# 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 slight problem is that sometime there are several optimal choices. E.g. for $U(x,y) = x + y$, $p_x = p_y$ the Engel-curve is a triangle. Is that a "straight line"? – Giskard Apr 12 '17 at 16:13
• Could you please elaborate on your counterexamples? I don't understand what you mean by "a homothetic function [that] does not start in the origin" (how is that possible?) and "where Engel lines are horizontal" (why is that a counterexample?). – Oliv Apr 12 '17 at 19:18
• @denesp If I get your example right, the isoquants are linear with the same slope as the MRS. As such, any combination is optimal. How can you compute the Engle curve? Would it be "indeterminate"? Or maybe, a linear Engle-curve would indeed be possible (and optimal), but not unique. It looks like a potential counterexample. – luchonacho Apr 13 '17 at 8:34
• @Oliv Regarding not starting in the origin, I am only taking the definition of homothetic function from the book. There is no mention at all of $P$ being the origin. In fact, homothetic seems to be mainly a geometrical definition, maybe independent of economics. Regarding your second point, maybe you are right. E.g. in the case denesp mentions, if $p_{x}<p_{y}$, optimal is a corner solution, with $x^*=I/p_{x}$ and $y^*=0$. Thus, the expansion path is vertical or horizontal (depending on the axis ordering), and naturally, still a straight line. So, it seems that is not a good counterexample. – luchonacho Apr 13 '17 at 8:40
• I specified $p_x = p_y$. And yes, my point was that the Engel-curve is ill-defined in this case. If it contains all $x$ values that may constitute an optimal bundle then the Engle-curve would be a mapping, not a function, and in my case it would look like a triangle. – Giskard Apr 13 '17 at 9:55

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.

• But @denesp's comment shows a counterexample where homotheticity does not yield linear Engle-curves, unless we stretch the definition of sufficiency so that a linear Engle-curve can be optimum, even if alternative, non-linear curves and be so at the same time. – luchonacho Apr 13 '17 at 13:14
• @luchonacho Granted. I just tend to discuss subjects here for the economically interesting cases. Denesp's example leads to indeterminacy of the optimal consumption bundle, and since all around us people do consume things, it follows that it is not a very useful description of economic reality, although it is very useful as an educational device. – Alecos Papadopoulos Apr 13 '17 at 13:54
• Entirely agree. But I am concerned that your conclusion in bold is not strictly true. Is that a conclusion by Gorman? I would add the disclaimer that, for economic significant cases, it is an accurate conclusion. – luchonacho Apr 13 '17 at 14:17