What is monotonic transformation? (economics)

[60051]

The textbook says it's a way of transforming a set of numbers into another set that preserves the order. But I don't understand what that means.

Here are a few examples. The question was: do these functions represent a monotonic transformation.


u = 2v - 13 (yes)

u = -1/v² (no)

u = ln(v) (yes)

u = v² (no)



I have no clue what you're supposed to look for.

 

[tiny-tim]

Hi 60051!

It means the graph is always going up (or always going down).

Look at the graph for the second and fourth examples …

if we include negative values of v, then the graph of u against v (u up and v across) comes down and then goes up …

for example, in the fourth one, -2 < 1 < 60051, but (-2)² > 1² < 60051², so the order isn't preserved.

 

[60051]

What if there are two variables?

 

[tiny-tim]

You mean if, say, you are told that "u is a monotonic function of x and y"?

I've never heard of "monotonic" being used for two (input) variables, but I suppose it would mean that, for each fixed value of y, u is a monotonic function of x, and for each fixed value of x, u is a monotonic function of y (and I expect they'd need to be either both monotonic increasing or both monotonic decreasing).

But that seems completely different from your textbook, which says that it's a way of transforming a set of numbers into another set that preserves the order …

that presumes that the original set of numbers has an order, and a two-variable set of numbers doesn't have any obvious order (eg is (3,7) before or after (5,5) ?).

 

[Mark44]

What if there are two variables?

All four examples you gave have two variables -- u and v. In each of those examples u is a function of v.

tiny-tim already answered for a situation in which there are three variables, where, for example, z is a function of two variables, x and y.

 

[60051]

Say you have an example like:

u = x²y²

So du/dx = 2xy², and du/dy = 2yx².

So how do you tell if the graph is always rising or falling?

 

[HallsofIvy]

Say you have an example like:

u = x²y²

So du/dx = 2xy², and du/dy = 2yx².

So how do you tell if the graph is always rising or falling?

As tiny-time said, "monotonic" is normally used for functions of several variables.

Here, it is clear that $x^2$ and $y^2$ are always non-negative so whether the du/dx is positive nor negative depends on the sign of y and whether du/dy is positive or negative depends on the sign of x.

In the first quadrant, where x> 0 and y> 0, du/dx and du/dy are both positive so the function increases as both x and y increase. In the third quadrant where both are negative, the function decreases as both x and y increase. In the second and fourth quadrants, whether the function vaue increases or decreases with increasing x and y depends on the precise values of x and y.

 

[tiny-tim]

Put y constant, and see whether du/dx is always positive (or always negative).

Put x constant, and see whether du/dy is always positive (or always negative).

EDIT: ooh, HallsofIvy beat me to it!

(but i think he meant ' "monotonic" is normally used for functions of one variable ')​

 

[Mark44]

The graph is not a single curve; it's a surface in three dimensions.

The textbook says it's a way of transforming a set of numbers into another set that preserves the order.

Your textbook seems to be talking about functions of a single variable. As tiny-tim pointed out, there is no obvious way of determining whether one ordered pair is "less than" another ordered pair.

BTW, if u = x²y², the partial derivatives have meaning, but not the ordinary derivatives.

IOW
$$\frac{\partial u}{\partial x} = 2xy^2$$
and

$$\frac{\partial u}{\partial y} = 2x^2y$$

 

[Mark44]

As tiny-time said, "monotonic" is normally used for functions of several variables.

I'm sure you meant functions of a single variable.