Euler’s Theorem and Product Exhaustion Theorem

2 years ago

Atiqur Rahman

12 minutes

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 the “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.

Solution of Product Exhaustion Theorem:

The three solutions proposed for the problem of the 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

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 𝑓(λx) = λ^t𝑓(𝑥), for all λ > 0 and 𝑥 = (𝑥₁, 𝑥₂, ….., 𝑥_𝑛).

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:

λ^t 𝑌 = 𝑓 [λK, λL]

Where,

Y is output

K is capital

t is the 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 the production function is homogeneous of degree t, then

The Marginal Rate of Technical Substitution is constant along rays extending from the origin and
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

𝑥₁. 𝜕𝑓(𝑥)/𝜕𝑥₁ + 𝑥₂. 𝜕𝑓(𝑥)/𝜕𝑥₂ + 𝑥₃. 𝜕𝑓(𝑥)/𝜕𝑥₃ ⋯ + 𝑥_𝑛. 𝜕𝑓(𝑥)/𝜕𝑥_𝑛 = 𝑡𝑓(𝑥)

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

𝑑𝑓(𝜆𝑥)/𝑑𝜆 = 𝑥₁. 𝑑𝑓(𝜆𝑥)/𝜕𝜆𝑥₁ + 𝑥₂. 𝑑𝑓(𝜆𝑥)/𝜕𝜆𝑥₂ + 𝑥₃. 𝑑𝑓(𝜆𝑥)/𝜕𝜆𝑥₃ ⋯ + 𝑥_𝑛. 𝑑𝑓(𝜆𝑥)/𝜕𝜆𝑥_𝑛

𝑑𝜆^𝑡𝑓(𝑥)/𝑑𝑡 = 𝑡.𝜆^𝑡−1𝑓(𝑥)

If setting λ=1, the theorem follows.

Euler’s Theorem and Production Function:

It is based on some postulations. These are:

It assumes a standardized linear production of the first degree, which implies invariable returns to scale.
It assumes that the factors are complementary, i.e. if a variable factor increases, it increases the marginal productivity of the fixed factor.
It assumes that factors of production are perfectly divisible.
The relative shares of the factors are invariable and independent of the level of the product.
There is a stationary, reckless economy where there are no profits.
There is perfect competition.
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 = 𝐾. 𝜕𝑌/𝜕𝐾 + 𝐿. 𝜕𝑌/𝜕𝐿

Where,

Y: Output

K: Capital

L: Labour

𝜕𝑌/𝜕𝐾: Marginal Product of Capital

𝜕𝑌/𝜕𝐿: Marginal Product of Labor

The marginal product of capital is the addition to the total output attributable to the addition of one more unit of capital. It is calculated by partially differentiating output with respect to capital, keeping labour constant.

Similarly, the Marginal product of labour is the addition to the total output attributable to the addition of one more unit of labour. It is calculated by partially differentiating output with respect to labour, keeping capital constant.

Euler’s theorem states that the marginal product of capital multiplied by the amount of capital plus the marginal product of labour multiplied by the amount of labour equals to the total product of the firm.

𝑌 = 𝑓(𝐾, 𝐿, 𝐴) = 𝐴𝐾^𝛼𝐿^1−𝛼

𝑀𝑃_𝐾 = 𝜕𝑌/𝜕𝐾 = 𝐴𝛼𝐾^𝛼−1𝐿^1−𝛼 = 𝐴𝛼 (𝐾/𝐿)^𝛼−1

𝑀𝑃_𝐿 = 𝜕𝑌/𝜕𝐿 = 𝐴(1 − 𝛼)𝐾^𝛼𝐿^−𝛼 = 𝐴(1 − 𝛼) (𝐾/𝐿)^𝛼

Putting the values in Euler’s theorem, Y=K. MP_k + L. MP_L, we get

𝑌 = 𝐴𝛼𝐾^𝛼−1𝐿^1−𝛼𝐾 + 𝐿𝐴(1 − 𝛼)𝐾^𝛼𝐿^−𝛼

𝑌 = 𝐴𝛼𝐾^𝛼𝐿^1−𝛼 + 𝐴(1 − 𝛼)𝐾^𝛼𝐿^1−𝛼

𝑌 = 𝐴𝐾^𝛼𝐿^1−𝛼

If the production function is homogeneous of degree 1, then marginal products are homogeneous of degree zero.

