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Let $U(x,y)$ be a utility function. Suppose I have an indifference curve for which $U(x,y) = \bar{U}$. Then $dU = 0$ along the curve and I can rearrange to find the MRS.

Suppose I have a monotonic transformation of my utility function. That is, some function $$W(x,y) = f(U(x,y))$$ $$f'(\cdot) > 0$$

Since it too is an indifference curve, $dW = 0$ and I can rearrange as follows

$$\frac{dy}{dx} = -\frac{W_x}{W_y} = -\frac{f'(U(x,y))U_x}{f'(U(x,y))U_y} =-\frac{U_x}{U_y}$$ Thus, we observe MRS is invariant under monotonic transformation.

My Question:

Can someone explain what this means graphically? I don't get why the slope remains the same. Why does MRS remain the same but MU changes?

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You're right that it's a bit counterintuitive that the shape of the indifference curves shouldn't change when you transform the utility function. The reason is that you are transforming along an axis that is perpendicular to the plane where the indifference curve lives.

Let's imagine we have two goods, x and y, and let's say that the original utility function is given by $U(x,y)=x^\frac{1}{3}y^\frac{2}{3}$. We can create a 3D plot of the utility function where $z = U(x,y)$. Any transformation $f(z)$ will work on the z dimension. In the following example, the transformation $f(\cdot)$ is multiplying the utility function by a positive scalar

Scalar multiplication of utility function

If you look down on the $x-y$ plane from above, you will notice that you can plot an indifference curve through a point on that plane and it will look identical before and after the transformation. Note that it will not be the same indifference curve because the utility level will change under the transformation. Going back to the multiplication example, let's look at the indifference curve (red line) that goes through the bundle $(x,y) = (10,10)$ (red dot). Regardless of the multiplier, the indifference curve maintains its shape but the associated utility level increases from 10 to 20.

Scalar multiplication, shifting indifference curves

I have also included other level curves, for utility levels 4, 8, 12, ... 40. Note that as the factor with which we multiply the original utility function gets larger, these level curves get squeezed into the lower left corner.

You can see the same thing happening with other monotonic transformations, for example translation (adding a constant) and power transformation.

Adding a constant

Power transformation

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  • $\begingroup$ Excellent use of graphics and animations! $\endgroup$ – Bill Clark Jun 13 at 14:34
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Preferences are primitive. MRS is defined solely in terms of the primitive preferences:

If I give up 1 apple (good on the horizontal axis), then MRS is the number of bananas (good on the vertical axis) I must be given to remain indifferent.

Clearly then, if preferences do not in any way change, then neither should the MRS.

Utility functions/representations are merely a (very) convenient tool to help us study these preferences. Under certain technical conditions, we can speak also of marginal utility. But these, it must always be remembered, are merely tools to help us study preferences.

So utility functions can be transformed monotonically. But such monotonic transformations will do nothing whatsoever to the preferences. And so of course the MRS will not change either. But the utility functions (and the MU) will.

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Graphically: you have to have in mind a 3D graph. On the basis (x,y) and for $z$ the level of your utility. The MRS must be read on the $(x,y)$ space: for a same utility level (z is constant), what is the trade off between goods 1 and 2. Now, when you do this kind of transformation, you simply adjust the level of the utility (the value of z). Btw, $f'<0$ is also ok. Hope it helps despite my English.

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There are four basic levels of measurement in statistical theory:

  • nominal (also referred to as "categorical")
  • ordinal
  • interval
  • ratio

(Note that interval and ratio are sometimes collectively referred to as "cardinal" data.)

They are sometimes defined in terms of which kind of mathematical transformations are allowed on a given dataset, without changing the relevant properties (distinguishability, order, and relative/absolute magnitude.)

Ordinal data allows for any monotonic transformation of the data — also called "order-preserving" transformations. If utility is taken to be of the ordinal type (which is typically — though not always, and not historically — the case) then such transformations are permitted, and will not impact the results of any allowed calculations on the data. This is not unique to utility; any ordinal data will allow for such transformations.

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