A system of linear equations is homogeneous if all of the constant terms are zero: A homogeneous system is equivalent to a matrix equation of the form. where A is an m × n matrix, x is a column vector with n entries, and 0 is the zero vector with m entries.

Still, oil is still relatively homogeneous compared to many other goods – relative to most manufactured goods, for example. It is sufficiently homogeneous to be traded on organized exchanges.

perfect competition

Homogeneous Differential Equations And both M(x, y) and N(x, y) are homogeneous functions of the same degree.

Multivariate functions that are “homogeneous” of some degree are often used in economic theory. 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.

A constant function is homogeneous of degree 0. If a function is homogeneous of degree 0, then it is constant on rays from the the origin. Linear functions are homogenous of degree one.

In mathematics, a homothetic function is a monotonic transformation of a function which is homogeneous; however, since ordinal utility functions are only defined up to an increasing monotonic transformation, there is a small distinction between the two concepts in consumer theory.

In microeconomics, they use homogeneous production functions, including the function of Cobb–Douglas, developed in 1928, the degree of such homogeneous functions can be negative which was interpreted as decreasing returns to scale.

we say that it is homogenous if and only if g(x)≡0. You can write down many examples of linear differential equations to check if they are homogenous or not. For example, y″sinx+ycosx=y′ is homogenous, but y″sinx+ytanx+x=0 is not and so on.

So let’s go:

1. Start with: dy dx = 1−y/x 1+y/x.
2. y = vx and dy dx = v + x dvdx v + x dv dx = 1−v 1+v.
3. Subtract v from both sides:x dv dx = 1−v 1+v − v.
4. Then:x dv dx = 1−v 1+v − v+v2 1+v.
5. Simplify:x dv dx = 1−2v−v2 1+v.

g(tx,ty)=−tx(f(tx,ty))2=−tx⋅(tx⋅ty)2=t5(−x(xy)2)=t5g(x,y). So, the degree of homogenity is 5. Put g(x,y)=−x(f(x,y))2.

This is Euler’s theorem. Euler’s theorem states that if a function f(ai, i = 1,2,…) is homogeneous to degree “k”, then such a function can be written in terms of its partial derivatives, as follows: kλk−1f(ai)=∑iai(∂f(ai)∂(λai))|λx. 15.6a.