Homothetic function versus straight Engle curve Homothetic functions can be defined as follow (Green, 1964, p.49):


  If two figures A and B are "homothetic", or "similarly placed" with reference to a point P [for example, the origin], then for any two straight lines PQR and PQ'R' through P, cutting A in Q and Q' and B in R and R', the ratios PQ/PR and PQ'/PR' are equal. 


Additionally, consider the case where "each set of points, in output of commodity space, at which marginal rates of substitution are constant, is a straight line". This is, a straight Engel curve.

Are these two definitions equivalent, in the sense that each implies the other?

Or to put it differently, can we define:


an homothetic function as a function which yields straight Engle curves? and
a function with straight Engle curve as an homothetic function?


I am a bit confused, because in a text I'm reading both are used at different times, without mention their direct relation. 

The answer to me seems to be yes. However, Maybe in special cases like when an homothetic function does not start in the origin, or where Engel lines are horizontal, the relation might break. If so, would these be the only exceptions, such that we could conclude that, for most of our purposes, they are indeed equivalent?
A general result using indirect utility functions, is that expenditure linear in income ("straight-line" Engel curves) is produced if and only if the indirect utility function has the "Gorman-polar form".

The original paper is

Gorman, W. M. (1961). On a class of preference fields. Metroeconomica, 13(2), 53-56.

...where we read that a linear Engel curve is produced when we have a homothetic utility function (Gorman does not use the word, he writes "homogeneous" and in a footnote, enhances that to "function of a homogeneous function", which is how a homothetic function is defined).

But the reverse does not hold:homotheticity is sufficient for linear Engel curves, but not necessary.

For example, linear Engel curves are also produced by a utility function that reflects "quasi-linear" preferences, $U(x,m) = u(x) + m$, which are not homothetic.



A homothetic function is properly defined mathematically as follows:

Let $f(\mathbf x)$ be a homogeneous function of some degree, and let $g$ be a function with non zero derivative. Then $g[f(\mathbf x)]$ is called a homothetic function. In economics, we usually require for the derivative of $g$ to be strictly positive, so that we remain inside economically meaningful territory.