# Homogeneous function

A homogeneous function is a function whose value scales by a fixed power when all of its arguments are scaled by the same factor: a function \( f \) defined on a set closed under multiplication by positive scalars is homogeneous of degree \( k \) if \( f(\lambda x) = \lambda^{k} f(x) \) for every \( x \) in the domain and every \( \lambda > 0 \) with \( \lambda x \) in the domain<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup>. The degree \( k \) need not be an integer, and the concept applies to functions of any number of variables. Homogeneous functions appear throughout multivariable calculus, differential equations, economics, and the theory of polynomials on vector spaces.

| Key fact | Detail |
|---|---|
| Defining property | \( f(\lambda x) = \lambda^{k} f(x) \) for all \( \lambda > 0 \); the domain is usually taken to be a cone, a set closed under multiplication by positive scalars<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup> |
| Euler's theorem | If \( f \) is differentiable and homogeneous of degree \( k \), then \( \sum_{i=1}^{n} x_i \, \partial f / \partial x_i = k f(x) \); for a function on \( \mathbb{R}^n_+ \) that is continuous and differentiable on \( \mathbb{R}^n_{++} \) the condition is also sufficient<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup><sup> • </sup><sup>[2](https://scipp.ucsc.edu/~haber/ph116A/ODEs.pdf)</sup> |
| Derivatives | Each partial derivative of a function homogeneous of degree \( k \) is homogeneous of degree \( k - 1 \)<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup> |
| Degree 0 | A constant function is homogeneous of degree 0, and any degree-0 homogeneous function is constant on rays from the origin<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup> |
| Cobb–Douglas example | \( f(x_1, \ldots, x_n) = x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n} \) is homogeneous of degree \( \alpha_1 + \cdots + \alpha_n \)<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup> |
| Smoothness caveat | For \( n \geq 2 \) variables, a homogeneous function of any degree need not be differentiable, so Euler's theorem can fail without a smoothness assumption<sup>[3](https://www.tandfonline.com/doi/full/10.1080/0020739X.2018.1452303)</sup> |

## Definition and basic examples

The definition constrains the domain: it must contain \( (t x_1, \ldots, t x_n) \) whenever it contains \( (x_1, \ldots, x_n) \) and \( t > 0 \)<sup>[4](https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/HOM/t)</sup>. In practice the domain is usually a cone such as \( \mathbb{R}^n_+ \), the vectors with nonnegative components, or \( \mathbb{R}^n_{++} \), the vectors with strictly positive components<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup>.

Several immediate consequences follow from the definition. If \( 0 \) belongs to the domain and \( f \) is homogeneous of degree \( k \neq 0 \), then \( f(0) = f(2 \cdot 0) = 2^{k} f(0) \), which forces \( f(0) = 0 \)<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup>. A constant function is homogeneous of degree 0, and any function homogeneous of degree 0 is constant on rays from the origin, since scaling the argument does not change its value<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup>. Linear functions are homogeneous of degree one<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup>.

For functions of two variables, homogeneity of degree \( n \) means \( F(\lambda x, \lambda y) = \lambda^{n} F(x, y) \), and setting \( \lambda = 1/x \) shows that any such function can be written as \( F(x, y) = x^{n} F(1, y/x) \), a function of the ratio \( y/x \) alone times a power of \( x \)<sup>[2](https://scipp.ucsc.edu/~haber/ph116A/ODEs.pdf)</sup>. Standard examples of degrees 3, 2, and 1 are, respectively,

\[ x y^{2}, \qquad \frac{x^{2} y + 2 y^{3}}{x + y}, \qquad x \cos(y/x). \]

The last example shows that a homogeneous function need not be a polynomial: the trigonometric factor depends only on the ratio \( y/x \)<sup>[2](https://scipp.ucsc.edu/~haber/ph116A/ODEs.pdf)</sup>.

