**ECONOMICS ** **PAPER No.: 5- ** **Advanced Microeconomics**

**MODULE No. : 6- **

**Elasticity of Technical Substitution, Technical Progress**

**and Factor Shares, Product Exhaustion Theorem**

**Subject ** **ECONOMICS **

**Paper No and Title ** **Paper-5 Advanced Microeconomics **

**Module No and Title ** **Module-6: Elasticity of Technical Substitution, Technical **
**Progress and Factor Shares, Product Exhaustion Theorem **

**Module Tag ** **ECO_P5_M6 **

**ECONOMICS ** **PAPER No.: 5- ** **Advanced Microeconomics**

**MODULE No. : 6- **

**Elasticity of Technical Substitution, Technical Progress**

**and Factor Shares, Product Exhaustion Theorem**

**Table of Contents**

**1.**

**Learning Outcomes**2.

**Introduction**

3. **Elasticity of Technical Substitution **
** 4. Technical Progress & Factor Shares **
** 5. Product Exhaustion Theorem **

**5.1. Euler’s Theorem of Product Exhaustion **

**5.2. Clark, Wicksteed & Walras Theorem of Product Exhaustion. **

** 6. Summary **

**ECONOMICS ** **PAPER No.: 5- ** **Advanced Microeconomics**

**MODULE No. : 6- **

**Elasticity of Technical Substitution, Technical Progress**

**and Factor Shares, Product Exhaustion Theorem**

## 1. **Learning Outcome**

** After studying this module **

You will understand how change in factor price affect the distributive shares of the factors.

You will know how to relate elasticity of technical substitution with shares of factors.

You will know different concepts of technical progress and how these affect the shares of factors.

You will understand the meaning and derivation of product exhaustion theorem.

## 2. **Introduction **

In this module we will derive the relation between factor shares and changes in relative factor prices.

The degree of this relationship is measured by the concept of elasticity of technical substitution or elasticity of factor substitution. Since technology of production changes continuously over time, the shares of factors may be affected by this technical progress. If the factors are paid according to their value of marginal productivity, the value of total output will be exhausted between the factors’

payment. Euler proved this by assuming linearly homogeneous [production function, while Clark, Wicksteed and Walras established this product exhaustion relation without considering the homogeneous production function. All the above concepts we analyze in details in the following sections.

## 3. **Elasticity of Technical Substitution **

In this section we analyze how changes in factor prices affect the share of factors and distribution of
factors’ income. When input prices change the firm will employ cheaper input in place of relatively
costly one. For example, if the cost of labour increases relative to the cost of capital, the firm will
substitute more capital in place of labour. This happens due to profit maximization of firm. In order to
maximize profit the firm has to change the ratio of capital to labour (𝐾/𝐿) due to the change factor
prices. This results in the change in the relative shares of the factors. The change in factor price
affects how much depend on the responsiveness of the change of 𝐾/𝐿 ratio to the change in factor
price. This responsiveness is measured by the concept of elasticity of substitution which is defined as
the percentage change in the 𝐾/𝐿 ratio due to the percentage change of the marginal rate of technical
substitution of labour for capital (𝑀𝑅𝑇𝑆_{𝐿𝐾}). That is,

**ECONOMICS ** **PAPER No.: 5- ** **Advanced Microeconomics**

**MODULE No. : 6- **

**Elasticity of Technical Substitution, Technical Progress**

**and Factor Shares, Product Exhaustion Theorem**

𝜃_{𝐿𝐾} = ^{∆(}

𝐾
𝐿)/(^{𝐾}_{𝐿})

∆(𝑀𝑅𝑇𝑆_{𝐿𝐾})/(𝑀𝑅𝑇𝑆_{𝐿𝐾}) (1)

In perfectly competitive factor market the profit maximization condition occurs when the marginal rate of technical substitution of factors equals to the ratio of factor prices, that is:

𝑀𝑅𝑇𝑆_{𝐿𝐾} = 𝑤/𝑟

We can therefore rewrite the elasticity of substitution as
𝜃_{𝐿𝐾} = ^{∆(}

𝐾
𝐿)/(^{𝐾}_{𝐿})

∆(𝑤/𝑟)/(𝑤/𝑟) (2)

