2.5 Homogeneous functions
Definition
Multivariate functions that are “homogeneous” of some degree are often used in economic theory. A function is homogeneous of degree k if, when each of its arguments is multiplied by any number t > 0, the value of the function is multiplied by t^{k}. For example, a function is homogeneous of degree 1 if, when all its arguments are multiplied by any number t > 0, the value of the function is multiplied by the same number t.Here is a precise definition. Because the definition involves the relation between the value of the function at (x_{1}, ..., x_{n}) and it values at points of the form (tx_{1}, ..., tx_{n}) where t is any positive number, it is restricted to functions for which (tx_{1}, ..., tx_{n}) is in the domain whenever t > 0 and (x_{1}, ..., x_{n}) is in the domain. (Some domains that have this property are the set of all real numbers, the set of nonnegative real numbers, the set of positive real numbers, the set of all ntuples (x_{1}, ..., x_{n}) of real numbers, the set of ntuples of nonnegative real numbers, and the set of ntuples of positive real numbers.)
 Definition

Let f be a function of n variables defined on a set S for which (tx_{1}, ..., tx_{n}) ∈ S whenever t > 0 and (x_{1}, ..., x_{n}) ∈ S. Then
f is homogeneous of degree k if
f(tx_{1}, ..., tx_{n}) = t^{k}f(x_{1}, ..., x_{n}) for all (x_{1}, ..., x_{n}) ∈ S and all t > 0.
 Example

For the function f(x_{1}, x_{2}) = Axa
1xb
2 with domain {(x_{1}, x_{2}): x_{1} ≥ 0 and x_{2} ≥ 0} we havef(tx_{1}, tx_{2}) = A(tx_{1})^{a}(tx_{2})^{b} = At^{a+b}xaso that f is homogeneous of degree a + b.
1xb
2 = t^{a+b}f(x_{1}, x_{2}),
 Example

Let f(x_{1}, x_{2}) = x_{1} + x2
2, with domain {(x_{1}, x_{2}): x_{1} ≥ 0 and x_{2} ≥ 0}. Thenf(tx_{1}, tx_{2}) = tx_{1} + t^{2}x2It doesn't seem to be possible to write this expression in the form t^{k}(x_{1} + x2
2.
2) for any value of k. But how do we prove that there is no such value of k? Suppose that there were such a value. That is, suppose that for some k we havetx_{1} + t^{2}x2Then in particular, taking t = 2, we have
2 = t^{k}(x_{1} + x2
2) for all (x_{1}, x_{2}) ≥ (0, 0) and all t > 0.2x_{1} + 4x_{2} = 2^{k}(x_{1} + x2
2) for all (x_{1}, x_{2}).2 = 2^{k} and 4 = 2^{k},which is not possible. Thus f is not homogeneous of any degree.
Suppose that a consumer's demand for goods, as a function of prices and her income, arises from her choosing, among all the bundles she can afford, the one that is best according to her preferences. Then we can show that this demand function is homogeneous of degree zero: if all prices and the consumer's income are multiplied by any number t > 0 then her demands for goods stay the same.
Partial derivatives of homogeneous functions
The following result is sometimes useful. Proposition
 Let f be a differentiable function of n variables that is homogeneous of degree k. Then each of its partial derivatives f'_{i} (for i = 1, ..., n) is homogeneous of degree k − 1.
 Proof

The homogeneity of f means that
f(tx_{1}, ..., tx_{n}) = t^{k}f(x_{1}, ..., x_{n}) for all (x_{1}, ..., x_{n}) and all t > 0.Now differentiate both sides of this equation with respect to x_{i}, to gettf'_{i}(tx_{1}, ..., tx_{n}) = t^{k}f'_{i}(x_{1}, ..., x_{n}),and then divide both sides by t to getf'_{i}(tx_{1}, ..., tx_{n}) = t^{k−1}f'_{i}(x_{1}, ..., x_{n}),so that f'_{i} is homogeneous of degree k − 1.
Application: level curves of homogeneous functions
This result can be used to demonstrate a nice result about the slopes of the level curves of a homogeneous function. As we have seen, the slope of the level curve of the function F through the point (x_{0}, y_{0}) at this point is− 

. 
− 

. 
− 

= − 

. 
In this figure, the red lines are two level curves, and the two green lines, the tangents to the curves at (x_{0}, y_{0}) and at (cx_{0}, xy_{0}), are parallel.
We may summarize this result as follows.
Let F be a differentiable function of two variables that is homogeneous of some degree. Then along any given ray from the origin, the slopes of the level curves of F are the same.
Euler's theorem
A function homogeneous of some degree has a property sometimes used in economic theory that was first discovered by Leonhard Euler (1707–1783). Proposition (Euler's theorem)

Let f be a differentiable function of n variables defined on an open set S for which (tx_{1}, ..., tx_{n}) ∈ S whenever t > 0 and
(x_{1}, ..., x_{n}) ∈ S. Then f is homogeneous of degree k if and only if
∑n
i=1x_{i}f'_{i}(x_{1}, ..., x_{n}) = kf(x_{1}, ..., x_{n}) for all (x_{1}, ..., x_{n}). (*)
 Proof

I first show that if f is homogeneous of degree k then (*) holds. If f is homogeneous of degree k then
f(tx_{1}, ..., tx_{n}) = t^{k}f(x_{1}, ..., x_{n}) for all (x_{1}, ..., x_{n}) and all t > 0.Differentiate each side of this equation with respect to t, to givex_{1}f'_{1}(tx_{1}, ..., tx_{n}) + x_{2}f'_{2}(tx_{1}, ..., tx_{n}) + ... + x_{n}f'_{n}(tx_{1}, ..., tx_{n}) = kt^{k−1}f(x_{1}, ..., x_{n}).Now set t = 1, to obtain (*).
I now show that if (*) holds then f is homogeneous of degree k. Suppose that (*) holds. Fix (x_{1}, ..., x_{n}) and define the function g of a single variable by
g(t) = t^{−k}f(tx_{1}, ..., tx_{n}) − f(x_{1}, ..., x_{n}). g'(t) = −kt^{−k−1}f(tx_{1}, ..., tx_{n}) + t^{−k}∑n
i=1 x_{i}f'_{i}(tx_{1}, ..., tx_{n}).∑n
i=1tx_{i}f_{i}'(tx_{1}, ..., tx_{n}) = kf(tx_{1}, ..., tx_{n}),
 Example

Let f(x_{1}, ..., x_{n}) be a firm's production function; suppose it is homogeneous of degree 1 (i.e. has “constant returns to scale”). Euler's theorem shows that if the price (in terms of units of output) of each input i is its “marginal product”
f'_{i}(x_{1}, ..., x_{n}), then the total cost, namely
∑nis equal to the total output, namely f(x_{1}, ..., x_{n}).
i=1x_{i}f'_{i}(x_{1}, ..., x_{n})