Homogeneity composes predictably. If \( f_1, \ldots, f_m \) are homogeneous of degree \( k \) and \( g \) is homogeneous of degree \( \ell \), then the composite \( h(x) = g(f_1(x), \ldots, f_m(x)) \) is homogeneous of degree \( k + \ell \)<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup>.

## Euler's theorem

The central result about homogeneous functions is credited to [Leonhard Euler](https://www.edgepedia.org/leonhard-euler) (1707–1783), who established a theorem on homogeneous functions known by his name<sup>[4](https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/HOM/t)</sup><sup> • </sup><sup>[5](https://en.wikisource.org/wiki/A_History_of_Mathematics_%281893%29/Modern_Europe/Euler%2C_Lagrange%2C_and_Laplace)</sup>. For a function \( F(x, y) \) homogeneous of degree \( n \), the theorem states

\[ x F_x + y F_y = n F. \]

The proof differentiates the identity \( F(\lambda x, \lambda y) = \lambda^{n} F(x, y) \) with respect to \( \lambda \) and then sets \( \lambda = 1 \)<sup>[2](https://scipp.ucsc.edu/~haber/ph116A/ODEs.pdf)</sup>. In \( n \) variables the statement is

\[ \sum_{i=1}^{n} x_i \frac{\partial f}{\partial x_i} = k f(x), \qquad \text{equivalently} \qquad x \cdot \nabla f(x) = k f(x). \]

For a function \( f : \mathbb{R}^n_+ \to \mathbb{R} \) that is continuous and differentiable on \( \mathbb{R}^n_{++} \), the converse holds as well: the differential equation above is satisfied if and only if \( f \) is homogeneous of degree \( k \)<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup>. An equivalent formulation says that homogeneous functions are eigenfunctions of the Euler operator \( \sum_i x_i \, \partial / \partial x_i \), with the degree of homogeneity as the eigenvalue<sup>[6](https://planetmath.org/eulerstheoremonhomogeneousfunctions)</sup>.

Two corollaries carry the scaling structure to the derivatives. First, each partial derivative \( D_j f \) of a function homogeneous of degree \( k \) is itself homogeneous of degree \( k - 1 \)<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup>. Second, for a twice-differentiable function homogeneous of degree 1, the [Hessian matrix](https://www.edgepedia.org/hessian-matrix) \( [D_{ij} f(x)] \) is singular at every point, because the degree-1 case gives \( [D_{ij} f(x)]\, x = 0 \) for \( x \neq 0 \)<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup>.

The smoothness hypothesis matters. A 2018 note in the *International Journal of Mathematical Education in Science and Technology* gives an example showing that when there are \( n \geq 2 \) variables, a homogeneous function of any degree need not be differentiable, so the conclusion of [Euler's theorem](https://www.edgepedia.org/eulers-theorem) fails for such a function; the same work emphasizes behavioral differences between the one-variable and multi-variable settings<sup>[3](https://www.tandfonline.com/doi/full/10.1080/0020739X.2018.1452303)</sup>.

## Applications in economics

Economists use homogeneous functions to model returns to scale. A production function homogeneous of degree 1 exhibits constant returns to scale: doubling all inputs doubles output. A function homogeneous of a degree greater than 1 exhibits increasing returns to scale, and one homogeneous of a degree less than 1 exhibits diminishing returns to scale<sup>[7](https://www.afdb.org/sites/default/files/documents/publications/wps_no_362_economic_growth_total_factor_productivity_and_output_gap_in_sierra_leone1.pdf)</sup>.