The sign of 𝜃_{𝐿𝐾} will be always positive except when 𝜃_{𝐿𝐾} = 0 as the numerator and denominator change
in same direction. This implies that if w/r increases, labour is relatively costlier than capital and the firm,
therefore, substitutes capital for labour leading to increase in 𝐾/𝐿. The values of 𝜃_{𝐿𝐾} lie between the
range from 0 to ∞. When 𝜃_{𝐿𝐾} = 0, there will be no substitution of one factor to another. This is a case of
fixed proportion production function. The factors are perfect substitutes to each other when 𝜃_{𝐿𝐾} = ∞. In
this situation, the isoquant will be negatively sloped straight line. The factors will be substitutable to each
other to a certain extent when 0 < 𝜃_{𝐿𝐾} < ∞ and the isoquant will be convex to origin. For Cobb-Douglas
type production function the value of elasticity of substitution equals to unity. We can compare the
degrees of substitutability by considering the following three cases:

(i) 𝜃_{𝐿𝐾} < 1 (Inelastic substitutability)
(ii) 𝜃_{𝐿𝐾} = 1 (Unitary elastic substitutability)
(iii) 𝜃_{𝐿𝐾} > 1 (Elastic Substitutability)

How the distribution of factors’ share is related to different values of elasticity of substitution analyzed below. The share of labour and capital can be written as:

𝑆ℎ𝑎𝑟𝑒 𝑜𝑓 𝑙𝑎𝑏𝑜𝑢𝑟 =_{𝑃}^{𝑤.𝐿}

𝑌𝑌 & 𝑆ℎ𝑎𝑟𝑒 𝑜𝑓 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 =_{𝑃}^{𝑟.𝐾}

𝑌𝑌

Hence, the relative factor share is

𝑆ℎ𝑎𝑟𝑒 𝑜𝑓 𝑙𝑎𝑏𝑜𝑢𝑟 𝑆ℎ𝑎𝑟𝑒 𝑜𝑓 𝐶𝑎𝑝𝑖𝑡𝑎𝑙=𝑤. 𝐿

𝑟. 𝐾 Or

𝑆ℎ𝑎𝑟𝑒 𝑜𝑓 𝑙𝑎𝑏𝑜𝑢𝑟

𝑆ℎ𝑎𝑟𝑒 𝑜𝑓 𝐶𝑎𝑝𝑖𝑡𝑎𝑙=^{𝑤/𝑟}_{𝐾/𝐿} (3)

The equation (3) traces the relationship between change in 𝑤/𝑟 ratio to the relative shares of labour and
capital. Now when 𝜃_{𝐿𝐾}< 1, a given percentage change in the 𝑤/𝑟 ratio induces a smaller change in the
𝐾/𝐿 rato. This results an increase in relative factor share. The 𝜃_{𝐿𝐾} = 1 implies that there will be equal

**ECONOMICS ** **PAPER No.: 5- ** **Advanced Microeconomics**

**MODULE No. : 6- **

**Elasticity of Technical Substitution, Technical Progress**

**and Factor Shares, Product Exhaustion Theorem**

percentage change in 𝐾/𝐿 ratio for a given percentage change of 𝑤/𝑟 and consequently the relative factor
share will be unchanged. However, in case of 𝜃_{𝐿𝐾} > 1, the relative factor share will decline as
percentage increase in 𝑤/𝑟 ratio causes relatively smaller percentage increase in 𝐾/𝐿 ratio. It is
important to note that there is a two-way causation between 𝑤/𝑟 ratio and 𝐾/𝐿. That is, change in 𝐾/𝐿
ratio induces change in the ratio of factor prices which in turn changes the shares of factors to output.

## 4. **Technical Progress and Factor Shares **

In a dynamic and growing economy technical progress happens continuously. This leads to a change in ratio of capital to labour (𝐾/𝐿) and elasticity of substitution with the shifts of production functions.

It is, therefore, important to examine how factor shares are affected by technical progress.

There are three different types of technical progress observed in production: (i) Capital deepening, (ii) labour deepening, and (iii) neutral. These various types of technical progress are shown by the downward shift of isoquants for a given level of output in figure-4 (a) – (c).

**(i) ** **Capital Deepening Technical Progress **

Capital deepening technical progress occurs when 𝑀𝑅𝑇𝑆_{𝐿𝐾} declines at a constant 𝐾/𝐿 ratio. This will
lead an increase in rental cost of capital (𝑟) relative to wage rate 𝑤 and consequently fall in
equilibrium 𝑤/𝑟 ratio. The resulting impact of this technical progress on factor shares is that the ratio
of factor shares falls. That is, share of labour declines while, the share of capital rises. This is shown
in Figure-4(a).