Y = K. MP_K + L. MP_L or K. F_K + L. F_L

Differentiate with respect to K,

K. F_KK + L. F_LK + F_K = F_K

K. F_KK + L. F_LK = 0. F_K = 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 the 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 the real world, i.e. production function is not homogeneous of degree one.

Edgeworth commented on 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”.

Economists pointed out that the production function is such that it yields a long-run average cost curve which is ‘U’ shaped. LAC curve is also known as the “envelope curve” as it envelopes short-run average cost curves.

The long-run average cost curve is U-shaped, i.e. it initially 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 beyond a 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 a surplus. So, 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 the production function exhibits constant returns to scale, then the shape of the long-run average cost curve would be a horizontal straight line parallel to the x-axis. The horizontal straight-line shape of the long-run average cost is not compatible with the perfectly competitive market structure, as the firm would not be able to determine the equilibrium position.

a. Wicksell, Walras and Barone’s Solution to Product Exhaustion Problem:

After Wicksteed, the more satisfactory solution to the problem of the 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 yielded a long-run average cost curve to be of a ‘U’ shape.

They pointed out the applicability of the product exhaustion theorem in the long run in a perfect competition market. In a 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 a minimum of the long-run average cost. In the long run, all the firms are earning zero economic profit, and no firm has the 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, and Walras pointed out the applicability of the product exhaustion theorem in the case of a long-run perfectly competitive equilibrium. And Wicksteed provided a solution to the product exhaustion theorem with the help of Euler’s theorem and assumed a 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 a perfectly competitive market structure and dealing with the 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 perfect competition.

In a perfect competition market structure, firms are earning zero economic profits. Thus the solution to the product exhaustion problem in the 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 long run.

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 the value of total output is equal to the total cost of production.

Let capital (K) and labour (L) be the two factors of production used by perfectly competitive firms 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 = 𝑉𝑀𝑃_𝐿 = 𝑃𝑥𝑀𝑃𝑃_𝐿

𝑟 = 𝑉𝑀𝑃_𝐾 = 𝑃𝑥𝑀𝑃𝑃_𝐾

Where VMP is the value of the marginal product.

Now substitute these values of r and w in equation (1), and we get

𝑃. 𝑄 = 𝐿. 𝑃. 𝑀𝑃𝑃_𝐿 + 𝐾. 𝑃. 𝑀𝑃𝑃_𝐾

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.

Euler’s Theorem and Product Exhaustion Theorem

Solution of Product Exhaustion Theorem:

a) Philip Wicksteed Solution: Euler’s Theorem

Euler’s Theorem:

Euler’s Theorem

Euler’s Theorem and Production Function:

Wicksteed’s Solution and Criticism

i. First drawback of Wicksteed’s Solution:

ii. Second drawback of Wicksteed’s Solution:

a. Wicksell, Walras and Barone’s Solution to Product Exhaustion Problem:

b. Hicks and Samuelson Solution to Product Exhaustion Theorem:

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Solution of Product Exhaustion Theorem:

a) Philip Wicksteed Solution: Euler’s Theorem

Euler’s Theorem:

Euler’s Theorem

Euler’s Theorem and Production Function:

Wicksteed’s Solution and Criticism

i. First drawback of Wicksteed’s Solution:

ii. Second drawback of Wicksteed’s Solution:

a. Wicksell, Walras and Barone’s Solution to Product Exhaustion Problem:

b. Hicks and Samuelson Solution to Product Exhaustion Theorem:

Read More- Microeconomics

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