Homothetic Functions and Monotonic Transformations

Question

Using the following definition of a homotheic function (taken from my Mathematical Economics course pack).

• A function $f: \mathbb{R^{n+}} \to \mathbb{R}$ is homothetic if it has the form:
• $f(x,y) = q(r(x,y))$

1. Where $r$ is a function that is homogenous of degree 1
2. $q$ is strictly increasing

For example: Let the following be a production function: $f(x, y) = x^a y^b$ with $a+b > 0$ We can rite this as:

$f(x,y) = h(r(x,y))= (x^qy^p)^{a+b}$

Where $q = \frac{a}{a+b}$ $p = \frac{b}{a+b}$

Hence we have written $f$ in terms of a monotonic transformation of a homogenous to degree $d = 1$ function $r$. And $h$ is strictly increasing.

Question 1):

• I want to clarify here that "Monotonic transformation" is related but different to "Montonic function".
• Montonic Function: A non-decreasing function (It could be constant?)
• Monotonic transformation: Preserves the ordering of the function e.g. If $f(h(x_2,y_2))$ is a monotonic transformation of $h$, and $h$ is strictly increasing, e.g. $h(x_1, y_1) > h(x_2,y_2)$ $\forall (x_i,y_i)>(x_j,y_j)$ i.e. then $f(h(x_1,y_1)) > f(h(x_2,y_2))$
• This would mean that a monotonic transformation would preserve the strictness I.e. if $h$ is strictly increasing $f(h(x_2,y_2))$ is a monotonic transformation and is strictly increasing?

Question 2):

As a result of the discussion above regarding the necessity for a strictly increasing function. How does the following definition from Hal R. Varian, "Microeconomic Analysis," 3rd Edition. fit with this? Why the subtle difference? Note, he uses "monotonic" rather than "strictly increasing:"

A homothetic function is a monotonic transformation of a function that is homogeneous of degree 1. In other words, $f(x)$ is homothetic if and only if it can be written as $f(x) = g(h(x))$ where $h(.)$ is homogeneous of degree 1 and $g(.)$ is a monotonic function

Essentially this is a hybrid between definition 2 and definition 2 in a longer post i have made about homothetic function definitions.

Answer 1

Question 1.

When you say monotonic transformation, it is assumed to be a positive and strictly monotonic transformation. This is mentioned by Varian as you can see in my other answer. If you don't assume it to be strict, then you lose the utility function (and also the order).

Example. Suppose $h(x,y) = x^a y^{1-a}$ such that $a \in (0,1)$ and $f(x) = 5$. Then $f(h(x,y))$ doesn't describe the same preferences as $h(x,y)$.

If $f$ is strictly increasing, then $h(x_1,y_1) > h(x_2, y_2) \iff f(h(x_1,y_1)) > f(h(x_2,y_2))$ which ensures that the order is preserved (or the new function $(f \cdot h)(x,y)$ describes the same preferences).

Question 2.

Answered here.