The weighted geometric mean with domain \( \mathbb{R}^n_+ \), known to economists as the Cobb–Douglas function,

\[ f(x_1, \ldots, x_n) = x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n}, \qquad \alpha_i > 0, \]

is homogeneous of degree \( \alpha = \alpha_1 + \cdots + \alpha_n \)<sup>[1](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)</sup>. Charles Cobb and Paul Douglas introduced this production function in 1928 using data from the US manufacturing sector for 1899 to 1922, and it has been widely used in economic theory since<sup>[8](https://arxiv.org/abs/1910.06739)</sup>. A 2024 note on arXiv adds a characterization: the Cobb–Douglas function is uniquely determined by the property that a constant-returns-to-scale firm whose labor share of cost stays constant under cost minimization for any output level must have a [Cobb–Douglas production function](https://www.edgepedia.org/cobb-douglas-production-function)<sup>[9](https://arxiv.org/abs/2411.08067)</sup>.

Euler's theorem enters economic analysis directly. For a production function homogeneous of degree 1, the theorem states that the sum of each input multiplied by its marginal product equals total output, a relationship used in the Solow growth model and in the theory of income distribution<sup>[4](https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/HOM/t)</sup>. Homogeneity also constrains empirical work: if a production process is characterized by a homogeneous production function, profit maximization implies that the demand for each input must be elastic with respect to the output price, a restriction that limits the usefulness of such functions in empirical analysis<sup>[10](https://ageconsearch.umn.edu/record/137430/?ln=en)</sup>.

The use of aggregate homogeneous production functions such as \( \mathrm{GDP} = F(K, L) \) has been criticized on theoretical grounds. A 2003 survey in *Metroeconomica* finds that the requirements for deriving a well-behaved [aggregate production function](https://www.edgepedia.org/aggregate-production-function) from micro production functions are so strict that real economies probably do not meet them, making it unlikely that GDP can be correctly regarded as a well-defined neoclassical function of aggregate capital and labor<sup>[11](https://doi.org/10.1111/1467-999X.00166)</sup>.

## Homogeneous polynomials and polarization

In algebra and analysis, homogeneity is defined for polynomials on vector spaces. A map \( P : X \to Y \) between Banach spaces is an \( m \)-homogeneous polynomial if there exists a symmetric \( m \)-linear mapping \( \check{P} \) with \( P(x) = \check{P}(x, \ldots, x) \) for all \( x \)<sup>[12](https://depot.univ-msila.dz/server/api/core/bitstreams/2cddb662-5d3a-4d14-8e86-733c4878a269/content)</sup>. The symmetric multilinear mapping representing a monomial is unique, and in the smooth setting the \( p \)-homogeneous maps are exactly the monomials of degree \( p \)<sup>[13](https://www.mat.univie.ac.at/~michor/apbookh.pdf)</sup>.

The correspondence between homogeneous polynomials and symmetric multilinear forms is made explicit by the polarization formula, which recovers \( \check{P}(x_1, \ldots, x_m) \) from the values of \( P \) on sums of the arguments with signs:

\[ \check{P}(x_1, \ldots, x_m) = \frac{1}{2^{m} m!} \sum_{\varepsilon_i = \pm 1} \varepsilon_1 \cdots \varepsilon_m \, P(\varepsilon_1 x_1 + \cdots + \varepsilon_m x_m). \]

This generalizes the classical correspondence between quadratic forms and symmetric bilinear forms<sup>[12](https://depot.univ-msila.dz/server/api/core/bitstreams/2cddb662-5d3a-4d14-8e86-733c4878a269/content)</sup><sup> • </sup><sup>[14](https://arxiv.org/abs/1509.05707)</sup>. Work by T. Velanga, a researcher in functional analysis, shows that \( (k, m) \)-linear mappings introduced by I. Chernega and A. Zagorodnyuk are particular cases of polynomials, with the corollary that every multilinear mapping is a homogeneous polynomial, and provides a multipolynomial polarization formula that unifies previous well-known formulas<sup>[15](https://arxiv.org/abs/1706.04703)</sup>.

