Metric so that each point of an ellipse has the same distance to the center

Question

In oeconomics one often handles with indifferene curves on $\mathbb R^2$. Given an individual preference of two goods the indifference curves is the set of points, who have the same euclidean distance to the individual preference. Usually this is a circle. However sometimes you prefere one good over another. Image you prefer good 1 (on the $x$-Axis) twice over good 2. This results in an ellipical-shaped indifference curve. Here is a picture of the given scenario.

In my example the individual preference is at the center (10,10). The individual wont prefer $(15,10)$ over $(10,20)$ or $(5,10)$ and $(10,0)$ and vice versa. He is indifferent to those options. How can I define a metric, so that each point has the same distance to the center (which is the understanding of being indifference in oeconomics in this scenario). Because on the euclidean metric the distance of $(15,10)$ is of course less to the center, than the distance from $(10,20)$ to the center.

The ellipse through the given points is described by: $$ x^2 + \frac 14 y^2 - 20x - 5y = -100.$$

Answer 1

The "deformed" distance $d'$ is given by the formula $$d'( (x_1,y_1),(x_2, y_2)) = d( (x_1, \frac{1}{2} y_1),(x_2, \frac{1}{2} y_2) )$$ where $d$ is the standard euclidean distance.

Answer 2

The mathematical premise of the problem is already flawed. I guess some economic modeling is missing in the explanation. You say the indifference curve if you like 2 things "just the same" is a circle. That's not correct, when you operate with numbers.

Say there are 2 kinds of sweets, they are red and green with different flavors you equally like. You currently have none of either (so your current position is at the origin $(0,0)$ of the plane).

You like getting 5 red the same as getting 5 green, so the indifference curve should be centered in $(0,0)$ and go through $(0,5)$ and $(5,0)$. If it really is a circle, it would be the circle centered in the origin with radius $5$.

The equation of that circle in the $(r,g)$ coordinate system is

$$r^2+g^2=25$$

On that circle lies the point $(4,3)$ as well (plug in the numbers to see that this is correct). But of course you would like to have $7$ sweets more than $5$ sweets, so this can't be the correct indifference curve.

Even more strangly, the point $(-5,0)$ lies on that circle as well. That means you value gaining $5$ red sweets (point $(5,0)$) and loosing 5 red sweets (point $(-5,0)$) the same, which is certainly not how I would assume this works.

If we take the simple idea that you like more sweets better then less sweets, and don't care about the color, then the curve of indifference is simply the curve where the number of sweets you have is constant, that would be

$$r+g=k,$$

for some integer $k$. That is not a circle but a line, plotted below for $k=10$:

Similiarly, if you like the red sweets twice as much as the green sweets, the curve of indifference is $2r+g=k$ for some constant $k$, another line with a different ascent than the one before.

Again, I don't doubt that the circle method you described is a good representation in some sense, but it isn't in the literal sense where you have a coordinate plane and one coordinate is the quanitity of one good and the other another good.