**(ii) ** **Labour Deepening Technical Progress **

**ECONOMICS ** **PAPER No.: 5- ** **Advanced Microeconomics**

**MODULE No. : 6- **

**Elasticity of Technical Substitution, Technical Progress**

**and Factor Shares, Product Exhaustion Theorem**

Technical progress is said to be labour deepening when 𝑀𝑅𝑇𝑆_{𝐿𝐾} rises for a given level of 𝐾/𝐿 ratio.

However, the equilibrium 𝑤/𝑟 ratio improves when technological progress happens shown in figure- 4(b). Consequently the share of labour increases, while the share of capital decreases.

**(iii) ** **Neutral Technical Progress **

In neutral technical progress the 𝑀𝑅𝑇𝑆_{𝐿𝐾} remains constant for a given level of 𝐾/𝐿 ratio. Therefore,
in equilibrium, 𝑤/𝑟 ratio will be constant shown in figure-4(c). This implies that the share labour and
capital will be unchanged.

**ECONOMICS ** **PAPER No.: 5- ** **Advanced Microeconomics**

**MODULE No. : 6- **

**Elasticity of Technical Substitution, Technical Progress**

**and Factor Shares, Product Exhaustion Theorem**

## 5. **Product Exhaustion Theorems **

In the context of income distribution, the marginal productivity theory of factor price determination suggests that the value of total product is distributed among the cost of labour and cost of capital. This accounting identity can be written as,

𝑃_{𝑌}. 𝑌 = 𝑤. 𝐿 + 𝑟. 𝐾 (1)

The above identity can also be written as in terms of factor shares. Dividing equation (1) both side by
𝑃_{𝑌}. 𝑌 , we will get the summation of factor shares equal to unity:

i.e. 1 =_{𝑃}^{𝑤.𝐿}

𝑌.𝑌+_{𝑃}^{𝑟.𝐾}

𝑌.𝑌 (2)

Where 𝑃_{𝑌}. 𝑌 = Value of output
𝑤. 𝐿 = Cost of labour

𝑟. 𝐾 = Cost of capital

_{𝑃}^{𝑤.𝐿}

𝑌.𝑌= Share of labour

𝑟.𝐾

𝑃𝑌.𝑌= Share of capital

This implies that the value of output should be exhausted by payments of factors used in production.

Now, it is important to examine whether the marginal productivity theory of factor price determination satisfy the income distribution identity given in equation (2).

That is, 𝑇𝑃 = 𝑇𝑃_{𝐿}+ 𝑇𝑃_{𝐾}

or 𝑌 = 𝑀𝑃_{𝐿}. 𝐿 + 𝑀𝑃_{𝐾}. 𝐾
And multiplying by 𝑃_{𝑌} both side we get

𝑃_{𝑌}. 𝑌 = 𝑃_{𝑌}. 𝑀𝑃_{𝐿}. 𝐿 + 𝑃_{𝑌}. 𝑀𝑃_{𝐾}. 𝐾 (3)

Where, 𝑃_{𝑌}. 𝑀𝑃_{𝐿}= Value of marginal productivity of labour
𝑃_{𝑌}. 𝑀𝑃_{𝐾}= Value of marginal productivity of capital

According to marginal productivity theory of factor price determination, factors are paid equal to their value of marginal productivity. That is,

𝑃_{𝑌}. 𝑀𝑃_{𝐿}= 𝑤 and 𝑃_{𝑌}. 𝑀𝑃_{𝐾}= 𝑟 (4)

**ECONOMICS ** **PAPER No.: 5- ** **Advanced Microeconomics**

**MODULE No. : 6- **

**Elasticity of Technical Substitution, Technical Progress**

**and Factor Shares, Product Exhaustion Theorem**

If the above relation is satisfied, total output is exhausted between factor payments. Hence we have
𝑃_{𝑌}. 𝑌 = 𝑤. 𝐿 + 𝑟. 𝐾

**5.1. Euler’s Theorem of Product Exhaustion **

Leonhard Euler examined the product exhaustion relation (eqation-3) and showed that the identity expressed in equation-3 is satisfied under the linearly homogenous production function. The Euler’s theorem states that when production function exhibits constant returns to scale(CRS) or linearly homogenous, the following relationship will be held:

𝑌 =𝜕𝑌

𝜕𝐿. 𝐿 +𝜕𝑌

𝜕𝐾. 𝐾

Or, 𝑌 = 𝑀𝑃_{𝐿}. 𝐿 + 𝑀𝑃_{𝐾}. 𝐾 [Where ^{𝜕𝑌}_{𝜕𝐿}= 𝑀𝑃_{𝐿} & _{𝜕𝐾}^{𝜕𝑌}= 𝑀𝑃_{𝐾} ]

Hence, if the marginal productivity theory of factor price determination holds, the product exhaustion relation stated in equation-2 can be obtained from the Euler’s relation. The formal proof of Euler’s theorem is given below:

Let us consider a production function 𝑌 = 𝑓(𝐾, 𝐿) which is homogeneous of degree 𝑚 if
𝑓(𝜌𝐾, 𝜌𝐿) = 𝜌^{𝑚}. 𝑓(𝐾, 𝐿)

Differentiating with respect to 𝜌 we have 𝐿.𝑑𝑓

𝑑𝐿+ 𝐾.𝑑𝑓

𝑑𝐾 = 𝑚. 𝜌^{𝑚−1}. 𝑓(𝐾. 𝐿)

By putting 𝑚 = 1, we obtain constant returns to scale production function and then we have
𝐿. 𝑀𝑃_{𝐿}+ 𝐾. 𝑀𝑃_{𝐾}= 𝑓(𝐾, 𝐿)

Or

𝑌 = 𝐿. 𝑀𝑃_{𝐿}+ 𝐾. 𝑀𝑃_{𝐾}
Multiplying both side of the above expression by price 𝑃_{𝑌}, we have

𝑃_{𝑌}𝑌 = 𝐿. 𝑃_{𝑌}𝑀𝑃_{𝐿}+ 𝐾. 𝑃_{𝑌}𝑀𝑃_{𝐾}

**ECONOMICS ** **PAPER No.: 5- ** **Advanced Microeconomics**

**MODULE No. : 6- **

**Elasticity of Technical Substitution, Technical Progress**

**and Factor Shares, Product Exhaustion Theorem**

This implies that by paying factors according to their value of marginal product, the value of output will be exhausted. Hence, we prove the Euler’s product exhaustion theorem. This theorem is an identity as it is satisfied for all values of the variables.

**5.2. ** **The Clark-Wicksteed-Walras Theorem of Product Exhaustion **

According to Clark, Wicksteed and Walras the product exhaustion relation based on the marginal productivity theory of factor price determination can be held for any production function under long run competitive equilibrium situation. Their theory implies that when firms operate at the long run competitive equilibrium point and factors are paid equal to their value of marginal productivity, total product will be exhausted among the factors used in the production. Here, homogenous production function is not necessary condition for satisfying the product exhaustion relation.

The Clark-Wicksteed-Walras proof of product exhaustion relation is not satisfied for all level of output. It only holds for that level of output at which long run average cost is minimum. That is long run competitive equilibrium level of output. Therefore their product exhaustion relation is not an identity.

The mathematical proof of their product exhaustion relation is given below:

The total product is exhausted when

𝑌 = 𝑀𝑃_{𝐿}. 𝐿 + 𝑀𝑃_{𝐾}. 𝐾 (1)

Taking total differentiation of the above expression, it gives that

𝑑𝑌 = 𝑀𝑃_{𝐿}. 𝑑𝐿 + 𝑀𝑃_{𝐾}. 𝑑𝐾 (2)

Suppose that capital and labour are increased by a constant proportion, say 𝜇

It follows that 𝜇 =^{𝑑𝐿}_{𝐿} =^{𝑑𝐾}_{𝐾} (3)

Or 𝑑𝐿 = 𝜇. 𝐿 & 𝑑𝐾 = 𝜇. 𝐾

Substituting these in equation-2 we have

𝑑𝑌 = 𝑀𝑃_{𝐿}. 𝜇. 𝐿 + 𝑀𝑃_{𝐾}. 𝜇. 𝐾
Or

𝑑𝑌 = 𝜇(𝑀𝑃_{𝐿}. 𝐿 + 𝑀𝑃_{𝐾}. 𝐾)
Or

𝑑𝑌

𝜇 = (𝑀𝑃_{𝐿}. 𝐿 + 𝑀𝑃_{𝐾}. 𝐾)

Multiplying and dividing the left hand side of the above expression by 𝑌 we obtain