## Related uses of the term

The word homogeneous appears in several distinct mathematical contexts that share the scaling idea. A first-order differential equation whose coefficient functions \( P(x, y) \) and \( Q(x, y) \) are both homogeneous of degree \( n \) is called a homogeneous coefficient equation; such equations are solved by a change of dependent variable that makes them separable<sup>[2](https://scipp.ucsc.edu/~haber/ph116A/ODEs.pdf)</sup>. In physics, dimensional analysis imposes a related constraint: every term in a physically meaningful equation must have the same dimensions, and the arguments of standard functions such as trigonometric functions, logarithms, and exponentials must be dimensionless<sup>[16](https://pressbooks.online.ucf.edu/osuniversityphysics/chapter/1-4-dimensional-analysis/)</sup>.

## References

1. [KC Border (2000, rev. 2017). Euler's Theorem for Homogeneous Functions. Caltech Division of the Humanities and Social Sciences.](https://healy.econ.ohio-state.edu/kcb/Notes/EulerHomogeneity.pdf)
2. [Homogeneous functions and Euler's theorem, in *Introduction to Methods of Applied Mathematics* (UCSC lecture notes).](https://scipp.ucsc.edu/~haber/ph116A/ODEs.pdf)
3. [On the smoothness condition in Euler's theorem on homogeneous functions, *International Journal of Mathematical Education in Science and Technology* 49(8).](https://www.tandfonline.com/doi/full/10.1080/0020739X.2018.1452303)
4. [Martin J. Osborne. Mathematical methods for economic theory: 2.5 Homogeneous functions. University of Toronto.](https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/HOM/t)
5. [A History of Mathematics (1893): Euler, Lagrange, and Laplace. Wikisource.](https://en.wikisource.org/wiki/A_History_of_Mathematics_%281893%29/Modern_Europe/Euler%2C_Lagrange%2C_and_Laplace)
6. [Euler's theorem on homogeneous functions. PlanetMath.](https://planetmath.org/eulerstheoremonhomogeneousfunctions)
7. [diseconomies of scale. a function homogeneous of a degree greater than 1 exhibits increasing returns to scale or economies of scale. a function homogeneous of degree less than 1 exhibits diminishing returns to scale or diseconomies of scale. for analytical purposes, this paper employs the cobb-douglas production function homogeneous of degree 1. due to the limited number of observations, this paper does not conduct econometric estimation of the elasticity of factor inputs in the production function.](https://www.afdb.org/sites/default/files/documents/publications/wps_no_362_economic_growth_total_factor_productivity_and_output_gap_in_sierra_leone1.pdf)
8. [The Cobb-Douglas production function revisited. arXiv:1910.06739.](https://arxiv.org/abs/1910.06739)
9. [A note on the Cobb-Douglas function. arXiv:2411.08067.](https://arxiv.org/abs/2411.08067)
10. [Price Elasticities Implied by Homogeneous Production Functions, *Journal of Agricultural Economics Research*.](https://ageconsearch.umn.edu/record/137430/?ln=en)
11. [Jesus Felipe (2003). Aggregation in Production Functions: What Applied Economists should Know. *Metroeconomica*.](https://doi.org/10.1111/1467-999X.00166)
12. [m-homogeneous polynomials and tensor products, University of M'sila thesis excerpt.](https://depot.univ-msila.dz/server/api/core/bitstreams/2cddb662-5d3a-4d14-8e86-733c4878a269/content)
13. [Andreas Kriegl and Peter W. Michor. Polynomials versus monomials, in *The Convenient Setting of Global Analysis*.](https://www.mat.univie.ac.at/~michor/apbookh.pdf)
14. [Aleš Drápal, Petr Vojtěchovský (2015). Symmetric multilinear forms and polarization of polynomials. arXiv:1509.05707.](https://arxiv.org/abs/1509.05707)
15. [T. Velanga (2017). Multilinear mappings versus homogeneous polynomials and a multipolynomial polarization formula. arXiv:1706.04703.](https://arxiv.org/abs/1706.04703)
16. [1.4 Dimensional Analysis, *University Physics Volume 1*, OpenStax.](https://pressbooks.online.ucf.edu/osuniversityphysics/chapter/1-4-dimensional-analysis/)

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