<p>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, $f'&lt;0$ is also ok. Hope it helps despite my English. </p>

CC BY-SA 3.0
2015-04-10T23:39:18.667Z
2015-04-10T23:39:18.667Z
Yann
<p>There are four basic <a href="https://en.wikipedia.org/wiki/Level_of_measurement" rel="nofollow noreferrer">levels of measurement</a> in statistical theory:  </p>

<ul>
<li>nominal (also referred to as "categorical")</li>
<li>ordinal</li>
<li>interval</li>
<li>ratio</li>
</ul>

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

<p>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.)</p>

<p>Ordinal data allows for any monotonic transformation of the data &mdash; also called "order-preserving" transformations.  If utility is taken to be of the ordinal type (which is typically &mdash; though not <em>always</em>, and not <em>historically</em> &mdash; 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; <em>any</em> ordinal data will allow for such transformations.</p>

CC BY-SA 4.0
2019-06-12T13:44:07.030Z
2019-06-12T13:44:07.030Z
Bill Clark
<p>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.</p>

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

<p><a href="https://i.sstatic.net/1Qf7H.gif" rel="noreferrer"><img src="https://i.sstatic.net/1Qf7H.gif" alt="Scalar multiplication of utility function"></a></p>

<p>If you look down on the <span class="math-container">$x-y$</span> 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 <span class="math-container">$(x,y) = (10,10)$</span> (red dot). Regardless of the multiplier, the indifference curve maintains its shape but the associated utility level increases from 10 to 20.</p>

<p><a href="https://i.sstatic.net/aDuUM.gif" rel="noreferrer"><img src="https://i.sstatic.net/aDuUM.gif" alt="Scalar multiplication, shifting indifference curves"></a></p>

<p>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.</p>

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

<p><a href="https://i.sstatic.net/mLNdy.gif" rel="noreferrer"><img src="https://i.sstatic.net/mLNdy.gif" alt="Adding a constant"></a></p>

<p><a href="https://i.sstatic.net/QSXL3.gif" rel="noreferrer"><img src="https://i.sstatic.net/QSXL3.gif" alt="Power transformation"></a></p>

CC BY-SA 4.0
2019-06-13T14:34:34.860Z
Excellent use of graphics and animations!
Bill Clark
CC BY-SA 4.0
2019-06-13T13:59:05.943Z
2019-06-13T13:59:05.943Z
Frederic Schneider
<p>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. </p>

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

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

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

<p><strong>My Question:</strong></p>

<p>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? </p>

<microeconomics><consumer-theory>
Can someone explain graphically why MRS is invariant under monotonic transformation?
Stan Shunpike
# Can someone explain graphically why MRS is invariant under monotonic transformation?

(Asked by: username_0 on 2015-04-09)

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?

**Tags:** microeconomics, consumer-theory


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## 3 answers

### Answer by username_3 on 2019-06-13. Score: 13 | Accepted answer

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](https://i.sstatic.net/1Qf7H.gif)](https://i.sstatic.net/1Qf7H.gif)

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](https://i.sstatic.net/aDuUM.gif)](https://i.sstatic.net/aDuUM.gif)

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](https://i.sstatic.net/mLNdy.gif)](https://i.sstatic.net/mLNdy.gif)

[![Power transformation](https://i.sstatic.net/QSXL3.gif)](https://i.sstatic.net/QSXL3.gif)

> **Comments:**
> **username_2**: Excellent use of graphics and animations!  

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### Answer by username_1 on 2015-04-10. Score: 3

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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### Answer by username_2 on 2019-06-12. Score: 0

There are four basic [levels of measurement](https://en.wikipedia.org/wiki/Level_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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