# Homothetic Functions and Monotonic Transformations

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.

• In the first question, can you please mention the behaviours of the functions $f$ and $h$ properly? You have used several notations for each letter (function).
– user43302
Apr 23 at 13:42
• Hello again, thanks for pointing that out. I've tired to make it more clear. If anything else isn't clear it might be more my lack of understanding, which needs clarifying in answer. But let me know anyway, cheers! Apr 23 at 14:02

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).