 𝜂

The increase in proportion of capital junked negatively affects the business

confidence thereby reducing the rate of investment.

5. Working of Domar’s Model

The Question Harrod asks is that what must be the rate of growth of investment if the
increase in productive capacity (due to change in capital stock) is to be exactly matched by
the increase in output (due to change in demand for investment), so that the economy
always remains in full employment (of labor) equilibrium over time.

Rewriting equation (2) above as follows gives us the rate of change of productive capacity.

𝑑𝑃
= 𝐼𝜎 … (4)
𝑑𝑡

Note that productive capacity increases so long as net investment and potential social
average investment productivity are positive.

The rate of growth of output is obtained by differentiating equation (1) with respect to time
as given below

𝑑𝑌 𝑑𝐼 1
= × … (5)
𝑑𝑡 𝑑𝑡 𝑠

For simplification, Domar assumes that the economy is in equilibrium to begin with i.e.
𝑃0 = 𝑌0 … (6)

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The equilibrium condition then is that change in productive capacity equals change in
output, i.e.
𝑑𝑃 𝑑𝑌
= … (7)
𝑑𝑡 𝑑𝑡

Substituting from (3) and (4), we obtain the equilibrium condition as follows
1 𝑑𝐼
× = 𝑠𝜎 … (8)
𝐼 𝑑𝑡

The above condition implies that equilibrium over time requires investment to grow at a
constant continuous rate equal to 𝑠𝜎. Equation (1), (5) and (6) together imply that

1 𝑑𝑌 1 𝑑𝐼
× = × … (9)
𝑌 𝑑𝑡 𝐼 𝑑𝑡

This implies that output grows as the same rate as investment, i.e. 𝑠𝜎. If the economy fails
to grow at this warranted rate then the capacity will be under-utilized implying less than
full employment of labor as well as capital.

Notice the similarity between the expression of the warranted (required) rate of growth of
𝑠
Domar and that of Harrod. The warranted rate of Harrod is𝐶 , where 𝐶𝑟 is reciprocal of the
𝑟
marginal output-capital ratio.

The consequences of an economy to grow at a rate different from the warranted rate i.e. 𝑠𝜎
are demonstrated by Domar under the following two conditions:

1. 𝝈 = 𝜼 (no junking)

This is a situation in which no junking of capital equipment occurs. Under such

situation, suppose the investment and output grow at a constant rate of 𝑟, i.e.

𝐼 = 𝐼0 𝑒 𝑟𝑡
𝑑𝐾
Now since 𝑑𝑡 = 𝐼, therefore capital stock at any point of time is nothing but the
continuous sum of all net investments, we can therefore write
𝑡
𝐼0 𝑟𝑡
𝐾 = 𝐾0 + ∫ 𝐼0 𝑒 𝑟𝑡 𝑑𝑡 = 𝐾0 + (𝑒 − 1)
𝑟
0

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𝐼 𝐼
As 𝑡 ⟶ ∞,𝐾0 + 𝑟0 (𝑒 𝑟𝑡 − 1) ⟶ 𝑟0 𝑒 𝑟𝑡 , as the other terms become relatively
insignificant. This implies that the capital will also grow at the rate approaching 𝑟.
𝐼 𝑃
Since 𝑌 = 𝑠 and 𝜂 = 𝜎 = 𝐾,

𝑌 𝐼0 𝑒 𝑟𝑡
lim = lim
𝑡→∞ 𝑃 𝑡→∞ 𝑠𝜎𝐾
𝐼0 𝑒 𝑟𝑡 1
= lim 𝐼0
= lim 𝐾0 1
𝑡→∞ (𝑒 𝑟𝑡 − 1)] 𝑡→∞
𝑠𝜎 [𝐾0 + 𝑠𝜎 [( 𝐼 − 1) 𝑒 −𝑟𝑡 + 𝑟 ]
𝑟 0
𝑟
=
𝑠𝜎
𝑟
The expression 𝑠𝜎 is termed by Domar as the coefficient of utilization and denoted
by 𝜃. When

i. 𝜃 < 1 ⟺ 𝑟 < 𝑠𝜎
The economy fails to utilize its capacity i.e. 𝑌 < 𝑃. The proportion of
capital stock unutilized is given by(1 − 𝜃). This not only creates unused
capital but unused labor force as well.

ii. 𝜃 = 1 ⟺ 𝑟 = 𝑠𝜎
Only under this situation wherein actual growth rate is equal to the
warranted growth rate, full capacity is utilized and the economy is in
equilibrium.

2. 𝝈 < 𝜂 (junking)
𝐼(𝜂−𝜎)
As seen above, whenever 𝜎 < 𝜂, the amount 𝜂 of capital is junked every year.
𝑑𝐾 𝐼(𝜂−𝜎) 𝐼𝜎
This implies that ≠ 𝐼 but rather = 𝐼 − = .
𝑑𝑡 𝜂 𝜂
Therefore
𝐼0 𝜎 𝑟𝑡
𝐾 = 𝐾0 + (𝑒 − 1)
𝑟𝜂
&

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𝑌 𝐼0 𝑒 𝑟𝑡
lim = lim
𝑡→∞ 𝑃 𝑡→∞ 𝑠𝜂𝐾
𝐼0 𝑒 𝑟𝑡 1
= lim 𝐼0 𝜎
= lim 𝐾 𝜎
𝑡→∞ 𝑡→∞
𝑠𝜂 [𝐾0 + (𝑒 𝑟𝑡 − 1)] 𝑠𝜂 [( 𝐼 0 − 1) 𝑒 −𝑟𝑡 + 𝑟𝜂]
𝑟𝜂 0
𝑟
=
𝑠𝜎

Thus Domar demonstrates that even under the case of junking, the capacity will be
fully utilized so long as the rate of growth is equal to the warranted rate of growth.

