Euler's Theorem: A Proof

by Klaus Hagendorf

For marginal analysis of production and more specific the relation of inputs and outputs there is a certain type of a functional relationship which is very convenient to assume. It is the equality of the proportional change of output resulting from a proportional change of all inputs. This is called a homogeneous functional relationship of degree 1 and in economics one speaks of the case of constant returns to scale as distinguished from decreasing returns to scale if the change of output is less or increasing returns to scale if the change of output is greater than the change of inputs.

The homogenity of a function is expressed as follows

(1)     f (λx₁, λx₂, ..., λx_n)   =   λ^k f (x₁, x₂, ..., x_n)

where x_i; i = 1 ... n are the inputs, f() is the output, and λ is a factor of change.

k is the degree of homogenity of the function. Constant returns to scale prevail if the function is homogeneous of degree 1, k = 1.

If this is the case the marginal products of the inputs sum up to the total product.

(2)     ∂f / ∂(λx₁)   ∂(λx₁) /∂λ  +  ∂f / ∂(λx₂)   ∂(λx₂) / ∂λ  +  ...  +  ∂f / ∂(λx_n)   ∂(λx_n) / ∂λ  =  λ f(x₁, x₂ ... x_n).

and as ∂ (λx_i) / ∂ λ = x_i

(3)     ∂f / ∂(λx₁)   x₁  +  ∂f / ∂(λx₂)   x₂  +  ...  +  ∂f / ∂(λx_n)   x_n  =  λ f(x₁, x₂ ... x_n).

and for λ = 1

(4)     ∂f / ∂(x₁)   x₁  +  ∂f / ∂(x₂)   x₂  +  ...  +  ∂f / ∂(x_n)   x_n  =  f(x₁, x₂ ... x_n).

which is the formular used by neoclassical economists to "explain" the functional distribution of income amongst the different factors of production (x_i). However, this function can also be used as a basis for the marginal labour theory of value (see below).

In its general form the relationship of homogeneous functions and their marginal products has been discovered by the Swiss mathematician Euler (1707 - 1783) and is called Euler's Theorem. According to Steedman (1987) it has been introduced to economics by the English critics of Marx Wicksteed (1994) and can be stated as follows:

If f(x) is homogeneous of degree k then

                                          
(5)     ∑ ∂f / ∂(x_i)   x_i  =  k f(x₁, x₂,..., x_n)
i = 1                                           

or

(6)     f'(X)x = k f(X)

In (6) the capital type X represents the vector of the x_i.

The proof:

(7)     f (λx₁, λx₂, ..., λx_n)   =   λ^k f (x₁, x₂, ..., x_n)

(7a)     f(λX) = λ^kf(X)

Both sides differentiated using the chaine rule for the left side:

(8)     ∂f / ∂(λx₁)   x₁  +  ∂f / ∂(λx₂)   x₂  +  ...  +  ∂f / ∂(λx_n)   x_n  =  k λ ^k-1 f(X).

and as ∂ (λx_i) / ∂ λ = x_i and λ = 1

(9)     ∂f / ∂(x₁)   x₁  +  ∂f / ∂(x₂)   x₂  +  ...  +  ∂f / ∂(x_n)   x_n  =  k λ^k-1 f(X).
or in short

(10)     f'(X)x = k f(X)

Q.E.D.

The inverse is also true. If the relationship (6) holds then the function is homogeneous of degree k.

We varify the validity of constant returns to scale for the Cobb-Douglas production function for which the sum of the production elasticities is 1.

(11)     f(K,L)   =   AK^αL^β   ,   α   +   β   =   1

(12)     λf(K,L)   =   A (λK)^α (λL)^β

(12a)     λf(K,L)   =   A K^α L^β λ^α+β

(12b)     λf(K,L)   =   λ A K^α L^β

(12c)     λ f(K,L)   =   λ f(K,L)

And the marginal products sum up to the total product as can be shown as follows:

(13)     ∂f(K,L)/∂K   =   α f(K,L)/K

(13a)     ∂f(K,L)/∂L   =   β f(K,L)/L

and

(14)     f(K,L)   =   α f(K,L)/K * K + β f(K,L)/L * L

(15)     f(K,L)   =   (α + β) f(K,L)

with α   +   β   =   1

(16)     f(K,L)   =   f(K,L)

Q.E.D.

The proof of Euler's theorem can be found in many textbooks on mathematical economics, for example in Archinard (1992).

In Hagendorf (2003) I have used Euler's Theorem to demonstrate that the marginal productivity theory is valid only if the labour theory of value holds, something neoclassical economists seem to prefer to overlook and Marxian economists and even Maurice Dobb (1937), p. 138 ff. and Ronald Meek (1977) p. 165 ff. seem to have been unaware of.

Certainly for the marginal productivity theory of labour the following statement by Maurice Dobb does not apply: "The fundamental objection to this, as to any form of productivity theory, was that, as Marx pointed out, it included the illicit link of imputing to the owner the "productivity" of the things he owned. "A social relation between men assumes the fantastic form of a relation between things" and the behaviour of things is not only represented animistically as due to some innate property in them, but imputed to the influence of those individuals who exercise rights of ownership over them." (ibid., p. 138).

Bibliographie

Archinar, Gabriel; Guerrien, Bernard;
Analyse mathematique pour economistes. Cous et exercices corrigés; 4th edition;
Paris: Economica; 1992.

Dobb, Maurice H.Political Economy and Capitalism.
Some Essays in Economic Tradition.
London: Routledge & Kegan Paul; 1937.

Hagendorf, Klaus
On the Relationship of the Marginal Productivity Theory and the Labour Theory of Value.
Amsterdam: EURODOS Publication; 2003.

Meek, Ronald L.
Smith, Marx & After. Ten Essays in the Development of Economic Thought.
London: Chapman & Hall; 1977.

Steedman, Ian
"Wicksteed, Philip Henry (1844 - 1927)". Article in: The New Palgrave. Dictionnary of Economics. John Eatwell, Murray Milgate, Peter Newman; London, New York, Tokyo: The Macmillan Press Limited, The Stockton Press, Maruzen Company Limited; 1987. Vol. 4.

Wicksteed, Philip Henry;
An Essay on the Co-ordination of the Laws of Distribution.
London: Macmillan, 1894.
republished as:
The Co-ordination of the laws of distribution.
with an introduction by Ian Steedman
Aldershot, Brookfield: E. Elgar; 1992.

Nanterre Université, 12.5.2004

Klaus Hagendorf