**ECONOMICS ** **PAPER No.: 5- ** **Advanced Microeconomics**

**MODULE No. : 6- **

**Elasticity of Technical Substitution, Technical Progress**

**and Factor Shares, Product Exhaustion Theorem**

𝑌^{𝑑𝑌}

𝜇𝑌= (𝑀𝑃_{𝐿}. 𝐿 + 𝑀𝑃_{𝐾}. 𝐾) (4)

By substituting equation (3) we have 𝑌 [𝑑𝑌/𝑌

𝑑𝐿/𝐿] = 𝑌 [𝑑𝑌/𝑌

𝑑𝐾/𝐾] = (𝑀𝑃_{𝐿}. 𝐿 + 𝑀𝑃_{𝐾}. 𝐾)

Here, [^{𝑑𝑌/𝑌}

𝑑𝐿/𝐿] = 𝜀_{𝑌𝐿} =labour elasticity of output,
[_{𝑑𝐾/𝐾}^{𝑑𝑌/𝑌}] = 𝜀_{𝑌𝐾} = labour elasticity of output

In general [^{𝑑𝑌/𝑌}_{𝜇} ] = 𝜀 = input elasticity of output.

Substituting 𝜀 in equation (4) we have
𝑌. 𝜀 = (𝑀𝑃_{𝐿}. 𝐿 + 𝑀𝑃_{𝐾}. 𝐾)

Multiplying both side by 𝑃_{𝑦} we have

𝑃_{𝑦}𝑌. 𝜀 = 𝑃_{𝑦}𝑀𝑃_{𝐿}. 𝐿 + 𝑃_{𝑦}𝑀𝑃_{𝐾}. 𝐾 (5)

When 𝜀 =1, the equation (5) gives the expression which is identical with Euler’s product exhaustion theorem. This implies that when 𝜀 =1, firm operates under constant returns to scale in which long run average cost curve in minimum and constant and it occurs only at long run competitive equilibrium. Thus equation (5) does not hold for any other values of 𝜀. The product exhaustion relation proved by Clark, Wicksteed & Walras is, therefore, not an identity holding for any values of the variables.

**ECONOMICS ** **PAPER No.: 5- ** **Advanced Microeconomics**

**MODULE No. : 6- **

**Elasticity of Technical Substitution, Technical Progress**

**and Factor Shares, Product Exhaustion Theorem**

## 6. **Summary **

Elasticity of factor substitution( 𝜃_{𝐿𝐾}) measures the degree of responsiveness of change in relative
factor price on relative shares of factors. It is defined as the percentage change in the 𝐾/𝐿 ratio
due to the percentage change of the marginal rate of technical substitution of labour for capital
(𝑀𝑅𝑇𝑆_{𝐿𝐾}). It lies between 0 and∞.

𝜃_{𝐿𝐾} < 1 implies that increase in 𝑤/𝑟 ratio will increase labour’s share relative to share of
capital.

If 𝜃_{𝐿𝐾}> 1, an increase in 𝑤/𝑟 ratio will reduce the ratio of share labour to capital.

When 𝜃_{𝐿𝐾} = 1, there will be no change of relative factor shares in response to change 𝑤/𝑟 ratio.

There is a two-way causation between 𝑤/𝑟 ratio and𝐾/𝐿. That is, change in 𝐾/𝐿 ratio induces change in the ratio of factor prices which in turn changes the shares of factors to output.

Technical progress affects the factor shares by shifting the production function.

Technical progress is classified into three types: capital deepening, labour deepening and neutral.

Capital deepening technical progress occurs when 𝑀𝑅𝑇𝑆_{𝐿𝐾} declines at a constant 𝐾/𝐿 ratio.

Technical progress is said to be labour deepening when 𝑀𝑅𝑇𝑆_{𝐿𝐾} rises for a given level of 𝐾/𝐿
ratio.

In neutral technical progress the 𝑀𝑅𝑇𝑆_{𝐿𝐾} remains constant for a given level of 𝐾/𝐿 ratio.

The product exhaustion means that the value of output is exhausted by payments of factors used in production when factors are paid according to their marginal product.

According to Euler, product exhaustion relation is satisfied under the assumption of linearly homogeneous production function.

Clark, Wicksteed & Walras theorem of product exhaustion states that when firms operate in the long run competitive equilibrium point and factors are paid equal to their value of marginal productivity, total product will be exhausted among the factors used in the production. In order to satisfy this theorem the production function is not required to be homogeneous.