Let us start with a definition:

Homogeneity: Let ｦ :R^{n}ｮ R be a real-valued function. Then ｦ (x_{1}, x_{2}...., x_{n}) ishomogeneous of degree kif l^{k}ｦ(x) = ｦ(l x) where l ｳ 0 (x is the vector [x_{1}...x_{n}]).

In other words, a function is called *homogeneous of degree k* if by multiplying *all*
arguments by a constant scalar l , we increase the value of the
function by l^{k}, i.e.

l

^{k}ｦ(x_{1}, x_{2},..., x_{n}) = ｦ(lx_{1}, lx_{2},...., lx_{n})

If k = 1, we call this a *linearly homogenous* function. If we
interpret ｦ(x) as a production
function, then k = 1 implies constant returns to scale (as l^{k}
= l), k > 1 implies increasing returns to scale (as l^{k} > l) and if 0 < k <
1, then we have decreasing returns to scale (as l^{k}
< l).

Phillip Wicksteed (1894) stated the
"product exhaustion" thesis implied by the marginal
productivity theory of distribution - namely, that if all agents were paid their
marginal product, then total costs would exhaust the entire product. Wicksteed assumed
constant returns to scale - and thus employed a linear homogeneous production function, a
function which was homogeneous of degree one. It was A.W. Flux (1894) who pointed out that Wicksteed's
"product exhaustion" thesis was merely a restatement of *Euler's Theorem*.
Euler’s Theorem states that under homogeneity of degree 1, a function ｦ (x) can be reduced to the sum of its arguments multiplied by their
first partial derivatives, in short:

Theorem: (Euler's Theorem) Given the function ｦ :R^{n}ｮ R, then if ｦ is positively homogeneous of degree 1 then:ｦ (x

_{1}, x_{2}, ...., x_{n}) = x_{1}[ｶ ｦ /ｶ x_{1}] + x_{2}[ｶ ｦ /ｶ x_{2}] + ...... + x_{n}[ｶ ｦ /dｶx_{n}]or simply:

ｦ (x) = ・/font>

^{n}_{i=1 }[dｦ (x)/dx_{i}]ｷx_{i}

Proof: By definition of homogeneity of degree k, letting k = 1, then l ｦ (x) = ｦ (l x) where x is a n-dimensional vector and l
is a scalar. Differentiating both sides of this expression with respect to x_{i }and
using the chain rule, we see that:

[ｶ l ｦ (x)/ｶ ｦ (x)]ｷ[ｶ ｦ (x)/ｶ x

_{i}] = [ｶ ｦ (l x)/ｶ (l x_{i})]ｷ[ｶ (l x_{i})/ｶ x_{i}]

as [ｶ l ｦ
(x)/ｶ ｦ (x)] = l
and ｶ (l x_{i})/ｶ x_{i} = l then l [ｶ ｦ (x)/ｶ x_{i}] = [ｶ ｦ (l x)/ｶ (l x_{i})]l then:

ｶ ｦ (x)/ｶ x

_{i}= ｶ ｦ (l x)/ｶ (l x_{i}) (E.1)

Now, differentiating both sides of the original expression l ｦ (x) = ｦ (l x) with respect to l , we get:

ｶ l ｦ (x)/ｶ l = ・/font>

^{n}_{i=1}[ｶ ｦ (l x)/ｶ (l x_{i})]ｷ[ｶ (l x_{i})/ｶ l ]

As ｶ l ｦ
(x_{i})/ｶ l = ｦ (x_{i}) and ｶ (l x_{i})/ｶ l
= x_{i} for all i = 1,..., n, then this expression reduces to:

ｦ (x) = ・/font>

^{n}_{i=1}[ｶ ｦ (l x)/ｶ (l x_{i})]ｷx_{i}

Now using the equality in (E.1), we can substitute ｶ ｦ (x)/ｶ x_{i} for ｶ ｦ (l x)/ｶ (l x_{i}). Thus, this becomes:

ｦ (x) = ・/font>

^{n}_{i=1}[ｶ ｦ (x)/ｶ x_{i}]ｷx_{i}

which is Euler’s Theorem.ｧ

One of the interesting results is that if ｦ(x) is a
homogeneous function of degree k, then the first derivatives, ｦ_{i}(x),
are themselves homogeneous functions of degree k-1. So, for the homogeneous of
degree 1 case, ｦ _{i}(x) is homogeneous of degree
zero. Consequently, there is a corollary to Euler's Theorem:

Corollary: if ｦ :R^{n}ｮ R is homogenous of degree 1, then ・/font>^{n}_{i=1}[ｶ^{2}ｦ(x)/ｶ x_{i}ｶx_{j}]x_{i}= 0 for any j.

Proof: By Euler’s Theorem, ｦ (x) = ・/font>
^{n}_{i=1}[ｶ ｦ
(x)/ｶ x_{i}]ｷx_{i }. Differentiating with
respect to x_{j} yields:

ｶ ｦ (x)/ｶ x

_{j}= [ｶ^{2}ｦ^{ }(x)/ｶ x_{1}ｶx_{j}]x_{1}+ ..... + [ｶ^{2}ｦ^{ }(x)/ｶ x_{j}ｶx_{j}]x_{j}+ ｶ ｦ (x)/ｶ x_{j}+ ..... + [ｶ^{2}ｦ^{ }(x)/ｶ x_{n}ｶx_{j}]x_{n}

or rewriting:

ｶ ｦ (x)/ｶ x

_{j}= ・/font>^{n}_{i=1}[ｶ^{2}ｦ^{ }(x)/ｶ x_{i}ｶx_{j}]x_{i}+ ｶ ｦ (x)/ｶ x_{j}

where, note, the summation expression sums from all i from 1 to n (including i = j).
Nonetheless, note that the expression on the extreme right, ｶ ｦ (x)/ｶ x_{j} appears on both
sides of the equation. Thus:

・/font>

^{n}_{i=1}[ｶ^{2}ｦ^{ }(x)/ｶ x_{i}ｶx_{j}]x_{i}= 0

which is what we sought.ｧ

Back | Top | Selected References | Next |