Under this case, although there is full employment of labor but less than full
employment of capital owing to premature death of capital. Entrepreneurs are
discouraged for further investments in such case.

Stability of Equilibrium in Domar’s model

Domar only considers the question of downward stability of his equilibrium, although not
explicitly.

According to Domar, even in the situation of no junking, if the investment ever fails to
grow at the warranted rate then the economy would progress towards depression because
investors would further reduce the rate of investment following developed unused capacity
due to fall of output below productive capacity.

The situation becomes even grimmer in the case of junking. Under this situation, the
business confidence would be negatively affected despite the fact the economy is on the
equilibrium path i.e. 𝑃 = 𝑌. This owes to the junking of capital following new investments.
Junking (i.e. unused capital stock due to shortfall in labor force or demand or both) leads
to lack of desire to invest further and thereby a reduction in rate of invest below the
warranted rate of growth even when the economy started with the latter rate of growth.
One way Domar points out that in which entrepreneurs will react to a high unused capacity
would be by decreasing the real wage rates and thereby increasing their share of profits.
However this move, since capitalists have a higher savings rate than the rest, would leads
to increase in the savings rate of the economy as a whole and thus the warranted rate of
growth, making it even more difficult to achieve the warranted rate and therefore further
building of the excess capacity.

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Conclusions of Domar’s model

1. Without Junking or unwanted scrapping of capital, the income, investment,

productive capacity and capital stock can grow at the warranted rate of growth 𝑠𝜎
ensuring the full employment of both labor and capital.
2. With junking of capital every year, the economy would move towards acute
depression.
3. It is the unused capital equipment that endangers the attainment of full employment
equilibrium in the economy.

6. Comparison between Harrod’s and Domar Model

As noted above, the two models are similar in the following ways:

1. For both the economists, the started point to make the Keynesian framework
dynamic.
2. Both have assumed, to some extent, constancy of capital-output ratio 𝐶𝑟 in case of
Harrod and its inverse, the related output-capital ratio 𝜎 in case of Domar.
3. Both obtain constant growth paths.
4. Both assume that output follows demand thereby implying that output can fall
below potential leading to entrepreneurial reaction of change in rate of investment.
5. Both conclude a grimmer future if ever the economy diverges from its unstable
equilibrium path.

Although similar on many broad counts, the two differences in models can be summarized
as below:

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Issue Domar Harrod

Rate of growth which equate output Equilibrium rate of growth for
to productive capacity, ensuring full entrepreneurs, which requires full
1. Key concept employment of labor but if junking employment of capital but is compatible
is present, it is possible with less with less than full employment of labor.
than full employment of capital.
Junking of capital decreasing the Labor shortage not allowing economy to
2. Long-run problem rate of investment below the reach warranted rate of growth
warranted
Unused capacity Less than full employment of labor.
3. Feature of economy
Shortages leads to junking of Determines the natural rate of growth
4. Role of Labor capital and thus negatively affects which is a cap on the rate of growth an
rate of investment economy can achieve
Continuously decreasing The process of adjustment which results
investment incentive due to in expected rate of growth diverge
5. Cause of instability
increasing unused capacity further and further from the warranted
rate of growth
Simplification assumption Fixed relative rate of interest and thus
6. Causes of fixed
fixed capital-labor ratio combined with
capital-output ratio
CRS

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7. Summary

 Keynesian analysis considers investment only to be an income generating (i.e. demand

side) instrument through the multiplier while ignoring the essential productive capacity
changing (i.e. supply side) role of investment
 Keynesian analysis, while giving over importance to the need of full employment of
labor, almost ignores the issue of unemployment of capital, which in turn becomes the
source of labor unemployment
 Productive capacity of an economy (or capital stock) is defined by Domar as its output
when the entire labor force is fully employed in some conventional sense
 The ratio of change in productive capacity of the economy (due to new investment) to
the amount of investment is termed by Domar as the “potential social average
investment productivity” and is denoted by 𝜎
 The incentives for investment are provided by the rate of growth of output and hurt by
the amount of junking.
+ −
Δ𝐼 ⏞ ⏞
Δ𝑌 𝜂−𝜎
= 𝑓( , )
I Y 𝜂

The increase in proportion of capital junked negatively affects the business confidence
thereby reducing the rate of investment.

 Without Junking or unwanted scrapping of capital, the income, investment, productive

capacity and capital stock can grow at the warranted rate of growth 𝑠𝜎 ensuring the full
employment of both labor and capital.
 With junking of capital every year, the economy would move towards acute depression.
 It is the unused capital equipment that endangers the attainment of full employment
equilibrium in the economy

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